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A strong convergence theorem for generalizedΦstrongly monotone maps, with applications
Fixed Point Theory and Applicationsvolume 2019, Article number: 11 (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.
Introduction
Let X be a real Banach space with dual space \(X^{*}\). Let \(A:D(A)\subset X \to X\) be a map, where \(D(A)\) denotes the domain of A. The map A is called accretive if, for each \(u,v\in D(A)\), there exists \(j(uv) \in J(uv)\) such that
where \(J:X\to 2^{X^{*}}\) is the normalized duality map defined, for each \(u\in X\), by
The map A is called strongly accretive if there exists \(k>0\) such that, for each \(u,v\in D(A)\), there exists \(j(uv) \in J(uv)\) such that
The map A is called stronglyΦaccretive if, for each \(u,v\in D(A)\), there exist \(j(uv) \in J(uv)\) and a strictly increasing function \(\varPhi:[0, \infty ) \to [0,\infty )\) with \(\varPhi (0)=0\) such that
The map A is called generalizedΦstrongly accretive if, for each \(u,v\in D(A)\), there exist \(j(uv) \in J(uv)\) and a strictly increasing function \(\varPhi:[0, \infty ) \to [0,\infty )\) with \(\varPhi (0)=0\) such that
The accretive map A is called maccretive if \(R(I+\lambda A)=X\) for all \(\lambda >0\) (see, e.g., Guan and Kartsatos [35], Reich [47], and the references therein). It is known that the class of generalizedΦstrongly accretive maps properly contains the class of Φstrongly accretive maps which, in turn, contains the class of strongly accretive maps. In Hilbert spaces, accretive maps are called monotone maps. The accretive maps were introduced independently in 1967 by Browder [11] and Kato [38]. Interest in such maps stems mainly from their firm connection with evolution equations (see, e.g., Berinde [4], Chidume [17], Reich [48], and the references contained in them). A fundamental problem in the study of accretive maps in Banach spaces is the following:
Several existence theorems have been established for equation (1.1) (see, e.g., Browder [8,9,10,11], Martin [40, 41]). It is well known that the class of generalizedΦstrongly accretive maps is the largest class of accretivetype maps for which, if a solution of equation (1.1) exists, it is always unique. Iterative algorithms for approximating solutions of equation (1.1) have been studied extensively by numerous authors. The first iterative method for approximating solutions of equation (1.1) in real Banach spaces more general than Hilbert spaces, as far as we know, was that by Chidume [16]. He proved that if \(X=L_{p}, p\geq 2\), and \(T:K \to K \) is a Lipschitz strongly pseudocontractive map, then the Mann iteration process converges strongly to \(u^{*}\in F(T)\), where K is a nonempty closed convex and bounded subset of X and \(F(T):=\{x\in K: Tx=x\}\). This result signalled the return to extensive research on iterative methods for approximating solutions of equation (1.1) in more general Banach spaces. This theorem of Chidume has been generalized in various directions by numerous authors. It has been extended to more general real Banach spaces and more general classes of nonlinear operators. The literature on this abounds, and most of these extensions and their applications can be found in any of the following monographs and journal papers: Berinde [4], Chidume [17], Goebel and Reich [34, 49]. Let \(A:D(A)\subset X\rightarrow X^{*}\) be a map, where \(D(A)\) denotes the domain of A. The map A is called monotone if
The map A is called strongly monotone if there exists \(k>0\) such that
The map A is called Φstrongly monotone if there exists a strictly increasing function \(\varPhi:[0,\infty ) \to [0,\infty ) \) with \(\varPhi (0)=0\) such that
The map A is called generalizedΦstrongly monotone if there exists a strictly increasing function \(\varPhi:[0,\infty ) \to [0,\infty ) \) with \(\varPhi (0)=0\) such that
It is easy to see that the class of generalizedΦstrongly monotone maps contains the class of Φstrongly monotone maps and the class of strongly monotone maps.
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.
