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Random fixed point theorems in Banach spaces applied to a random nonlinear integral equation of the Hammerstein type

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Abstract

The purpose of this paper is to define a new random operator called the generalized ϕ-weakly contraction of the rational type. This new random operator includes those studied by Khan et al. (Filomat 31(12):3611–3626, 2017) and Zhang et al. (Appl. Math. Mech. 32(6):805–810, 2011) as special cases. We prove some convergence, existence, and stability results in separable Banach spaces. Moreover, we produce some numerical examples to demonstrate the applicability of our analytical results. Furthermore, we apply our results in proving the existence of a solution of a nonlinear integral equation of the Hammerstein type.

Introduction

The process of solving some real life problems is characterized with uncertainties, ambiguities, and difficulties. To develop an approach for solving probabilistic models, probabilistic functional analysis has emerged as one of the indispensable mathematical disciplines and tools. Consequently, it has attracted the attention of well-known mathematicians over the years in view of its applications in diverse areas from pure mathematics to engineering and applied sciences. Random nonlinear analysis, which is an important branch of probabilistic functional analysis, deals with the solution of several classes of random operator equations and related problems. Interestingly, the development of random methods has revolutionized the financial markets and related sectors in many world economies (see, e.g., [27]). Whenever the mathematical models or equations used to describe certain phenomena in the physical, engineering, and biological systems that contain some parameters or coefficients that have specific interpretations, but whose values are not known, then it is more realistic to study such equations as random operator equations (see, e.g., Graef et al. [16]). Random fixed point theorems are stochastic generalizations of classical or deterministic fixed point theorems and are required for the theory of stochastic dynamic programming, random equations, random matrices, random partial differential equations, and various classes of random operators arising in physical systems (see, e.g., [20, 27]). Random fixed point theory was initiated in 1950s by Prague school of probabilists. Spacek [35] and Hans [17] established a stochastic analogue of the Banach fixed point theorem in a separable complete metric space. Itoh [19] in 1979 generalized and extended Spacek and Han’s theorem to a multivalued contraction random operator. The survey article by Bharucha-Reid [12] in 1976, in which he established the sufficient conditions for a stochastic analogue of Schauder’s fixed point theorem for random operators, gave wings to random fixed point theory. This area of research over the years has received the attention of several well-known mathematicians leading to the development of several interesting techniques to obtaining the solution of nonlinear random systems (see, e.g., [7,8,9,10, 15,16,17, 19, 20, 22, 23, 27,28,29,30, 33,34,35, 37]).

In 2011, Zhang et al. [37] proved some stability and convergence results for some random iterative sequence generated by a ϕ-weakly contractive random operator. Okeke and Abbas [27] introduced the concept of generalized ϕ-weakly contraction random operators and proved some interesting random fixed point theorems for this kind of random operators in separable Banach spaces. There results unify and generalize the results of Zhang et al. [37]. Recently, Khan et al. [22] introduced a generalized random operator and proved some convergence and stability results for those kinds of random operators in separable Banach spaces. Their results generalize and improve several known random fixed point results in the literature, including the results of Okeke and Abbas [27].

In 2012, Chugh et al. [13] proposed a new three step iterative scheme, called the CR iterative scheme. They proved that the CR iterative scheme is equivalent to and converges faster than all of Picard, Mann [25], Ishikawa [18], Agarwal et al. [2], Noor [26], and SP [32] iterative schemes for certain contractive operators in the sense of Berinde [11]. In 2013, Karahan and Ozdemir [21] proposed a new three step iterative scheme. They proved that this iteration process is faster than all of Picard, Mann [25], and S [2] iteration processes in the sense of Berinde [11].

Motivated by the results above, we introduce a random operator, called the generalized ϕ-weakly contraction of the rational type. This new random operator includes those studied by Khan et al. [22] and Zhang et al. [37] as special cases. We introduce the random versions of the CR iterative scheme and the Karahan–Ozdemir iterative scheme. We prove some existence, convergence, and stability results for the generalized ϕ-weakly contraction of the rational type via these random iteration schemes. Some numerical examples are given to demonstrate the applicability of our analytical results. Furthermore, we apply our results in proving the existence of a solution of a nonlinear integral equation of the Hammerstein type. Our results unify, extend, and generalize several deterministic fixed point theorems in stochastic version, including the results of Akewe et al. [4], Akewe and Okeke [3], Chugh et al. [13], and Karahan and Ozdemir [21] among others.

Preliminaries

Let \((\varOmega ,\varSigma ,\mu )\) be a complete probability measure space and \((E,B(E))\) be a measurable space, where E is a separable Banach space, \(B(E)\) is Borel sigma algebra of E, \((\varOmega ,\varSigma )\) is a measurable space (Σ—sigma algebra), and μ is a probability measure on Σ, that is, a measure with total measure one. A mapping \(\xi :\varOmega \rightarrow E\) is called (a) E-valued random variable if ξ is \((\varSigma ,B(E))\)-measurable; (b) strongly μ-measurable if there exists a sequence \(\{\xi _{n}\}\) of μ-simple functions converging to ξ μ-almost everywhere. Due to the separability of a Banach space E, the sum of two E-valued random variables is an E-valued random variable. A mapping \(T:\varOmega \times E\rightarrow E\) is called a random operator if, for each fixed e in E, the mapping \(T(\cdot,e):\varOmega \rightarrow E\) is measurable. Throughout this study, we assume that \((\varOmega ,\xi , \mu )\) is a complete probability measure space and E is a nonempty subset of a separable Banach space X.

The following definitions will be needed in this study

Definition 2.1

([20])

A random variable \(x:\varOmega \to E\) is Bochner integrable if, for each

$$ \omega \in \varOmega , \quad \int _{\varOmega } \bigl\Vert x(\omega ) \bigr\Vert \,d\mu (\omega )< \infty , $$
(2.1)

i.e., \(\Vert x(\omega )\Vert \in L^{1}(\varOmega ,\xi ,\mu )\).

Note: The Bochner integral is a natural generalization of the Lebesgue integral for vector-valued set functions.

Proposition 2.1

([20])

A random variable \(x:\varOmega \to E\) is Bochner integrable if and only if there exists a sequence of random variables \(\{x_{n}\} _{n=1}^{\infty }\) converging strongly to x almost surely such that

$$ \lim_{n\to \infty } \int _{\varOmega } \bigl\Vert x_{n}(\omega )-x(\omega ) \bigr\Vert \,d\mu (\omega )=0. $$
(2.2)

Definition 2.2

([37])

Let \((\varOmega ,\xi ,\mu )\) be a complete probability measure space and E be a nonempty subset of a separable Banach space X. Let \(T:\varOmega \times E\to E\) be a random operator. Denote by \(\operatorname{RF}(T)=\{x^{\ast }(\omega )\in E:T(\omega ,x^{\ast }( \omega ))=x^{\ast }(\omega ), \omega \in \varOmega \}\) the random fixed point set of T. For any given arbitrary measurable mapping \(x_{0}:\varOmega \to E\), let \(\{x_{n}(\omega )\}_{n=0}^{\infty }\) be a sequence of measurable mapping from Ω to E, and

$$ x_{n+1}(\omega )=f\bigl(T,x_{n}(\omega )\bigr), \quad n=0,1,2,\ldots , $$
(2.3)

where f is some function measurable in the second variable.

Let \(x^{\ast }(\omega )\) be a random fixed point of T and Bochner integrable with respect to \(\{x_{n}(\omega )\}_{n=0}^{\infty }\). Let \(\{y_{n}(\omega )\}_{n=0}^{\infty }\subset \varOmega \times E\) be an arbitrary sequence of measurable mapping. Denote

$$ \varepsilon _{n}(\omega )= \bigl\Vert y_{n+1}(\omega )-f \bigl(T,y_{n}(\omega )\bigr) \bigr\Vert , $$

and assume that \(\Vert \varepsilon _{n}(\omega )\Vert \in L^{1}(\varOmega ,\xi ,\mu )\), \(n=0,1,2,\ldots \) Then the iterative scheme (2.3) is T-stable almost surely (or the iterative scheme (2.3) is stable with respect to T almost surely) if and only if

$$ \lim_{n\to \infty } \int _{\varOmega } \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert \,d\mu (\omega )=0 $$

implies that \(x^{\ast }(\omega )\) is Bochner integrable with respect to \(\{y_{n}(\omega )\}_{n=0}^{\infty }\).

