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Iterative algorithms for solutions of Hammerstein equations in real Banach spaces

Abstract

Let B be a uniformly convex and uniformly smooth real Banach space with dual space \(B^{*}\). Let \(F:B\to B^{*}\), \(K:B^{*} \to B\) be maximal monotone mappings. An iterative algorithm is constructed and the sequence of the algorithm is proved to converge strongly to a solution of the Hammerstein equation \(u+KFu=0\). This theorem is a significant improvement of some important recent results which were proved in real Hilbert spaces under the assumption that F and K are maximal monotone continuous and bounded. The continuity and boundedness restrictions on K and F have been dispensed with, using our new method, even in the more general setting considered in our theorems. Finally, numerical experiments are presented to illustrate the convergence of the sequence of our algorithm.

Introduction

Let B be a real Banach space with a strictly convex dual space \(B^{*}\). Consider the Hammerstein equation

$$ (I +KF)u=0, $$
(1.1)

where \(F:B \to B^{*}\) is a nonlinear mapping and \(K: B^{*} \to B\) is a linear map such that \(R(F) \subset D(K)\). If Ω denotes a domain of σ-finite measure dy in \(\Bbb {R}^{N}\), and \(\kappa: \varOmega \times \varOmega \to \Bbb {R}\) and \(f: \varOmega \times \Bbb {R}\to \Bbb {R}\) are measurable real-valued functions on Ω, one can define a linear integral operator K by \(Kv:= \int _{\varOmega }\kappa ( \cdot,y) v(y)\,dy \) and an operator F by the Nemitskyi or superposition operator given by \(Fu:= f (\cdot, u(\cdot ) )\) to obtain equation (1.1).

Numerous problems in differential equation theory can, as a rule, be modeled by a Hammerstein equation (see, e.g., Pascali and Sburlan [40]). For example, the amplitude of oscillation \(v(t)\) is a solution of the problem

$$ \textstyle\begin{cases} \frac{d^{2}v}{dt^{2}}+a^{2}\sin v(t)=z(t), \quad t\in [0,1], \\ v(0)=v(1)=0, \end{cases} $$
(1.2)

where the driving force \(z(t)\) is periodical and odd. The constant \(a\neq 0\) depends on the length of the pendulum and on gravity. Since the Green’s function of the problem

$$ v''(t)=0, \qquad v(0)=v(1)=0 $$

is the triangular function

$$ k(t,x)= \textstyle\begin{cases} t(1-x), & 0\leq t \leq x, \\ x(1-t), & x\leq t \leq 1, \end{cases} $$
(1.3)

problem (1.2) is equivalent to the nonlinear integral equation

$$ v(t)=- \int _{0}^{1} k(t,x)\bigl[z(x)-a^{2}\sin v(x)\bigr]\,dx. $$
(1.4)

If \(\int _{0}^{1} k(t,x)z(x)\,dx=g(t)\) and \(v(t)+g(t)=u(t)\), then (1.4) can be written as the Hammerstein integral equation

$$ u(t)+ \int _{0}^{1} k(t,x)f\bigl(x,u(x)\bigr)\,dx=0, $$

where \(f(x,u(x))=a^{2}\sin [u(x)-g(x)]\).

The Hammerstein equation also plays a crucial role in the theory of optimal control systems, in automation and in network theory. For example, in studying automatic control, Narendra and Gallman [36], using a Hammerstein model, proposed an iterative method for the identification of nonlinear systems, for samples of inputs and outputs in the presence of noise. For more on problems in optimal control, automation and network systems that can be modeled as Hammerstein equations, see, e.g., Dolezal [30].

Several existence and uniqueness theorems have been proved for equations of Hammerstein type (see, e.g., Brezis and Browder [3, 4], Browder and Gupta [6], Chepanovich [7], De Figueiredo and Gupta [28]).

A map \(A: D(A)\subset B \to B^{*}\) is called monotone if for all \(x,y \in B\), \(\langle Ax-Ay, x-y \rangle \geq 0\). The map A is called maximal monotone if A is monotone, and in addition, \(R(J+\lambda A)\) is \(B^{*}\), for all \(\lambda >0\). Monotone mappings were studied in Hilbert spaces by Zarantonello [46], Minty [35], and a host of other authors. For more recent studies of monotone mappings and Hammerstein equations, the reader may see, for example, Berinde [2], Chidume [810], Chidume and Chidume [14], Goebel and Reich [31], Zeidler [47].

In general, equations of Hammerstein type are nonlinear and thus, there are no closed-form solutions of such equations. Consequently, methods for approximating such equations are of interest. Several attempts have been made to approximate solutions of equations of Hammerstein type. An early method introduced by Brezis and Browder [5] was in the case where one of the operators is angle bounded (see, e.g., Pascali and Sburlan [40]). They proved strong convergence of a suitably defined Galerkin approximation to a solution of (1.1) as the following theorem.

Theorem 1.1

LetHbe a separable Hilbert space andCbe a closed subspace ofH. Let\(K: H \to C\)be a bounded continuous monotone operator and\(F: C\to H \)be an angle-bounded and weakly compact mapping. For a given\(f\in C\), consider the Hammerstein equation

$$ ( I + KF ) u = f $$
(1.5)

and itsnth Galerkin approximation given by

$$ ( I + K_{n} F_{n} ) u^{n} = P_{n}^{*}f, $$
(1.6)

with\(K_{n} = P_{n}^{*} KP_{n}: H\to C\)and\(F_{n} = P_{n} F P_{n} ^{*}: C_{n} \to H\), where the symbols have their usual meanings (see [40]). Then, for each\(n \in N\), the Galerkin approximation (1.6) admits a unique solution\(u^{n} \in C_{n}\)and\(\{ u^{n}\}\)converges strongly inHto the unique solution\(u \in C\)of the equation (1.5).

The first iterative methods for approximating solutions of Hammerstein equations, in real Banach spaces more general than Hilbert spaces, as far as we know, were obtained by Chidume and Zegeye [25] (see also Chidume [10], Chap. 13).

Let B be a real Banach space and \(F,K:B \to B\) be accretive-type mappings. Let \(E:=B\times B\). Then, define \(T:E \to E\) by

$$ T[u,v]=[Fu-v,Kv+u], \quad\text{for } [u,v]\in E. $$

We note that \(T[u,v]=0 \Longleftrightarrow u\) solves (1.1) and \(v=Fu\). With this, Chidume and Zegeye [25] were able to prove strong convergence of an iterative algorithm defined in the Cartesian product space E to a solution of the Hammerstein equation (1.1). Extensions of these early results of Chidume and Zegeye [25] were obtained by several authors (see, e.g., Chidume and Zegeye [26, 27], Chidume and Djitte [1517], Chidume and Ofoedu [21], Chidume and Shehu [22, 23], Chidume et al. [12, 20] Djitte and Sene [29], Ofoedu and Onyi [39], Ofoedu and Malonza [38], Shehu [43], Minjibir and Mohammed [34], and the references contained therein).

In 2015, Chidume and Shehu [24] introduced a \(new\) explicit iterative algorithm in the setting of a real Hilbert space and proved the following strong convergence theorem.

Theorem 1.2

LetHbe a real Hilbert space. Let\(K,F:H \to H\)be bounded, continuous and monotone mappings. Suppose that\(u^{*}\in H\)is a solution of\(u+KFu=0\). Let\(\{u^{n}\}\)and\(\{v^{n}\}\)be sequences inHdefined iteratively from arbitrary\(u^{1},v^{1}\in H\)by

$$ \textstyle\begin{cases} u^{n+1}=u^{n}-\beta _{n}^{2}(Fu^{n}-v^{n})-\beta _{n}\alpha _{n}(u^{n}-u ^{1}), \\ v^{n+1}=v^{n}-\beta _{n}^{2}(Kv^{n}+u^{n})-\beta _{n}\alpha _{n}(v^{n}-v ^{1}), \end{cases} $$
(1.7)

where\(\{\alpha _{n}\}\)and\(\{\beta _{n}\}\)are real sequences in\((0,1)\)such that

$$ \sum_{n=1}^{\infty }\beta _{n}^{2}< \infty\quad \textit{and}\quad \sum_{n=1}^{ \infty }\alpha _{n}\beta _{n}=\infty. $$

Then, there exists a real constant\(\epsilon _{0}>0\)such that if\(\beta _{n}<\epsilon _{0}\alpha _{n}\), \(\forall n\geq n_{0}\), for some\(n_{0}\in \Bbb {N}\), the sequence\(\{u^{n}\}\)converges to\(u^{*}\).

Recently, Chidume and Bello [13], studied a coupled explicit algorithm different from given by (1.7) and proved the following strong convergence theorem in \(L_{p}\) spaces, \(1< p<2\).

Theorem 1.3

Let\(E=L_{p}\), \(1< p<2\). Let\(F:E\to E^{*}\)and\(K:E^{*} \to E\)be strongly monotone and bounded maps. For\((u^{0},v^{0})\in E\times E ^{*}\), define the sequences\(\{u^{n}\}\)and\(\{v^{n}\}\)inEand\(E^{*}\)respectively by

$$ \textstyle\begin{cases} u^{n+1} =J^{-1}(Ju^{n}-\alpha _{n}(Fu^{n}-v^{n})), \\ v^{n+1} =J(J^{-1}v^{n}-\alpha _{n}(Kv^{n}+u^{n})), \quad n\geq 0, \end{cases} $$
(1.8)

where\(\{\alpha _{n}\}\subset (0,1)\)satisfies the following conditions: \(\sum_{n=1}^{\infty }\alpha _{n}=\infty \), \(\sum_{n=1}^{\infty }\alpha _{n}^{2} <\infty \)and\(\sum_{n=1}^{\infty }(\alpha _{n})^{ \frac{q}{q-1}}<\infty \), withqbeing such that\(\frac{1}{q}+ \frac{1}{p}=1\). Assume that the equation\(u+KFu=0\)has a solution. Then there exists\(\gamma _{0}>0\)such that if\(\alpha _{n} \leq \gamma _{0}\)for all\(n\geq 1\), the sequences\(\{ u^{n} \}\)and\(\{ v^{n} \}\)converge strongly touandv, respectively, whereuis the solution of\(u + KFu = 0\)with\(v = F u\).

In 2017, Uba et al. [44] proved the following theorem:

Theorem 1.4

Let E be a uniformly convex and uniformly smooth real Banach space and\(F:E\to E^{*}\), \(K:E^{*}\to E\)be maximal monotone and bounded maps. For\(u_{1} \in E, v_{1}\in E^{*}\)define the sequences\(\{u_{n}\}\)and\(\{v^{n}\}\)inEand\(E^{*}\), respectively, by

$$ \textstyle\begin{cases} u^{n+1} =J^{-1} (Ju^{n}-\lambda _{n}(Fu^{n}-v^{n})-\lambda _{n}\theta _{n}(Ju^{n}-Ju^{1}) ), \\ v^{n+1} =J (J^{-1}v^{n}-\lambda _{n}(Kv^{n}+u^{n})-\lambda _{n} \theta _{n}(J^{-1}v^{n}-J^{-1}v^{1}) ), \quad n\geq 1. \end{cases} $$
(1.9)

Assume that the equation\(u + KF u = 0\)has a solution. Then, the sequences\(\{u^{n}\}\)and\(\{v^{n}\}\)converge strongly to\(u^{*}\)and\(v^{*}\), respectively, where\(u^{*}\)is the solution of\(u+KF u = 0\)with\(v^{*} = Fu^{*}\), where\(\{\lambda _{n}\}\)and\(\{\theta _{n}\}\)are sequences in\((0,1)\)satisfying some appropriate conditions.

Remark 1

We observe that in Theorems 1.2, 1.3, and 1.4, the operators K and F are required to be bounded. This is a serious drawback on these theorems.

