PPF dependent fixed point theorems for {\alpha }_c-admissible rational type contractive mappings in Banach spaces

In this paper, we prove some PPF dependent fixed point theorems in the Razumikhin class for some rational type contractive mappings involving {\alpha }_c-admissible mappings where the domain and range of the mappings are not the same. As applications of these results, we derive some PPF dependent fixed point theorems for these nonself-contractions whenever the range space is endowed with a graph. Our results extend and generalize some results in the literature.

MSC:46N40, 47H10, 54H25, 46T99.

The fixed point theory in Banach spaces plays an important role and is useful in mathematics. In fact, fixed point theory can be applied for solving equilibrium problems, variational inequalities and optimization problems. In particular, a very powerful tool is the Banach fixed point theorem, which was generalized and extended in various directions (see [1–37]). In 1977, Bernfeld et al. [2] introduced the concept of PPF dependent fixed point or the fixed point with PPF dependence which is a fixed point for mappings that have different domains and ranges. They also proved the existence of PPF dependent fixed point theorems in the Razumikhin class for Banach type contraction mappings. Very recently, some authors established the existence and uniqueness of PPF dependent fixed point for different types of contractive mappings and generalized some results of Bernfeld et al. [2] (see [1, 4, 12, 15, 20], and [33]).

In order to generalize the Banach contraction principle, Geraghty [9] proved the following theorem.

Theorem 1 (Geraghty [9])

Let (X,d) be a complete metric space and T:X\to X be an operator. Suppose that there exists \beta :[0,+\mathrm{\infty })\to [0,1) satisfying the condition

If T satisfies the following inequality:

then T has a unique fixed point.

Throughout this paper, let (E,{\parallel \cdot \parallel }_E) be a Banach space, I denotes a closed interval [a,b] in ℝ and E_0=C(I,E) denotes the set of all continuous E-valued functions on I equipped with the supremum norm {\parallel \cdot \parallel }_{E_0} defined by

For a fixed element c\in I, the Razumikhin or minimal class of functions in E_0 is defined by

Clearly, every constant function from I to E is a member of {\mathcal{R}}_c. It is easy to see that the class {\mathcal{R}}_c is algebraically closed with respect to difference, i.e., \varphi -\xi \in {\mathcal{R}}_c when \varphi ,\xi \in {\mathcal{R}}_c. Also the class {\mathcal{R}}_c is topologically closed if it is closed with respect to the topology on E_0 generated by the norm {\parallel \cdot \parallel }_{E_0}.

Definition 1 ([2])

A mapping \varphi \in E_0 is said to be a PPF dependent fixed point or a fixed point with PPF dependence of mapping T:E_0\to E if T\varphi =\varphi (c) for some c\in I.

Definition 2 ([2])

The mapping T:E_0\to E is called a Banach type contraction if there exists k\in [0,1) such that

for all \varphi ,\xi \in E_0.

Samet in 2012 introduced the concepts of α-ψ-contractive and α-admissible mappings. Karapınar and Samet generalized these notions to obtain other fixed point results. Many authors generalized these notions to obtain fixed point results (see [18, 19, 21–23], and [32]).

Samet et al. [31], defined the notion of α-admissible mappings as follows:

Definition 3 ([31])

Let T be a self-mapping on X and \alpha :X\times X\to [0,\mathrm{\infty }) be a function. We say that T is an α-admissible mapping if

Definition 4 ([17])

Let f:X\to X and \alpha :X\times X\to [0,+\mathrm{\infty }). We say that f is a triangular α-admissible mapping if

(T1) \alpha (x,y)\ge 1 implies \alpha (fx,fy)\ge 1, x,y\in X,

(T2) \{\begin{array}{l}\alpha (x,z)\ge 1,\\ \alpha (z,y)\ge 1\end{array} implies \alpha (x,y)\ge 1, x,y,z\in X.

The concept of {\alpha }_c-admissible mapping was introduced by Agarwal et al. in 2013 (see [1]).

Definition 5 ([1])

Let c\in I, T:E_0\to E, and \alpha :E\times E\to [0,\mathrm{\infty }). We say T is an {\alpha }_c-admissible mapping if for \varphi ,\xi \in E_0

Definition 6 ([4])

Let c\in I, T:E_0\to E, and \alpha :E\times E\to [0,\mathrm{\infty }). We say T is a triangular {\alpha }_c-admissible mapping if

(T1) \alpha (\varphi (c),\xi (c))\ge 1 implies \alpha (T\varphi ,T\xi )\ge 1,

(T2) \alpha (\varphi (c),\mu (c))\ge 1 and \alpha (\mu (c),\xi (c))\ge 1 implies \alpha (\varphi (c),\xi (c))\ge 1

for \varphi ,\xi ,\mu \in E_0.

