Approximation of zeros of bounded maximal monotone mappings, solutions of Hammerstein integral equations and convex minimization problems
 Charles E Chidume^{1}Email author and
 Kennedy O Idu^{1}
https://doi.org/10.1186/s1366301605828
© Chidume and Idu 2016
Received: 23 June 2016
Accepted: 9 September 2016
Published: 21 October 2016
Abstract
Let E be a real normed space with dual space \(E^{*}\) and let \(A:E\rightarrow2^{E^{*}}\) be any map. Let \(J:E\rightarrow2^{E^{*}}\) be the normalized duality map on E. A new class of mappings, Jpseudocontractive maps, is introduced and the notion of Jfixed points is used to prove that \(T:=(JA)\) is Jpseudocontractive if and only if A is monotone. In the case that E is a uniformly convex and uniformly smooth real Banach space with dual \(E^{*}\), \(T: E\rightarrow2^{E^{*}}\) is a bounded Jpseudocontractive map with a nonempty Jfixed point set, and \(JT :E\rightarrow2^{E^{*}}\) is maximal monotone, a sequence is constructed which converges strongly to a Jfixed point of T. As an immediate consequence of this result, an analog of a recent important result of Chidume for bounded maccretive maps is obtained in the case that \(A:E\rightarrow2^{E^{*}}\) is bounded maximal monotone, a result which complements the proximal point algorithm of Martinet and Rockafellar. Furthermore, this analog is applied to approximate solutions of Hammerstein integral equations and is also applied to convex optimization problems. Finally, the techniques of the proofs are of independent interest.
Keywords
Jfixed points Jpseudocontractive mapping monotone mapping strong convergenceMSC
47H04 47H05 46N10 47H06 47J251 Introduction
Furthermore, the equation \(0\in Au\) when A is a monotone map from a real Hilbert space to itself also appears in evolution systems. Consider the evolution equation \(\frac{du}{dt} + Au=0\) where A is a monotone map from a real Hilbert space to itself. At an equilibrium state, \(\frac{du}{dt}=0\) so that \(Au=0\), whose solutions correspond to the equilibrium state of the dynamical system.
Thus, approximating zeros of equation (1.4) is equivalent to the approximation of solutions of the diffusion equation (1.2) at equilibrium state.
The notion of monotone mapping has been extended to real normed spaces. We now briefly examine two wellstudied extensions of Hilbert space monotonicity to arbitrary normed spaces.
1.1 Accretivetype mappings
In a Hilbert space, the normalized duality map is the identity map, and so, in this case, inequality (1.5) and inequality (1.1) coincide. Hence, accretivity is one extension of Hilbert space monotonicity to general normed spaces.
Accretive operators have been studied extensively by numerous mathematicians (see, e.g., the following monographs: Berinde [4], Browder [5], Chidume [6], Reich [7], and the references therein).
1.2 Monotonetype mappings in arbitrary normed spaces
The extension of the monotonicity condition from a Banach space into its dual has been the starting point for the development of nonlinear functional analysis…. The monotone mappings appear in a rather wide variety of contexts, since they can be found in many functional equations. Many of them appear also in calculus of variations, as subdifferential of convex functions (Pascali and Sburian [8], p.101).
In studying the equation \(0\in Au\), where A is a multivalued accretive operator on a Hilbert space H, Browder introduced an operator T defined by \(T:= IA\) where I is the identity map on H. He called such an operator pseudocontractive. It is clear that solutions of \(0\in Au\), if they exist, correspond to fixed points of T.
Within the past 35 years or so, methods for approximating solutions of equation (1.8) when A is an accretivetype operator have become a flourishing area of research for numerous mathematicians. Numerous convergence theorems have been published in various Banach spaces and under various continuity assumptions. Many important results have been proved, thanks to geometric properties of Banach spaces developed from the mid1980s to the early 1990s. The theory of approximation of solutions of the equation when A is of the accretivetype reached a level of maturity appropriate for an examination of its central themes. This resulted in the publication of several monographs which presented indepth coverage of the main ideas, concepts, and most important results on iterative algorithms for appropriation of fixed points of nonexpansive and pseudocontractive mappings and their generalizations, approximation of zeros of accretivetype operators; iterative algorithms for solutions of Hammerstein integral equations involving accretivetype mappings; iterative approximation of common fixed points (and common zeros) of families of these mappings; solutions of equilibrium problems; and so on (see, e.g., Agarwal et al. [17]; Berinde [4]; Chidume [6]; Reich [18]; Censor and Reich [19]; William and Shahzad [20], and the references therein). Typical of the results proved for solutions of equation (1.8) is the following theorem.