Definition 2.3

([37])

Let \((\varOmega ,\varSigma ,\mu )\) be a complete probability measure space and E be a nonempty subset of a separable Banach space X. A random operator \(T:\varOmega \times E\rightarrow E\) is called a ϕ-weakly contractive type random operator if there exists a continuous and nondecreasing function \(\phi :\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) with \(\phi (t)>0\) for each \(t\in (0,\infty )\) and \(\phi (0)=0 \) such that, for each \(x,\varsigma \in E\), \(\omega \in \varOmega \), we have

$$ \int _{\varOmega } \bigl\Vert T(\omega ,x)-T(\omega ,\varsigma ) \bigr\Vert \,d\mu ( \omega )\leq \int _{\varOmega } \Vert x-\varsigma \Vert \,d\mu (\omega )- \phi \biggl( \int _{\varOmega } \Vert x-\varsigma \Vert \,d\mu (\omega ) \biggr) . $$
(2.4)

Definition 2.4

([27])

Let \((\varOmega ,\xi ,\mu )\) be a complete probability measure space and E be a nonempty subset of a separable Banach space X. A random operator \(T:\varOmega \times E\rightarrow E\) is the generalized ϕ-weakly contractive type if there exist \(L(\omega )\geq 0\) and a continuous and nondecreasing function \(\phi : \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) with \(\phi (t)>0\) for each \(t\in (0,\infty )\) and \(\phi (0)=0\) such that, for each \(x,\varsigma \in E\), \(\omega \in \varOmega \),

$$ \begin{aligned}[b] &\int _{\varOmega } \bigl\Vert T(\omega ,x)-T(\omega ,\varsigma ) \bigr\Vert \,d\mu ( \omega )\\ &\quad \leq e^{L(\omega ) \Vert x-\varsigma \Vert } \biggl[ \int _{ \varOmega } \Vert x-\varsigma \Vert \,d\mu (\omega )-\phi \biggl( \int _{\varOmega } \Vert x-\varsigma \Vert \,d\mu (\omega )\biggr) \biggr] . \end{aligned} $$
(2.5)

Definition 2.5

([22])

A random operator \(T:\varOmega \times C\to C\) is generalized ϕ-weakly contractive type if there exist \(L(\omega )\geq 0\) and a continuous and nondecreasing function \(\phi : \mathbb{R}^{+} \to \mathbb{R}^{+}\) with \(\phi (t)>0\) for each \(t\in (0,\infty )\), \(\phi (0)=0\) and for each \(x,y\in C\), \(\omega \in \varOmega \), we have

$$ \begin{aligned}[b] &\int _{\varOmega } \bigl\Vert T(\omega ,x)-T(\omega ,y) \bigr\Vert \,d\mu (\omega ) \\ &\quad \leq e^{L(\omega ) \Vert x-T(\omega , x) \Vert } \biggl[ \int _{\varOmega } \Vert x-y \Vert \,d\mu (\omega )-\phi \biggl( \int _{\varOmega } \Vert x-y \Vert \,d\mu (\omega ) \biggr) \biggr]. \end{aligned} $$
(2.6)

Motivated by the results above, we introduce the following generalized ϕ-weakly contraction of the rational type.

Definition 2.6

A random operator \(T:\varOmega \times C\to C\) is a generalized ϕ-weakly contraction of the rational type if there exist \(L(\omega ), M(\omega )\geq 0\) and a continuous and nondecreasing function \(\phi :\mathbb{R}^{+} \to \mathbb{R}^{+}\) with \(\phi (t)>0\) for each \(t\in (0,\infty )\), \(\phi (0)=0\), and for each \(x,y\in C\), \(\omega \in \varOmega \), we have

$$ \begin{aligned}[b] \int _{\varOmega } \bigl\Vert T(\omega ,x)-T(\omega ,y) \bigr\Vert \,d\mu (\omega )& \leq e^{L(\omega ) \Vert x-T(\omega ,x) \Vert } \biggl[ \int _{\varOmega }\frac{ \Vert x-y \Vert }{1+M(\omega ) \Vert x-T(\omega ,x) \Vert }\,d\mu (\omega ) \\ &\quad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert x-y \Vert }{1+M(\omega ) \Vert x-T(\omega ,x) \Vert }\,d\mu (\omega ) \biggr)\biggr]. \end{aligned} $$
(2.7)

Remark 2.1

Observe that if \(M(\omega )=0\), then relation (2.7) reduces to relation (2.6). Clearly, the generalized ϕ-weakly contraction of the rational type (2.7) includes (2.4) and (2.6) as special cases.

Let \(T:\varOmega \times C\to C\) be a random operator where C is a nonempty convex subset of X. Suppose that \(u_{0}:\varOmega \to C\) and \(p_{0}:\varOmega \to C\) are arbitrary measurable mappings, the random versions of the CR iterative scheme [13] and the Karahan–Ozdemir iterative scheme [21] are given as follows:

$$\begin{aligned}& \textstyle\begin{cases} u_{0}(\omega )\in C, \\ u_{n+1}(\omega )=(1-\alpha _{n})v_{n}(\omega )+\alpha _{n}T(\omega ,v _{n}(\omega )), \\ v_{n}(\omega )=(1-\beta _{n})T(\omega ,u_{n}(\omega ))+\beta _{n}T( \omega ,y_{n}(\omega )), \\ y_{n}(\omega )=(1-\gamma _{n})u_{n}(\omega )+\gamma _{n}T(\omega ,u_{n}( \omega )), \quad n\in \mathbb{N}, \end{cases}\displaystyle \end{aligned}$$
(2.8)
$$\begin{aligned}& \textstyle\begin{cases} p_{0}(\omega )\in C, \\ p_{n+1}(\omega )=(1-\alpha _{n})T(\omega ,p_{n}(\omega ))+\alpha _{n}T( \omega ,q_{n}(\omega )), \\ q_{n}(\omega )=(1-\beta _{n})T(\omega ,p_{n}(\omega ))+\beta _{n}T( \omega ,r_{n}(\omega )), \\ r_{n}(\omega )=(1-\gamma _{n})p_{n}(\omega )+\gamma _{n}T(\omega ,p_{n}( \omega )), \quad n\in \mathbb{N}. \end{cases}\displaystyle \end{aligned}$$
(2.9)

Definition 2.7

([11])

Let \(\{a_{n}\}_{n=0}^{\infty }\), \(\{b_{n}\}_{n=0} ^{\infty }\) be two sequences of positive numbers that converge to a, respectively b. Assume that there exists

$$ l=\lim_{n\to \infty }\frac{ \vert a_{n}-a \vert }{ \vert b_{n}-b \vert }. $$
(2.10)
  1. 1.

    If \(l=0\), then it is said that the sequence \(\{a_{n}\}_{n=0}^{ \infty }\) converges to a faster than the sequence \(\{b_{n}\}_{n=0} ^{\infty }\) to b;

  2. 2.

    If \(0< l<\infty \), then we say that the sequences \(\{a_{n}\}_{n=0} ^{\infty }\) and \(\{b_{n}\}_{n=0}^{\infty }\) have the same rate of convergence.

Lemma 2.1

([5])

Let \(\{\zeta _{n}\}\) and \(\{\lambda _{n}\}\) be two sequences of nonnegative real numbers. Let \(\{\sigma _{n}\}\) be a sequence of positive numbers satisfying the conditions \(\sum_{n=1} ^{\infty }\sigma _{n}=\infty \) and \(\lim_{n\to \infty }\frac{\zeta _{n}}{ \sigma _{n}}=0\). If the following condition is satisfied:

$$ \lambda _{n+1}\leq \lambda _{n}-\sigma _{n}\phi ( \lambda _{n})+\zeta _{n},\quad \forall n\geq 1, $$

where \(\phi :\mathbb{R}^{+}\to \mathbb{R}^{+}\) is a continuous and strictly increasing function with \(\phi (0)=0\), then \(\{\lambda _{n}\}\) converges to 0 as \(n\to \infty \).

Convergence theorems in separable Banach spaces

We begin this section by proving some convergence and existence of fixed point results in separable Banach spaces. Our results unify, generalize, and extend several known deterministic fixed point theorems in stochastic version.

Theorem 3.1

Let C be a nonempty closed and convex subset of a separable Banach space X, \(T:\varOmega \times C\to C\) be a random generalized ϕ-weakly contraction of the rational type satisfying condition (2.7) with \(\operatorname{RF}(T)\neq \emptyset \). Suppose that \(x^{\ast }(\omega )\) is the random fixed point of T and \(\{u_{n}( \omega )\}\) is the random CR-iteration process defined by (2.8), where \(\{\alpha _{n}\}\), \(\{\beta _{n}\}\), and \(\{\gamma _{n}\}\) are real sequences in \((0,1)\) such that \(\sum_{n=1}^{\infty }\beta _{n}\gamma _{n}=\infty \). Then the random fixed point \(x^{\ast }(\omega )\) of T is Bochner integrable.

Proof

To prove that \(x^{\ast }(\omega )\) is Bochner integrable, it suffices to prove that

$$ \lim_{n\to \infty } \int _{\varOmega } \bigl\Vert u_{n}(\omega )-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega )=0. $$
(3.1)

Using relations (2.7) and (2.8), we have

$$\begin{aligned}& \int _{\varOmega } \bigl\Vert u_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega ) \\& \quad \leq (1-\alpha _{n}) \int _{\varOmega } \bigl\Vert v_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,v_{n}(\omega )\bigr)-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega ) \\& \quad = (1-\alpha _{n}) \int _{\varOmega } \bigl\Vert v_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,x^{\ast }(\omega ) \bigr)-T\bigl( \omega ,v_{n}(\omega )\bigr) \bigr\Vert \,d\mu (\omega ) \\& \quad \leq (1-\alpha _{n}) \int _{\varOmega } \bigl\Vert v_{n}(\omega )-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n}e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-v_{n}( \omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{ \ast }(\omega )) \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi\biggr( \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-v_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\,d\mu (\omega )\biggl)\biggr] \\& \quad = (1-\alpha _{n}) \int _{\varOmega } \bigl\Vert v_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n}e^{L(\omega ) \Vert 0 \Vert } \biggl[ \int _{\varOmega }\frac{ \Vert v_{n}(\omega )-x^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d\mu (\omega )-\phi \biggl( \int _{\varOmega }\frac{ \Vert v_{n}( \omega )-x^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d\mu ( \omega ) \biggr) \biggr] \\& \quad = (1-\alpha _{n}) \int _{\varOmega } \bigl\Vert v_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n} \int _{\varOmega } \bigl\Vert v_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )-\phi \biggl( \int _{\varOmega } \bigl\Vert v_{n}(\omega )-x ^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \biggr) \\& \quad \leq \int _{\varOmega } \bigl\Vert v_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d \mu (\omega ) \\& \quad \leq (1-\beta _{n}) \int _{\varOmega } \bigl\Vert T\bigl(\omega ,u_{n}(\omega )\bigr)-x ^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,y_{n}(\omega )\bigr)-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega ) \\& \quad \leq (1-\beta _{n})e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x ^{\ast }(\omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-u_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }(\omega )) \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-u_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\,d\mu (\omega )\biggr)\biggr] \\& \qquad {}+\beta _{n}e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-y_{n}( \omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{ \ast }(\omega )) \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-y_{n}( \omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{ \ast }(\omega )) \Vert }\,d\mu (\omega ) \biggr)\biggr] \\& \quad = (1-\beta _{n})e^{L(\omega ) \Vert 0 \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert u_{n}(\omega )-x^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d\mu (\omega )-\phi \biggl( \int _{\varOmega }\frac{ \Vert u_{n}(\omega )-x ^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d\mu (\omega )\biggr)\biggr] \\& \qquad {}+\beta _{n}e^{L(\omega ) \Vert 0 \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert y _{n}(\omega )-x^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d \mu (\omega )-\phi \biggl( \int _{\varOmega }\frac{ \Vert y_{n}(\omega )-x ^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d\mu (\omega ) \biggr)\biggr] \\& \quad = (1-\beta _{n}) \biggl[ \int _{\varOmega } \bigl\Vert u_{n}(\omega )-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega )-\phi \biggl( \int _{\varOmega } \bigl\Vert u_{n}(\omega )-x ^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )\biggr) \biggr] \\& \qquad {}+\beta _{n} \biggl[ \int _{\varOmega } \bigl\Vert y_{n}(\omega )-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega )-\phi \biggl( \int _{\varOmega } \bigl\Vert y_{n}(\omega )-x ^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )\biggr) \biggr] \\& \quad \leq (1-\beta _{n}) \int _{\varOmega } \bigl\Vert u_{n}(\omega )-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega )+\beta _{n} \int _{\varOmega } \bigl\Vert y_{n}( \omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ). \end{aligned}$$
(3.2)