It is our purpose in this paper to first prove a significant improvement of Theorem 1.2. We introduce a new recursive formula, and extend Theorem 1.2 to uniformly convex and uniformly smooth real Banach spaces and, at the same time, dispense with the requirement in Theorem 1.2 that the mappings K and F be continuous and bounded. Furthermore, using our new recursive formula, we are able to prove strong convergence without imposing any boundedness assumption on the mappings K and F. This makes our theorem much more applicable than Theorem 1.4. Finally, we give numerical experiments to illustrate the convergence of the sequence of our theorem.

Preliminaries

In this section, we present definitions of some terms, and results that will be needed in the proof of our main theorem.

Definition 2.1

Let E be a smooth real Banach space. The Lyapunov functional \(\phi:E\times E \to \Bbb {R}\) is defined by

$$ \phi (u,v)= \Vert u \Vert ^{2}-2\langle u, Jv \rangle + \Vert v \Vert ^{2}, \quad\forall u,v \in E. $$
(2.1)

It was introduced by Alber and has been studied by many authors (see, e.g., Alber [1]; Chidume et al. [11, 19]; Kamimura and Takahashi [32]; Nilsrakoo and Saejung [37]; and the references therein). It is easy to see that from the definition of ϕ,

$$ \bigl( \Vert u \Vert - \Vert v \Vert \bigr)^{2} \leq \phi (u,v)\leq \bigl( \Vert u \Vert + \Vert v \Vert \bigr)^{2},\quad \forall u,v \in E. $$
(2.2)

Definition 2.2

Let E be a normed linear space and consider the map \(V:E\times E ^{*} \to \Bbb {R}\) defined by

$$ V\bigl(u,u^{*}\bigr)= \Vert u \Vert ^{2}-2\bigl\langle u, u^{*}\bigr\rangle + \bigl\Vert u^{*} \bigr\Vert ,\quad \forall u \in E, u^{*}\in E^{*}. $$

Observe that

$$ V\bigl(u,u^{*}\bigr)=\phi \bigl(u,J^{-1}u^{*} \bigr), \quad\forall u\in E, u^{*}\in E^{*}. $$

Lemma 2.3

(Alber and Ryazantseva [1])

LetEbe a reflexive strictly convex and smooth Banach space with\(E^{*}\)as its dual. Then,

$$ V\bigl(u,u^{*}\bigr)+2\bigl\langle J^{-1}u^{*}-u,v^{*} \bigr\rangle \leq V\bigl(u,u^{*}+v^{*}\bigr), $$
(2.3)

for all\(u\in E\)and\(u^{*},v^{*}\in E^{*}\).

Lemma 2.4

(Alber and Ryazantseva [1])

LetEbe a reflexive strictly convex and smooth Banach space with dual space\(E^{*}\). Let\(W:E\times E \to \Bbb {R}\)be defined by\(W(u,v)= \frac{1}{2}\phi (v,u)\). Then, \(\forall u,v,s \in E\),

$$ W(u,v)-W(s,u)\geq \langle s-v, Ju-Js \rangle, \quad\textit{i.e.}, \phi (v,u)-\phi (u,s) \geq 2 \langle s-v, Ju-Js\rangle, $$

and also, \(W(u,v)\leq \langle u-v, Ju-Jv \rangle \).

Lemma 2.5

(Chidume and Idu [18])

LetEbe a smooth real Banach space with dual space\(E^{*}\). Let\(\phi: E\times E \to \Bbb {R}\)be the Lyapunov functional. Then,

$$ \phi (v,u)=\phi (u,v)-\langle u+v, Ju-Jv\rangle +2\bigl( \Vert u \Vert ^{2}- \Vert v \Vert ^{2}\bigr),\quad \forall u,v \in E. $$

Lemma 2.6

(Rockafellar [42]; see also, Pascali and Sburlan [40])

A monotone mapping\(T:E\to 2^{E^{*}}\)is locally bounded at the interior points of its domain.

Remark 2

Let \(E^{*}\) be a real strictly convex dual space with a Fréchet differentiable norm. Let \(A: E \to E^{*}\) be a maximal monotone operator with no monotone extension. Let \(s\in E^{*}\) be fixed. Then for every \(\rho >0\), there exists a unique \(u_{\rho } \in E\) such that \(Ju_{\rho }+\rho Au_{\rho } \ni s\) (see Reich [41]). Setting \(J_{\rho } s=u_{\rho }\), we have the resolvent \(J_{\rho }:=(J+\rho A)^{-1}: E^{*}\to E\) of A for every \(\rho >0\). The following is a celebrated result of Reich.

Lemma 2.7

(Reich [41])

Let\(E^{*}\)be a real strictly convex dual space with a Fréchet differentiable norm, and letAbe a maximal monotone operator fromEto\(E^{*}\)such that\(A^{-1}0 \neq \emptyset \). Let\(s\in E^{*}\)be arbitrary but fixed. For each\(\rho >0\)there exists a unique\(u_{\rho } \in E\)such that\(Ju_{\rho }+\rho Au_{\rho } \ni s\). Furthermore, \(u_{\rho }\)converges strongly to a unique point\(p\in A^{-1}0\).

Remark 3

From Lemma 2.7, setting \(\rho _{n}=\frac{\beta _{n}}{\alpha _{n}}\) where \(\frac{\alpha _{n}}{\beta _{n}} \to 0\), as \(n\to \infty, s=Jv\) for some \(v\in E\), and \(y^{n}:= (J+\frac{\beta _{n}}{\alpha _{n}}A )^{-1}s\), we obtain

$$ Ay^{n}=\frac{\alpha _{n}}{\beta _{n}} \bigl(Jv-Jy^{n} \bigr), $$
(2.4)

\(y^{n} \to y^{*} \in A^{-1}0\), where \(A:E\to E^{*}\) is maximal monotone.

Remark 4

Let \(r>0\) such that \(\|v\|\leq r,\|y^{n}\|\leq r\), for all \(n \geq 1\). We observe that equation (2.4) yields

$$ Jy^{n-1}-Jy^{n}+\frac{\beta _{n}}{\alpha _{n}} \bigl(Ay^{n-1}-Ay^{n}\bigr)= \biggl( \frac{\alpha _{n-1}\beta _{n}-\alpha _{n}\beta _{n-1}}{\alpha _{n}\beta _{n-1}} \biggr) \bigl(Jv-Jy^{n-1}\bigr). $$
(2.5)

From (2.5), applying Cauchy–Schwarz inequality, and using the fact that A is monotone, we obtain

$$ \bigl\langle y^{n-1}-y^{n}, Jy^{n-1}-Jy^{n} \bigr\rangle \leq \biggl( \frac{ \alpha _{n-1}\beta _{n}-\alpha _{n}\beta _{n-1}}{\alpha _{n}\beta _{n-1}} \biggr) \bigl\Vert Jv-Jy^{n-1} \bigr\Vert \bigl\Vert y^{n-1}-y^{n} \bigr\Vert . $$

It follows that if E is uniformly convex and uniformly smooth, using a result of Alber and Ryazantseva [1] (see, e.g., Chidume and Idu [18, Lemma 2.3]), we obtain

$$\begin{aligned} (2L)^{-1}\delta _{E}\bigl(c_{2}^{-1} \bigl\Vert y^{n-1}-y^{n} \bigr\Vert \bigr) &\leq \biggl( \frac{ \alpha _{n-1}\beta _{n}-\alpha _{n}\beta _{n-1}}{\alpha _{n}\beta _{n-1}} \biggr) \bigl\Vert Jv-Jy^{n-1} \bigr\Vert \bigl\Vert y^{n-1}-y^{n} \bigr\Vert \\ &\leq 2r\sup \bigl\{ \bigl\Vert Jv-Jy^{n-1} \bigr\Vert \bigr\} \biggl( \frac{\alpha _{n-1} \beta _{n}-\alpha _{n}\beta _{n-1}}{\alpha _{n}\beta _{n-1}} \biggr) , \end{aligned}$$
(2.6)

which gives

$$ \bigl\Vert y^{n-1}-y^{n} \bigr\Vert \leq c_{2}\delta _{E}^{-1} \biggl( \frac{\alpha _{n-1} \beta _{n}-\alpha _{n}\beta _{n-1}}{\alpha _{n}\beta _{n-1}} D \biggr). $$
(2.7)

Similarly, using a result of Alber and Ryazantseva [1] (see, e.g., Chidume and Idu [18, Lemma 2.4]), we obtain

$$ \bigl\Vert Jy^{n-1}-Jy^{n} \bigr\Vert \leq c_{2}\delta _{E^{*}}^{-1} \biggl( \frac{\alpha _{n-1}\beta _{n}-\alpha _{n}\beta _{n-1}}{\alpha _{n}\beta _{n-1}} D \biggr). $$
(2.8)

Lemma 2.8

(Kamimura and Takahashi [32])

LetEbe a uniformly convex and smooth real Banach space, and let\(\{u^{n}\}\)and\(\{v^{n}\}\)be two sequences ofE. If either\(\{u^{n}\}\)or\(\{v^{n}\}\)is bounded and\(\phi (u^{n},v^{n} )\to 0\)then\(\Vert u^{n}-v^{n}\Vert \to 0 \).

Lemma 2.9

(Chidume and Idu [18])

LetX, Ybe real uniformly convex and uniformly smooth spaces. Then\(E=X\times Y\)is uniformly convex and uniformly smooth.

Lemma 2.10

(Chidume and Idu [18])

LetEbe a uniformly convex and uniformly smooth real Banach space and\(F: E\to E^{*}, K:E^{*} \to E\)be maximal monotone. Define\(A: E \times E^{*} \to E^{*} \times E\)by

$$ A[u,v]=[Fu-v, Kv+u],\quad \forall [u,v] \in E\times E^{*}. $$

Then, Ais maximal monotone.

Remark 5

From Lemma 2.7, setting \(\rho _{n}:=\frac{\beta _{n}}{\alpha _{n}}\) where \(\frac{\alpha _{n}}{\beta _{n}} \to 0\), as \(n \to \infty, z=[z^{1},z^{2}]=J_{E\times E^{*}}[u^{1},v^{1}]\), for some \([u^{1},v ^{1}]\in E\times E^{*}\), and \([y^{n},(y^{n})^{*}]:=(J_{E\times E^{*}}+ \frac{\beta _{n}}{\alpha _{n}}A)^{-1}[z^{1},z^{2}]\), we obtain that

$$\begin{aligned} &Jy^{n}+\frac{\beta _{n}}{\alpha _{n}}\bigl(Fy^{n}- \bigl(y^{n}\bigr)^{*}\bigr)=z^{1},\quad \forall n\geq 1,\quad \text{and} \end{aligned}$$
(2.9)
$$\begin{aligned} & J^{-1}\bigl(y^{n}\bigr)^{*}+ \frac{\beta _{n}}{\alpha _{n}}\bigl(K\bigl(y^{n}\bigr)^{*}+y^{n} \bigr)=z ^{2},\quad \forall n\geq 1. \end{aligned}$$
(2.10)

Remark 6

Let \(y^{n}\to u^{*}\) and \((y^{n})^{*}\to v^{*}\). From Lemma 2.7 we have that \([y^{n},(y^{n})^{*}]\) converges to a point in \(A^{-1}0\). This implies that \([u^{*},v^{*}]\in A^{-1}0\). Consequently, \(A[u^{*},v^{*}]=0\), that is, \(Fu^{*}-v^{*}=0\) and \(Kv^{*}+u^{*}=0\). Hence, \(v^{*}=Fu^{*}\) and \(u^{*}+KFu^{*}=0\).

