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

In this paper, we define a new notion of J^@h-proximal mapping for a nonconvex, lower semicontinuous, @h-subdifferentiable proper functional in Banach spaces. The existence and Lipschitz continuity of J^@h-proximal mapping of a lower semicontinuous, @h-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.

[1]  Xie Ping Ding,et al.  Perturbed Proximal Point Algorithms for Generalized Quasivariational Inclusions , 1997 .

[2]  Xie Ping Ding,et al.  A minimax inequality with applications to existence of equilibrium point and fixed point theorems , 1992 .

[3]  R. Ahmad.  Generalized Multivalued Nonlinear Quasi-variational like Inclusions , 2002 .

[4]  Xie Ping Ding,et al.  A new class of completely generalized quasi-variational inclusions in Banach spaces , 2002 .

[5]  Jen-Chih Yao,et al.  A perturbed algorithm for strongly nonlinear variational-like inclusions , 2000, Bulletin of the Australian Mathematical Society.

[6]  W. Petryshyn,et al.  A characterization of strict convexity of banach spaces and other uses of duality mappings , 1970 .

[7]  Y. Alber.  Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications , 1993, funct-an/9311001.

[8]  Abdellatif Moudafi,et al.  A Perturbed Algorithm for Variational Inclusions , 1994 .

[9]  Qamrul Hasan Ansari,et al.  An iterative algorithm for generalized nonlinear variational inclusions , 2000, Appl. Math. Lett..

[10]  Ding Xieping.  Proximal point algorithm with errors for generalized strongly nonlinear quasivariational inclusions , 1998 .

[11]  Xie Ping Ding,et al.  Generalized quasi-variational-like inclusions with nonconvex functionals , 2001, Appl. Math. Comput..

[12]  Xie Ping Ding,et al.  General algorithm of solutions for nonlinear variational inequalities in Banach space , 1997 .

[13]  Xie Ping Ding,et al.  Perturbed proximal point algorithms for general quasi-variational-like inclusions , 2000 .

[14]  S. