On the convergence of a generalized modified Krasnoselskii iterative process for generalized strictly pseudocontractive mappings in uniformly convex Banach spaces
- Ali Mohamed Saddeek^{1}Email author
https://doi.org/10.1186/s13663-016-0549-9
© Saddeek 2016
Received: 25 August 2015
Accepted: 25 April 2016
Published: 10 May 2016
Abstract
This paper aims to study the strong convergence of generalized modified Krasnoselskii iterative process for finding the minimum norm solutions of certain nonlinear equations with generalized strictly pseudocontractive, demiclosed, coercive, bounded, and potential mappings in uniformly convex Banach spaces. An application to nonlinear pseudomonotone equations is provided. The results extend and improve recent work in this direction.
Keywords
MSC
1 Introduction and preliminaries
Let H be a real Hilbert space with norm \(\|\cdot\|_{H}\) and inner product \((\cdot,\cdot)\). Let C be a nonempty closed and convex subset of H. Let T be a nonlinear mapping of H into itself. Let I denote the identity mapping on H. Denote by \(\mathfrak{F}(T)\) the set of fixed points of T.
Moreover, the symbols ⇀ and → stand for weak and strong convergence, respectively.
For \(r(x,y)=\lambda\in(0,1)\) (resp., \(r(x,y)=1\)) such mappings are said to be λ-contractive (resp., nonexpansive) mappings.
If \(r(x,y)=\lambda> 0\), then the class of generalized Lipschitzian mappings coincide with the class of λ-Lipschitzian mappings.
The class of λ-strictly pseudocontractive mappings has been studied recently by various authors (see, for example, [4–9]).
It worth noting that the class of generalized strictly pseudocontractive mappings includes generalized Lipschizian mappings, λ-strictly pseudocontractive mappings, λ-Lipschitzian mappings, pseudocontractive mappings, nonexpansive (or 0-strictly pseudocontractive) mappings.
These mappings appear in nonlinear analysis and its applications.
Definition 1.1
- (i)
demiclosed at 0 (see, for example, [10]) if \(Tx=0\) whenever \(\{x_{n}\}\subset H\) with \(x_{n}\rightharpoonup x\) and \(Tx_{n}\rightarrow0\), as \(n \rightarrow \infty\);
- (ii)pseudomonotone (see, for example, [11]) if it is bounded and \(x_{n}\rightharpoonup x \in H\) and$$ \limsup_{n\rightarrow \infty} ( Tx_{n},x_{n}-x) \leq0 \quad \Longrightarrow\quad \liminf_{n\rightarrow\infty} ( Tx_{n},x_{n}-y) \geq (Tx,x-y); $$
- (iii)coercive (see, for example, [12]) if$$ (Tx,x) \geq \rho\bigl(\Vert x\Vert _{H}\bigr)\|x \|_{H},\qquad \lim_{\xi\rightarrow+\infty} \rho (\xi)=+\infty; $$
- (iv)potential (see, for example, [13]) if$$ \int^{1}_{0}\bigl(\bigl(T\bigl(t(x+y),x+y\bigr) \bigr)-\bigl(T(tx),x\bigr)\bigr) \,dt= \int^{1}_{0}\bigl( T(x+ty),y\bigr) \,dt; $$
- (v)hemicontinuous (see, for example, [12]) if$$ \lim_{t\rightarrow 0} \bigl( T(x+ty),z\bigr)=(Tx,z); $$
- (vi)demicontinuous (see, for example, [12]) if$$ \lim_{\|x_{n}-x\|_{H}\rightarrow 0} ( Tx_{n},y)=(Tx,y); $$
- (vii)uniformly monotone (see, for example, [11]) if there exist \(p\geq2\), \(\alpha>0\) such that$$ ( Tx-Ty,x-y) \geq\alpha\|x-y\|_{H}^{p}; $$
- (viii)bounded Lipschitz continuous (see, for example, [13]) if there exist \(p\geq2\), \(M>0\) such that$$ \|Tx-Ty\|_{H} \leq M \bigl(\Vert x\Vert _{H}+\|y \|_{H}\bigr)^{p-2} \|x-y\|_{H}. $$
It should be noted that any demicontinuous mapping is hemicontinuous and every uniformly monotone is monotone (i.e., \((Tx-Ty,x-y)\geq 0\), \(\forall x, y\in H\)) and every monotone hemicontinuous is pseudomonotone.