The analytical representations of duality maps are known in \(L_{p}(\varOmega )\) and \(L_{q}(\varOmega )\) spaces, for \(p>1\) and \(q>1\) such that \(\frac{1}{p}+\frac{1}{q}=1\):

$$\begin{aligned} &Jz= \Vert z \Vert _{L_{p}}^{2-p} \bigl\vert z(s) \bigr\vert ^{p-2}z(s) \in L_{q}(\varOmega ),\quad s\in \varOmega, \\ &J^{-1}z= \Vert z \Vert _{L_{q}}^{2-q} \bigl\vert z(s) \bigr\vert ^{q-2}z(s) \in L_{p}(\varOmega ),\quad s\in \varOmega \end{aligned}$$

(see, e.g., Lindenstrauss and Tzafriri [33]).

Main result

Definition 3.1

A mapping \(A:E\to E^{*}\) is quasi-bounded if for any \(\sigma >0\) there exists \(\tau >0\) such that whenever \(\langle y, Ay \rangle \leq \sigma \|y\|\) and \(\|y\|\leq \sigma \) then \(\|Ay\|\leq \tau \).

We first prove the following lemma (see Pascali and Sburlan [40, Chap. III, Lemma 3.6]).

Lemma 3.2

LetEbe a real normed space with dual space\(E^{*}\). Any monotone map\(A: D(A) \subset E\to E^{*}\)with\(0\in \operatorname{Int} D(A)\)is quasi-bounded.

Proof

By Lemma 2.6, A is locally bounded at 0. Now, A is locally bounded at \(0 \Rightarrow \exists r>0\) such that

$$ \Vert Ax \Vert \leq \sigma, \quad \forall x\in B(0,r), \text{ for some } \sigma >0. $$

Using this \(\sigma >0\), suppose \(\langle Ay,y \rangle \leq \sigma \|y \|\) and \(\|y\|\leq \sigma \). Then, by monotonicity of A, we have

$$ \langle Ay,y \rangle \geq \langle Ay,x\rangle +\langle Ax,y-x\rangle ,\quad \forall x \in B(0,r). $$

Observe that

$$\begin{aligned} \langle Ax,y-x\rangle &\leq \bigl\vert \langle Ax,y-x\rangle \bigr\vert \\ &\leq \Vert Ax \Vert \bigl( \Vert y \Vert + \Vert x \Vert \bigr) \leq \sigma \bigl( \Vert y \Vert +r\bigr). \end{aligned}$$

Thus,

$$\begin{aligned} \langle Ay, x \rangle &\leq \langle Ay, y \rangle +\langle Ax,x-y \rangle \\ &\leq \sigma \Vert y \Vert +\sigma \bigl( \Vert y \Vert +r\bigr),\quad \forall x\in B(0,r). \end{aligned}$$

This implies that

$$ \bigl\vert \langle Ay, x \rangle \bigr\vert \leq \sigma \Vert y \Vert + \sigma \bigl( \Vert y \Vert +r\bigr), \quad\forall x\in B(0,r). $$

Thus,

$$ \sup_{ \Vert x \Vert \leq r} \bigl\vert \langle Ay, x \rangle \bigr\vert \leq \sigma \Vert y \Vert +\sigma \bigl( \Vert y \Vert +r\bigr). $$

And therefore,

$$ \Vert Ay \Vert \leq \frac{\sigma }{r} \Vert y \Vert +\frac{\sigma }{r} \bigl( \Vert y \Vert +r\bigr), $$

implying easily the quasi-boundedness of A. □

In Lemma 3.3 below, the sequences \(\{\alpha _{n}\}, \{\beta _{n}\}\subset (0,1)\) satisfy the following conditions:

  1. (i)

    \(\lim_{n\to \infty } \alpha _{n}=0, \lim_{n \to \infty }\frac{\alpha _{n}}{\beta _{n}}=0, \sum_{n=1}^{\infty }\alpha _{n}\beta _{n}=\infty \),

  2. (ii)

    \(\delta _{E}^{-1}(\beta _{n} M_{0})\leq \alpha _{n}\gamma _{0}\); \(\delta _{E^{*}}^{-1}(\beta _{n} M_{0}^{*})\leq \alpha _{n}\gamma _{0}; \beta _{n} S^{*}\leq \alpha _{n}\gamma _{0}\), \(\forall n\geq 1\), for some constants, \(M_{0},M_{0}^{*},S^{*},\gamma _{0}>0\).

Lemma 3.3

LetEbe a uniformly convex and uniformly smooth real Banach space. Let\(F: E \to E^{*}, K:E^{*}\to E\)be maximal monotone mappings. For arbitrary\(x\in E, y\in E^{*}\), define the sequences\(\{u^{n}\}\)and\(\{v^{n}\}\)inEand\(E^{*}\), respectively, for\(u^{1}\in E, v^{1} \in E^{*}\)by

$$\begin{aligned} &u^{n+1}=J^{-1} \bigl(Ju^{n}-\beta _{n}^{2}\bigl(Fu^{n}-v^{n}\bigr)-\alpha _{n}\beta _{n}\bigl(Ju^{n}-Jx\bigr) \bigr), \end{aligned}$$
(3.1)
$$\begin{aligned} &v^{n+1}=J \bigl(J^{-1}v^{n}-\beta _{n}^{2}\bigl(Kv^{n}+u^{n}\bigr)-\alpha _{n} \beta _{n}\bigl(J^{-1}v^{n}-J^{-1}y \bigr) \bigr) , \end{aligned}$$
(3.2)

where\(\{\alpha _{n}\}\)and\(\{\beta _{n}\}\)are sequences in\((0,1)\)satisfying conditions (i) and (ii). Assume that the equation\(u+KFu=0\)has a solution. Then, the sequences\(\{u^{n}\}\)and\(\{v^{n}\}\)are bounded.

Proof

Now, to show that sequences \(\{u^{n}\}\) and \(\{v^{n}\}\) are bounded, set

$$ w^{n}=\bigl(u^{n},v^{n}\bigr),\qquad w^{*}= \bigl(u^{*},v^{*}\bigr)\in W= E\times E^{*}, $$

where \(u^{*}\) is a solution (1.1) with \(v^{*}=Fu^{*}\).

Define \(\varPhi: W\times W \to \Bbb {R}\) by

$$ \varPhi \bigl(w^{1},w^{2}\bigr)=\phi \bigl( u^{1},u^{2}\bigr)+\phi \bigl(v^{1},v^{2} \bigr), $$

where \(w^{1}=(u^{1},v^{1})\) and \(w^{2}=(u^{2},v^{2})\). Let W be endowed with the norm

$$ \bigl\Vert (u,v) \bigr\Vert _{W}= \bigl( \Vert u \Vert ^{2}_{E}+ \Vert v \Vert ^{2}_{E^{*}} \bigr)^{\frac{1}{2}}. $$

It suffices to show that \(\{w^{n}\}\) is bounded. We show this by induction. Let \(w^{1}\in W\). Then there exists \(r>0\) such that \(\varPhi (w^{*},w^{1})\leq \frac{r}{8}\). Let \(B:= \{w=(u,v)\in W: \varPhi (w^{*},w)\leq r \}\). It suffices to show that \(\varPhi (w^{*},w ^{n})\leq r\), for all \(n\geq 1\). Let \(w,w^{1}\in B\) and \(\alpha, \beta \in (0,1)\). Then,

$$ \varPhi \bigl(w^{*},w\bigr)\leq r, \quad\text{i.e.}, \phi \bigl(u^{*},u\bigr)+\phi \bigl(v^{*},v\bigr)\leq r. $$

Therefore,

$$ \phi \bigl(u^{*},u\bigr)\leq r \quad\text{and}\quad \phi \bigl(v^{*},v \bigr)\leq r. $$

Now, using inequality (2.2),

$$ \phi \bigl(u^{*},u\bigr)\leq r \quad\Longrightarrow \quad \Vert u \Vert \leq \bigl\Vert u^{*} \bigr\Vert +\sqrt{r}. $$

Since F is also locally bounded at u, there exists \(k_{1}>0\) such that

$$ \langle u,Fu \rangle \leq k_{1} \Vert u \Vert . $$

Define \(\sigma:=\max \{k_{1},\|u^{*}\|+\sqrt{r}\}\). Hence, \(\langle u, Fu \rangle \leq \sigma \|u\|\) and \(\|u\|\leq \sigma \). By Lemma 3.2, F is quasi-bounded. Thus, there exists \(\tau _{1}>0\) such that

$$ \Vert Fu \Vert \leq \tau _{1},\quad \forall (u,v)\in B. $$

Similarly, there exists \(\tau _{2}>0\) such that

$$ \Vert Kv \Vert \leq \tau _{2},\quad \forall (u,v)\in B. $$

Define

S=max{sup{β(Fuv)+α(JuJx)}+1,sup{JuJx}+1,sup{J1(Juβ2(Fuv)αβ(JuJx))u}+1,sup{β(Kv+u)+α(J1vJ1y)}+1,sup{J1vJ1y}+1,sup{J(J1vβ2(Kv+u)αβ(J1vJ1y))v}+1},

where all the suprema are taken over \(u,v\in B\). The boundedness of \(K,F,J\) on bounded sets guarantees that \(S^{*}\) is well defined. Let \(M:=\max \{c_{2}S^{*}, S^{*} \}\), \(M_{0}=M_{0}^{*}:=2L(S^{*})^{2}\)\(\gamma _{0}:=\min \{1, \frac{r}{32M} \}\). Then, for \(n=1\), by construction \(\varPhi (w^{*},w ^{1})\leq r\). Assume \(\varPhi (w^{*},w^{n})\leq r\), for some \(n\geq 1\), i.e.,

$$ \phi \bigl(u^{*},u^{n}\bigr)+\phi \bigl(v^{*},v^{n} \bigr)\leq r, \quad\text{for some } n\geq 1. $$

We show that \(\varPhi (w^{*},w^{n+1})\leq r\). For contradiction, suppose \(r<\varPhi (w^{*},w^{n+1})\). Using a result of Alber and Ryazantseva [1] (see, e.g., Chidume and Idu, [18, Lemma 2.3]) and recurrence relation (3.1), we have

$$\begin{aligned} &(2L)^{-1}\delta _{E}\bigl(c_{2}^{-1} \bigl\Vert u^{n+1}-u^{n} \bigr\Vert \bigr) \\ &\quad\leq \bigl\langle Ju^{n+1}-Ju ^{n}, u^{n+1}-u^{n}\bigr\rangle \\ &\quad\leq \bigl\Vert Ju^{n+1}-Ju^{n} \bigr\Vert \bigl\Vert u^{n+1}-u^{n} \bigr\Vert \\ &\quad\leq \beta _{n} \bigl[\sup \bigl\{ \bigl\Vert \beta _{n} \bigl(Fu^{n}-v^{n}\bigr)+\alpha _{n}\bigl(Ju ^{n}-Jx\bigr) \bigr\Vert \bigr\} +1 \bigr] \bigl\Vert u^{n+1}-u^{n} \bigr\Vert . \end{aligned}$$

Thus,

$$ \bigl\Vert u^{n+1}-u^{n} \bigr\Vert \leq c_{2}\delta _{E}^{-1}(\beta _{n} M_{0}). $$
(3.3)

Similarly, using a result of Alber and Ryazantseva [1] (see, e.g., Chidume and Idu, [18, Lemma 2.4]) and recurrence relation (3.2), we obtain

$$ \bigl\Vert v^{n+1}-v^{n} \bigr\Vert \leq c_{2}\delta _{E^{*}}^{-1}\bigl(\beta _{n} M_{0}^{*}\bigr). $$
(3.4)