Next, we obtain the following estimate:

$$\begin{aligned}& \int _{\varOmega } \bigl\Vert y_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega ) \\& \quad \leq (1-\gamma _{n}) \int _{\varOmega } \bigl\Vert u_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\gamma _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,u_{n}(\omega )\bigr)-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega ) \\& \quad \leq (1-\gamma _{n}) \int _{\varOmega } \bigl\Vert u_{n}(\omega )-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\gamma _{n}e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-u_{n}( \omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{ \ast }(\omega )) \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-u_{n}( \omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{ \ast }(\omega )) \Vert }\,d\mu (\omega ) \biggr)\biggr] \\& \quad = (1-\gamma _{n}) \int _{\varOmega } \bigl\Vert u_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )+\gamma _{n}e^{L(\omega ) \Vert 0 \Vert }\biggl[ \int _{ \varOmega }\frac{ \Vert u_{n}(\omega )-x^{\ast }(\omega ) \Vert }{1+M( \omega ) \Vert 0 \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert u_{n}(\omega )-x^{\ast }( \omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d\mu (\omega ) \biggr)\biggr] \\& \quad = (1-\gamma _{n}) \int _{\varOmega } \bigl\Vert u_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )+\gamma _{n} \int _{\varOmega } \bigl\Vert u_{n}(\omega )-x ^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}-\gamma _{n}\phi \biggl( \int _{\varOmega } \bigl\Vert u_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )\biggr) \\& \quad = \int _{\varOmega } \bigl\Vert u_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega )-\gamma _{n} \phi \biggl( \int _{\varOmega } \bigl\Vert u_{n}(\omega )-x ^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \biggr). \end{aligned}$$
(3.3)

Using relation (3.3) in (3.2), we have

$$\begin{aligned}& \int _{\varOmega } \bigl\Vert u_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega ) \\& \quad \leq (1-\beta _{n}) \int _{\varOmega } \bigl\Vert u_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n} \int _{\varOmega } \bigl\Vert u_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )-\beta _{n} \gamma _{n}\phi \biggl( \int _{\varOmega } \bigl\Vert u _{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )\biggr) \\& \quad = \int _{\varOmega } \bigl\Vert u_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega )-\beta _{n} \gamma _{n}\phi \biggl( \int _{\varOmega } \bigl\Vert u_{n}( \omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \biggr). \end{aligned}$$
(3.4)

If we take \(\lambda _{n}=\int _{\varOmega }\Vert u_{n}(\omega )-x^{\ast }( \omega )\Vert \,d\mu (\omega )\), \(\sigma _{n}=\beta _{n}\gamma _{n}\), and \(\zeta _{n}=0\) in Lemma 2.1, by the assumptions of Theorem 3.1, we see that all the conditions of Lemma 2.1 are satisfied. Therefore, we have

$$ \lim_{n\to \infty } \int _{\varOmega } \bigl\Vert u_{n}(\omega )-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega )=0. $$
(3.5)

The proof of Theorem 3.1 is completed. □

Theorem 3.2

Let C be a nonempty closed and convex subset of a separable Banach space X, \(T:\varOmega \times C\to C\) be a random generalized ϕ-weakly contraction of the rational type satisfying condition (2.7) with \(\operatorname{RF}(T)\neq \emptyset \). Suppose \(x^{\ast }( \omega )\) is the random fixed point of T and \(\{p_{n}(\omega )\}\) is the random Karahan–Ozdemir iteration process defined by (2.9), where \(\{\alpha _{n}\}\), \(\{\beta _{n}\}\), and \(\{\gamma _{n}\}\) are real sequences in \((0,1)\) such that \(\beta _{n}(1-\gamma _{n})\leq \beta _{n}\), \(\alpha _{n}\beta _{n}\gamma _{n}\leq \gamma _{n}\), and \(\sum_{n=1}^{ \infty }\beta _{n}\gamma _{n}=\infty \). Then the random fixed point \(x^{\ast }(\omega )\) of T is Bochner integrable.

Proof

To prove that \(x^{\ast }(\omega )\) is Bochner integrable, it suffices to prove that

$$ \lim_{n\to \infty } \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega )=0. $$
(3.6)

Using relations (2.7) and (2.9), we have

$$\begin{aligned}& \int _{\varOmega } \bigl\Vert p_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega ) \\& \quad \leq (1-\alpha _{n}) \int _{\varOmega } \bigl\Vert T\bigl(\omega ,p_{n}( \omega )\bigr)-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,q_{n}(\omega )\bigr)-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega ) \\& \quad = (1-\alpha _{n}) \int _{\varOmega } \bigl\Vert T\bigl(\omega ,x^{\ast }(\omega ) \bigr)-T\bigl( \omega ,p_{n}(\omega )\bigr) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,x^{\ast }(\omega ) \bigr)-T\bigl( \omega ,q_{n}(\omega )\bigr) \bigr\Vert \,d\mu (\omega ) \\& \quad \leq (1-\alpha _{n})e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x ^{\ast }(\omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-p_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }(\omega )) \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-p_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\,d\mu (\omega )\biggr)\biggr] \\& \qquad {}+\alpha _{n}e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-q_{n}( \omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{ \ast }(\omega )) \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-q_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\,d\mu (\omega )\biggr)\biggr] \\& \quad \leq (1-\alpha _{n})e^{L(\omega ) \Vert 0 \Vert } \int _{\varOmega }\frac{ \Vert p_{n}(\omega )-x^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d\mu (\omega ) \\& \qquad {}+\alpha _{n}e^{L(\omega ) \Vert 0 \Vert } \int _{\varOmega }\frac{ \Vert q _{n}(\omega )-x^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d \mu (\omega ) \\& \quad = (1-\alpha _{n}) \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )+\alpha _{n} \int _{\varOmega } \bigl\Vert q_{n}(\omega )-x ^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ). \end{aligned}$$
(3.7)

Next, using the assumption that \(\beta _{n}(1-\gamma _{n}) \leq \beta _{n}\), we have

$$\begin{aligned}& \int _{\varOmega } \bigl\Vert q_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega ) \\& \quad \leq (1-\beta _{n}) \int _{\varOmega } \bigl\Vert T\bigl(\omega ,p_{n}( \omega )\bigr)-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,r_{n}(\omega )\bigr)-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega ) \\& \quad = (1-\beta _{n}) \int _{\varOmega } \bigl\Vert T\bigl(\omega ,x^{\ast }(\omega ) \bigr)-T\bigl( \omega ,p_{n}(\omega )\bigr) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,x^{\ast }(\omega ) \bigr)-T\bigl(\omega ,r_{n}(\omega )\bigr) \bigr\Vert \,d\mu (\omega ) \\& \quad \leq (1-\beta _{n})e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x ^{\ast }(\omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-p_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }(\omega )) \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-p_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\,d\mu (\omega )\biggr)\biggr] \\& \qquad {}+\beta _{n}e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-r_{n}( \omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{ \ast }(\omega )) \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-r_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\,d\mu (\omega )\biggr)\biggr] \\& \quad \leq (1-\beta _{n})e^{L(\omega ) \Vert 0 \Vert } \int _{\varOmega }\frac{ \Vert p_{n}(\omega )-x^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d\mu (\omega ) \\& \qquad {}+\beta _{n}e^{L(\omega ) \Vert 0 \Vert } \int _{\varOmega }\frac{ \Vert r _{n}(\omega )-x^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d \mu (\omega ) \\& \quad= (1-\beta _{n}) \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )+\beta _{n} \int _{\varOmega } \bigl\Vert r_{n}(\omega )-x ^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \quad \leq (1-\beta _{n}) \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n}\biggl[(1-\gamma _{n}) \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\gamma _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,p_{n}(\omega )\bigr)-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega )\biggr] \\& \quad = (1-\beta _{n}) \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n}(1-\gamma _{n}) \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n}\gamma _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,x^{\ast }(\omega ) \bigr)-T\bigl( \omega ,p_{n}(\omega )\bigr) \bigr\Vert \,d\mu (\omega ) \\& \quad \leq (1-\beta _{n}) \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n} \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n}\gamma _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,x^{\ast }(\omega ) \bigr)-T\bigl( \omega ,p_{n}(\omega )\bigr) \bigr\Vert \,d\mu (\omega ) \\& \quad = \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega )+\beta _{n} \gamma _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,x^{\ast }( \omega )\bigr)-T\bigl(\omega ,p_{n}(\omega )\bigr) \bigr\Vert \,d\mu (\omega ) \\& \quad \leq \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d \mu (\omega ) \\& \qquad {}+\beta _{n}\gamma _{n}e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x ^{\ast }(\omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-p_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }(\omega )) \Vert } \,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-p_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert } \,d\mu (\omega )\biggr)\biggr] \\& \quad = \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega )+\beta _{n} \gamma _{n} \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \quad {}-\beta _{n}\phi \biggl( \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )\biggr). \end{aligned}$$
(3.8)