Theorem 1.1
(Chidume [21])
 (i)
\(\lim_{n\rightarrow\infty}\theta_{n} =0\), \(\{\theta_{n}\}\) is decreasing;
 (ii)
\(\sum\lambda_{n}\theta_{n} = \infty\); \(\sum\rho_{E}(\lambda_{n}M_{1})<\infty\), for some constant \(M_{1} > 0\);
 (iii)
\(\lim_{n\rightarrow\infty} \frac{ [\frac{\theta _{n1}}{\theta _{n}}1 ]}{\lambda_{n}\theta_{n}}=0\).
Unfortunately, developing algorithms for approximating solutions of equations of type (1.8) when \(A:E\rightarrow2^{E^{*}}\) is of monotone type has not been very fruitful. Part of the difficulty seems to be that all efforts made to apply directly the geometric properties of Banach spaces developed from the mid 1980s to the early 1990s proved abortive. Furthermore, the technique of converting the inclusion (1.8) into a fixed point problem for \(T:= IA : E\rightarrow E\) is not applicable since, in this case when A is monotone, A maps E into \(E^{*}\), and the identity map does not make sense.
Fortunately, Alber [22] (see also, Alber and Ryazantseva [23]) recently introduced a Lyapunov functional \(\phi:E\times E\rightarrow\mathbb{R}\), which signaled the beginning of the development of new geometric properties of Banach spaces which are appropriate for studying iterative methods for approximating solutions of (1.8) when \(A:E\rightarrow2^{E^{*}}\) is of monotone type. Geometric properties so far obtained have rekindled enormous research interest on iterative methods for approximating solutions of equation (1.8) where A is of monotone type, and other related problems (see, e.g., Alber [22]; Alber and GuerreDelabriere [24]; Chidume [21, 25]; Chidume et al. [26]; Diop et al.[27]; Moudafi [28], Moudafi and Tera [29]; Reich [30]; Sow et al. [31]; Takahashi [32]; Zegeye [33] and the references therein).
It is our purpose in this paper to apply the notion of Jfixed points (which has also been defined as a semifixed point (see, e.g., Zegeye [33]), a duality fixed point (see, e.g., Liu [34]) and, as far as we know, a new class of mappings called Jpseudocontractive maps introduced here to prove that \(T:=(JA)\) is Jpseudocontractive if and only if A is monotone; and in the case that E is a uniformly convex and a uniformly smooth real Banach space with dual \(E^{*}\), \(T: E\rightarrow2^{E^{*}}\) is a bounded Jpseudocontractive map with a nonempty Jfixed point set, and \(JT :E\rightarrow2^{E^{*}}\) is maximal monotone, a sequence is constructed which converges strongly to a Jfixed point of T. As an immediate application of this result, an analog of Theorem 1.1 for bounded maximal monotone maps is obtained, which is also a complement of the proximal point algorithm of Martinet [35] and Rockafellar [36], which has also been studied by numerous authors (see, e.g., Bruck [37]; Chidume [38]; Chidume [21]; Chidume and Djitte [39]; Kamimura and Takahashi [40]; Lehdili and Moudafi [41]; Reich [42]; Reich and Sabach [43, 44]; Solodov and Svaiter [45]; Xu [46] and the references therein). Furthermore, this analog is applied to approximate solutions of Hammerstein integral equations and is also applied to convex optimization problems. Finally, our techniques of proofs are of independent interest.
2 Preliminaries
Lemma 2.1
(Alber and Ryazantseva [23])
Lemma 2.2
(Alber and Ryazantseva [23], p.50)
Lemma 2.3
(Alber and Ryazantseva [23], p.45)
Lemma 2.4
(Alber and Ryazantseva [23], p.46)
Let \(E^{*}\) be a real strictly convex dual Banach space with a Fréchet differentiable norm. Let \(A:E\rightarrow2^{E^{*}}\) be a maximal monotone operator with no monotone extension. Let \(z\in E^{*}\) be fixed. Then for every \(\lambda>0\), there exists a unique \(x_{\lambda}\in E\) such that \(Jx_{\lambda}+\lambda Ax_{\lambda}\ni z\) (see Reich [7], p. 342). Setting \(J_{\lambda}z=x_{\lambda}\), we have the resolvent \(J_{\lambda}:=(J+\lambda A)^{1} :E^{*}\rightarrow E\) of A for every \(\lambda>0\). The following is a celebrated result of Reich.