Interest in monotone maps stems mainly from their usefulness in numerous applications. Consider, for example, the following: Let \(f:X\rightarrow \mathbb{R}\cup \{\infty \}\) be a proper lower semicontinuous and convex function. The subdifferential of f, \(\partial f: X\rightarrow 2^{X^{*}}\) is defined, for each \(u\in X\), by
It is easy to see that the ∂f is a monotone map on X and that \(0\in \partial f(u)\) if and only if u is a minimizer of f. Setting \(\partial f\equiv A\), then solving the equation \(Au=0\) is equivalent to solving for a minimizer of f. Several existence theorems have been established for the equation \(Au=0\) when the map A is of monotone type (see, e.g., Deimling [32]; Pascali and Shurian [46], and the references contained in them). Iterative methods for approximating solutions of \(Au=0\), where \(A: X \to X^{*}\) is of monotone type, have been studied by various authors. Unfortunately, not much has been achieved. Part of the problem is that in real Banach spaces more general than Hilbert spaces, since the map A maps X to \(X^{*}\), the recursion formulas containing \(u_{n}\) and \(Au_{n}\) used for accretivetype maps may not be well defined in this setting. Several attempts have been made to overcome this difficulty in the recursion formulas for approximating zeros of monotonetype maps (see, e.g., Chidume et al. [26], Kamimura and Takahashi [37], Reich and Sabach [51], Reich [48], Chidume et al. [28], and the references contained in them). In 2015, Diop et al. [33] studied an iterative scheme of Mann type to approximate the zero of a strongly monotone bounded map in a 2uniformly convex real Banach space with a uniformly Gâteaux differentiable norm. They proved the following theorem.
Theorem 1.1
(Diop et al. [33])
Let X be a 2uniformly convex real Banach space with uniformly Gâteaux differentiable norm and \(X^{*}\) be its dual space. Let \(A:X\to X^{*}\) be a bounded and kstrongly monotone map such that \(A^{1}(0)\neq \emptyset \). For arbitrary \(u_{1} \in X\), let \(\{u_{n}\}\) be the sequence defined iteratively by
where J is the normalized duality map on X and \(\{\alpha _{n}\} \subset (0,1)\) is a real sequence satisfying the following conditions: (i) \(\sum \alpha _{n}=\infty \), (ii) \(\sum \alpha _{n}^{2}<\infty \). Then there exists \(\gamma _{0}>0\) such that \(\alpha _{n}<\gamma _{0}\), the sequence \(\{u_{n}\}\) converges strongly to the solution of the equation \(Au=0\).
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.
Preliminaries
Let X be a smooth real Banach space with dual space \(X^{*}\). The map \(\psi:X\times X\to \mathbb{R}\), defined by \(\psi (u,v)=\u\^{2}2 \langle u,Jv\rangle +\v\^{2},\forall u,v\in X \), will play a central role in what follows. The map ψ was introduced by Alber [1] and has been studied by Alber [1], Kamimura and Takahashi [37], Reich [51], Chidume [17], Berinde [4], Chidume and Monday [23], and a host of other authors. It is easy to see from the definition of the map ψ that
Let \(V:X\times X^{*}\to \mathbb{R}\) be a map defined by \(V(u,u^{*})=\u\^{2}2\langle u,u^{*}\rangle +\u^{*} \^{2},\forall u\in X, u^{*}\in X\). Observe that \(V(u,u^{*})= \phi (u,J^{1}(u^{*})),\forall u\in X,u^{*}\in X^{*}. \) The following lemmas will be needed in the sequel.
Lemma 2.1
(Alber [1])
Let X be a reflexive strictly convex and smooth Banach space with \(X^{*}\) as its dual. Then
Lemma 2.2
(Chidume [18])
Let X be a uniformly convex real Banach space. For arbitrary \(r>0\), let \(B_{r}(0):=\{u\in X: \Vert u \Vert \le r\}\). Then, for arbitrary \(u,v\in B_{r}(0)\), the following inequality holds:
Lemma 2.3
(Tan and Xu [55])
Let \(\{a_{n}\}\) and \(\{\sigma \}\) be sequences of nonnegative real numbers. For some \(N_{o}\in \mathbb{N}\), the following relation holds:
(a) If \(\sum \sigma _{n}<\infty \), then \(\lim a_{n}\) exists. (b) If, in addition, the sequence \(\{a_{n}\}\) has a subsequence that converges to 0, then \(\{a_{n}\}\) converges to 0.