Using (3.8) in (3.7) and the assumption that \(\alpha _{n}\beta _{n} \gamma _{n}\leq \gamma _{n}\), we obtain

$$\begin{aligned}& \int _{\varOmega } \bigl\Vert p_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega ) \\& \quad \leq (1-\alpha _{n}) \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n}\biggl[ \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )+\beta _{n} \gamma _{n} \int _{\varOmega } \bigl\Vert p_{n}( \omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}-\beta _{n}\phi \biggl( \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )\biggr)\biggr] \\& \quad = (1-\alpha _{n}) \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n} \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )+\alpha _{n} \beta _{n}\gamma _{n} \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}-\alpha _{n}\beta _{n}\phi \biggl( \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega )\biggr) \\& \quad = \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega )+\alpha _{n} \beta _{n}\gamma _{n} \int _{\varOmega } \bigl\Vert p_{n}( \omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}-\alpha _{n}\beta _{n}\phi \biggl( \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega )\biggr) \\& \quad \leq \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d \mu (\omega )-\alpha _{n} \beta _{n}\phi \biggl( \int _{\varOmega } \bigl\Vert p_{n}( \omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )\biggr) \\& \qquad {}+\gamma _{n} \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ). \end{aligned}$$
(3.9)

If we take \(\lambda _{n}=\int _{\varOmega }\Vert p_{n}(\omega )-x^{\ast }(\omega )\Vert \,d\mu (\omega )\), \(\sigma _{n}=\alpha _{n}\beta _{n}\), and \(\zeta _{n}=\gamma _{n}\int _{\varOmega }\Vert p_{n}(\omega )-x ^{\ast }(\omega )\Vert \,d\mu (\omega )\) in Lemma 2.1, by the assumptions of Theorem 3.2, we see that all the conditions of Lemma 2.1 are satisfied. Therefore, we have

$$ \lim_{n\to \infty } \int _{\varOmega } \bigl\Vert p_{n}(\omega )-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega )=0. $$
(3.10)

The proof of Theorem 3.2 is completed. □

Next, we obtain the following theorem which is a generalization of the results of Zhang et al. [37] among others.

Theorem 3.3

Let C be a nonempty closed and convex subset of a separable Banach space X, \(T:\varOmega \times C\to C\) be a random generalized ϕ-weakly contraction of the rational type satisfying condition (2.7) with \(\operatorname{RF}(T)\neq \emptyset \). Suppose that \(x^{\ast }(\omega )\) is the random fixed point of T and \(\{x_{n}( \omega )\}\) is the random Mann-iteration process defined by

$$ \textstyle\begin{cases} x_{0}(\omega )\in C, \\ x_{n+1}(\omega )=(1-\alpha _{n})x_{n}(\omega )+\alpha _{n}T(\omega ,x _{n}(\omega )), \quad n\in \mathbb{N}, \end{cases} $$
(3.11)

where \(\{\alpha _{n}\}\) is a real sequence in \((0,1)\) such that \(\sum_{n=1}^{\infty }\alpha _{n}=\infty \). Then the random fixed point \(x^{\ast }(\omega )\) of T is Bochner integrable.

Proof

The proof of Theorem 3.3 follows similar lines as in the proof of Theorem 3.1. □

Remark 3.1

Theorem 3.1, Theorem 3.2, and Theorem 3.3 generalize several known results in the literature, including the results of Khan et al. [22], Okeke and Abbas [27], and Zhang et al. [37]. Moreover, our results extend and generalize several deterministic fixed point theorems in stochastic version, including the results of Chugh et al. [13] and Karahan and Ozdemir [21] among others.

Next, we prove the following existence results in separable Banach spaces.

Theorem 3.4

Suppose that X is a separable Banach space and \((\varOmega ,\varSigma , \mu )\) is a complete probability measure space. Let \(T:\varOmega \times X\rightarrow X\) be a continuous random operator such that

$$ \bigl\Vert T(\omega ,x_{1})-T(\omega ,x_{2}) \bigr\Vert \leq \Vert x_{1}-x_{2} \Vert -\phi \bigl( \Vert x_{1}-x_{2} \Vert \bigr) $$
(3.12)

almost surely for all \(x_{1}, x_{2}\in X\), where ϕ is a continuous and nondecreasing function \(\phi :\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) with \(\phi (t)>0\) for each \(t\in (0,\infty )\) and \(\phi (0)=0 \) almost surely. Then T has a unique random fixed point.

Proof

Suppose

$$\begin{aligned}& A=\bigl\{ \omega \in \varOmega : T(\omega , x) {\text{ is a continuous function of }} x\bigr\} , \end{aligned}$$
(3.13)
$$\begin{aligned}& C_{x_{1},x_{2}}=\bigl\{ \omega \in \varOmega : \bigl\Vert T(\omega , x_{1})-T( \omega , x_{2}) \bigr\Vert \leq \Vert x_{1}-x_{2} \Vert -\phi \bigl( \Vert x_{1}-x _{2} \Vert \bigr)\bigr\} \end{aligned}$$
(3.14)

and

$$ \begin{aligned}[b] B&=\bigl\{ \phi :\mathbb{R}^{+} \to \mathbb{R}^{+} \text{ such that } \phi{\text{ is continuous and nondecreasing, with }} \\ &\quad \phi (t)>0, \forall t\in (0,\infty ) {\text{ and }} \phi (0)=0\bigr\} . \end{aligned} $$
(3.15)

Suppose that H is a countable dense subset of X. Then we show that

$$ \bigcap_{x_{1},x_{2}\in X}(C_{x_{1},x_{2}}\cap A\cap B)= \bigcap _{h_{1},h_{2}\in H}(C_{h_{1},h_{2}}\cap A\cap B). $$
(3.16)

Now, we show that

$$ \bigcap_{h_{1},h_{2}\in H}C_{h_{1},h_{2}}\cap A\cap B\subset \bigcap_{x_{1},x_{2}\in X}C_{x_{1},x_{2}}\cap A\cap B. $$
(3.17)

Let \(\omega \in \bigcap_{h_{1},h_{2}\in H}(C_{h_{1},h_{2}}\cap A \cap B)\), then for each \(h_{1},h_{2}\in H\), we have

$$ \bigl\Vert T(\omega ,h_{1})-T(\omega ,h_{2}) \bigr\Vert \leq \Vert h_{1}-h_{2} \Vert -\phi \bigl( \Vert h_{1}-h_{2} \Vert \bigr). $$
(3.18)

Suppose \(x_{1},x_{2}\in X\), we obtain

$$\begin{aligned} &\bigl\Vert T(\omega ,x_{1})-T(\omega ,x_{2}) \bigr\Vert \\ &\quad \leq \bigl\Vert T(\omega ,x _{1})-T(\omega ,h_{1}) \bigr\Vert + \bigl\Vert T(\omega ,h_{1})-T(\omega ,h_{2}) \bigr\Vert + \bigl\Vert T(\omega ,h_{2})-T( \omega ,x_{2}) \bigr\Vert \\ &\quad \leq \bigl\Vert T(\omega ,x_{1})-T(\omega ,h_{1}) \bigr\Vert + \bigl\Vert T(\omega ,h _{2})-T(\omega ,x_{2}) \bigr\Vert + \Vert h_{1}-h_{2} \Vert -\phi \bigl( \Vert h _{1}-h_{2} \Vert \bigr) \\ &\quad \leq \bigl\Vert T(\omega ,x_{1})-T(\omega ,h_{1}) \bigr\Vert + \bigl\Vert T(\omega ,h _{2})-T(\omega ,x_{2}) \bigr\Vert + \Vert h_{1}-x_{1} \Vert \\ &\qquad {}+ \Vert x_{1}-x_{2} \Vert + \Vert x_{2}-h_{2} \Vert -\phi \bigl( \Vert x_{1}-x _{2} \Vert \bigr). \end{aligned}$$
(3.19)

Since for each \(\omega \in \varOmega \), \(T(\omega ,x)\) is a continuous function of x, this means that for arbitrary \(\epsilon >0\), there exists \(\delta _{i}(x_{i})>0\) \((i=1,2)\) such that \(\Vert T(\omega ,x _{1})-T(\omega ,h_{1})\Vert <\frac{\epsilon }{2}\) whenever \(\Vert x _{1}-h_{1}\Vert < \delta _{1}(x_{1})\) and \(\Vert T(\omega ,h_{2})-T( \omega ,x_{2})\Vert <\frac{\epsilon }{2}\) whenever \(\Vert h_{2}-x_{2} \Vert <\delta _{2}(x_{2})\).