Lemma 2.5
(Reich, [7]; see also, Kido, [49])
Let \(E^{*}\) be a strictly convex dual Banach space with a Fréchet differentiable norm, and let A be a maximal monotone operator from E to \(E^{*}\) such that \(A^{1}0\ne\emptyset\). Let \(z\in E^{*}\) be arbitrary but fixed. For each \(\lambda>0\) there exists a unique \(x_{\lambda}\in E\) such that \(Jx_{\lambda}+ \lambda Ax_{\lambda}\ni z\). Furthermore, \(x_{\lambda}\) converges strongly to a unique \(p\in A^{1}0\).
Lemma 2.6
Remark 1
Remark 2
Lemma 2.7
(Kamimura and Takahashi [48])
Let X be a real smooth and uniformly convex Banach space, and let \(\{ x_{n}\}\) and \(\{y_{n}\}\) be two sequences of X. If either \(\{x_{n}\}\) or \(\{ y_{n}\}\) is bounded and \(\phi(x_{n},y_{n})\to0\) as \(n\to\infty\), then \(\ x_{n}y_{n}\ \to0\) as \(n\to\infty\).
Lemma 2.8
(Xu [50])
 (i)
\(\{\sigma_{n}\}_{n=1}^{\infty}\subset[0,1]\), \(\sum_{n=1}^{\infty}\sigma_{n}=\infty\), or equivalently, \(\prod_{n=1}^{\infty}(1\sigma_{n})=0\);
 (ii)
\(\limsup_{n\rightarrow\infty}b_{n}\le0\);
 (iii)
\(c_{n}\ge0\) (\(n\ge0\)), \(\sum_{n=1}^{\infty}c_{n}<\infty\).
Definition 2.9
(Jfixed point)
Let E be an arbitrary normed space and \(E^{*}\) be its dual. Let \(T:E\rightarrow2^{E^{*}}\) be any mapping. A point \(x\in E\) will be called a Jfixed point of T if and only if there exists \(\eta\in Tx\) such that \(\eta\in Jx\).
3 Main results
We introduce the following definition.
Definition 3.1
(Jpseudocontractive mappings)
Example 1
If \(E=H\), a real Hilbert space, then J is the identity map on H. Consequently, every pseudocontractive map on H is Jpseudocontractive.
For our next example, we need the following characterization of the normalized duality map on \(l_{p}\), \(1< p<\infty\).
Example 2
Remark 4
We observe that, assuming existence, a zero of a monotone mapping \(A:E\rightarrow2^{E^{*}}\) corresponds to a Jfixed point of a Jpseudocontractive mapping, T.
The following lemma asserts that \(A:E\rightarrow2^{E^{*}}\) is monotone if and only if \(T:=(JA):E\rightarrow2^{E^{*}}\) is Jpseudocontractive.
Lemma 3.2
Let E be an arbitrary real normed space and \(E^{*}\) be its dual space. Let \(A:E\rightarrow2^{E^{*}}\) be any mapping. Then A is monotone if and only if \(T:=(JA): E\rightarrow2^{E^{*}}\) is Jpseudocontractive.
Proof
Hence, A is monotone. □
We now prove the following lemma, which will be crucial in the sequel.
Lemma 3.3
Proof
 (i)
\(\sum_{n=1}^{\infty}\lambda_{n}=\infty\);
 (ii)
\(\lambda_{n}M_{0}^{*}\le\gamma_{0}\theta_{n}\); \(\delta ^{1}_{E}(\lambda _{n}M_{0}^{*}) \leq\gamma_{0}\theta_{n}\),
Theorem 3.4
Proof
 (i)
\(\sum_{n=1}^{\infty}\lambda_{n}\theta_{n}=\infty\),
 (ii)
\(\lambda_{n}M_{0}^{*}\le\gamma_{0}\theta_{n}\); \(\delta ^{1}_{E}(\lambda_{n}M_{0}^{*}) \leq\gamma_{0}\theta_{n}\),
 (iii)
\(\frac{\delta^{1}_{E} (\frac{\theta_{n1}\theta _{n}}{\theta_{n}}K )}{\lambda_{n}\theta_{n}} \rightarrow0\), \(\frac{\delta ^{1}_{E^{*}} (\frac{\theta_{n1}\theta_{n}}{\theta_{n}}K )}{\lambda_{n}\theta_{n}} \rightarrow0\), as \(n\rightarrow\infty\),
 (iv)
\(\frac{1}{2} (\frac{\theta_{n1}\theta_{n}}{\theta _{n}}K )\in(0,1)\),
Theorem 3.5
Proof
Example 3
We now verify that, with these prototypes, the conditions (i)(iii) of Theorem 3.5 are satisfied. Clearly (i) and the first part of (ii) are easily verified.