If T is uniformly monotone (resp. bounded Lipschitz continuous) with \(p=2\), then T is called strongly monotone (resp. M-Lipschitzian).
Recently, in a real Hilbert space setting, Saddeek and Ahmed [1] proved that the Krasnoselskii iterative sequence given by (1.3) converges weakly to a fixed point of T under the basic assumptions that \(I-T\) is generalized Lipschitzian, demiclosed at 0, coercive, bounded, and potential. Moreover, they also applied their result to the stationary filtration problem with a discontinuous law.
Saddeek [2] obtained some strong convergence theorems of the iterative algorithm (1.4) for finding the minimum norm solutions of certain nonlinear operator equations.
The class of uniformly convex Banach spaces play an important role in both the geometry of Banach spaces and relative topics in nonlinear functional analysis (see, for example, [16, 17]).
Let X be a real Banach space with its dual \(X^{\ast}\). Denote by \(\langle\cdot,\cdot\rangle\) the duality pairing between \(X^{\ast}\) and X. Let \(\|\cdot\|_{X}\) be a norm in X, and \(\|\cdot\|_{X^{\ast}}\) be a norm in \(X^{\ast}\).
A Banach space X is said to be strictly convex if \(\|x+y\|_{X}<2\) for every \(x, y \in X\) with \(\|x\|_{X}\leq1\), \(\|y\|_{X}\leq1\) and \(x\neq y\).
A Banach space X is said to be uniformly convex if for every \(\varepsilon>0\), there exists an increasing positive function \(\delta(\varepsilon)\) with \(\delta(0)=0\) such that \(\|x\|_{X}\leq 1\), \(\|y\|_{X}\leq1\) with \(\|x-y\|_{X}\geq\varepsilon\) imply \(\|x+y\|_{X}\leq2(1-\delta(\varepsilon))\) for every \(x, y \in X\).
It is well known that every Hilbert space is uniformly convex and every uniformly convex Banach space is reflexive and strictly convex.
Hilbert spaces, \(L^{p}\) (or \(l_{p}\)) spaces, and Sobolev spaces \(W_{p}^{1}\) (\(1< p<\infty\)) are uniformly convex and have a uniformly Gateaux differentiable norm.
Definition 1.1 above can easily be stated for mappings T from C to \(X^{\ast}\). The only change here is that one replaces the inner product \((\cdot,\cdot)\) by the bilinear form \(\langle\cdot,\cdot\rangle\).
It is well known (see, for example, [12, 20, 21]) that if A is pseudomonotone and coercive, then \(\operatorname{VI}(C,A)\) is a nonempty, closed, and convex subset of X. Further, if \(A=j_{p}-T\), then \(\tilde{\mathfrak{F}}(j_{p},T)=\{x \in C: j_{p}x=Tx\}=A^{-1}0\). In addition, there exists also a unique element \(z=\operatorname{proj}_{A^{-1}0}(0) \in \operatorname{VI}(A^{-1}0,j_{p})\), called the minimum norm solution of variational inequality (1.5) (or the metric projection of the origin onto \(A^{-1}0\)). If \(X=H\), then \(j_{p}=I\) and hence \(\tilde{\mathfrak{F}}=\mathfrak{F}\).
Example 1.1
Let Ω be a bounded domain in \(\mathbb{R}^{n}\) with Lipschitz continuous boundary. Let us consider \(p\geq2\), \(\frac{1}{p}+\frac{1}{q}=1\), and \(X=\mathring{W}_{p}^{(1)}(\Omega)\), \(X^{\ast }={W}_{q}^{(-1)}(\Omega)\). The p-Laplacian is the mapping \(-\Delta_{p}:\mathring{W}_{p}^{(1)}(\Omega)\rightarrow {W}_{q}^{(-1)}(\Omega)\), \(\Delta_{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\) for \(u\in \mathring{W}_{p}^{(1)}(\Omega)\).
It is well known that the p-Laplacian is in fact the generalized duality mapping \(j_{p}\) (more specifically, \(j_{p}=-\Delta_{p}\)), i.e., \(\langle j_{p}u,v\rangle=\int_{\Omega}{|\nabla u|^{p-2}(\nabla u, \nabla u) \,dx}\), \(\forall u, v \in \mathring{W}_{p}^{(1)}(\Omega)\).
