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A strong convergence theorem for generalizedΦstrongly monotone maps, with applications
 C. E. Chidume^{1}Email author,
 M. O. Nnakwe^{1} and
 A. Adamu^{1}
https://doi.org/10.1186/s1366301906609
© The Author(s) 2019
 Received: 14 December 2018
 Accepted: 20 May 2019
 Published: 17 June 2019
Abstract
Let X be a uniformly convex and uniformly smooth real Banach space with dual space \(X^{*}\). In this paper, a Manntype iterative algorithm that approximates the zero of a generalizedΦstrongly monotone map is constructed. A strong convergence theorem for a sequence generated by the algorithm is proved. Furthermore, the theorem is applied to approximate the solution of a convex optimization problem, a Hammerstein integral equation, and a variational inequality problem. This theorem generalizes, improves, and complements some recent results. Finally, examples of generalizedΦstrongly monotone maps are constructed and numerical experiments which illustrate the convergence of the sequence generated by our algorithm are presented.
Keywords
 GeneralizedΦstrongly monotone map
 Optimization problem
 Hammerstein integral equation
 Variational inequality problem
 Strong convergence
MSC
 47H09
 47H05
 47J25
 47J05
1 Introduction
Remark 1
The class of generalizedΦstrongly monotone maps is the largest class of monotone maps for which, if a solution of equation (1.1) exists, it is always unique.
Theorem 1.1
(Diop et al. [33])
It is our purpose in this paper to first prove a strong convergence theorem for a generalizedΦstrongly monotone map using a Manntype iterative algorithm and without imposing the restriction that the operator be bounded. Then, the convergence theorem proved is applied to approximate the solution of a convex minimization problem, a Hammerstein integral equation, and a variational inequality problem over the set of common fixed points of a finite family of quasiΦnonexpansive maps. Our theorems are improvements of the results of Diop et al. [33], Chidume and Bello [20], Chidume [18], Chidume et al. [24, 26], and a host of other results in the literature (see Remark 5 below). Finally, we construct examples of generalizedΦstrongly monotone maps and also give numerical experiments to illustrate the convergence of the sequence generated by our algorithm.
2 Preliminaries
Lemma 2.1
(Alber [1])
Lemma 2.2
(Chidume [18])
Lemma 2.3
(Tan and Xu [55])
Lemma 2.4
(Kamimura and Takahashi [37])
Let X be a uniformly convex and uniformly smooth real Banach space and \(\{u_{n}\},\{v_{n}\}\) be sequences in X such that either \(\{u_{n}\} or\{v_{n}\}\) is bounded. If \(\underset{n\rightarrow \infty }{\lim }\psi (u_{n},v_{n})=0\), then \(\underset{n\rightarrow \infty }{\lim } \Vert u_{n}v_{n} \Vert =0\).
Remark 2
It is easy to see that the converse of Lemma 2.4 is also true whenever \(\{u_{n}\} and\{v_{n}\}\) are bounded.
Lemma 2.5
(Alber and Ryazantseva [2])
Lemma 2.6
(Alber and Ryazantseva [2])
Lemma 2.7
(Rockafellar [52], see also Pascali and Sburlin [46])
A monotone map \({A:X\to X^{*}}\) is locally bounded at the interior points of its domain.
Definition 2.8
A map \(A:X\to X^{*}\) is quasibounded if, for every \(\mu >0\), there exists \(\gamma >0\) such that whenever \(\langle v,Av \rangle \le \mu \Vert v \Vert \) and \(\Vert v \Vert \le \mu \), then \(\Vert Av \Vert \le \gamma \).
The following lemma has been proved. However, for completeness, we present the proof here (see, e.g., Pascali and Sburlan [46], chapter III, Lemma 3.6).
Lemma 2.9
Let X be a real normed space with dual space \(X^{*}\). Every monotone map \(A:D(A)\subset X\to X^{*}\) with \(0\in \operatorname{Int}D(A)\) is quasibounded.
Proof
3 Main result
In Theorem 3.1 below, the sequence \(\{\beta _{n}\}\subset (0,1)\) is assumed to satisfy the following conditions: (\(C_{1}\)) \(\sum \beta _{n}=\infty,\lim \beta _{n}=0\); (\(C_{2}\)) \(2\sum \delta _{X}^{1}(\beta _{n}M)M<\infty \); (\(C_{3}\)) \(2\delta _{X}^{1}(\beta _{n}M) \le \gamma _{0}\) for some \(M>0\), \(\gamma _{0}>0\), where \(\delta _{X}\) is the modulus of convexity (see, e.g., Chidume [17], pp. 5, 6).
Theorem 3.1
Proof
Claim. \(\liminf \varPhi ( \Vert v_{n}v^{*} \Vert )=0\).
This completes the proof. □
4 Application to convex optimization problem
In this section, we apply Theorem 3.1 in solving the problem of finding minimizers of convex functions defined on real Banach spaces. First, we begin with the following known results.
Lemma 4.1
(See, e.g., Diop et al. [33])
Let X be a real Banach space and \(g:X\to \mathbb{R}\) be a convex and differentiable function. Let \(dg: X\to X^{*}\) denote the differential map associated with g. Then \(v\in X\) is a minimizer of g if and only if \(dg(v) = 0\).
Lemma 4.2
(Xu [56], see also Chidume [17], p. 43)
Lemma 4.3
(Chidume et al. [26])
Remark 3
If for any \(R > 0\) and for any \(u,v \in X\) such that \(\Vert u \Vert \le R, \Vert v \Vert \le R\), then the map \(dg:X\to X^{*}\) is generalizedΦstrongly monotone. This can easily be seen from Lemmas 4.2 and 4.3.
We now prove the following theorem.
Theorem 4.4
Proof
Since g is a lower semicontinuous, convex, proper, and coercive function, then g has a minimizer \(v^{*} \in X\). Furthermore, \(dg:X\to X^{*}\) is generalizedΦstrongly monotone. Hence, the conclusion follows from Theorem 3.1. □
5 Application to Hammerstein integral equation
Lemma 5.1
Let X be a uniformly convex and uniformly smooth real Banach space with dual space \(X^{*}\) and \(E=X\times X^{*}\). Let \(F:X\to X^{*}\) and \(K:X^{*}\to X\) be generalized\(\varPhi _{1}\)strongly monotone and generalized\(\varPhi _{2}\)strongly monotone maps, respectively. Let \(A:E\to E^{*}\) be defined by \(A([u,v])=[Fuv,Kv+u]\). Then A is a generalizedΦstrongly monotone map.
Proof
Remark 4
For A defined in Lemma 5.1, \([u^{*},v^{*}]\) is a zero of A if and only if \(u^{*}\) solves (5.3), where \(v^{*} = Fu\).
 (\(C_{1}\)):