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])
Let X be a uniformly convex Banach space with dual space \(X^{*}\). Then, for any \(R>0\) and for any \(u,v\in X^{*}\) such that \(\Vert u \Vert \le R\), \(\Vert v \Vert \le R\), the following inequality holds:
where \(c_{2}=2\max \{1,R\},1< L<1.7\).
Lemma 2.6
(Alber and Ryazantseva [2])
Let X be a uniformly convex Banach space with dual space \(X^{*}\). Then, for any \(R>0\) and for any \(u,v\in X\) such that \(\Vert u \Vert \le R\), \(\Vert v \Vert \le R\), the following inequality holds:
where \(c_{2}=2\max \{1,R\},1< L<1.7\).
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
By Lemma 2.7, A is locally bounded at 0, i.e., there exists \(r>0\) such that
Now, using this \(\mu >0\), suppose \(\langle v,Av\rangle \le \mu \Vert v \Vert \) and \(\Vert v \Vert \le \mu \). Then, by the monotonicity of A, we have that
Observe that
Thus,
This implies that
Thus,
Therefore,
Hence, A is quasibounded. □
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
Let X be a uniformly convex and uniformly smooth real Banach space with dual space \(X^{*}\). Let \(A: D(A)=X\to X^{*}\) be a generalizedΦstrongly monotone map, where \(D(A)\) is the domain of A and \(A^{1}(0)\neq \emptyset \). For arbitrary \(v_{1}\in X\), let \(\{v_{n}\}\) be a sequence generated iteratively by
where J is the normalized duality map on X, and the sequence \(\{\beta _{n}\}\subset (0,1)\) satisfies conditions \(C_{1}\), \(C_{2}\), and \(C_{3}\). Then the sequence \(\{v_{n}\}\) converges strongly to \(v^{*} \in A^{1}(0)\).
Proof
First, we observe that if the equation \(Au=0\) has a solution, it is necessarily unique. If \(y^{*}\) is a solution of the equation \(Au=0\), then, from inequality (1.5), we have that
Suppose that \(u^{*}\neq y^{*}\) is another solution of the equation \(Au=0\), substituting \(u^{*}\) in inequality (3.2), we have
which implies, by the properties of Φ, that \(u^{*}=y^{*}\). This contradiction yields the uniqueness of the solution. The remainder of the proof is now in two steps.
Step 1. We show that the sequence \(\{v_{n}\}\) is bounded. Let \(v^{*} \in A^{1}(0)\). Let \(\mu >0\) be arbitrary but fixed. Then there exists \(r>0\) such that
Define \(B:=\{v\in X:\psi (v^{*},v)\le r\}\). It suffices to show that \(\{\psi (v^{*}, v_{n})\}\) is bounded for each \(n\in \mathbb{N}\). We proceed by induction. For \(n=1\), by construction, we have that \(\psi (v^{*}, v_{1})\le r\). Assume that \(\psi (v^{*}, v_{n})\le r\) for some \(n\ge 1\). Using inequality (2.1), we have that \(\Vert v_{n} \Vert \le \Vert v^{*} \Vert +\sqrt{r}\). Now, we show that \(\psi (v^{*}, v _{n+1})\le r\). Suppose by contradiction that \(\psi (v^{*}, v_{n+1}) \le r\) does not hold. Then \(\psi (v^{*}, v_{n+1})> r\). Since \(A:X\to X^{*}\) is locally bounded at \(v\in X\), there exist \(r_{v}>0\) and \(m>0\) such that
Define \(M_{0}:=\max \{m, \Vert v^{*} \Vert +\sqrt{r} \}\). Then \(\langle v,Av \rangle \le M_{0} \Vert v \Vert \) and \(\Vert v \Vert \le M_{0}\). By Lemma 2.9, there exists \(M>0\) such that \({ \Vert Av \Vert \le M,\forall v\in B.}\) Define \(\gamma _{0}:=\min \{1, \frac{\varPhi (\mu )}{M},\frac{\mu }{M}\}\). Using Lemma 2.1, we compute as follows:
Using the fact that A is a generalizedΦstrongly monotone map and Lemma 2.5, it follows from inequality (3.4) that
But from recursion formula (3.1), we have that
Applying Lemma 2.5 and inequality (3.6), we have that
Thus, from inequality (3.7), we obtain that
From Lemma 2.2, we have that
Using inequality (3.3), we have that
Hence,
From inequalities (3.7), (3.8), and the definition of \(\gamma _{0}\), we have that
Since Φ is strictly increasing, we have that
From inequality (3.5) and the definition of \(\gamma _{0}\), we have that
This is a contradiction. Hence, \(\{\psi (v^{*},v_{n})\}\) is bounded. Consequently, \(\{v_{n}\}\) is bounded.
