Proximal point algorithm for generalized multivalued nonlinear quasi-variational-like inclusions in Banach spaces

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Abstract

In this paper, we define a new notion of Jη-proximal mapping for a nonconvex, lower semicontinuous, η-subdifferentiable proper functional in Banach spaces. The existence and Lipschitz continuity of Jη-proximal mapping of a lower semicontinuous, η-subdifferentiable proper functional are proved. By applying this notion, we introduce and study generalized multivalued nonlinear quasi-variational-like inclusions in reflexive Banach spaces and propose a proximal point algorithm for finding the approximate solutions of this class of variational inclusions. The convergence criteria of the iterative sequences generated by our algorithm is discussed. Several special cases are also given.

Introduction

In 1994, Hassouni and Moudafi [1] introduced a perturbed method for solving a new class of variational inequalities, known as variational inclusions. A useful and important generalization of the variational inclusions is called the quasi-variational inclusion. Quasi-variational inclusions are being used as mathematical programming models to study a large number of equilibrium problems arising in finance, economics, transportation, optimization, operation research, and engineering sciences.

Adly [2], Huang [3], Ding [4], [5], Ahmad and Ansari [6] have obtained some important extensions of the results in [1] in various different directions. Recently, Cohen [7] and Ding [8], [9] have extended the auxiliary principle technique to suggest and analyze an innovative and novel iterative algorithm for computing the solution of mixed variational inequalities in reflexive Banach spaces. Chang et al. [10] and Chang [11] have studied some classes of set-valued variational inclusions with m-accretive operator and φ-strongly accretive operators in uniformly Banach spaces.

Iterative algorithms have played a central role in the approximation solvability, especially of nonlinear variational inequalities as well as nonlinear equations in several fields such as applied mathematics, mathematical programming, mathematical finance, control theory and optimization, engineering sciences and others. In general we cannot use resolvent operator or proximal mapping technique for studying a perturbed algorithm for finding the approximate solutions of variational-like inequalities.

In this paper, we define a new notion of Jη-proximal mapping for a lower semicontinuous, η-subdifferentiable, proper (may not be convex) functional on Banach spaces. The existence and Lipschitz continuity of the Jη-proximal mapping of the functional are proved under suitable conditions in reflexive Banach spaces. By using this notion, we introduce and study generalized multivalued nonlinear quasi-variational-like inclusions in reflexive Banach spaces and propose a proximal point algorithm for finding the approximate solutions of our inclusions. The convergence of the iterative sequences generated by our algorithm is discussed. Several special cases are also discussed.

Section snippets

Preliminaries

Let E be a Banach space with the dual space E, 〈u,x〉 be the dual pairing between uE and xE and CB(E) be the family of all nonempty closed bounded subset of E. H(.,.) is the Hausdörff metric on CB(E) defined byH(A,B)=maxsupu∈Ad(u,B),supv∈Bd(A,v)forallA,B∈CB(E),where d(u,B)=infvBd(u,v) and d(A,v)=infuAd(u,v).

We extend the concept of η-subdifferentiability of a functional defined by Lee et al. [12] in Hilbert spaces to a Banach space setting.

Let η:E×EE and φ:ER∪{+∞}. A vector wE is

Proximal point algorithm

Definition 3.1

The mapping N:E×EE is said to be

  • (i)

    Lipschitz continuous with respect to the first argument, if there exists a constant λN1>0 such that∥N(u1,.)−N(u2,.)∥⩽λN1∥u1−u2for all u1T(x1), u2T(x2) and x1,x2E.

  • (ii)

    Lipschitz continuous with respect to the second argument, if there exists a constant λN2>0 such that∥N(.,v1)−N(.,v2)∥⩽λN2∥v1−v2for all v1A(x1), v2A(x2) and x1,x2E.


The following Lemma plays an important role in proving our main result.

Lemma 3.1

[19]

Let E be a real Banach space and F:E→2E be the

Acknowledgements

For the rest of his work, A.H. Siddiqi would like to thank King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent facilities.

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This work was partially done during the visit of first and second author to Mathematics section, the Abdus Salam, ICTP, Trieste, Italy.

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