\(\sum \beta _{n}=\infty \); \(\lim \beta _{n}=0\).
 (\(C_{2}\)):

\(2\sum (\delta _{X}^{1}(\beta _{n}M_{1})M_{1}+ \delta _{X} ^{1}(\beta _{n}M_{2})M_{2})<\infty \).
 (\(C_{3}\)):

\(2\max \{\delta _{X}^{1}(\beta _{n}M_{1})M_{1},\delta _{X^{*}} ^{1}(\beta _{n}M_{2})M_{2}\}\le \gamma _{0}\) for some \(M_{1}>0\), \(M_{2}\), \(\gamma _{0}>0\).
 (\(C_{4}\)):

\(\gamma _{0}=\min \{1,\frac{\varPhi (\mu )}{2M_{1}},\frac{ \varPhi (\mu )}{2M_{2}}\}\), \(\delta _{X}\) is the modulus of convexity (see, e.g., Chidume [17], pp. 5, 6). We now prove the following theorem.
Theorem 5.2
6 Application to variational inequality problems
Theorem 6.1
Proof
The proof is in two steps.
Step 1. We show that the sequence \(\{v_{n}\}\) is bounded.
7 Examples
Example 1
Example 2
8 Numerical illustration
In this section, we present numerical examples to illustrate the convergence of the sequence generated by our algorithm.
Example 3
Algorithm.
Step 0: Choose any \(v_{1}\in \mathbb{R}^{2}\) and set a tolerance \(\epsilon _{0}>0\). Let \(k=1\) and set the maximum number of iterations, n.
Step 1: If \(\v_{k}\\leq \epsilon _{0}\) or \(k>n\), STOP. Otherwise, set \(\beta _{n}=\frac{1}{k+1}\).
Numerical illustration for the zero of a generalizedϕstrongly monotone map
Initial points  Num. of iter  Approx. solution 