Now choose \(\delta _{1}=\min \{\delta _{1}(x_{1}),\frac{ \epsilon }{2}\}\) and \(\delta _{2}=\min \{\delta _{2}(x_{2}),\frac{ \epsilon }{2}\}\). Using the choice of \(\delta _{1}\), \(\delta _{2}\), we see that (3.19) becomes

$$ \begin{aligned}[b] \bigl\Vert T(\omega ,x_{1})-T(\omega ,x_{2}) \bigr\Vert &\leq \frac{\epsilon }{2}+\frac{\epsilon }{2}+ \frac{\epsilon }{2}+ \Vert x _{1}-x_{2} \Vert + \frac{\epsilon }{2}-\phi \bigl( \Vert x_{1}-x_{2} \Vert \bigr) \\ &\leq \epsilon + \Vert x_{1}-x_{2} \Vert -\phi \bigl( \Vert x_{1}-x_{2} \Vert \bigr). \end{aligned} $$
(3.20)

Since \(\epsilon >0\) is arbitrary, it follows from (3.20) that

$$ \bigl\Vert T(\omega ,x_{1})-T(\omega ,x_{2}) \bigr\Vert \leq \Vert x_{1}-x_{2} \Vert -\phi \bigl( \Vert x_{1}-x_{2} \Vert \bigr). $$
(3.21)

This means that \(\omega \in \bigcap_{x_{1},x_{2}\in X}C_{x_{1},x_{2}} \cap A\cap B\), this implies that

$$ \bigcap_{h_{1},h_{2} \in H}C_{h_{1},h_{2}}\cap A\cap B\subset \bigcap_{x_{1},x_{2}\in X}C_{x_{1},x_{2}}\cap A\cap B. $$
(3.22)

Similarly, we can easily show that

$$ \bigcap_{x_{1},x_{2}\in X}C_{x_{1},x_{2}}\cap A\cap B\subset \bigcap_{h_{1},h_{2} \in H}C_{h_{1},h_{2}}\cap A\cap B. $$
(3.23)

Hence, by (3.22) and (3.23), we have

$$ \bigcap_{x_{1},x_{2}\in X}C_{x_{1},x_{2}}\cap A\cap B= \bigcap _{h_{1},h_{2} \in H}C_{h_{1},h_{2}}\cap A\cap B. $$
(3.24)

Suppose

$$ M^{\prime }=\bigcap_{h_{1},h_{2}\in H}C_{h_{1},h_{2}}\cap A \cap B, $$
(3.25)

then \(\mu (M^{\prime })=1\).

Therefore, for all \(\omega \in M^{\prime }\), \(T(\omega ,x)\) is a deterministic operator satisfying relation (3.12). Hence T has a unique random fixed point in X. The proof of Theorem 3.4 is completed. □

Example 3.1

Suppose that \(\varOmega =[0,1]\) and Σ is the sigma algebra of the Lebesgue measurable subsets of Ω. Let \(X=\mathbb{R}\), \(C=[0,2]\) and define the generalized ϕ-weakly contraction of the rational type operator \(T:\varOmega \times C\to C\) as \(T(\omega ,x)=\frac{ \omega -x}{6}\). Then the measurable mapping \(x^{\ast }:\varOmega \to X\) defined by \(x^{\ast }(\omega )=\frac{\omega }{7}\) for every \(\omega \in \varOmega \) is a random fixed point of T. Let \(\phi (t)= \frac{t}{3}\) for each \(t\in (0,\infty )\), \(L(\omega )=7\), and \(M(\omega )=5\), then we have

$$ \begin{aligned}[b] &\int _{\varOmega } \bigl\Vert T(\omega ,x)-T(\omega ,y) \bigr\Vert \,d\mu (\omega )\\ &\quad = \biggl[ \int _{\varOmega }\frac{ \Vert x-y \Vert }{1+5 \Vert x-T(\omega ,x) \Vert }\,d\mu (\omega )-\phi \biggl( \int _{\varOmega }\frac{ \Vert x-y \Vert }{1+5 \Vert x-T(\omega ,x) \Vert }\,d\mu (\omega ) \biggr) \biggr] \\ &\quad \leq e^{7 \Vert x-T(\omega ,x) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x-y \Vert }{1+5 \Vert x-T(\omega ,x) \Vert }\,d\mu (\omega ) \\ &\qquad{} -\frac{1}{3} \int _{\varOmega }\frac{ \Vert x-y \Vert }{1+5 \Vert x-T(\omega ,x) \Vert }\,d\mu (\omega )\biggr]. \end{aligned} $$
(3.26)

Clearly, T satisfies condition (2.7). Choose the prototype sequences \(\alpha _{n}=\beta _{n}=\frac{n^{2}}{1+n^{2}}\), \(\gamma _{n}=\frac{n^{3}}{1+n ^{3}}\). Then \(\sum_{n=1}^{\infty }\beta _{n}\gamma _{n}=\sum_{n=1}^{ \infty }\frac{n^{5}}{(1+n^{2})(1+n^{3})}=\infty \). Hence, all the conditions of Theorem 3.1 are satisfied; therefore, the random fixed point \(x^{\ast }(\omega )=\frac{\omega }{7}\) of \(T(\omega ,x)\) is Bochner integrable.

Stability theorems in separable Banach spaces

In this section, we establish some stability results in separable Banach spaces. Our results unify, generalize, and extend several known deterministic fixed point theorems in stochastic version. Furthermore, we give a numerical example to demonstrate the applicability of our analytical results.

Theorem 4.1

Let C be a nonempty closed and convex subset of a separable Banach space X, \(T:\varOmega \times C\to C\) be a random generalized ϕ-weakly contraction of the rational type satisfying condition (2.7) with \(\operatorname{RF}(T)\neq \emptyset \). Suppose that \(x^{\ast }(\omega )\) is the random fixed point of T and \(\{u_{n}( \omega )\}\) is the random CR-iteration process defined by (2.8), where \(\{\alpha _{n}\}\), \(\{\beta _{n}\}\), and \(\{\gamma _{n}\}\) are real sequences in \((0,1)\) such that \(0<\alpha \leq \alpha _{n}\), \(0<\beta \leq \beta _{n}\), and \(0<\gamma \leq \gamma _{n}\). Then \(\{u_{n}(\omega )\}\) is T-stable almost surely.

Proof

Suppose that \(\{h_{n}(\omega )\}\) is an arbitrary sequence of random variables in E and

$$ \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert = \bigl\Vert h_{n+1}(\omega )-(1-\alpha _{n})m_{n}(\omega )+ \alpha _{n}T\bigl(\omega ,m_{n}(\omega )\bigr) \bigr\Vert , $$
(4.1)

where

$$ \textstyle\begin{cases} m_{n}(\omega )=(1-\beta _{n})T(\omega ,h_{n}(\omega ))+\beta _{n}T( \omega ,s_{n}(\omega )), \\ s_{n}(\omega )=(1-\gamma _{n})h_{n}(\omega )+\gamma _{n}T(\omega ,h_{n}( \omega )) \end{cases} $$
(4.2)

and \(\lim_{n\to \infty }\int _{\varOmega }\Vert \varepsilon _{n}(\omega ) \Vert \,d\mu (\omega )=0\).

Next, we prove that \(x^{\ast }(\omega )\) is Bochner integrable with respect to the sequence \(\{h_{n}(\omega )\}\). It follows from (4.1) that

$$\begin{aligned}& \int _{\varOmega } \bigl\Vert h_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega ) \\& \quad \leq \int _{\varOmega } \bigl\Vert h_{n+1}(\omega )-(1-\alpha _{n})m _{n}(\omega )-\alpha _{n}T\bigl(\omega ,m_{n}(\omega )\bigr) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+(1-\alpha _{n}) \int _{\varOmega } \bigl\Vert m_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,m_{n}(\omega )\bigr)-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega ) \\& \quad = \int _{\varOmega } \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert \,d\mu (\omega )+(1- \alpha _{n}) \int _{\varOmega } \bigl\Vert m_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,x^{\ast }(\omega ) \bigr)-T\bigl( \omega ,m_{n}(\omega )\bigr) \bigr\Vert \,d\mu (\omega ) \\& \quad \leq \int _{\varOmega } \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert \,d\mu (\omega )+(1-\alpha _{n}) \int _{\varOmega } \bigl\Vert m_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n}e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-m_{n}( \omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{ \ast }(\omega )) \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-m_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\,d\mu (\omega )\biggr)\biggr] \\& \quad \leq \int _{\varOmega } \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert \,d\mu (\omega )+(1-\alpha _{n}) \int _{\varOmega } \bigl\Vert m_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n}e^{L(\omega ) \Vert 0 \Vert } \int _{\varOmega }\frac{ \Vert m _{n}(\omega )-x^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d \mu (\omega ) \\& \quad = \int _{\varOmega } \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert \,d\mu (\omega )+(1- \alpha _{n}) \int _{\varOmega } \bigl\Vert m_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n} \int _{\varOmega } \bigl\Vert m_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \quad = \int _{\varOmega } \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert \,d\mu (\omega )+ \int _{\varOmega } \bigl\Vert m_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega ) \\& \quad \leq \int _{\varOmega } \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert \,d\mu (\omega )+(1-\beta _{n}) \int _{\varOmega } \bigl\Vert T\bigl(\omega ,h_{n}(\omega )\bigr)-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,s_{n}(\omega )\bigr)-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega ) \\& \quad = \int _{\varOmega } \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert \,d\mu (\omega )+(1- \beta _{n}) \int _{\varOmega } \bigl\Vert T\bigl(\omega ,x^{\ast }(\omega ) \bigr)-T\bigl(\omega ,h_{n}(\omega )\bigr) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,x^{\ast }(\omega ) \bigr)-T\bigl( \omega ,s_{n}(\omega )\bigr) \bigr\Vert \,d\mu (\omega ) \\& \quad \leq \int _{\varOmega } \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+ (1-\beta _{n})e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{ \ast }(\omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-h _{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x ^{\ast }(\omega )) \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-h_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\,d\mu (\omega )\biggr)\biggr] \\& \qquad {}+\beta _{n}e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-s_{n}( \omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{ \ast }(\omega )) \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-s_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\,d\mu (\omega )\biggr)\biggr] \\& \quad \leq \int _{\varOmega } \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert \,d\mu (\omega )+(1-\beta _{n})e^{L(\omega ) \Vert 0 \Vert } \int _{\varOmega }\frac{ \Vert h _{n}(\omega )-x^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d \mu (\omega ) \\& \qquad {}+\beta _{n}e^{L(\omega ) \Vert 0 \Vert } \int _{\varOmega }\frac{ \Vert s _{n}(\omega )-x^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d \mu (\omega ) \\& \quad = \int _{\varOmega } \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert \,d\mu (\omega )+(1- \beta _{n}) \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n} \int _{\varOmega } \bigl\Vert s_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ). \end{aligned}$$
(4.3)