Step 2. We show that the sequence \(\{v_{n}\}\) converges strongly to a point \(v^{*}\in A^{1}(0)\). Using inequality (3.5), we have that
By Lemma 2.3, we get that \(\{\psi (v^{*},v_{n})\}\) is convergent. Furthermore, we have that
Claim. \(\liminf \varPhi ( \Vert v_{n}v^{*} \Vert )=0\).
Suppose by contradiction that \(\liminf \varPhi ( \Vert v_{n}v^{*} \Vert )=0\) does not hold. Then \(\liminf \varPhi ( \Vert v_{n}v^{*} \Vert )=s>0\). Hence, there exists \(N_{1} \in \mathbb{N}\) such that
Using inequality (3.17), conditions \(C_{1}\) and \(C_{2}\), we have that
This is a contradiction. Hence, \(\liminf \varPhi ( \Vert v_{n}v^{*} \Vert )=0\). Thus, there exists a subsequence \(\{v_{n_{k}}\}\) of \(\{v_{n}\}\) such that
Using the property of Φ, it follows that \(\underset{k\to \infty }{\lim } \Vert v_{n_{k}}v^{*} \Vert =0\). By Remark 2, we have that
Consequently, by Lemma 2.3, we have that \(\underset{n\to \infty }{\lim }\psi (v^{*},v_{n})=0\). Hence, by Lemma 2.4, we have that \(\underset{n\to \infty }{ \lim } \Vert v_{n}v^{*} \Vert =0\).
This completes the proof. □
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)
Let X be a uniformly convex real Banach space. For arbitrary \(r > 0\), let \(B_{r}(0):= \{v \in X: \Vert v \Vert \le r\}\). Then there exists a continuous strictly increasing convex function \(\varPhi: [0,\infty )\to [0,\infty ),\varPhi (0) = 0\) such that, for every \(u,v \in B_{r}(0)\), the following inequality holds:
where J is the singlevalued normalized duality map on X.
Lemma 4.3
(Chidume et al. [26])
Let X be a uniformly convex and uniformly smooth real Banach space. Let \(g:X\to \mathbb{R}\) be a differentiable convex function. Then the differential map \({dg:X\to X^{*}}\) satisfies the following inequality:
where J is the singlevalued normalized duality map on X.
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
Let X be a uniformly convex and uniformly smooth real Banach space with dual space \(X^{*}\). Let \(g:X\to \mathbb{R}\cup \{\infty \}\) be a differentiable, convex, proper, and coercive function such that \((dg)^{1}(0)\neq \emptyset \). For arbitrary \(v_{1}\in X\), let the sequence \(\{v_{n}\}\) be generated by
where J is the normalized duality map on X. Assume that \(\{\beta _{n}\}\subset (0,1)\) satisfies conditions \(C_{1}\), \(C_{2}\), and \(C_{3}\) of Theorem 3.1. Then g has a unique minimizer \(v^{*}\in X\) and the sequence \(\{v_{n}\}\) converges strongly to \(v^{*}\).
