Sensitivity analysis for a system of parametric general quasi-variational-like inequality problems in uniformly smooth Banach spaces
Introduction
In recent years, much attention has been given to develop general for the sensitivity analysis of solution set of various classes of variational inequality (inclusion) problems. From the mathematical and engineering point of view, sensitivity properties of various classes of variational inequality problems can provide new insight concerning the problems being studied and can stimulate ideas for solving problems. The sensitivity analysis of solution set for variational inequality (inclusion) problems has been studied extensively by many authors using quite different methods. By using the projection technique, Defermos [4], Mukherjee and Verma [19], Noor [21] and Yen [26] studied the sensitivity analysis of solution of some classes of variational inequality problems with single-valued mappings. By using the implicit function approach that makes use of normal mappings, Robinson [25] studied the sensitivity analysis of solutions for variational inequality problems in finite-dimensional spaces. By using proximal-point (resolvent) mapping technique, Adly [1] and Agarwal et al. [2] studied the sensitivity analysis of solution of some classes of quasi-variational inclusions with single-valued mappings. Also, by using projection and proximal-point techniques, Ding and Luo [7], Liu et al. [18], Park and Jeong [23] and Ding [6], studied the behaviour and sensitivity analysis of solution set for some classes of generalized variational inequality (inclusion) problems with set-valued mappings. It is worth mentioning that most of the results in this direction have been obtained in the setting of Hilbert space.
Recently, Kazmi and Khan [12], [14], [15] and Kazmi et al. [16] have studied behavior and sensitivity analysis of solution set of system of some classes of variational inequality (inclusion) problems by using --proximal-point mappings in the setting of Banach spaces involving set-valued mappings.
Motivated by recent research work in this direction, we consider a system of parametric general quasi-variational-like inequality problems (in short, SPGQVLIP) in uniformly smooth Banach spaces. Further by using, --proximal point mappings introduced by Kazmi and Bhat [11], we study the existence and sensitivity analysis of the solution set of SPGQVLIP. Furthermore, the Lipschitz continuity of the solution set of SPSQVLIP is proved under some suitable conditions. The theorems presented in this paper improve and generalize the corresponding results of [1], [2], [4], [12], [14], [15], [16], [21], [24].
Section snippets
Preliminaries
Let be a real Banach space equipped with the norm . Let denote the dual pair between and its dual space ; let denote the family of all nonempty compact subsets of ; let denote the power set of ; let be the Hausdorff metric on defined byand let be the normalized duality mapping defined byIt is well known that if is smooth, then is single-valued and if ,
System of parametric generalized quasi-variational inequality problems
Throughout rest of this paper, unless otherwise stated, for each , we assume that is uniformly smooth Banach space with norm , and denote the duality pairing between and its dual by .
Let and be nonempty open subsets of and , respectively, in which parameters and takes the values; let ; be single-valued mappings such that and and for all ; let and be
Sensitivity analysis of solution set
Now, we shall study the behavior and sensitivity analysis of the solution set of SPGQVLIP (3.1) and further, under suitable conditions, we shall discuss Lipschitz continuity of the solution set with respect to the parameters. Theorem 4.1 For each , let be uniformly smooth Banach space with for some ; let the mappings and such that is -strongly accretive and -mixed Lipschitz continuous and be -mixed Lipschitz continuous; let the
References (26)
Perturbed algorithms and sensitivity analysis for a general class of variational inclusions
J. Math. Anal. Appl.
(1996)Parametric completely generalized mixed implicit quasi-variational inclusions involving -maximal monotone mappings
J. Comput. Appl. Math.
(2005)- et al.
Perturbed proximal point algorithms for general quasi-variational-like inclusions
J. Comput. Appl. Math.
(2000) - et al.
A new class of completely generalized quasi-variational inclusions in Banach spaces
J. Comput. Appl. Math.
(2002) - et al.
Convergence and stability of iterative algorithms of generalized set-valued variational-like inclusions in Banach spaces
Appl. Math. Comput.
(2005) - et al.
Iterative approximation of a unique solution of a system of variational-like inclusions in real -uniformly smooth Banach spaces
Nonlinear Anal.
(2007) On fixed point stability for set-valued contractive mappings with applications to generalized differential equations
J. Math. Anal. Appl.
(1985)- et al.
Sensitivity analysis for parametric completely generalized nonlinear implicit quasi-variational inclusions
J. Math. Anal. Appl.
(2003) - et al.
Sensitivity analysis of generalized variational inequalities
J. Math. Anal. Appl.
(1992) Sensitivity analysis framework for general quasi-variational inclusions
Comput. Math. Appl.
(2002)
Sensitivity analysis for quasivariational inclusions
J. Math. Anal. Appl.
Parametric generalized mixed variational inequalities
Appl. Math. Lett.
Sensitivity analysis for parametric completely generalized strongly nonlinear implicit quasi-inclusions
Comput. Math. Appl.
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2011, Nonlinear Analysis, Theory, Methods and ApplicationsSolution sensitivity of generalized nonlinear parametric [InlineEquation not available: see fulltext.]-proximal operator system of equations in Hilbert spaces
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Supported by NBHM, Department of Atomic Energy, Government of India under Grants-in-aid for Post-doctoral fellowship (Reference no. 40/2/2007-R&D II/3910).