Now, using recurrence relation (3.1), Lemma 2.3, and inequality (3.3), we have

$$\begin{aligned} &\phi \bigl(u^{*},u^{n+1}\bigr) \\ &\quad= V \bigl(u^{*},Ju^{n}- \beta _{n}^{2}\bigl(Fu^{n}-v^{n}\bigr)- \alpha _{n}\beta _{n}\bigl(Ju^{n}-Jx\bigr) \bigr) \\ &\quad\leq V\bigl(u^{*},Ju^{n}\bigr)-2\bigl\langle u^{n+1}-u^{*},\beta _{n}^{2} \bigl(Fu^{n}-v ^{n}\bigr)+\alpha _{n}\beta _{n}\bigl(Ju^{n}-Jx\bigr) \bigr\rangle \\ &\quad= \phi \bigl(u^{*},u^{n}\bigr)-2\bigl\langle u^{n}-u^{*},\beta _{n}^{2} \bigl(Fu^{n}-v ^{n}\bigr)+\alpha _{n}\beta _{n}\bigl(Ju^{n}-Jx\bigr) \bigr\rangle \\ &\qquad{} -2\bigl\langle u^{n+1}-u^{n},\beta _{n}^{2} \bigl(Fu^{n}-v^{n}\bigr)+\alpha _{n}\beta _{n}\bigl(Ju^{n}-Jx\bigr) \bigr\rangle \\ &\quad\leq \phi \bigl(u^{*},u^{n}\bigr)-2\bigl\langle u^{n}-u^{*},\beta _{n}^{2} \bigl(Fu^{n}-v ^{n}\bigr)+\alpha _{n}\beta _{n}\bigl(Ju^{n}-Jx\bigr) \bigr\rangle \\ &\qquad{} + 2\beta _{n} \bigl\Vert u^{n+1}-u^{n} \bigr\Vert \bigl\Vert \beta _{n}\bigl(Fu^{n}-v^{n} \bigr)+\alpha _{n}\bigl(Ju ^{n}-Jx\bigr) \bigr\Vert \\ &\quad\leq \phi \bigl(u^{*},u^{n}\bigr)-2\beta _{n}^{2}\bigl\langle u^{n}-u^{*},Fu^{n}-v ^{n}\bigr\rangle -2\alpha _{n}\beta _{n}\bigl\langle u^{n}-u^{*},Ju^{n}-Jx\bigr\rangle \\ &\qquad{} +2\beta _{n}c_{2}S^{*}\delta _{E}^{-1}(\beta _{n}M_{0}). \end{aligned}$$
(3.5)

Observe that, by monotonicity of F and the fact that \(v^{*}=Fu^{*}\), we have

$$ \bigl\langle u^{n}-u^{*},Fu^{n}-v^{n} \bigr\rangle \geq \bigl\langle u^{n}-u^{*},v ^{*}-v^{n}\bigr\rangle . $$

Thus, substituting this in inequality (3.5), we have

$$\begin{aligned} \phi \bigl(u^{*},u^{n+1}\bigr)\leq{}& \phi \bigl(u^{*},u^{n}\bigr)-2\beta _{n}^{2} \bigl\langle u ^{n}-u^{*},v^{*}-v^{n}\bigr\rangle \\ &{} -2\alpha _{n}\beta _{n}\bigl\langle u^{n}-u^{n+1},Ju^{n}-Jx \bigr\rangle -2\alpha _{n}\beta _{n}\bigl\langle u^{n+1}-u^{*},Ju^{n}-Ju^{n+1}\bigr\rangle \\ & {}-2\alpha _{n}\beta _{n}\bigl\langle u^{n+1}-u^{*},Ju^{n+1}-Jx \bigr\rangle +2\beta _{n}c_{2}S^{*}\delta _{E}^{-1}(\beta _{n}M_{0}). \end{aligned}$$
(3.6)

Using Lemma 2.4, we have

$$ -2\alpha _{n}\beta _{n}\bigl\langle u^{n+1}-u^{*},Ju^{n+1}-Jx \bigr\rangle \leq \alpha _{n}\beta _{n}\phi \bigl(u^{*},x\bigr)-\alpha _{n}\beta _{n}\phi \bigl(u^{*},u ^{n+1}\bigr). $$

Substituting this into inequality (3.6), we obtain

$$\begin{aligned} &\phi \bigl(u^{*},u^{n+1}\bigr) \\ &\quad\leq \phi \bigl(u^{*},u^{n}\bigr)-\alpha _{n}\beta _{n} \phi \bigl(u^{*},u^{n+1}\bigr)+ \alpha _{n}\beta _{n}\phi \bigl(u^{*},x\bigr) \\ &\qquad{} -2\beta _{n}^{2}\bigl\langle u^{n}-u^{*},v^{*}-v^{n} \bigr\rangle - 2\alpha _{n} \beta _{n}\bigl\langle u^{n}-u^{n+1},Ju^{n}-Jx\bigr\rangle \\ & \qquad{}-2\alpha _{n}\beta _{n}\bigl\langle u^{n+1}-u^{*},Ju^{n}-Ju^{n+1} \bigr\rangle +2 \beta _{n}c_{2}S^{*}\delta _{E}^{-1}(\beta _{n}M_{0}) \\ &\quad\leq \phi \bigl(u^{*},u^{n}\bigr)-\alpha _{n} \beta _{n}\phi \bigl(u^{*},u^{n+1}\bigr)+ \alpha _{n}\beta _{n}\phi \bigl(u^{*},x\bigr) \\ &\qquad{} -2\beta _{n}^{2}\bigl\langle u^{n}-u^{*},v^{*}-v^{n} \bigr\rangle + 2\alpha _{n} \beta _{n} \bigl\Vert u^{n}-u^{n+1} \bigr\Vert \bigl\Vert Ju^{n}-Jx \bigr\Vert \\ &\qquad{} + 2\alpha _{n}\beta _{n} \bigl\Vert u^{n+1}-u^{*} \bigr\Vert \bigl\Vert Ju^{n}-Ju^{n+1} \bigr\Vert +2\beta _{n}c_{2}S^{*}\delta _{E}^{-1}( \beta _{n}M_{0}) \\ &\quad\leq \phi \bigl(u^{*},u^{n}\bigr)-\alpha _{n} \beta _{n}\phi \bigl(u^{*},u^{n+1}\bigr)+ \alpha _{n}\beta _{n}\phi \bigl(u^{*},x\bigr) \\ &\qquad{} -2\beta _{n}^{2}\bigl\langle u^{n}-u^{*},v^{*}-v^{n} \bigr\rangle +2\alpha _{n} \beta _{n}c_{2}S^{*} \delta _{E}^{-1}(\beta _{n}M_{0}) \\ &\qquad{} + 2\alpha _{n}\beta _{n}\bigl(\beta _{n}S^{*} \bigr)S^{*} +2\beta _{n}c_{2}S^{*} \delta _{E}^{-1}(\beta _{n}M_{0}). \end{aligned}$$
(3.7)

Similarly, using recurrence relation (3.2), Lemma 2.3, inequality (3.4), monotonicity of K, the fact that \(Kv^{*}=-u^{*}\), and Lemma 2.4, we obtain

$$\begin{aligned} \phi \bigl(v^{*},v^{n+1}\bigr)\leq{}& \phi \bigl(v^{*},v^{n}\bigr)-\alpha _{n}\beta _{n} \phi \bigl(v^{*},v^{n+1}\bigr)+\alpha _{n}\beta _{n} \phi \bigl(v^{*},y\bigr) \\ & {}-2\beta _{n}^{2}\bigl\langle v^{n}-v^{*},u^{n}-u^{*} \bigr\rangle +2\alpha _{n} \beta _{n}c_{2}S^{*} \delta _{E^{*}}^{-1}\bigl(\beta _{n}M_{0}^{*} \bigr) \\ &{} +2\alpha _{n}\beta _{n}\bigl(\beta _{n}S^{*} \bigr)S^{*} +2\beta _{n}c_{2}S^{*} \delta _{E^{*}}^{-1}\bigl(\beta _{n}M_{0}^{*} \bigr). \end{aligned}$$
(3.8)

Thus, adding inequalities (3.7) and (3.8), we obtain

$$\begin{aligned} r< {}& \varPhi \bigl(w^{*},w^{n+1}\bigr) \\ ={}&\phi \bigl(u^{*},u^{n+1} \bigr)+\phi \bigl(v^{*},v^{n+1}\bigr) \\ \leq {}&\varPhi \bigl(w^{*},w^{n}\bigr)-\alpha _{n} \beta _{n}\varPhi \bigl(w^{*},w^{n+1}\bigr)+ \alpha _{n}\beta _{n}\varPhi \bigl(w^{*},w^{1} \bigr) \\ &{} +2\alpha _{n}\beta _{n}c_{2}S^{*} \delta _{E}^{-1}(\beta _{n}M_{0})+ 2 \alpha _{n}\beta _{n}\bigl(\beta _{n}S^{*} \bigr)S^{*} \\ & {}+2\beta _{n}c_{2}S^{*}\delta _{E}^{-1}(\beta _{n}M_{0})+2 \alpha _{n}\beta _{n}c_{2}S^{*}\delta _{E^{*}}^{-1}\bigl(\beta _{n}M_{0}^{*} \bigr) \\ &{} +2\alpha _{n}\beta _{n}\bigl(\beta _{n}S^{*} \bigr)S^{*}+2\beta _{n}c_{2}S^{*} \delta _{E^{*}}^{-1}\bigl(\beta _{n}M_{0}^{*} \bigr) \\ \leq{}& \varPhi \bigl(w^{*},w^{n}\bigr)-\alpha _{n} \beta _{n}\varPhi \bigl(w^{*},w^{n+1}\bigr)+ \alpha _{n}\beta _{n}\varPhi \bigl(w^{*},w^{1} \bigr) \\ & {}+2\alpha _{n}^{2}\beta _{n}c_{2}S^{*} \gamma _{0}+2\alpha _{n}^{2}\beta _{n}S^{*}\gamma _{0}+ 2\alpha _{n}\beta _{n}c_{2}S^{*}\gamma _{0} \\ &{} + 2\alpha _{n}^{2}\beta _{n}c_{2}S^{*} \gamma _{0}+2\alpha _{n}^{2}\beta _{n}S^{*}\gamma _{0}+ 2\alpha _{n}\beta _{n}c_{2}S^{*}\gamma _{0} \\ \leq{}& \varPhi \bigl(w^{*},w^{n}\bigr)-\alpha _{n} \beta _{n}\varPhi \bigl(w^{*},w^{n+1}\bigr)+\alpha _{n}\beta _{n}\varPhi \bigl(w^{*},w^{1} \bigr)\ \\ &{} +2\alpha _{n}\beta _{n}M\gamma _{0} +2\alpha _{n}\beta _{n}M \gamma _{0}+2\alpha _{n} \beta _{n}M\gamma _{0} \\ &{} +2\alpha _{n}\beta _{n}M\gamma _{0} +2\alpha _{n}\beta _{n}M \gamma _{0}+2\alpha _{n} \beta _{n}M\gamma _{0} \\ \leq{}& \varPhi \bigl(w^{*},w^{n}\bigr)-\alpha _{n} \beta _{n}\varPhi \bigl(w^{*},w^{n+1}\bigr)+ \frac{r \alpha _{n}\beta _{n}}{8} +\frac{3r\alpha _{n}\beta _{n}}{8} \\ \leq {}&r-r\alpha _{n}\beta _{n}+\frac{r\alpha _{n}\beta _{n}}{2}=r- \frac{r \alpha _{n}\beta _{n}}{2}< r. \end{aligned}$$

This is a contradiction. Hence, \(\varPhi (w^{*},w^{n+1})\leq r\). Thus, \(\varPhi (w^{*},w^{n})\leq r\), for all \(n\geq 1\). Consequently, we have \(\phi (u^{*},u^{n})\leq r\) and \(\phi (v^{*},v^{n})\leq r\), for all \(n \geq 1\). Therefore, using inequality (2.2), we deduce that \(\{u^{n}\}\) and \(\{v^{n}\}\) are bounded. □

In Theorem 3.4 below, the sequences \(\{\alpha _{n}\}, \{\beta _{n}\}\subset (0,1)\) satisfy the following conditions:

  1. (i)

    \(\lim_{n\to \infty } \alpha _{n}=0, \lim_{n \to \infty }\frac{\alpha _{n}}{\beta _{n}}=0, \sum_{n=1}^{\infty }\alpha _{n}\beta _{n}=\infty \),

  2. (ii)

    \(\delta _{E}^{-1}(\beta _{n} M_{0})\leq \alpha _{n}\gamma _{0}\); \(\delta _{E^{*}}^{-1}(\beta _{n} M_{0}^{*})\leq \alpha _{n}\gamma _{0}; \beta _{n} S^{*}\leq \alpha _{n}\gamma _{0}\), \(\forall n\geq 1\), for some constants, \(M_{0},M_{0}^{*},S^{*},\gamma _{0}>0\),

  3. (iii)

    \(\delta _{E}^{-1}(\eta _{n}) \to 0\); \(\delta _{E ^{*}}^{-1}(\eta _{n})\to 0\),

  4. (iv)

    \(\frac{\delta _{E}^{-1}(\eta _{n})}{\alpha _{n}\beta _{n}}\to 0\); \(\frac{\delta _{E^{*}}^{-1}(\eta _{n})}{\alpha _{n}\beta _{n}} \to 0\),

\(\eta _{n}=\frac{\alpha _{n-1}\beta _{n}-\alpha _{n}\beta _{n-1}}{\alpha _{n}\beta _{n-1}}\).