Remark 5
We remark, following Lindenstrauss and Tzafriri [47], that in applications, we do not often use the precise value of the modulus of convexity but only a power type estimate from below.
A uniformly convex space X has modulus of convexity of power type p if, for some \(0< K<\infty\), \(\delta_{X}(\epsilon)\ge K\epsilon^{p}\). For instance, \(L_{p}\) spaces have modulus of convexity of power type 2, for \(1< p\le2\), and of power type p, for \(p>2\) (see, e.g., [47], p.63). We observe that the condition for modulus of convexity of power type p corresponds to that of puniformly convex spaces. However, we see that \(L_{p}\) spaces are puniformly convex, for \(1< p< 2\), and are 2uniformly convex, for \(p\ge2\). These lead us to prove the following corollary of Theorem 3.4, which will be crucial in several applications.
Corollary 3.6
Proof
 (i)^{∗} :

\(\lambda_{n}\le\gamma_{0}\theta_{n}\),
 (ii)^{∗} :

\(\sum_{n=1}^{\infty}\lambda_{n}\theta_{n}=\infty\),
 (iii)^{∗} :

\((\frac{\theta_{n1}\theta_{n}}{\theta_{n}} )^{1/p}\rightarrow0\), \(\frac{M^{*} (\frac{\theta_{n1}\theta _{n}}{\theta_{n}} )^{1/p}}{\lambda_{n}\theta_{n}}\rightarrow0\), \(\frac{ (\lambda _{n}^{(1/p)}M_{0}^{**} )}{\theta_{n}} \rightarrow0\), as \(n\rightarrow\infty\), for some \(M_{0}^{**}, M^{*}>0\),
Example 4
4 Application to zeros of maximal monotone maps
Corollary 4.1
5 Complement to proximal point algorithm
This was resolved in the negative by Güler [51] who produced a proper closed convex function g in the infinite dimensional Hilbert space \(l_{2}\) for which the proximal point algorithm converges weakly but not strongly (see also Bauschke et al. [52]). Several authors modified the proximal point algorithm to obtain strong convergence (see, e.g., Bruck [37]; Kamimura and Takahashi [40]; Lehdili and Moudafi [41]; Reich [42]; Solodov and Svaiter [45]; Xu [46]). We remark that in every one of these modifications, the recursion formula developed involved either the computation of \((I+\lambda_{k} A)^{1}(x_{k})\) at each point of the iteration process or the construction, at each iteration, of two subsets of the space, intersecting them and projecting the initial vector onto the intersection. As far as we know, the first iteration process to approximate a solution of \(0\in Au\) in real Banach spaces more general than Hilbert spaces and which does not involve either of these setbacks was given by Chidume and Djitte [39] who proved a special case of Theorem 1.1 in which the space E is a 2uniformly smooth real Banach space. These spaces include \(L_{p}\) spaces, \(2\le p<\infty\), but do not include \(L_{p}\) spaces, \(1< p<2\). This result of Chidume and Djitte has recently been proved in uniformly convex and uniformly smooth real Banach spaces (which include \(L_{p}\) spaces, \(1< p<\infty\)) (Chidume (Theorem 1.1) above).
Corollary 4.1 of this paper is an analog of Theorem 1.1 for maximal monotone maps when \(A:E\rightarrow2^{E^{*}}\) is a maximal monotone and bounded map, a result which complements the proximal point algorithm, under this setting, in the sense that it yields strong convergence to a solution of \(0\in Au\) and without requiring either the computation of \((J+\lambda A)^{1}(z_{n})\) at each iteration process, or the construction of two subsets of E, and projection of the initial vector onto their intersection, at each stage of the iteration process.