Next, we have the following estimate:

$$\begin{aligned}& \int _{\varOmega } \bigl\Vert s_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega ) \\& \quad \leq (1-\gamma _{n}) \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega )+\gamma _{n} \int _{\varOmega } \bigl\Vert T\bigl( \omega ,h_{n}(\omega )\bigr)-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \quad = (1-\gamma _{n}) \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\gamma _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,x^{\ast }(\omega ) \bigr)-T\bigl( \omega ,h_{n}(\omega )\bigr) \bigr\Vert \,d\mu (\omega ) \\& \quad \leq (1-\gamma _{n}) \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\gamma _{n}e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-h_{n}( \omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{ \ast }(\omega )) \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-h_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\,d\mu (\omega )\biggr)\biggr] \\& \quad = (1-\gamma _{n}) \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )+\gamma _{n}e^{L(\omega ) \Vert 0 \Vert }\biggl[ \int _{ \varOmega }\frac{ \Vert h_{n}(\omega )-x^{\ast }(\omega ) \Vert }{1+M( \omega ) \Vert 0 \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert h_{n}(\omega )-x^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d\mu (\omega )\biggr)\biggr] \\& \quad = (1-\gamma _{n}) \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )+\gamma _{n} \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x ^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}-\gamma _{n}\phi \biggl( \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )\biggr) \\& \quad = \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega )-\gamma _{n} \phi \biggl( \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega )\biggr). \end{aligned}$$
(4.4)

Substituting (4.4) in (4.3), we have

$$\begin{aligned}& \int _{\varOmega } \bigl\Vert h_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega ) \\& \quad \leq \int _{\varOmega } \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert \,d \mu (\omega )+ (1-\beta _{n}) \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n}\biggl[ \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )-\gamma _{n}\phi \biggl( \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )\biggr)\biggr] \\& \quad = \int _{\varOmega } \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert \,d\mu (\omega )+ (1- \beta _{n}) \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n} \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )-\beta _{n} \gamma _{n}\phi \biggl( \int _{\varOmega } \bigl\Vert h _{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )\biggr) \\& \quad = \int _{\varOmega } \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert \,d\mu (\omega )+ \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu ( \omega ) \\& \qquad {}-\beta _{n}\gamma _{n}\phi \biggl( \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega )\biggr). \end{aligned}$$
(4.5)

Using the assumptions that \(\lim_{n\to \infty }\int _{\varOmega }\Vert \varepsilon _{n}(\omega )\Vert \,d\mu (\omega )=0\), \(0<\alpha \leq \alpha _{n}\), \(0<\beta \leq \beta _{n}\), and \(0<\gamma \leq \gamma _{n}\) for all \(n\in \mathbb{N}\), we have

$$ \lim_{n\to \infty }\frac{\int _{\varOmega } \Vert \varepsilon _{n}(\omega ) \Vert \,d\mu (\omega )}{\beta _{n}\gamma _{n}}\leq \lim_{n\to \infty } \frac{ \int _{\varOmega } \Vert \varepsilon _{n}(\omega ) \Vert \,d\mu (\omega )}{ \beta \gamma }=0. $$
(4.6)

If in Lemma 2.1 we take \(\lambda _{n}=\int _{\varOmega }\Vert h_{n}(\omega )-x^{\ast }(\omega )\Vert \,d\mu (\omega )\), \(\sigma _{n}=\beta _{n}\gamma _{n}\), and \(\zeta _{n}= [4]\int _{\varOmega }\Vert \varepsilon _{n}(\omega ) \Vert \,d\mu (\omega )\), we see that all the conditions of Lemma 2.1 are satisfied. Therefore, we have

$$ \lim_{n\to \infty } \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega )=0. $$
(4.7)

Conversely, suppose that \(x^{\ast }(\omega )\) is Bochner integrable with respect to the sequence \(\{h_{n}(\omega )\}\), then we have

$$\begin{aligned}& \int _{\varOmega } \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \quad = \int _{\varOmega } \bigl\Vert h_{n+1}(\omega )-(1-\alpha _{n})m_{n}(\omega )- \alpha _{n}T\bigl(\omega ,m_{n}(\omega )\bigr) \bigr\Vert \,d\mu (\omega ) \\& \quad \leq \int _{\varOmega } \bigl\Vert h_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d \mu (\omega )+(1-\alpha _{n}) \int _{\varOmega } \bigl\Vert x^{\ast }(\omega )-m _{n}(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n} \int _{\varOmega } \bigl\Vert x^{\ast }(\omega )-T\bigl(\omega ,m_{n}( \omega )\bigr) \bigr\Vert \,d\mu (\omega ) \\& \quad \leq \int _{\varOmega } \bigl\Vert h_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d \mu (\omega )+(1-\alpha _{n}) \int _{\varOmega } \bigl\Vert m_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n}e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-m_{n}( \omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{ \ast }(\omega )) \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-m_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\,d\mu (\omega )\biggr)\biggr] \\& \quad = \int _{\varOmega } \bigl\Vert h_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d \mu (\omega )+(1-\alpha _{n}) \int _{\varOmega } \bigl\Vert m_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n}e^{L(\omega ) \Vert 0 \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert m _{n}(\omega )-x^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d \mu (\omega ) -\phi \biggl( \int _{\varOmega }\frac{ \Vert m_{n}(\omega )-x^{ \ast }(\omega ) \Vert }{1+M(\omega ) \Vert 0 \Vert }\,d\mu (\omega )\biggr)\biggr] \\& \quad \leq \int _{\varOmega } \bigl\Vert h_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d \mu (\omega )+(1-\alpha _{n}) \int _{\varOmega } \bigl\Vert m_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\alpha _{n} \int _{\varOmega } \bigl\Vert m_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \quad = \int _{\varOmega } \bigl\Vert h_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d \mu (\omega )+ \int _{\varOmega } \bigl\Vert m_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \quad \leq \int _{\varOmega } \bigl\Vert h_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d \mu (\omega )+(1-\beta _{n}) \int _{\varOmega } \bigl\Vert T\bigl(\omega ,h_{n}(\omega )\bigr)-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {} +\beta _{n} \int _{\varOmega } \bigl\Vert T\bigl(\omega ,s_{n}(\omega )\bigr)-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega ) \\& \quad \leq \int _{\varOmega } \bigl\Vert h_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d \mu (\omega ) \\& \qquad {}+(1-\beta _{n})e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{ \ast }(\omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-h _{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x ^{\ast }(\omega )) \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-h_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\,d\mu (\omega )\biggr)\biggr] \\& \qquad {}+\beta _{n}e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-s_{n}( \omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{ \ast }(\omega )) \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert x^{\ast }(\omega )-s_{n}(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\,d\mu (\omega )\biggr)\biggr] \\& \quad \leq \int _{\varOmega } \bigl\Vert h_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d \mu (\omega )+(1-\beta _{n}) \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n} \int _{\varOmega } \bigl\Vert s_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \quad \leq \int _{\varOmega } \bigl\Vert h_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d \mu (\omega )+(1-\beta _{n}) \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n}\biggl[(1-\gamma _{n}) \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega )+\gamma _{n} \int _{\varOmega } \bigl\Vert T\bigl( \omega ,h_{n}(\omega )\bigr)-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )\biggr] \\& \quad \leq \int _{\varOmega } \bigl\Vert h_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d \mu (\omega )+(1-\beta _{n}) \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{ \ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n} \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n}\gamma _{n}e^{L(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }(\omega )) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert h_{n}(\omega )-x ^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T( \omega ,x^{\ast }(\omega )) \Vert }\,d\mu (\omega ) \\& \qquad {}-\phi \biggl( \int _{\varOmega }\frac{ \Vert h_{n}(\omega )-x^{\ast }(\omega ) \Vert }{1+M(\omega ) \Vert x^{\ast }(\omega )-T(\omega ,x^{\ast }( \omega )) \Vert }\,d\mu (\omega )\biggr)\biggr] \\& \quad = \int _{\varOmega } \bigl\Vert h_{n+1}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d \mu (\omega )+ \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega ) \\& \qquad {}+\beta _{n}\gamma _{n} \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }( \omega ) \bigr\Vert \,d\mu (\omega ) -\beta _{n} \gamma _{n}\phi \biggl( \int _{\varOmega } \bigl\Vert h_{n}(\omega )-x^{\ast }(\omega ) \bigr\Vert \,d\mu (\omega )\biggr). \end{aligned}$$
(4.8)

Therefore, we obtain

$$ \lim_{n\to \infty } \int _{\varOmega } \bigl\Vert \varepsilon _{n}(\omega ) \bigr\Vert \,d\mu (\omega )=0. $$
(4.9)

This means that the random CR-iteration process \(\{u_{n}(\omega )\}\) defined by (2.8) is T-stable almost surely. The proof of Theorem 4.1 is completed. □

Theorem 4.2

Let C be a nonempty closed and convex subset of a separable Banach space X, \(T:\varOmega \times C\to C\) be a random generalized ϕ-weakly contraction of the rational type satisfying condition (2.7) with \(\operatorname{RF}(T)\neq \emptyset \). Suppose that \(x^{\ast }(\omega )\) is the random fixed point of T and \(\{p_{n}( \omega )\}\) is the random Karahan–Ozdemir iteration process defined by (2.9), where \(\{\alpha _{n}\}\), \(\{\beta _{n}\}\), and \(\{\gamma _{n}\}\) are real sequences in \((0,1)\) such that \(0<\alpha \leq \alpha _{n}\), \(0<\beta \leq \beta _{n}\), and \(0<\gamma \leq \gamma _{n}\). Then \(\{p_{n}(\omega )\}\) is T-stable almost surely.