Theorem 3.4

LetEbe a uniformly convex and uniformly smooth real Banach space. Let\(F: E \to E^{*}, K:E^{*}\to E\)be maximal monotone mappings. For arbitrary\(x\in E, y\in E^{*}\), define the sequences\(\{u^{n}\}\)and\(\{v^{n}\}\)inEand\(E^{*}\), respectively, for\(u^{1}\in E, v^{1} \in E^{*}\), by

$$ \textstyle\begin{cases} u^{n+1}=J^{-1} (Ju^{n}-\beta _{n}^{2}(Fu^{n}-v^{n})-\alpha _{n}\beta _{n}(Ju^{n}-Jx) ), \\ v^{n+1}=J (J^{-1}v^{n}-\beta _{n}^{2}(Kv^{n}+u^{n})-\alpha _{n}\beta _{n}(J^{-1}v^{n}-J^{-1}y) ), \end{cases} $$
(3.9)

where\(\{\alpha _{n}\}\)and\(\{\beta _{n}\}\)are sequences in\((0,1)\)satisfying conditions (i)(iv). Assume that the equation\(u+KFu=0\)has a solution. Then, the sequences\(\{u^{n}\}\)and\(\{v^{n}\}\)converge strongly to\(u^{*}\)and\(v^{*}\), respectively, where\(u^{*}\)is a solution of\(u+KFu=0\)with\(v^{*}=Fu^{*}\).

Proof

Using Lemmas 2.3 and 2.5, we have

$$\begin{aligned} \phi \bigl(y^{n},u^{n+1}\bigr)={}&V \bigl(y^{n}, Ju^{n}-\beta _{n}^{2}\bigl(Fu^{n}-v^{n} \bigr)- \alpha _{n}\beta _{n}\bigl(Ju^{n}-Jx\bigr) \bigr) \\ \leq{}& V\bigl(y^{n},Ju^{n}\bigr)-2\bigl\langle u^{n+1}-y^{n}, \beta _{n}^{2} \bigl(Fu^{n}-v ^{n}\bigr)+\alpha _{n}\beta _{n}\bigl(Ju^{n}-Jx\bigr) \bigr\rangle \\ ={}&\phi \bigl(y^{n},u^{n}\bigr)-2\beta _{n}^{2} \bigl\langle u^{n+1}-y^{n}, Fu^{n}-v ^{n} \bigr\rangle -2\alpha _{n}\beta _{n}\bigl\langle u^{n+1}-y^{n},Ju^{n}-Jx \bigr\rangle \\ ={}&\phi \bigl(u^{n},y^{n}\bigr)-2\bigl\langle u^{n}+y^{n},Ju^{n}-Jy^{n}\bigr\rangle +2 \bigl( \bigl\Vert u ^{n} \bigr\Vert ^{2}- \bigl\Vert y^{n} \bigr\Vert ^{2}\bigr) \\ &{} -2\beta _{n}^{2}\bigl\langle u^{n+1}-y^{n},Fu^{n}-v^{n} \bigr\rangle -2\alpha _{n}\beta _{n}\bigl\langle u^{n+1}-y^{n},Ju^{n}-Jx \bigr\rangle . \end{aligned}$$
(3.10)

Observe that

$$\begin{aligned} \phi \bigl(u^{n},y^{n}\bigr) &=V\bigl(u^{n},Jy^{n} \bigr)=V\bigl(u^{n},Jy^{n-1}+Jy^{n}-Jy^{n-1} \bigr) \\ & \leq V\bigl(u^{n},Jy^{n-1}\bigr)-2\bigl\langle y^{n}-u^{n},Jy^{n-1}-Jy^{n} \bigr\rangle . \end{aligned}$$

Thus, substituting this into inequality (3.10) and using Lemma 2.5, we obtain

$$\begin{aligned} \phi \bigl(y^{n},u^{n+1}\bigr) \leq{}& V\bigl(u^{n},Jy^{n-1} \bigr)-2\bigl\langle y^{n}-u^{n},Jy ^{n-1}-Jy^{n} \bigr\rangle \\ &{} -2\bigl\langle u^{n}+y^{n}, Ju^{n}-Jy^{n} \bigr\rangle +2\bigl( \bigl\Vert u^{n} \bigr\Vert ^{2}- \bigl\Vert y^{n} \bigr\Vert ^{2}\bigr) \\ &{} -2\beta _{n}^{2}\bigl\langle u^{n+1}-y^{n}, Fu^{n}-v^{n}\bigr\rangle -2\alpha _{n}\beta _{n}\bigl\langle u^{n+1}-y^{n},Ju^{n}-Jx \bigr\rangle \\ ={}&\phi \bigl(y^{n-1},u^{n}\bigr)-2\bigl\langle y^{n-1}+u^{n},Jy^{n-1}-Ju^{n}\bigr\rangle +2\bigl( \bigl\Vert y^{n-1} \bigr\Vert ^{2}- \bigl\Vert u^{n} \bigr\Vert ^{2}\bigr) \\ &{} -2\bigl\langle y^{n}-u^{n},Jy^{n-1}-Jy^{n} \bigr\rangle -2\bigl\langle u^{n}+y^{n}, Ju^{n}-Jy^{n} \bigr\rangle +2\bigl( \bigl\Vert u^{n} \bigr\Vert ^{2}- \bigl\Vert y^{n} \bigr\Vert ^{2}\bigr) \\ &{} -2\beta _{n}^{2}\bigl\langle u^{n+1}-y^{n}, Fu^{n}-v^{n}\bigr\rangle -2\alpha _{n}\beta _{n}\bigl\langle u^{n+1}-y^{n},Ju^{n}-Jx \bigr\rangle \\ ={}&\phi \bigl(y^{n-1},u^{n}\bigr)+2\bigl( \bigl\Vert y^{n-1} \bigr\Vert ^{2}- \bigl\Vert y^{n} \bigr\Vert ^{2}\bigr)+2\bigl\langle y ^{n-1}+u^{n},Ju^{n}-Jy^{n-1} \bigr\rangle \\ &{} -2\bigl\langle y^{n}-u^{n},Jy^{n-1}-Jy^{n} \bigr\rangle -2\bigl\langle u^{n}+y^{n}, Ju ^{n}-Jy^{n}\bigr\rangle \\ &{} -2\beta _{n}^{2}\bigl\langle u^{n+1}-y^{n}, Fu^{n}-v^{n}\bigr\rangle \,\boxed{ -2 \alpha _{n} \beta _{n}\bigl\langle u^{n+1}-y^{n},Ju^{n}-Jx \bigr\rangle } . \end{aligned}$$
(3.11)

Now, we estimate the boxed term using Lemma 2.4 as

$$\begin{aligned} &{-}2\alpha _{n}\beta _{n}\bigl\langle u^{n+1}-y^{n},Ju^{n}-Jx \bigr\rangle \\ &\quad= -2\alpha _{n}\beta _{n}\bigl\langle u^{n+1}-u^{n}, Ju^{n}-Jx \bigr\rangle -2 \alpha _{n}\beta _{n} \bigl\langle u^{n}-y^{n-1},Ju^{n}-Jy^{n-1} \bigr\rangle \\ & \qquad{}-2\alpha _{n}\beta _{n}\bigl\langle u^{n}-y^{n-1}, Jy^{n-1}-Jx \bigr\rangle -2 \alpha _{n}\beta _{n} \bigl\langle y^{n-1}-y^{n}, Ju^{n}-Jx \bigr\rangle \\ &\quad\leq -2\alpha _{n}\beta _{n}\bigl\langle u^{n+1}-u^{n}, Ju^{n}-Jx \bigr\rangle -\alpha _{n}\beta _{n}\phi \bigl(y^{n-1},u^{n} \bigr) \\ & \qquad{}-2\alpha _{n}\beta _{n}\bigl\langle u^{n}-y^{n-1}, Jy^{n-1}-Jx \bigr\rangle -2\alpha _{n}\beta _{n} \bigl\langle y^{n-1}-y^{n}, Ju^{n}-Jx \bigr\rangle . \end{aligned}$$

Therefore, substituting this into inequality (3.11), we have

$$\begin{aligned} &\phi \bigl(y^{n},u^{n+1}\bigr) \\ &\quad\leq \phi \bigl(y^{n-1},u^{n}\bigr)+2\bigl( \bigl\Vert y^{n-1} \bigr\Vert ^{2}- \bigl\Vert y ^{n} \bigr\Vert ^{2}\bigr)+2\bigl\langle y^{n-1}+u^{n},Ju^{n}-Jy^{n-1} \bigr\rangle \\ & \qquad{}-2\bigl\langle y^{n}-u^{n},Jy^{n-1}-Jy^{n} \bigr\rangle -2\bigl\langle u^{n}+y^{n}, Ju ^{n}-Jy^{n}\bigr\rangle \\ & \qquad{}-2\beta _{n}^{2}\bigl\langle u^{n+1}-y^{n}, Fu^{n}-v^{n}\bigr\rangle -2\alpha _{n}\beta _{n}\bigl\langle u^{n+1}-u^{n}, Ju^{n}-Jx \bigr\rangle \\ & \qquad{}-\alpha _{n}\beta _{n}\phi \bigl(y^{n-1},u^{n} \bigr) -2\alpha _{n} \beta _{n}\bigl\langle u^{n}-y^{n-1}, Jy^{n-1}-Jx \bigr\rangle \\ & \qquad{}-2\alpha _{n}\beta _{n}\bigl\langle y^{n-1}-y^{n}, Ju^{n}-Jx \bigr\rangle \\ &\quad=\phi \bigl(y^{n-1},u^{n}\bigr)+2\bigl( \bigl\Vert y^{n-1} \bigr\Vert ^{2}- \bigl\Vert y^{n} \bigr\Vert ^{2}\bigr)+2\bigl\langle y ^{n-1}-y^{n},Ju^{n}-Jy^{n-1} \bigr\rangle \\ &\qquad{} +2\bigl\langle u^{n}+y^{n},Ju^{n}-Jy^{n-1} \bigr\rangle -2\bigl\langle y^{n}-u^{n},Jy ^{n-1}-Jy^{n} \bigr\rangle \\ &\qquad{} -2\bigl\langle u^{n}+y^{n}, Ju^{n}-Jy^{n} \bigr\rangle -2\beta _{n}^{2}\bigl\langle u ^{n+1}-y^{n}, Fu^{n}-v^{n}\bigr\rangle \\ &\qquad {}-2\alpha _{n}\beta _{n}\bigl\langle u^{n+1}-u^{n}, Ju^{n}-Jx \bigr\rangle -\alpha _{n}\beta _{n}\phi \bigl(y^{n-1},u^{n}\bigr) \\ &\qquad{} -2\alpha _{n}\beta _{n}\bigl\langle u^{n}-y^{n-1}, Jy^{n-1}-Jx \bigr\rangle -2 \alpha _{n}\beta _{n} \bigl\langle y^{n-1}-y^{n}, Ju^{n}-Jx \bigr\rangle \\ &\quad=\phi \bigl(y^{n-1},u^{n}\bigr)-\alpha _{n}\beta _{n}\phi \bigl(y^{n-1},u^{n}\bigr)+2\bigl( \bigl\Vert y ^{n-1} \bigr\Vert ^{2}- \bigl\Vert y^{n} \bigr\Vert ^{2}\bigr) \\ &\qquad{} +2\bigl\langle y^{n-1}-y^{n},Ju^{n}-Jy^{n-1} \bigr\rangle -2 \bigl\langle u^{n}+y^{n},Jy^{n-1}-Jy^{n} \bigr\rangle \\ &\qquad{} -2\bigl\langle y^{n}-u^{n},Jy^{n-1}-Jy^{n} \bigr\rangle \,\boxed{ -2\beta _{n} ^{2}\bigl\langle u^{n+1}-y^{n}, Fu^{n}-v^{n}\bigr\rangle } \\ & \qquad{}-2\alpha _{n}\beta _{n}\bigl\langle u^{n+1}-u^{n}, Ju^{n}-Jx \bigr\rangle \,\boxed{ -2\alpha _{n}\beta _{n}\bigl\langle u^{n}-y^{n-1}, Jy^{n-1}-Jx \bigr\rangle } \\ &\qquad{} -2\alpha _{n}\beta _{n}\bigl\langle y^{n-1}-y^{n}, Ju^{n}-Jx \bigr\rangle . \end{aligned}$$
(3.12)