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. □
Application to Hammerstein integral equation
Let \(\varOmega \subset {\mathbb{R}}^{n}\) be Lebesgue measurable. Let \(k:\varOmega \times \varOmega \to \mathbb{R}\) and \({f:\varOmega \times \mathbb{R} \to \mathbb{R}}\) be measurable realvalued functions. An integral equation of Hammerstein type has the form
where the unknown function u and inhomogeneous function w lie in a Banach space X of measurable realvalued functions. Define a linear map K by
on Ω and denote by F the superposition or Nemitskyi operator corresponding to f, i.e., \(Fu(y)=f(y,u(y))\). Then equation (5.1) can be put in the form
where, without loss of generality, we have taken \(w\equiv 0\). Interest in Hammerstein integral equations stems mainly from the fact that several problems that arise in differential equations, for instance, elliptic boundary value problems whose linear parts possess Green’s function can, as a rule, be put in the form (5.1) (see, e.g., Pascali and Sburian [46], chapter p. 164). Several existence and uniqueness theorems have been proved for equations of Hammerstein type (see, e.g., Brezis and Browder [5, 6], Chepanovich [15], Browder and Gupta [12], De Figueiredo and Gupta [31], and the references contained in them). In general, equations of Hammerstein type are nonlinear and there is no known method to find closed form solutions for them. Consequently, methods for approximating solutions of such equations are of interest. For earlier and more recent works on approximation of solutions of equations of Hammerstein type, the reader may consult any of the following: Brezis and Browder [5, 6], Chidume and Shehu [27], Chidume and Ofoedu [25], Chidume and Zegeye [29], Chidume and Djitte [22], Ofoedu and Onyi [45], Ofoedu and Malonza [44], Zegeye and Malonza [58], Chidume and Bello [20], Minjibir and Mohammed [42], and the references contained in them. We now apply Theorem 3.1 to approximate a solution of equation (5.3). The following lemma would be needed in the proof of Theorem 5.2 below.
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
Let \([u_{1},v_{1}],[u_{2},v_{2}]\in E\). Then
□
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 \(v^{*} = Fu\).
In Theorem 5.2 below, the sequence \(\{\beta _{n}\}\subset (0,1)\) is assumed to satisfy the following conditions:
 (\(C_{1}\)):

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

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

\(2\max \{\delta _{X}^{1}(\beta _{n}M_{1})M_{1},\delta _{X^{*}} ^{1}(\beta _{n}M_{2})M_{2}\}\le \gamma _{0}\) 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
Let X be a uniformly convex and uniformly smooth real Banach space with dual space \(X^{*}\). Let \(F:D(F)=X\to X^{*}\) and \(K:D(K)=X^{*} \to X\) be generalized\(\varPhi _{1}\)strongly monotone and generalized\(\varPhi _{2}\)strongly monotone maps, respectively, where \(D(F)\) and \(D(K)\) denote the domains of F and K, respectively, and such that equation (5.3) has a solution. For arbitrary \((u_{1},v_{1}) \in X\times X^{*}\), define the sequences \(\{u_{n}\}\) and \(\{v_{n}\}\) by
Assume that the sequence \(\{\beta _{n}\}\subset (0,1)\) satisfies conditions \(C_{1}\), \(C_{2}\), and \(C_{3}\) of Theorem 3.1. Then the sequences \(\{u_{n}\}\) and \(\{v_{n}\}\) converge strongly to \(u^{*}\) and \(v^{*}\), respectively, where \(u^{*}\) is a solution of the equations \(u+KFu=0\) and \(v^{*}=Fu^{*}\).