Lemma 1 ([4])

Let T:E_0\to E be a triangular {\alpha }_c-admissible mapping. Define the sequence \{{\varphi }_n\} in the following way:

for all n\in \mathbb{N}, where {\varphi }_0\in {\mathcal{R}}_c is such that \alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1. Then

Let ℱ denotes the class of all functions \beta :[0,+\mathrm{\infty })\to [0,1) satisfying the following condition:

Definition 7 Let T:E_0\to E be a nonself-mapping and \alpha :E\times E\to [0,\mathrm{\infty }) be a function. We say T is a rational Geraghty contraction of type I if there exist \beta \in \mathcal{F} and c\in I such that

for all \varphi ,\xi \in E_0, where

Theorem 2 Let T:E_0\to E and \alpha :E\times E\to [0,\mathrm{\infty }) be two mappings satisfying the following assertions:

1. (a)

   there exists c\in I such that {\mathcal{R}}_c is topologically closed and algebraically closed with respect to difference,

2. (b)

   T is an {\alpha }_c-admissible,

3. (c)

   T is a rational Geraghty contractive mapping of type I,

4. (d)

   if \{{\varphi }_n\} is a sequence in E_0 such that {\varphi }_n\to \varphi as n\to \mathrm{\infty } and \alpha ({\varphi }_n(c),T{\varphi }_n)\ge 1, then \alpha (\varphi (c),T\varphi )\ge 1 for all n\in \mathbb{N},

5. (e)

   there exists {\varphi }_0\in {\mathcal{R}}_c such that \alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1.

Then T has a unique PPF dependent fixed point {\varphi }^{\ast }\in {\mathcal{R}}_c. Moreover, for a fixed {\varphi }_0\in {\mathcal{R}}_c, if the sequence \{{\varphi }_n\} of iterates of T is defined by T{\varphi }_{n-1}={\varphi }_n(c) for all n\in \mathbb{N}, then \{{\varphi }_n\} converges to {\varphi }^{\ast }\in {\mathcal{R}}_c.

Proof Let {\varphi }_0 is a point in {\mathcal{R}}_c\subset E_0 such that \alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1. Since T{\varphi }_0\in E, there exists x_1\in E such that T{\varphi }_0=x_1. Choose {\varphi }_1\in {\mathcal{R}}_c such that x_1={\varphi }_1(c). Since {\varphi }_1\in {\mathcal{R}}_c\subset E_0 and, by hypothesis, we get T{\varphi }_1\in E. This implies that there exists x_2\in E such that T{\varphi }_1=x_2. Thus, we can choose {\varphi }_2\in {\mathcal{R}}_c such that x_2={\varphi }_2(c). Continuing this process, by induction, we can build the sequence \{{\varphi }_n\} in {\mathcal{R}}_c\subset E_0 such that T{\varphi }_{n-1}={\varphi }_n(c) for all n\in \mathbb{N}. It follows from the fact that {\mathcal{R}}_c is algebraically closed with respect to difference

Since T is {\alpha }_c-admissible and \alpha ({\varphi }_0(c),{\varphi }_1(c))=\alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1, we deduce that

By continuing this process, we get \alpha ({\varphi }_{n-1}(c),T{\varphi }_{n-1})\ge 1 for all n\in \mathbb{N}. Since T is a rational Geraghty contraction of type I, we have

On the other hand,

If

from (2.2) we have

which is a contradiction. So,

By (2.2) we conclude

for all n\in \mathbb{N}. This implies that the sequence \{{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\} is decreasing in {\mathbb{R}}_{+}. So, it is convergent. Suppose that there exists r\ge 0 such that {lim}_{n\to +\mathrm{\infty }}{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}=r. Assume that r>0. Taking the limit as n\to +\mathrm{\infty } from (2.4) we conclude

which implies 1\le {lim}_{n\to +\mathrm{\infty }}\beta ({\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}). So,

and since \beta \in \mathcal{F}, {lim}_{n\to +\mathrm{\infty }}{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}=0, which is a contradiction. Hence, r=0. This means

We prove that the sequence \{{\varphi }_n\} is a Cauchy sequence in {\mathcal{R}}_c. Assume that \{{\varphi }_n\} is not a Cauchy sequence, then

Since T is a rational Geraghty contraction of type I, we have

Taking the limit when m,n\to \mathrm{\infty } in the above inequality and applying (2.5) we deduce

where

Letting m,n\to \mathrm{\infty } in the above inequality and applying (2.5), we get