6 Application to solutions of Hammerstein integral equations
Definition 6.1
Among the first early results on the approximation of solution of Hammerstein equations is the following result of Brézis and Browder.
Theorem 6.2
(Brézis and Browder [53])
It is obvious that if an iterative algorithm can be developed for the approximation of solutions of equation of Hammersteintype (6.3), this will certainly be preferred.
Attempts have been made to approximate solutions of equations of Hammersteintype using Manntype iteration scheme. However, the results obtained were not satisfactory (see, e.g., [54]). The recurrence formulas used in early attempts involved \(K^{1}\) which is also required to be strongly monotone, and this, apart from limiting the class of mappings to which such iterative schemes are applicable, it is also not convenient in applications. Part of the difficulty is the fact that the composition of two monotone operators need not to be monotone.
Some typical results obtained using the recursion formulas described above in approximating solutions of nonlinear Hammerstein equations involving monotone maps in Hilbert spaces can be found in [57, 58].
In real Banach space X more general than Hilbert spaces, where \(F,K: X\rightarrow X\) are of accretivetype, Chidume and Zegeye considered an operator \(A:E\rightarrow E\) where \(E:= X\times X\) and were able to successfully approximate solutions of Hammerstein equations using recursion formulas described above. These schemes have now been employed by Chidume and other authors to approximate solutions of Hammerstein equations in various Banach spaces under various continuity assumptions (see, e.g., [27, 31, 55–71]). This success has not carried over to the case of monotonetype mappings in Banach spaces where K and F map a space into its dual. In this section, we introduce a new iterative scheme and prove that a sequence of our scheme converges strongly to a solution of a Hammerstein equation under this setting. For this purpose, we begin with the following preliminaries and lemmas.
We now prove the following lemmas.
Lemma 6.3
 (a)
E is uniformly smooth and uniformly convex,
 (b)
\(j_{q}^{E}\) is singlevalued duality mapping on E.
Proof
The following lemma will be needed in the following.
Lemma 6.4
(Browder [73])
Let X be a strictly convex reflexive Banach space with a strictly convex conjugate space \(X^{*}\), \(T_{1}\) a maximal monotone mapping from X to \(X^{*}\), \(T_{2}\) a hemicontinuous monotone mapping of all of X into \(X^{*}\) which carries bounded subsets of X into bounded subsets of \(X^{*}\). Then the mapping \(T=T_{1}+T_{2}\) is a maximal monotone map of X into \(X^{*}\).
Using Lemma 6.4, we prove the following important lemma which will be used in the sequel.
Lemma 6.5
Proof
Observe that S is monotone. Let \(h=[h_{1},h_{2}]\in E^{*}\times E\). Since F, K are maximal monotone, take \(u=(J+\lambda F)^{1}h_{1}\) and \(v=(J_{*}+\lambda K)^{1}h_{2}\). Then \((J+\lambda S)w=h\), where \(w=[u,v]\). Hence, S is maximal monotone.
Clearly, T is bounded and monotone. Furthermore it is continuous. Hence, it is hemicontinuous. Therefore by Lemma 6.4, \(A=S+T\) is maximal monotone. □
Lemma 6.6
Let E be a uniformly convex and uniformly smooth real Banach space. Let \(F:E\rightarrow E^{*}\) and \(K:E^{*}\rightarrow E\) be monotone mappings with \(D(F)=D(K)=E\). Let \(T:E\times E^{*}\rightarrow E^{*}\times E\) be defined by \(T[u,v]=[JuFu+v,J_{*}vKvu]\) for all \((u,v)\in E\times E^{*}\), then T is Jpseudocontractive. Moreover, if the Hammerstein equation \(u+KFu=0\) has a solution in E, then \(u^{*}\) is a solution of \(u+KFu=0\) if and only if \((u^{*},v^{*})\in F_{E}^{J}(T)\), where \(v^{*}=Fu^{*}\).
Proof
Using the monotonicity of F and K, we easily obtain \(\langle Tw_{1}Tw_{2},w_{1}w_{2}\rangle\le\langle Jw_{1}Jw_{2},w_{1}w_{2}\rangle\) for all \(w_{1}=[u_{1},v_{1}], w_{2}=[u_{2},v_{2}]\in E\times E^{*}\).
We now prove the following theorem.