We now estimate the boxed terms. Using equation (2.9) and the fact that F is monotone, we obtain

$$\begin{aligned} &{-}2\beta _{n}^{2}\bigl\langle u^{n+1}-y^{n}, Fu^{n}-v^{n}\bigr\rangle -2\alpha _{n}\beta _{n}\bigl\langle u^{n}-y^{n-1}, Jy^{n-1}-Jx \bigr\rangle \\ &\quad=-2\beta _{n}^{2}\bigl\langle u^{n+1}-u^{n},Fu^{n}-v^{n} \bigr\rangle -2\beta _{n}^{2}\bigl\langle u^{n}-y^{n}, Fu^{n}-v^{n} \bigr\rangle \\ &\qquad{} -2\alpha _{n}\beta _{n}\bigl\langle y^{n}-y^{n-1},Jy^{n-1}-Jx \bigr\rangle -2\alpha _{n}\beta _{n}\bigl\langle u^{n}-y^{n}, Jy^{n-1}-Jy^{n} \bigr\rangle \\ &\qquad{} +2\beta _{n} \bigl\langle u^{n}-y^{n},-\alpha _{n}\bigl(Jy^{n}-Jx\bigr)\bigr\rangle \\ &\quad=-2\beta _{n}^{2}\bigl\langle u^{n+1}-u^{n},Fu^{n}-v^{n} \bigr\rangle +2\beta _{n}^{2}\bigl\langle u^{n}-y^{n}, v^{n}- Fu^{n} \bigr\rangle \\ &\qquad{} -2\alpha _{n}\beta _{n}\bigl\langle y^{n}-y^{n-1},Jy^{n-1}-Jx \bigr\rangle -2 \alpha _{n}\beta _{n}\bigl\langle u^{n}-y^{n}, Jy^{n-1}-Jy^{n}\bigr\rangle \\ &\qquad{} +2\beta ^{2}_{n} \bigl\langle u^{n}-y^{n},Fy^{n}- \bigl(y^{n}\bigr)^{*}\bigr\rangle \\ &\quad\leq -2\beta _{n}^{2}\bigl\langle u^{n+1}-u^{n},Fu^{n}-v^{n} \bigr\rangle -2 \alpha _{n}\beta _{n}\bigl\langle u^{n}-y^{n}, Jy^{n-1}-Jy^{n}\bigr\rangle \\ &\qquad{} -2\alpha _{n}\beta _{n}\bigl\langle y^{n}-y^{n-1},Jy^{n-1}-Jx \bigr\rangle +2\beta _{n}^{2} \bigl\langle u^{n}-y^{n},v^{n}-\bigl(y^{n} \bigr)^{*}\bigr\rangle . \end{aligned}$$

Thus, substituting this into inequality (3.12), and using inequalities (2.7), (2.8), and (3.3), we obtain

$$\begin{aligned} &\phi \bigl(y^{n},u^{n+1}\bigr) \\ &\quad\leq \phi \bigl(y^{n-1},u^{n}\bigr)-\alpha _{n}\beta _{n} \phi \bigl(y^{n-1},u^{n}\bigr)+2\bigl( \bigl\Vert y^{n-1} \bigr\Vert ^{2}- \bigl\Vert y^{n} \bigr\Vert ^{2}\bigr) \\ & \qquad{}+2\bigl\langle y^{n-1}-y^{n},Ju^{n}-Jy^{n-1} \bigr\rangle -2\bigl\langle u^{n}+y^{n},Jy ^{n-1}-Jy^{n}\bigr\rangle \\ &\qquad{} -2\bigl\langle y^{n}-u^{n},Jy^{n-1}-Jy^{n} \bigr\rangle -2\alpha _{n}\beta _{n} \bigl\langle u^{n+1}-u^{n}, Ju^{n}-Jx \bigr\rangle \\ & \qquad{}-2\alpha _{n}\beta _{n}\bigl\langle y^{n-1}-y^{n}, Ju^{n}-Jx \bigr\rangle -2\beta _{n}^{2}\bigl\langle u^{n+1}-u^{n},Fu^{n}-v^{n}\bigr\rangle \\ &\qquad{} -2\alpha _{n}\beta _{n}\bigl\langle u^{n}-y^{n}, Jy^{n-1}-Jy ^{n}\bigr\rangle -2\alpha _{n}\beta _{n}\bigl\langle y^{n}-y^{n-1},Jy^{n-1}-Jx \bigr\rangle \\ &\qquad{} +2\beta ^{2}_{n} \bigl\langle u^{n}-y^{n},v^{n}- \bigl(y^{n}\bigr)^{*}\bigr\rangle \\ &\quad \leq (1-\alpha _{n}\beta _{n})\phi \bigl(y^{n-1},u^{n} \bigr)+2N_{1}\bigl( \bigl\Vert y^{n-1}-y ^{n} \bigr\Vert + \bigl\Vert Jy^{n-1}-Jy^{n} \bigr\Vert \bigr) \\ &\qquad{} +2\alpha _{n}\beta _{n}N_{2}\bigl( \bigl\Vert u^{n+1}-u^{n} \bigr\Vert + \bigl\Vert y^{n-1}-y^{n} \bigr\Vert + \bigl\Vert Jy ^{n-1}-Jy^{n} \bigr\Vert \bigr) \\ &\qquad{} +2\beta _{n}^{2}N_{3} \bigl\Vert u^{n+1}-u^{n} \bigr\Vert +2\beta _{n}^{2} \bigl\langle u^{n}-y ^{n},v^{n}- \bigl(y^{n}\bigr)^{*}\bigr\rangle \\ &\quad\leq (1-\alpha _{n}\beta _{n})\phi \bigl(y^{n-1},u^{n} \bigr) +2N_{1} \bigl(c_{2} \delta _{E}^{-1} \eta _{n} +c_{2}\delta _{E^{*}}^{-1}\eta _{n} \bigr) \\ &\qquad{} +2\alpha _{n}\beta _{n}N_{2} \bigl( c_{2}\delta _{E}^{-1} (\beta _{n}M _{0} ) +c_{2}\delta _{E}^{-1}\eta _{n} +c_{2}\delta _{E^{*}}^{-1}\eta _{n} \bigr) \\ & \qquad{}+2\beta _{n}^{2}N_{3}c_{2}\delta _{E}^{-1}(\beta _{n}M_{0})+2\beta _{n} ^{2}\bigl\langle u^{n}-y^{n},v^{n}- \bigl(y^{n}\bigr)^{*}\bigr\rangle \\ &\quad\leq (1-\alpha _{n}\beta _{n})\phi \bigl(y^{n-1},u^{n} \bigr) +\alpha _{n}\beta _{n}\widehat{N} \biggl( c_{2}\delta _{E}^{-1} (\beta _{n}M_{0} ) +c _{2}\delta _{E}^{-1}\eta _{n} +c_{2}\delta _{E^{*}}^{-1}\eta _{n} \\ & \qquad{}+\beta _{n}c_{2}\gamma _{0} + c_{2} \delta _{E}^{-1} \biggl( \frac{\eta _{n}}{ \alpha _{n}\beta _{n}} \biggr) +c_{2}\delta _{E^{*}}^{-1} \biggl( \frac{\eta _{n}}{\alpha _{n}\beta _{n}} \biggr) \biggr) \\ &\qquad{} +2\beta _{n}^{2}\bigl\langle u^{n}-y^{n},v^{n}- \bigl(y^{n}\bigr)^{*}\bigr\rangle , \end{aligned}$$
(3.13)

for some \(N_{1},N_{2},N_{3}>0\), and \(\widehat{N}=\max \{N_{1},N_{2},N _{3}\}\).

Similarly, using Lemmas 2.3, 2.5 and 2.4, as well as equation (2.10), we obtain

$$\begin{aligned} &\phi \bigl(\bigl(y^{n}\bigr)^{*},v^{n+1} \bigr)\\ &\quad \leq \phi \bigl(\bigl(y^{n-1}\bigr)^{*},v^{n} \bigr)- \alpha _{n}\beta _{n}\phi \bigl(\bigl(y^{n-1} \bigr)^{*},v^{n}\bigr) +2\bigl( \bigl\Vert \bigl(y^{n-1}\bigr)^{*} \bigr\Vert ^{2}- \bigl\Vert \bigl(y^{n}\bigr)^{*} \bigr\Vert ^{2} \bigr) \\ &\qquad{} +2\bigl\langle \bigl(y^{n-1}\bigr)^{*}- \bigl(y^{n}\bigr)^{*}, J^{-1}v^{n}-J^{-1}y ^{n-1}\bigr\rangle +2\bigl\langle \bigl(y^{n} \bigr)^{*}+v^{n},J^{-1}y^{n}-J^{-1} \bigl(y^{n-1}\bigr)^{*} \bigr\rangle \\ & \qquad{}-2\bigl\langle \bigl(y^{n}\bigr)^{*}-v^{n},J^{-1} \bigl(y^{n-1}\bigr)^{*}-J^{-1}\bigl(y^{n} \bigr)^{*} \bigr\rangle -2\alpha _{n}\beta _{n}\bigl\langle v^{n+1}-v^{n},J^{-1}v^{n}-J^{-1}y \bigr\rangle \\ &\qquad{} -2\alpha _{n}\beta _{n}\bigl\langle \bigl(y^{n-1} \bigr)^{*}-\bigl(y^{n}\bigr)^{*},J^{-1}v^{n}-J ^{-1}y\bigr\rangle -2\beta _{n}^{2}\bigl\langle v^{n+1}-v^{n},Kv^{n}+u^{n}\bigr\rangle \\ &\qquad{} -2\alpha _{n}\beta _{n}\bigl\langle v^{n}- \bigl(y^{n}\bigr)^{*},J^{-1}y^{n-1}-J^{-1} \bigl(y ^{n}\bigr)^{*}\bigr\rangle \\ &\qquad{} -2\alpha _{n}\beta _{n}\bigl\langle \bigl(y^{n-1} \bigr)^{*}-\bigl(y^{n}\bigr)^{*}, J^{-1}y^{n-1}-J ^{-1}y\bigr\rangle +2\beta _{n}^{2}\bigl\langle v^{n}-\bigl(y^{n} \bigr)^{*},y^{n}-u^{n} \bigr\rangle . \end{aligned}$$