The generalized duality mapping \(j_{p}=-\Delta_{p}\) is bounded, demicontinuous (and hence hemicontinuous) and monotone, and hence \(j_{p}\) is pseudomonotone.
From the definition of \(j_{p}\), it follows that \(j_{p}\) is coercive.
Since \(j_{p}u\in{W}_{q}^{(-1)}(\Omega)\), \(\forall u\in \mathring{W}_{p}^{(1)}(\Omega)\) is the subgradient of \(\frac{1}{p}\|u\|_{\mathring{W}_{p}^{(1)}(\Omega)}^{p}\), it follows that \(j_{p}\) is potential.
Since \(j_{p}\) is pseudomonotone and coercive (it is surjective), then \(j_{p}\) is demiclosed at 0 (see Saddeek [2] for an explanation).
The mapping \(j_{p}\) is generalized strictly pseudocontractive with \(r_{1}(x,y)=1\).
The following two lemmas play an important role in the sequel.
Lemma 1.1
([23])
Lemma 1.2
([24])
This algorithm can also be regarded as a modification of algorithm (3) in [1]. We shall call this algorithm the generalized modified Krasnoselskii iterative algorithm.
Obviously, (1.6) and (1.7) reduce to (1.4) and (1.2), respectively, when X is a Hilbert space.
The main purpose of this paper is to extend the results in [2] to uniformly convex Banach spaces and to generalized modified iterative processes with generalized strictly pseudocontractive mappings.
2 Main results
Now we are ready to state and prove the results of this paper.
Theorem 2.1
Proof
An immediate consequence of Theorem 2.1 is the following corollary.
Corollary 2.1
A special case of Corollary 2.1 is the following theorem due to Saddeek [2], who proved it under the condition that T is generalized Lipschitzian, which in turn, is a generalization of Theorem 2 of Saddeek and Ahmed [1].
Corollary 2.2
Except for the M-Lipschitzian condition for the mapping T, let all the other assumptions of Corollary 2.1 be satisfied and \(r_{2}=0\). Then the sequence \(\{x_{n}\}\) defined by (2.20) with \(\sum_{n=0}^{\infty} h(x_{n})= \infty\), \(0<\tau=\min\{1,\frac{1}{\lambda}\}\), and \(\sup_{x, y \in C} [r_{1}(x,y)]= (\lambda)^{2}<\infty\), converges strongly to \(\bar{x}=\operatorname{proj}_{S_{{h(\bar{x})}}^{-1}0}(0)\).
Remark 2.1
All conditions imposed in Theorem 2.1 on the mapping \(S_{h(x)}\) are quintessential to prove the main theorem, more precisely for the existence solution of \(S_{h(x)}x=0\), and to ensure the strong convergence of the generalized modified Krasnoselski iterative algorithm.
3 Application to nonlinear pseudomonotone equations
To ensure the existence of solutions of (3.1), we shall assume that A is pseudomonotone and coercive on \(\mathring{W}_{p}^{(1)}(\Omega)\) (\(1< p<\infty\)) (see, for example, [12]). Such nonlinear equations occur, in particular, in descriptions of a stabilized filtration and in problems of finding the equilibria of soft shells (see, for example, [25, 26]).
Theorem 3.1
Proof
Define \(S_{h(x)} : C\rightarrow X^{\ast}\) by \(S_{h(x)}x= Ax-f\), \(\forall x \in C\). Since (3.1) has at least one solution, then \(S_{{h(x)}}^{-1}0\neq\phi\). On the other hand, condition (3.2) with the bounded Lipschitz continuity of \(j_{p}\) clearly imply that A is bounded Lipschitz continuous and the potentiality of \(j_{p}\) imply that \(s_{h(x)}\) is potential.
Finally, the pseudomonotonicity of A implies that \(S_{{h(x)}}\) is demiclosed at 0 can be proved by proceeding as in the proof of Theorem 5.1 of [2]. Now we apply Theorem 2.1 to yield the desired result. □
4 Conclusion
In this work, we introduce a generalized modified Krasnoselskii iterative process involving a pair of a generalized strictly pseudocontractive mapping and a generalized duality mapping and prove some strong convergence theorems of the proposed iterative process to the minimum norm solutions of certain nonlinear equations in the framework of uniformly convex Banach spaces. These results improve and generalize recent work in this direction.
Declarations
Acknowledgements
The author would like to thank the editor and the reviewers for their valuable suggestions and comments.
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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