(1,0)  88  9.6598 × 10^{−7} 
(0,1)  95  9.3690 × 10^{−7} 
(2,1)  103  9.9756 × 10^{−7} 
(1,4)  120  9.5080 × 10^{−7} 
\((\frac{1}{2},\frac{1}{2})\)  86  9.3020 × 10^{−7} 
\((1,\frac{1}{2})\)  92  9.6662 × 10^{−7} 
Example 4
Algorithm.
Step 0: Choose any \(u_{1},v_{1}\in \mathbb{R}^{2}\) and set a tolerance \(\epsilon _{0}>0\). Let \(k=1\) and set the maximum number of iterations, n.
Step 1: If \(\u_{k}\\leq \epsilon _{0}\) or \(k>n\), STOP. Otherwise, set \(\beta _{k}=\frac{1}{(k+1)}\).
Numerical illustration for the solution of Hammerstein integral equation
Initial points  Num. of iter  Approx. sol. (\(\u_{n+1}\\)) 

(1,0), (0,1)  45  9.7064 × 10^{−7} 
(1,1), (2,3)  49  9.4440 × 10^{−7} 
(2,3), (1,1)  49  9.9188 × 10^{−7} 
\((\frac{1}{2},\frac{1}{2})\), \((\frac{1}{2},\frac{1}{2})\)  36  9.6055 × 10^{−7} 
\((\frac{1}{2},1)\), \((\frac{1}{2},2)\)  38  9.4539 × 10^{−7} 
(3,5), (2,1)  55  9.7373 × 10^{−7} 
Example 5
Algorithm.
Step 0: Choose any \(v_{1}\in \mathbb{R}^{2}\) and set a tolerance \(\epsilon _{0}>0\). Let \(k=1\) and set the maximum number of iterations, n.
Step 1: If \(\v_{k}\\leq \epsilon _{0}\) or \(k>n\), STOP. Otherwise, set \(\beta _{n}=\frac{1}{k+1}\).
Numerical illustration for the solution of variational inequality problem
Initial points  Num. of iter  Approx. solution 

(1,0)  24  8.2377 × 10^{−7} 
(1,1)  24  9.6812 × 10^{−7} 
(2,3)  25  9.6103 × 10^{−7} 
(−2,1)  25  9.3095 × 10^{−7} 
\((\frac{1}{2},\frac{1}{2})\)  22  7.1434 × 10^{−7} 
\((\frac{1}{10},1)\)  92  9.6662 × 10^{−7} 
(5,8)  27  8.3144 × 10^{−7} 
Remark 5
 (1)
Theorems 3.1 and 5.2 are proved in a more general real Banach space which contains the space of 2uniformly convex space and \(L_{P}\) spaces, \(1< p<\infty \).
 (2)
The class of strongly monotone maps studied in Diop et al. [33], Chidume and Bello [20] is extended to the more general class of generalizedΦstrongly monotone maps in Theorems 3.1 and 5.2, respectively.
 (3)
The requirement that the maps A, K, and F be bounded, which is assumed in Theorems 1.1 and 3.1 of Diop et al. [33], Chidume and Bello [20], respectively, and in the theorem of Chidume et al. [24, 26] and Chidume [18], is dispensed with in our theorems.
9 Conclusion
In this paper, a Manntype iterative algorithm that approximates the zero of a generalizedΦstrongly monotone map is presented. A strong convergence theorem of the sequence generated by the algorithm is proved. Furthermore, the theorem proved is applied to approximate solutions of a convex minimization problem, a Hammerstein integral equation, and a variational inequality problem. The theorem proved generalizes, extends, and improves the results of Diop et al. [33], Chidume and Bello [20], Chidume [18], Chidume et al. [26], Chidume et al. [24], and other recent important related results in the literature. Finally, examples of generalizedΦstrongly monotone maps are constructed and numerical experiments which illustrate the convergence of the sequence generated by our algorithm, are presented.
Declarations
Acknowledgements
The authors appreciate the support of their institution. They also thank the anonymous referees for their very useful remarks which helped to improve the final version of this paper. Finally, they thank ACBF and AfDB for their financial support.
Availability of data and materials
Data sharing is not applicable to this article.
Funding
This work is supported from ACBF and AfDB Research Grant Funds to AUST.
Authors’ contributions
All the authors contributed equally in the writing of this paper. They read and approved the final manuscript.
Competing interests
The authors declare that they have no conflict of interest.
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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