Since \(E^{*}\) is uniformly convex and uniformly smooth, using inequalities (2.7), (2.8), and (3.4), we obtain that, for some \(\widehat{N}^{*}>0\),

$$\begin{aligned} &\phi \bigl(\bigl(y^{n}\bigr)^{*},v^{n+1}\bigr) \\ &\quad\leq (1-\alpha _{n}\beta _{n})\phi \bigl(\bigl(y^{n-1} \bigr)^{*},v ^{n}\bigr) +\alpha _{n}\beta _{n}\widehat{N}^{*} \biggl( c_{2}\delta _{E^{*}} ^{-1}\bigl(\beta _{n}M_{0}^{*} \bigr) +2c_{2}\delta _{E}^{-1}\eta _{n} \\ &\qquad{} +\beta _{n}c_{2}\gamma _{0} + c_{2} \delta _{E^{*}}^{-1} \biggl(\frac{\eta _{n}}{2\alpha _{n}\beta _{n}} \biggr) +c_{2}\delta _{E}^{-1} \biggl(\frac{\eta _{n}}{\alpha _{n}\beta _{n}} \biggr) \biggr) \\ &\qquad{} +2\beta _{n}^{2}\bigl\langle v^{n}- \bigl(y^{n}\bigr)^{*},y^{n}-u^{n}\bigr\rangle . \end{aligned}$$
(3.14)

Let \(p^{n}=(y^{n},(y^{n})^{*})\), \(w^{n}=(u^{n},v^{n})\). Adding inequalities (3.13) and (3.14), we obtain

$$\begin{aligned} &\varPhi \bigl(p^{n},w^{n+1}\bigr) \\ &\quad\leq (1-\alpha _{n} \beta _{n})\varPhi \bigl(p^{n-1},w^{n}\bigr)+ \alpha _{n}\beta _{n}N \biggl( c_{2}\delta _{E}^{-1} (\beta _{n}M_{0} ) +2\beta _{n}c_{2}\gamma _{0} +2c_{2}\delta _{E}^{-1}\eta _{n} \\ & \qquad{}+2c_{2}\delta _{E^{*}}^{-1}\eta _{n} + 2c_{2}\delta _{E}^{-1} \biggl( \frac{ \eta _{n}}{\alpha _{n}\beta _{n}} \biggr) +2c_{2}\delta _{E^{*}}^{-1} \biggl( \frac{ \eta _{n}}{\alpha _{n}\beta _{n}} \biggr) \\ &\qquad{} + c_{2}\delta _{E^{*}}^{-1}\bigl(\beta _{n}M_{0}^{*}\bigr) \biggr), \end{aligned}$$
(3.15)

where \(N=\max \{\widehat{N},\widehat{N}^{*}\}\). Now, setting

$$ a_{n}:=\varPhi \bigl(p^{n-1},w^{n}\bigr), \qquad\sigma _{n}:=\alpha _{n}\beta _{n},\qquad c_{n}\equiv 0, $$

and

$$\begin{aligned} b_{n}:={}& \biggl( c_{2}\delta _{E}^{-1} (\beta _{n}M_{0} ) +2\beta _{n}c_{2}\gamma _{0} +2c_{2}\delta _{E}^{-1}\eta _{n} +2c_{2}\delta _{E ^{*}}^{-1}\eta _{n} + 2c_{2}\delta _{E}^{-1} \biggl( \frac{\eta _{n}}{\alpha _{n}\beta _{n}} \biggr) \\ &{}+2c_{2}\delta _{E^{*}}^{-1} \biggl( \frac{\eta _{n}}{\alpha _{n}\beta _{n}} \biggr) + c_{2}\delta _{E^{*}}^{-1} \bigl( \beta _{n}M_{0}^{*}\bigr) \biggr) \end{aligned}$$
(3.16)

inequality (3.15) becomes

$$ a_{n+1}\leq (1-\sigma _{n})a_{n}+\sigma _{n}b_{n}+c_{n}, \quad n\geq 1. $$

It follows from a well known result of Xu [45] that \(a_{n}\to 0\) as \(n\to \infty \), which implies that \(\varPhi (p^{n-1},w ^{n})\to 0\), as \(n\to \infty \). By Lemma 2.8, we have \(\|w^{n}-p^{n-1}\|_{W} \to 0\). Consequently, \(\|u^{n}-y^{n-1} \| \to 0\). Furthermore, using Remark 6, since \([y^{n},(y^{n})^{*}] \to [u^{*},v^{*}]\in A^{-1}0\), we have that \(\{u^{n}\}\) converges strongly to a solution of the Hammerstein equation (1.1) with \(v^{*}=Fu^{*}\). This completes the proof. □

Corollary 3.5

Let\(E=L_{p}\) (or\(W_{p}^{m} (\varOmega )\)), \(1< p<\infty \). Let\(F: E \to E^{*}, K:E^{*}\to E\)be maximal monotone mappings. For arbitrary\(x\in E, y\in E^{*}\), define the sequences\(\{u^{n}\}\)and\(\{v^{n}\}\)inEand\(E^{*}\), respectively, for\(u^{1}\in E, v^{1} \in E^{*}\), by

$$ \begin{aligned} &u^{n+1}=J^{-1} \bigl(Ju^{n}-\beta _{n}^{2}\bigl(Fu^{n}-v^{n} \bigr)-\alpha _{n}\beta _{n}\bigl(Ju^{n}-Jx\bigr) \bigr), \\ &v^{n+1}=J \bigl(J^{-1}v^{n}-\beta _{n}^{2}\bigl(Kv ^{n}+u^{n}\bigr)-\alpha _{n}\beta _{n}\bigl(J^{-1}v^{n}-J^{-1}y \bigr) \bigr), \end{aligned} $$
(3.17)

where\(\{\alpha _{n}\}\)and\(\{\beta _{n}\}\)are sequences in\((0,1)\)satisfying conditions (i)(iv). Assume that the equation\(u+KFu=0\)has a solution. Then, the sequences\(\{u^{n}\}\)and\(\{v^{n}\}\)converge strongly to\(u^{*}\)and\(v^{*}\), respectively, where\(u^{*}\)is a solution of\(u+KFu=0\)with\(v^{*}=Fu^{*}\).

Proof

\(L_{p}\)\((\text{or } W_{p}^{m} (\varOmega ) ), 1< p<\infty \) are uniformly convex and uniformly smooth. Hence, the conclusion follows from Theorem 3.4. □

Corollary 3.6

LetHbe a real Hilbert space. Let\(F: H \to H, K:H\to H\)be maximal monotone mappings. For arbitrary\(x,y\in H\), define the sequences\(\{u^{n}\}\)and\(\{v^{n}\}\)inH, by: \(u^{1},v^{1}\in H\),

$$ \begin{aligned}&u^{n+1} = u^{n}- \beta _{n}^{2}\bigl(Fu^{n}-v^{n}\bigr)- \alpha _{n}\beta _{n}\bigl(u^{n}-x\bigr), \\ &v ^{n+1}=v^{n}-\beta _{n}^{2} \bigl(Kv^{n}+u^{n}\bigr)-\alpha _{n}\beta _{n}\bigl(v^{n}-y\bigr) , \end{aligned} $$
(3.18)

where\(\{\alpha _{n}\}\)and\(\{\beta _{n}\}\)are sequences in\((0,1)\)satisfying conditions (i)(iv). Assume that the equation\(u+KFu=0\)has a solution. Then, the sequences\(\{u^{n}\}\)and\(\{v^{n}\}\)converge strongly to\(u^{*}\)and\(v^{*}\), respectively, where\(u^{*}\)is a solution of\(u+KFu=0\)with\(v^{*}=Fu^{*}\).

Remark 7

If \(E=L_{p}\ (1< p<\infty )\), real sequences that satisfy the hypothesis of the theorem and corollaries above are \(\alpha _{n}=(n+1)^{-a}\) and \(\beta _{n}=(n+1)^{-b}\) with \(0< b<\frac{1}{r} a\) and \(a+b<\frac{1}{r}\), where \(r=\max \{p,q\}\). In particular, without loss of generality, let \(r = p\). Then, one can choose \(a:= \frac{1}{p+1}\) and \(b:= \min \{ \frac{1}{2D}, \frac{1}{2p(p+1)}\}\), where D is the constant appearing in Remark 4. The verifications basically follow as in Chidume and Idu [18]. However, for completeness, we verify conditions (ii)–(iv) of our theorem for \(L_{p}\) spaces, \(1< p<\infty \). For \(p>1, q>1\), \(E=L_{p}\), so that \(E^{*}=L_{q}\) (\(\frac{1}{p}+\frac{1}{q}=1\)). Then (see, e.g., [1, p. 47]):

$$ \delta _{E^{*}}(\epsilon )=1- \biggl[1- \biggl(\frac{\epsilon }{2} \biggr)^{q} \biggr]^{\frac{1}{q}}\quad (0< \epsilon \leq 2), $$

and so we obtain \(\delta _{E^{*}}^{-1}(\epsilon )=2 (1-(1-\epsilon )^{q} )^{\frac{1}{q}} \leq 2q^{\frac{1}{q}}\epsilon ^{\frac{1}{q}}, \text{since } (1-\epsilon )^{q} >1-q\epsilon \), for \(q>1\). We verify the following:

  1. (ii)

    \(\delta _{E}^{-1}(\beta _{n}M_{0})\leq \alpha _{n} \gamma _{0}\);

  2. (iii)

    \(\delta _{E}^{-1}(\eta _{n})=\delta _{E}^{-1} (\frac{ \alpha _{n-1}\beta _{n}}{\alpha _{n}\beta _{n}}-1 ) \to 0\), as \(n \to \infty \);

  3. (iv)

    \(\frac{\delta _{E}^{-1} (\frac{\alpha _{n-1}\beta _{n}}{ \alpha _{n}\beta _{n}}-1 )}{\alpha _{n}\beta _{n}}\to 0\), as \(n\to \infty \).