Proof

The proof of Theorem 4.2 follows similar lines as in the proof of Theorem 4.1. □

Remark 4.1

Theorem 4.1 and Theorem 4.2 generalize several known results in the literature, including the results of Khan et al. [22], Okeke and Abbas [27], and Zhang et al. [37]. Moreover, our results extend and generalize several deterministic fixed point theorems in stochastic version, including the results of Chugh et al. [13] and Karahan and Ozdemir [21] among others.

Next, we give the following numerical example to demonstrate the applicability of our results.

Example 4.1

Suppose that \(\varOmega =[0,1]\) and Σ is the sigma algebra of the Lebesgue measurable subsets of Ω. Let \(X=\mathbb{R}\), \(C=[0,2]\) and define the generalized ϕ-weakly contraction of the rational type operator \(T:\varOmega \times C\to C\) as \(T(\omega ,x)=\frac{ \omega -x}{6}\). Then the measurable mapping \(x^{\ast }:\varOmega \to X\) defined by \(x^{\ast }(\omega )=\frac{\omega }{7}\) for every \(\omega \in \varOmega \) is a random fixed point of T. Let \(\phi (t)= \frac{t}{3}\) for each \(t\in (0,\infty )\), \(L(\omega )=7\), and \(M(\omega )=5\), then we have

$$\begin{aligned}& \int _{\varOmega } \bigl\Vert T(\omega ,x)-T(\omega ,y) \bigr\Vert \,d\mu (\omega ) \\& \quad = \biggl[ \int _{\varOmega }\frac{ \Vert x-y \Vert }{1+5 \Vert x-T(\omega ,x) \Vert }\,d\mu (\omega )-\phi \biggl( \int _{\varOmega }\frac{ \Vert x-y \Vert }{1+5 \Vert x-T(\omega ,x) \Vert }\,d\mu (\omega ) \biggr) \biggr] \\& \quad \leq e^{7 \Vert x-T(\omega ,x) \Vert }\biggl[ \int _{\varOmega }\frac{ \Vert x-y \Vert }{1+5 \Vert x-T(\omega ,x) \Vert }\,d\mu (\omega ) \\& \qquad {}-\frac{1}{3} \int _{\varOmega }\frac{ \Vert x-y \Vert }{1+5 \Vert x-T(\omega ,x) \Vert }\,d\mu (\omega )\biggr]. \end{aligned}$$
(4.10)

Clearly, T satisfies condition (2.7). Suppose that \(\{u_{n}(\omega )\}\) is the random CR-iteration process defined by (2.8), where \(\alpha _{n}=\beta _{n}=\frac{n^{2}}{1+n^{2}}\), \(\gamma _{n}=\frac{n^{3}}{1+n ^{3}}\). Then \(0<\alpha \leq \alpha _{n}\), \(0<\beta \leq \beta _{n}\), and \(0<\gamma \leq \gamma _{n}\). Let \(\{y_{n}(\omega )\}=\frac{\omega }{n ^{2}}\) be an arbitrary sequence of measurable mapping in C. Hence, all the conditions of Theorem 4.1 are satisfied. Therefore, \(\{u_{n}( \omega )\}\) is T-stable almost surely.

Application to random nonlinear integral equation of the Hammerstein type

In this section, we shall apply Theorem 3.4 to prove the existence of a solution in a Banach space of a random nonlinear integral equation of the form

$$ x(t;\omega )=h(t;\omega )+ \int _{S}k(t,s;\omega )f\bigl(s,x(s;\omega )\bigr)\,d\mu _{0}(s), $$
(5.1)

where

  1. (i)

    S is a locally compact metric space with a metric d on \(S\times S\) equipped with a complete σ-finite measure \(\mu _{0}\) defined on the collection of Borel subsets of S;

  2. (ii)

    \(\omega \in \varOmega \), where ω is a supporting element of a set of probability measure space \((\varOmega ,\beta ,\mu )\);

  3. (iii)

    \(x(t;\omega )\) is the unknown vector-valued random variable for each \(t\in S\);

  4. (iv)

    \(h(t;\omega )\) is the stochastic free term defined for \(t\in S\);

  5. (v)

    \(k(t,s;\omega )\) is the stochastic kernel defined for t and s in S and

  6. (vi)

    \(f(t,x)\) is a vector-valued function of \(t\in S\) and x.

The integral Eq. (5.1) is interpreted as a Bochner integral (see Padgett [31]). Furthermore, we shall assume that S is the union of a countable family of compact sets \(\{C_{n}\}\) having the properties that \(C_{1}\subset C_{2}\subset \cdots \) and that for any other compact set S there is \(C_{i}\) which contains it (see Arens [6]).

Definition 5.1

([14])

We define the space \(C(S,L_{2}(\varOmega ,\beta ,\mu ))\) to be the space of all continuous functions from S into \(L_{2}(\varOmega ,\beta ,\mu )\) with the topology of uniform convergence on compacta, i.e., for each fixed \(t\in S\), \(x(t;\omega )\) is a vector-valued random variable such that

$$ \bigl\Vert x(t;\omega ) \bigr\Vert _{L_{2}(\varOmega ,\beta ,\mu )}^{2} = \int _{ \varOmega } \bigl\vert x(t;\omega ) \bigr\vert ^{2}\,d\mu (\omega )< \infty . $$

Note that \(C(S,L_{2}(\varOmega ,\beta ,\mu ))\) is a locally convex space, whose topology is defined by a countable family of semi-norms (see Yosida [36]) given by

$$ \bigl\Vert x(t;\omega ) \bigr\Vert _{n}=\sup_{t\in C_{n}} \bigl\Vert x(t;\omega ) \bigr\Vert _{L_{2}(\varOmega ,\beta ,\mu )}, \quad n=1,2,\ldots. $$

Moreover, \(C(S,L_{2}(\varOmega ,\beta ,\mu ))\) is complete relative to this topology, since \(L_{2}(\varOmega ,\beta ,\mu )\) is complete.

We define \(\mathrm{BC}=\mathrm{BC}(S,L_{2}(\varOmega ,\beta ,\mu ))\) to be the Banach space of all bounded continuous functions from S into \(L_{2}(\varOmega ,\beta ,\mu )\) with the norm

$$ \bigl\Vert x(t;\omega ) \bigr\Vert _{\mathrm{BC}}=\sup_{t\in S} \bigl\Vert x(t;\omega ) \bigr\Vert _{L_{2}(\varOmega ,\beta ,\mu )}. $$

The space \(\mathrm{BC}\subset C\) is the space of all second order vector-valued stochastic process defined on S, which is bounded and continuous in mean square. We will consider the function \(h(t;\omega )\) and \(f(t,x(t;\omega ))\) to be in the space \(C(S,L_{2}(\varOmega , \beta ,\mu ))\) with respect to the stochastic kernel. We assume that for each pair \((t,s)\), \(k(t,s;\omega )\in L_{\infty }(\varOmega ,\beta , \mu )\) and denote the norm by

$$ \bigl\Vert k(t,s;\omega ) \bigr\Vert = \bigl\Vert k(t,s;\omega ) \bigr\Vert _{L_{\infty }(\varOmega ,\beta ,\mu )}=\mu -\operatorname{ess} \sup_{\omega \in \varOmega } \bigl\vert k(t,s;\omega ) \bigr\vert . $$

Suppose that \(k(t,s;\omega )\) is such that \(|\Vert k(t,s;\omega ) \Vert |. \Vert x(s;\omega )\Vert _{L_{2}(\varOmega ,\beta ,\mu )}\) is \(\mu _{0}\)-integrable with respect to s for each \(t\in S\) and \(x(s;\omega )\) in \(C(S,L_{2}(\varOmega ,\beta ,\mu ))\), and there exists a real-valued function G defined \(\mu _{0}\)-a.e. on S, so that \(G(S)\Vert x(s;\omega )\Vert _{L_{2}(\varOmega ,\beta ,\mu )}\) is \(\mu _{0}\)-integrable and for each pair \((t,s)\in S\times S\),

$$ \bigl|\bigl\Vert k(t,u;\omega )-k(s,u;\omega )\bigr\Vert \bigr|. \bigl\Vert x(u,\omega ) \bigr\Vert _{L_{2}(\varOmega ,\beta ,\mu )}\leq G(u) \bigl\Vert x(u,\omega ) \bigr\Vert _{L_{2}(\varOmega ,\beta ,\mu )} $$

\(\mu _{0}\)-a.e. Furthermore, for almost all \(s\in S\), \(k(t,s;\omega )\) will be continuous in t from S into \(L_{\infty }(\varOmega ,\beta , \mu )\).