Proof
Set \(E=X\times X^{*}\) and \(A:E\to E^{*}\) by \(A([u,v])=[Fuv,Kv+u]\). Then by Lemma 5.1, A is a generalizedΦstrongly monotone map. Hence, by Theorem 3.1 and Remark 4, the result is immediate. □
Application to variational inequality problems
Let X be a real normed space with dual space \(X^{*}\). Let \(A: C\subset X\to X^{*}\) be a nonlinear map. The classical variational inequality problem is the following:
The set of solutions of problem (6.1) is denoted by \(V I(A, C)\). Variational inequality problems were first introduced and studied by Stampacchia [54] in 1964 and have been found to have numerous applications in the study of nonlinear analysis (see, e.g., Shi [53], Noor [43], Yao [57], Stampacchia [54], and the references contained in them). Several existence results for problem (6.1) have been proved when A is a monotonetype map defined on certain Banach spaces (see, e.g., Hartman and Stampacchia [36], Browder [7], Barbu and Precupanu [3], and the references contained in them). Iterative approximation of solutions of problem (6.1), assuming existence, has been studied extensively. For earlier and recent works on variational inequality problems, the reader may consult any of the following: Stampacchia [54], Korpelevich [39], Censor et al. [13], Chidume et al. [19, 21], and the references contained in them. We now prove the following theorem.
Theorem 6.1
Let X be a uniformly convex and uniformly smooth real Banach space with dual space \(X^{*}\), and let C be a nonempty closed and convex subset of X. Let \(A:D(A)=X\to X^{*}\) be a generalizedΦstrongly monotone map, where \(D(A)\) is the domain of A. Let \({T_{i}:C\to X}\), \({i=1,2,\ldots, N}\), be a finite family of quasiϕnonexpansive maps such that \(P:=\bigcap _{i=1}^{N}F(T_{i})\neq \emptyset \). For arbitrary \(v_{1}\in X\), define the sequence \(\{v_{n}\}\) generated by
Assume that \(VI(A,P)\neq \emptyset \), and the sequence \(\{\beta _{n}\} \subset (0,1)\) satisfies conditions \(C_{1}\), \(C_{2}\), and \(C_{3}\) of Theorem 3.1. Then the sequence \(\{v_{n}\}\) converges strongly to \(v^{*}\in VI(A,P)\).
Proof
The proof is in two steps.
Step 1. We show that the sequence \(\{v_{n}\}\) is bounded.
Let \(v^{*} \in G^{1}(0)\). Let \(\mu >0\) be arbitrary but fixed. Then there exists \(r>0\) such that
Define \(B=\{v\in X:\psi (v^{*},v)\le r\}\). It suffices to show that \(\{\psi (v^{*}, v_{n})\}\) is bounded for each \(n\in \mathbb{N}\). We proceed by induction. For \(n=1\), by construction, \(\psi (v^{*}, v_{1}) \le r\). Assume that \(\psi (v^{*}, v_{n})\le r\) for some \(n\ge 1\). Applying the definition of the map ψ, we have that \(\Vert v_{n} \Vert \le \Vert v^{*} \Vert +\sqrt{r}\). Now, we show that \(\psi (v^{*}, v_{n+1}) \le r\). Suppose not, i.e., suppose \(\psi (v^{*}, v_{n+1})> r\). By Lemma 2.9, A is quasibounded. Thus, there exists \(M>0\) such that \(\Vert Av \Vert \le M,\forall v\in B\). Define \(\gamma _{0}:=\min \{1, \frac{\varPhi (\mu )}{M},\frac{\mu }{M} \}\). Using Lemma 2.1, we compute as follows:
Using the fact that A is a generalizedΦstrongly monotone map and Lemma 2.5, it follows from inequality (6.4) that
But from recursion formula (6.2), we have that
Applying Lemma 2.5 and inequality (6.6), we have that
Thus, from inequality (6.7), we obtain that
From Lemma 2.2, we have that
Using inequality (6.3), we have that
Hence,
From inequalities (6.8), (6.10), and the definition of \(\gamma _{0}\), we have that
Since Φ is strictly increasing, we have that
From inequality (6.5) and the definition of \(\gamma _{0}\), we have that
This is a contradiction. Hence, \(\{\psi (v^{*},v_{n})\}\) is bounded. Consequently, \(\{v_{n}\}\) is bounded. The remaining part of the proof follows from the proof of Theorem 3.1. □
Examples
Example 1
Let \(X=l_{p},1< p< 2\), and let \(A:l_{p}\to l_{p}^{*}\) be a map defined by
where J is the normalized duality map on X. Then
Hence, A is generalizedΦstrongly monotone map with \(\varPhi (t)=(p1)t^{2}\) (see, e.g., Chidume [17], p. 55).