Now, to verify condition (ii), we have

$$\begin{aligned} \frac{ \delta _{E}^{-1}(\beta _{n}M_{0})}{\alpha _{n}} &= \frac{2 (1-(1- \beta _{n}M_{0}) )^{\frac{1}{p}}}{\alpha _{n}} \\ &\leq \frac{2p^{\frac{1}{p}}(\beta _{n}M_{0})^{\frac{1}{p}}}{\alpha _{n}} \\ &=\frac{2(pM_{0})^{\frac{1}{p}}(n+1)^{-b}}{(n+1)^{-a}}=2(pM_{0})^{ \frac{1}{p}}(n+1)^{a-b}\to 0, \quad\text{as } n\to \infty. \end{aligned}$$

Next we show that condition (iii) is satisfied:

$$\begin{aligned} \delta _{E}^{-1} \biggl( \frac{\alpha _{n-1}\beta _{n}}{\alpha _{n}\beta _{n}}-1 \biggr) &=2 \biggl[1- \biggl(1-\frac{\alpha _{n-1}\beta _{n}}{\alpha _{n}\beta _{n-1}}+1 \biggr)^{p} \biggr]^{\frac{1}{p}} \\ &=2 \biggl[1- \biggl(2- \frac{\alpha _{n-1}\beta _{n}}{\alpha _{n}\beta _{n-1}} \biggr)^{p} \biggr]^{ \frac{1}{p}} =2 \biggl[ 1- \biggl(2-\frac{n^{-a}(n+1)^{-b}}{(n+1)^{-a}n ^{-b}} \biggr)^{p} \biggr]^{\frac{1}{p}} \\ &=2 \biggl[1- \biggl(2- \biggl(\frac{n+1}{n} \biggr)^{a} \biggl( \frac{n}{n+1} \biggr)^{b} \biggr)^{p} \biggr]^{\frac{1}{p}} \\ &\leq 2 \biggl[1- \biggl(2- \biggl(1+\frac{1}{n} \biggr)^{a} \biggr)^{p} \biggr]^{ \frac{1}{p}} \leq 2 \biggl[1- \biggl(1- \frac{a}{n} \biggr)^{p} \biggr]^{ \frac{1}{p}} \\ &\leq 2p^{\frac{1}{p}} \biggl(\frac{a}{n} \biggr)^{\frac{1}{p}} \to 0, \quad\text{as } n\to \infty. \end{aligned}$$

Finally, we verify (iv). Using (iii), we have

$$\begin{aligned} \frac{\delta _{E}^{-1} (\frac{\alpha _{n-1}\beta _{n}}{\alpha _{n} \beta _{n}}-1 )}{\alpha _{n}\beta _{n}} &\leq \frac{2p^{\frac{1}{p}} (\frac{a}{n} )^{\frac{1}{p}}}{(n+1)^{-(a+b)}} \\ &= 2p^{\frac{1}{p}} \biggl(\frac{a}{n} \biggr)^{\frac{1}{p}}(n+1)^{(a+b)} \\ &\leq 2^{a+b+1}(pa)^{\frac{1}{p}} n^{a+b-\frac{1}{p}} \to 0, \quad\text{as } n\to \infty. \end{aligned}$$

Numerical illustrations

In this section, we present numerical examples to compare the convergence of the sequence generated by our algorithm (3.9) with those produced by algorithms (1.8) and (1.9).

Example 1

In Theorems 1.3, 1.4 and 3.4, set \(E=\Bbb {R}^{2}\) then \(E^{*}=\Bbb {R}^{2}\). Let

$$ Fu= \begin{pmatrix} 3&-1 \\ 1&5 \end{pmatrix} \begin{pmatrix} u_{1} \\ u_{2} \end{pmatrix} ,\qquad Kv= \begin{pmatrix} 7&2 \\ -2&5 \end{pmatrix} \begin{pmatrix} v_{1} \\ v_{2} \end{pmatrix} . $$

Then, it is easy to see that F and K are continuous and strongly monotone. Thus, K and F are maximal monotone and vector \(u^{*}=(0,0)^{T}\) is the only solution of the equation \(u+KFu=0\). In Theorem 1.3, we take \(\alpha _{n}=\frac{1}{n}\), in Theorem 1.4, we take \(\lambda _{n}=\theta _{n}=\frac{1}{(n+1)^{ \frac{1}{2}}}\), and in Theorem 3.4, we take \(\alpha _{n}=\frac{1}{(n+1)^{ \frac{1}{2}}}, \beta _{n}=\frac{1}{(n+1)^{\frac{1}{4}}}, n=1,2,\dots \), as our parameters. Clearly, these parameters satisfy the hypothesis of Theorems 1.3, 1.4, and 3.4. Setting a tolerance of 10−8 and maximum number of iterations \(n=1000\), we obtain the iterates (see Table 1 and Fig. 1).

Figure 1
figure1

Graph of the first 41 iterates of algorithms (1.8), (1.9) and (3.9), choosing \(u^{1}= (1,0)^{T} \), \(v^{1}=(2, 1)^{T}\)

Table 1 Values choosing \(u^{1}= (1,0)^{T} \), \(v^{1}=(2, 1)^{T}\)

In Figs. 1, 2, 3 and 4, the y-axis represents the values of \(\|u^{n+1}-\mathbf{0}\|\) while the x-axis represents the number of iterations n.

Figure 2
figure2

Graph of the first 80 iterates of algorithms (1.8), (1.9), and (3.9), choosing \(u^{1}=(0,-\frac{1}{4})^{T} \), \(v^{1}= (-\frac{1}{2}, 1)^{T}\)

Figure 3
figure3

Graph of the first 10 iterates of algorithms (1.9) and (3.9), choosing \(u^{1}(t)=t ^{2}+1\) and \(v^{1}(t)=\cos t \exp (-t)\)

Figure 4
figure4

Graph of the first 10 iterates of algorithms (1.9) and (3.9), choosing \(u^{1}(t)=\frac{1}{1+x\sin x}\) and \(v^{1}(t)=\exp ( t)\)

Example 2

In Theorems 1.3, 1.4, and 3.4, set \(E=\Bbb {R}^{2}\) then \(E^{*}=\Bbb {R}^{2}\). Let

$$ Fu= (3u_{1}-u_{2}+\sin u_{1}, u_{1}+7u_{2}+ \sin u_{2}),\qquad Kv= (5v_{1}-5v_{2},3v_{1}+6v_{2}). $$

Then, it is easy to see that F and K are continuous and strongly monotone. Thus, K and F are maximal monotone and vector \(u^{*}=(0,0)^{T}\) is the only solution of the equation \(u+KFu=0\). In Theorem 1.3, we take \(\alpha _{n}=\frac{1}{n}\), in Theorem 1.4, we take \(\lambda _{n}=\theta _{n}=\frac{1}{(n+1)^{ \frac{1}{2}}}\), and in Theorem 3.4, we take \(\alpha _{n}=\frac{1}{(n+1)^{ \frac{1}{2}}}, \beta _{n}=\frac{1}{(n+1)^{\frac{1}{4}}}, n=1,2,\dots \), as our parameters. Clearly, these parameters satisfy the hypothesis of Theorems 1.3, 1.4, and 3.4. Setting a tolerance of 10−8 and maximum number of iterations \(n=1000\), we obtain the iterates (see Table 2 and Fig. 2).

Table 2 Values choosing \(u^{1}=(0,-\frac{1}{4})^{T} \), \(v^{1}= (- \frac{1}{2}, 1)^{T}\)

In Theorems 1.4 and 3.4, set \(E=L_{p}([0,1])\),  \(E ^{*}=L_{q}([0,1])\), \(1< p< q<\infty \), and \(\frac{1}{p}+\frac{1}{q}=1\).

$$\begin{aligned} F:L_{p}\bigl([0,1]\bigr) & \to L_{q}\bigl([0,1]\bigr), \\ u &\longmapsto (Fu) (t)=Ju(t). \end{aligned}$$

It is well known that the normalized duality map J is monotone and uniformly continuous on bounded subsets of \(L_{p}\) and thus, maximal monotone. Let

$$\begin{aligned} K: L_{q}\bigl([0,1]\bigr) &\to L_{p}\bigl([0,1]\bigr), \\ v &\longmapsto (Kv) (t)=tv(t). \end{aligned}$$

Since \(L_{q}([0,1]) \subset L_{p}([0,1])\), K is well-defined. Observe that by definition K is linear. Next we show that K is monotone. Let \(v,w\in L_{q}([0,1])\), then

$$\begin{aligned} \bigl\langle (Kv) (t)-(Kw) (t), v(t)-w(t) \bigr\rangle &= \int _{0}^{1} \bigl(tv(t)-tw(t) \bigr) \bigl(v(t)-w(t) \bigr)\,dt \\ &= \int _{0}^{1} t \bigl(v(t)-w(t) \bigr)^{2} \,dt \geq 0. \end{aligned}$$

Hence, K is monotone. Furthermore, since K is continuous, K is maximal monotone and the function \(u^{*}(t)=( 0,0 )^{T} \) is the only solution of the equation \(u+KFu=0\). For the numerical experiments, in Examples 3 and 4 below, in Theorem 1.4, we take \(\lambda _{n}= \theta _{n}=\frac{1}{(n+1)^{\frac{1}{2}}}\), and in Theorem 3.4, we take \(\alpha _{n}=\frac{1}{(n+1)^{\frac{1}{2}}}, \beta _{n}=\frac{1}{(n+1)^{\frac{1}{4}}}, n=1,2,\dots \), as our parameters. Clearly, these parameters satisfy the hypothesis of Theorems 1.4 and 3.4. We set a tolerance of 10−2 and maximum number of iterations \(n=10\).

Example 3

Taking \(p=1.5\), we have \(E=L_{1.5}([0,1]), E^{*}=L_{3}([0,1])\) (see Table 3 and Fig. 3)

$$\begin{aligned} &F:L_{1.5}\bigl([0,1]\bigr) \to L_{3}\bigl([0,1]\bigr),\\ &\phantom{AL_{1.5}\bigl([0,1]\bigr)}u \longmapsto (Fu) (t)=Ju(t),\\ &K: L_{3}\bigl([0,1]\bigr) \to L_{1.5}\bigl([0,1]\bigr),\\ & \phantom{AL._{3}\bigl([0,1]\bigr) }v \longmapsto (Kv) (t)=tv(t). \end{aligned}$$
Table 3 Values choosing \(u^{1}(t)=t^{2}+1\) and \(v^{1}(t)=\cos t \exp (-t)\)

Example 4

Taking \(p=\frac{5}{3}\), we have \(E=L_{\frac{5}{3}}([0,1]), E^{*}=L _{2.5}([0,1])\) (see Table 4 and Fig. 4)

$$\begin{aligned} &F:L_{\frac{5}{3}}\bigl([0,1]\bigr) \to L_{2.5}\bigl([0,1]\bigr),\\ &\phantom{AL_{\frac{5}{3}}\bigl([0,1]\bigr)}u \longmapsto (Fu) (t)=Ju(t),\\ &K:L_{2.5}\bigl([0,1]\bigr) \to L_{\frac{5}{3}}\bigl([0,1]\bigr),\\ &\phantom{AL._{2.5}\bigl([0,1]\bigr)}v \longmapsto (Kv) (t)=tv(t). \end{aligned}$$
Table 4 Values choosing \(u^{1}(t)=\frac{1}{1+x\sin x}\) and \(v^{1}(t)=\exp ( t)\)

Remark 8

From the numerical experiments above, we see that algorithm (3.9) is more robust and efficient than algorithms (1.8) and (1.9), and converges faster in terms of number of iterations and CPU time in all the problems tested.

Conclusion

In this paper, an iterative algorithm that extends the results of Chidume and Shehu [24], and complements the results of Uba et al. [44] is constructed. Strong convergence of the sequence generated by the algorithm is proved in a uniformly convex and uniformly smooth real Banach space. The theorem proved is a significant improvement of the results of Chidume and Shehu [24] which was proved in real Hilbert spaces under the assumption that F and K are continuous and bounded. These restrictions on K and F have been dispensed with even in the more general setting considered here. Finally, numerical experiments are presented to demonstrate the convergence of the sequence of the proposed algorithm.

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Acknowledgements

The authors appreciate the support of their institute and AfDB for the Research Grant that enabled this work to be carried out. The authors wish to thank the referees for their esteemed comments and suggestions.

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This work is supported from AfDB Research Grant Funds to AUST.

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CEC formulated the problem and suggested the method of proof of the Theorem to AA and LCO. The computations using the method suggested by CEC were carried out by AA and LCO. The analysis of the computations to arrive at the proof of the theorem was done jointly by CEC, AA and LCO. All authors read and approved the final manuscript.

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Correspondence to Charles E. Chidume.

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Chidume, C.E., Adamu, A. & Okereke, L.C. Iterative algorithms for solutions of Hammerstein equations in real Banach spaces. Fixed Point Theory Appl 2020, 4 (2020). https://doi.org/10.1186/s13663-020-0670-7

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MSC

  • 47H09
  • 47H05
  • 47J25
  • 47J05

Keywords

  • Fixed point
  • Maximal monotone
  • Uniformly smooth
  • Uniformly convex
  • Hammerstein
  • Quasi-bounded