Now, we define the random integral operator T on \(C(S,L_{2}(\varOmega ,\beta ,\mu ))\) by

$$ (Tx) (t;\omega )= \int _{S} k(t,s;\omega )x(s;\omega )\,d\mu _{0}(s), $$
(5.2)

where the integral is a Bochner integral. Moreover, we have that for each \(t\in S\), \((Tx)(t;\omega )\in L_{2}(\varOmega ,\beta ,\mu )\) and that \((Tx)(t;\omega )\) is continuous in mean square by Lebesgue dominated convergence theorem. So \((Tx)(t;\omega )\in C(S,L_{2}(\varOmega ,\beta , \mu ))\).

Definition 5.2

([1, 24])

Let B and D be two Banach spaces. The pair \((B,D)\) is said to be admissible with respect to a random operator \(T(\omega )\) if \(T(\omega )(B)\subset D\).

Lemma 5.1

([20])

The linear operator T defined by (5.2) is continuous from \(C(S,L_{2}(\varOmega , \beta ,\mu ))\) into itself.

Lemma 5.2

([20, 24])

If T is a continuous linear operator from \(C(S,L_{2}(\varOmega ,\beta ,\mu ))\) into itself and \(B,D\subset C(S,L _{2}(\varOmega ,\beta ,\mu ))\) are Banach spaces stronger than \(C(S,L_{2}(\varOmega ,\beta ,\mu ))\) such that \((B,D)\) is admissible with respect to T, then T is continuous from B into D.

Remark 5.1

([31])

The operator T defined by (5.2) is a bounded linear operator from B into D. It is to be noted that a random solution of Eq. (5.1) will mean a function \(x(t;\omega )\) in \(C(S,L_{2}( \varOmega ,\beta ,\mu ))\) which satisfies Eq. (5.1) μ-a.e.

We now prove the following theorem.

Theorem 5.1

We consider the stochastic integral Eq. (5.1) subject to the following conditions:

  1. (a)

    B and D are Banach spaces stronger than \(C(S,L_{2}(\varOmega , \beta ,\mu ))\) such that \((B,D)\) is admissible with respect to the integral operator defined by (5.2);

  2. (b)

    \(x(t;\omega )\to f(t,x(t;\omega ))\) is an operator from the set

    $$ Q(\rho )=\bigl\{ x(t;\omega ): x(t;\omega )\in D, \bigl\Vert x(t;\omega ) \bigr\Vert _{D}\leq \rho \bigr\} $$

    into the space B satisfying

    $$ \begin{aligned}[b] &\bigl\Vert f\bigl(t,x_{1}(t;\omega )\bigr)-f\bigl(t,x_{2}(t; \omega )\bigr) \bigr\Vert _{B}\\ &\quad \leq \bigl\Vert x_{1}(t; \omega )-x_{2}(t;\omega ) \bigr\Vert _{D} -\phi \bigl( \bigl\Vert x_{1}(t; \omega )-x_{2}(t;\omega ) \bigr\Vert _{D}\bigr) \end{aligned} $$
    (5.3)

    for all \(x_{1}(t;\omega ), x_{2}(t;\omega )\in Q(\rho )\), where \(\phi :\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) is a continuous and nondecreasing function such that \(\phi (t)>0\) for each \(t\in (0, \infty )\) and \(\phi (0)=0 \).

  3. (c)

    \(h(t;\omega )\in D\).

Then there exists a unique random solution of (5.1) in \(Q(\rho )\), provided

$$ \bigl\Vert h(t;\omega ) \bigr\Vert _{D}+c(\omega ) \bigl\Vert f(t;0) \bigr\Vert _{B}\leq \rho \bigl(1-c(\omega ) \bigr), $$
(5.4)

where \(c(\omega )\) is the norm of \(T(\omega )\).

Proof

We define the operator \(U(\omega )\) from \(Q(\rho )\) into D as follows:

$$ (Ux) (t;\omega )=h(t;\omega )+ \int _{S} k(t,s;\omega )f\bigl(s,x(s;\omega )\bigr)\,d \mu _{0}(s). $$
(5.5)

Next we have

$$ \begin{aligned}[b] \bigl\Vert (Ux) (t;\omega ) \bigr\Vert _{D}&\leq \bigl\Vert h(t;\omega ) \bigr\Vert _{D}+c( \omega ) \bigl\Vert f\bigl(t,x(t;\omega )\bigr) \bigr\Vert _{B} \\ &\leq \bigl\Vert h(t;\omega ) \bigr\Vert _{D}+c(\omega ) \bigl\Vert f(t;0) \bigr\Vert _{B}+c( \omega ) \bigl\Vert f\bigl(t,x(t; \omega )\bigr)-f(t;0) \bigr\Vert _{B}. \end{aligned} $$
(5.6)

Using the condition of (5.3), we have

$$ \begin{aligned}[b] \bigl\Vert f\bigl(t,x(t;\omega )\bigr)-f(t;0) \bigr\Vert _{B}&\leq \bigl\Vert x(t;\omega ) \bigr\Vert _{D}-\phi \bigl( \bigl\Vert x(t;\omega ) \bigr\Vert _{D}\bigr) \\ &\leq \rho -\phi (\rho ) \\ &\leq \rho . \end{aligned} $$
(5.7)

Using (5.7) in (5.6), we have

$$ \begin{aligned}[b] \bigl\Vert (Ux) (t;\omega ) \bigr\Vert _{D}&\leq \bigl\Vert h(t;\omega ) \bigr\Vert _{D}+c( \omega ) \bigl\Vert f(t;0) \bigr\Vert _{B}+c(\omega )\rho \\ &\leq \rho . \end{aligned} $$
(5.8)

This means that \((Ux)(t;\omega )\in Q(\rho )\). Then, for each \(x_{1}(t;\omega ), x_{2}(t;\omega )\in Q(\rho )\), we have by using assumption (b) that

$$ \begin{aligned}[b] &\bigl\Vert (Ux_{1}) (t;\omega )-(Ux_{2}) (t;\omega ) \bigr\Vert _{D}\\ &\quad = \biggl\Vert \int _{S} k(t,s;\omega )\bigl[f\bigl(s,x_{1}(s; \omega )\bigr)-f\bigl(s,x_{2}(s;\omega )\bigr)\bigr]\,d\mu _{0}(s) \biggr\Vert _{D} \\ &\quad \leq \bigl\Vert x_{1}(t;\omega )-x_{2}(t;\omega ) \bigr\Vert _{D} -\phi \bigl( \bigl\Vert x_{1}(t;\omega )-x_{2}(t;\omega ) \bigr\Vert _{D}\bigr). \end{aligned} $$
(5.9)

Since \(\phi :\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) is a continuous and nondecreasing function such that \(\phi (t)>0\) for each \(t\in (0,\infty )\) and \(\phi (0)=0\), it follows that \(U(\omega )\) is a nonlinear contractive operator on \(Q(\rho )\). Therefore, by Theorem 3.4 there exists a unique random fixed point \(x^{\ast }(t,\omega )\) of \(U(\omega )\), which is the random solution of Eq. (5.1). The proof of Theorem 5.1 is completed. □

The following example demonstrates the applicability of Theorem 5.1.

Example 5.1

We consider the following nonlinear stochastic integral equation:

$$ x(t;\omega )= \int _{0}^{\infty }\frac{e^{-t-s}}{16(1+ \vert x(s;\omega ) \vert )}\,ds- \frac{1}{2} \int _{0}^{\infty }\frac{e^{-t-s}}{16(1+ \vert x(s;\omega ) \vert )}\,ds. $$
(5.10)

By comparing relation (5.10) with (5.1), we observe that

$$ h(t;\omega )=0, \qquad k(t,s;\omega )=\frac{1}{4}e^{-t-s}, \qquad f\bigl(s,x(s;\omega )\bigr)=\frac{1}{4(1+ \vert x(s;\omega ) \vert )}. $$
(5.11)

By the usual computation, we clearly see that (5.3) is satisfied with \(\phi (j)=\frac{j}{2}\) and \(\phi (0)=\frac{0}{2}=0\) for all \(j\in (0,\infty )\). Hence, all the conditions of Theorem 5.1 are satisfied. Therefore, there exists a unique random fixed point \(x^{\ast }(t,\omega )\) of the integral operator T satisfying (5.2).

Conclusion

In this research, we defined a new random operator called the generalized ϕ-weakly contraction of the rational type. This new random operator includes those studied by Khan et al. [22] and Zhang et al. [37] as special cases. We also introduced the random versions of some known faster fixed point iterative schemes (see [13, 21]). We proved some convergence and stability results for our newly introduced random operator via these random fixed point iterations. An existence result was also established for a generalized random operator. Moreover, we produced some numerical examples to demonstrate the applicability of our analytical results. Furthermore, we applied our results in proving the existence of a solution of a nonlinear integral equation of the Hammerstein type. Our results generalize several known results in the literature, including the results of Khan et al. [22] and Zhang et al. [37]. Moreover, our results unify, extend, and generalize several deterministic fixed point theorems in stochastic version, including the results of Akewe et al. [4], Akewe and Okeke [3], Chugh et al. [13], and Karahan and Ozdemir [21] among others.

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Acknowledgements

The authors wish to thank the editor and the anonymous referees for their comments and useful suggestions.

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The first author’s research is supported by the Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan through grant number ASSMS/2018-2019/452.

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Correspondence to Godwin Amechi Okeke.

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Okeke, G.A., Bishop, S.A. & Akewe, H. Random fixed point theorems in Banach spaces applied to a random nonlinear integral equation of the Hammerstein type. Fixed Point Theory Appl 2019, 15 (2019) doi:10.1186/s13663-019-0665-4

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MSC

  • 47H09
  • 47H10
  • 49M05
  • 54H25

Keywords

  • Random fixed point
  • Bochner integrable
  • Generalized ϕ-weakly contraction of the rational type
  • Random iterative process
  • Almost sure T-stability
  • Nonlinear integral equation of the Hammerstein type