So, by (2.7) and (2.9), we have

and hence from (2.6) we get 1\le {lim\hspace{0.17em}sup}_{m,n\to +\mathrm{\infty }}\beta ({\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}). This means

and since \beta \in \mathcal{F}, we conclude

which is a contradiction. Consequently,

and hence \{{\varphi }_n\} is a Cauchy sequence in {\mathcal{R}}_c\subseteq E_0. By Completeness of E_0, we find that \{{\varphi }_n\} converges to a point {\varphi }^{\ast }\in E_0, this means {\varphi }_n\to {\varphi }^{\ast }, as n\to +\mathrm{\infty }. Since {\mathcal{R}}_c is topologically closed, we deduce, {\varphi }^{\ast }\in {\mathcal{R}}_c. By condition b, we have \alpha ({\varphi }^{\ast }(c),T{\varphi }^{\ast })\ge 1. Now, since T is a rational Geraghty contraction of type I, we have

Taking the limit as n\to \mathrm{\infty } in the above inequality, we get

But

Therefore, from (2.10) and (2.11), we deduce

that is,

which implies that {\varphi }^{\ast } is a PPF dependent fixed point of T in {\mathcal{R}}_c. Now, we show that T has a unique PPF dependent fixed point in {\mathcal{R}}_c. Suppose on the contrary that {\varphi }^{\ast } and {\phi }^{\ast } are two PPF dependent fixed points of T in {\mathcal{R}}_c such that {\varphi }^{\ast }\ne {\phi }^{\ast }. Then

where

Therefore,

which is a contradiction. Hence, {\varphi }^{\ast }={\phi }^{\ast }. Then T has a unique PPF dependent fixed point in {\mathcal{R}}_c. □

Definition 8 Let \alpha :E\times E\to [0,\mathrm{\infty }) and T:E_0\to E. We say that T is a rational Geraghty contraction of type II if there exist \beta \in \mathcal{F} and c\in I such that

for all \varphi ,\xi \in E_0, where

Theorem 3 Let T:E_0\to E and \alpha :E\times E\to [0,\mathrm{\infty }) be two mappings satisfying the following assertions:

1. (a)

   there exists c\in I such that {\mathcal{R}}_c is topologically closed and algebraically closed with respect to difference,

2. (b)

   T is an {\alpha }_c-admissible,

3. (c)

   T is a rational Geraghty contractive mapping of type II,

4. (d)

   if \{{\varphi }_n\} is a sequence in E_0 such that {\varphi }_n\to \varphi as n\to \mathrm{\infty } and \alpha ({\varphi }_n(c),T{\varphi }_n)\ge 1, then \alpha (\varphi (c),T\varphi )\ge 1 for all n\in \mathbb{N},

5. (e)

   there exists {\varphi }_0\in {\mathcal{R}}_c such that \alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1.

Then T has a unique PPF dependent fixed point {\varphi }^{\ast }\in {\mathcal{R}}_c. Moreover, for a fixed {\varphi }_0\in {\mathcal{R}}_c, if the sequence \{{\varphi }_n\} of iterates of T is defined by T{\varphi }_{n-1}={\varphi }_n(c) for all n\in \mathbb{N}, then \{{\varphi }_n\} converges to {\varphi }^{\ast }\in {\mathcal{R}}_c.

Proof Suppose that {\varphi }_0 is a point in {\mathcal{R}}_c\subset E_0 such that \alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1. Since T{\varphi }_0\in E, there exists x_1\in E such that T{\varphi }_0=x_1. Choose {\varphi }_1\in {\mathcal{R}}_c such that x_1={\varphi }_1(c). Since {\varphi }_1\in {\mathcal{R}}_c\subset E_0 and, by hypothesis, we get T{\varphi }_1\in E. This implies that there exists x_2\in E such that T{\varphi }_1=x_2. Thus, we can choose {\varphi }_2\in {\mathcal{R}}_c such that x_2={\varphi }_2(c). Continuing this process, by induction, we can build the sequence \{{\varphi }_n\} in {\mathcal{R}}_c\subset E_0 such that T{\varphi }_{n-1}={\varphi }_n(c) for all n\in \mathbb{N}. It follows from the fact that {\mathcal{R}}_c is algebraically closed with respect to difference

Since T is {\alpha }_c-admissible and \alpha ({\varphi }_0(c),{\varphi }_1(c))=\alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1, we deduce that

Continuing this process, we get \alpha ({\varphi }_{n-1}(c),T{\varphi }_{n-1})\ge 1 for all n\in \mathbb{N}. Since T is a rational Geraghty contraction of type II, we have

On the other hand,

From (2.12) we conclude

for all n\in \mathbb{N}. So, the sequence \{{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\} is decreasing in {\mathbb{R}}_{+} and there exists r\ge 0 such that {lim}_{n\to +\mathrm{\infty }}{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}=r. Reviewing the proof of Theorem 2, we can show that r=0, i.e.,

Now, we prove that the sequence \{{\varphi }_n\} is Cauchy in {\mathcal{R}}_c. If not, then

From the fact that T is a rational Geraghty contraction of type II, we have

Letting m,n\to \mathrm{\infty } in the above inequality and applying (2.14) we deduce

where