Theorem 6.7
Proof
From Lemma 6.6 we see that \(T:E\times E^{*}\rightarrow E^{*}\times E\) defined by \(T[u,v]=[JuFu+v,J_{*}vKvu]\) for all \((u,v)\in E\times E^{*}\) is Jpseudocontractive, and \(A:=(JT)\) is maximal monotone.
Applying Theorem 3.4 where \(X=E\times E^{*}\), from Lemma 6.3, X is uniformly convex and uniformly smooth. We obtain (6.8) and (6.9) and the proof follows. □
7 Application to convex optimization problem
The following lemma is well known (see, e.g., [74], p.23, for similar proof in the Hilbert space case).
Lemma 7.1
Let X be a normed space. Let \(f:X\rightarrow\mathbb{R}\) be a convex function that is bounded on bounded subsets of X. Then the subdifferential, \(\partial f:X\rightarrow2^{X^{*}}\) is bounded on bounded subsets of E.
We now prove the following strong convergence theorem.
Theorem 7.2
Proof
Since f is convex and bounded, we see that ∂f is bounded. By Rockafellar [75, 76] (see also, e.g., Minty [2], Moreau [77]), we see that \((\partial f)\) is maximal monotone mapping from \(E^{*}\) into E and \(0\in(\partial f)^{1}v\) if and only if \(f(v)=\min_{x\in E}f(x)\). Since f is convex and bounded, from Lemma 7.1 we see that ∂f is bounded, hence, the conclusion follows from Corollary 4.1. □
Remark 6
8 Conclusion
Let E be a uniformly convex and uniformly smooth real Banach space with dual \(E^{*}\). Approximation of zeros of accretivetype maps of E to itself, assuming existence, has been studied extensively within the past 40 years or so (see, e.g., Agarwal et al. [17]; Berinde [4]; Chidume [6]; Reich [18]; Censor and Reich [19]; William and Shahzad [20], and the references therein). The key tool for this study has been the study of fixed points of pseudocontractivetype maps.
Unfortunately, for approximating zeros of monotonetype maps from E to \(E^{*}\), the normal fixed point technique is not applicable. This motivated the study of the notion of Jpseudocontractive maps introduced in this paper. The main result of this paper is Theorem 3.5 which provides an easily applicable iterative sequence that converges strongly to a Jfixed point of T, where \(T:E\rightarrow2^{E^{*}}\) is a Jpseudocontractive and bounded map such that \(JT\) is maximal monotone. The two parameters in the recursion formula of the theorem, \(\theta_{n}\) and \(\lambda_{n}\), are easily chosen in any possible application of the theorem (see Example 4 above).
The theorem is, in particular, applicable in \(L_{p}\) and \(l_{p}\) spaces, \(1< p<\infty\). In these spaces, the normalized duality maps J and \(J^{1}\) which appear in the recursion formula of the theorem are precisely known (see Remark 6 above).
Consequently, while the proof of the theorem is very technical and nontrivial, with the simple choices of the iteration parameters and the exact explicit formula for J and \(J^{1}\), the recursion formula of the theorem which does not involve the resolvent operator, \((J+\lambda A)^{1}\), is extremely attractive and user friendly.
Theorem 3.5 is applicable in numerous situations. In this paper, it has been applied to approximate a zero of a bounded maximal monotone map \(A:E\rightarrow2^{E^{*}}\) with \(A^{1}(0)\ne\emptyset\).
Furthermore, the theorem complements the proximal point algorithm by providing strong convergence to a zero of a maximal monotone operator A and without involving the resolvent \(J_{r}:=(J+rA)^{1}\) in the recursion formula. In addition, it is applied to approximate solutions of Hammerstein integral equations and also to approximate solutions of convex optimization problems. Theorem 3.5 continues to be applicable in approximating solutions of nonlinear equations. It has recently been applied to approximate a common zero of an infinite family of Jnonexpansive maps, \(T_{i}: E\rightarrow2^{E^{*}}\), \(i\ge1\) (see Chidume et al. [78]). In the case that \(E=H\) is a real Hilbert space, the result obtained in Chidume et al. [78] is a significant improvement of important known results. We strongly believed that the results of this paper will continue to be applied to approximate solutions of equilibrium problems in nonlinear operator theory.
Declarations
Acknowledgements
Research was supported from ACBF Research Grant Funds to AUST.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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