Example 2
Let \(X=l_{p},2\le p< \infty \), and let \(A:l_{p}\to l_{p}^{*}\) be a map defined by
Then
Hence, A is a generalizedΦstrongly monotone map with \(\varPhi (t)=p^{1}c_{p}t^{p}\) (see, e.g., Chidume [17], p. 54).
Numerical illustration
In this section, we present numerical examples to illustrate the convergence of the sequence generated by our algorithm.
Example 3
In Theorem 3.1, set \(X=\mathbb{R}^{2}\) so that \(X^{*}=\mathbb{R}^{2}\),
Then it is easy to see that A is a generalizedΦstrongly monotone map and the vector \(v^{*}=(0,0)\) is the unique solution of the equation \(Av=0\). Take \(\beta _{n}=\frac{1}{n+1},n=1,2,\ldots \) , as our parameter in Theorem 3.1. With this, we now give the following algorithm which is a specialized version of Theorem 3.1.
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 \(\beta _{n}=\frac{1}{k+1}\).
Step 2: Compute
Step 3: Set \(k=k+1\) and go to Step 1.
Table 1 gives our test results using 10^{−6} tolerance.
The numerical result for the initial point (1,\(\frac{1}{2}\)) is sketched below where the yaxis represents the values of \(\v_{n+1}{0}\\) while the xaxis represents the number of iterations n (see Fig. 1).
Example 4
In Theorem 5.2, set \(X=\mathbb{R}^{2}\) so that \(X^{*}=\mathbb{R}^{2}\),
Then it is easy to see that F and K are generalizedΦstrongly monotone maps and the vector \(u^{*}=(0,0)\) is the unique solution of the equation \(u+KFu=0\). Take \(\beta _{n}=\frac{1}{(n+1)},n=1,2,\ldots \) , as our parameters in Theorem 5.2. With this, we now give the following algorithm which is a specialized version of Theorem 5.2.
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 \(\beta _{k}=\frac{1}{(k+1)}\).
Step 2: Compute
Step 3: Set \(k=k+1\) and go to Step 1.
Table 2 gives our test results using 10^{−6} tolerance.
The numerical result for the initial point \((3,5)\), \((2,1)\) is sketched below where the yaxis represents the values of \(\u_{n+1}{0}\\), while the xaxis represents the number of iterations n (see Fig. 2).
Example 5
In Theorem 6.1, set \(X=\mathbb{R}^{2}\) so that \(X^{*}=\mathbb{R}^{2}\),
Then it is easy to see that A is a generalizedΦstrongly monotone map, T is quasiΦnonexpansive, and the vector \(v^{*}=(0,0)\) is the common solution. We take \(\beta _{n}= \frac{1}{n+1},n=1,2,\ldots \) , as our parameter in Theorem 6.1. With this, we now give the following algorithm which is a specialized version of Theorem 6.1.
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 \(\beta _{n}=\frac{1}{k+1}\).
Step 2: Compute
Step 3: Set \(k=k+1\) and go to Step 1.
Table 3 gives our test results using 10^{−6} tolerance.
The numerical result for the initial point \((5,8)\) is sketched below where the yaxis represents the values of \(\v_{n+1}{\mathbf{0}}\\), while the xaxis represents the number of iterations n (see Fig. 3).
Remark 5
Our theorem is a significant improvement of the results of Diop et al. [33], Chidume and Bello [20], Chidume [18], Chidume et al. [26], and Chidume et al. [24] in the following sense:

(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.
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.
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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.
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MSC
 47H09
 47H05
 47J25
 47J05
Keywords
 GeneralizedΦstrongly monotone map
 Optimization problem
 Hammerstein integral equation
 Variational inequality problem
 Strong convergence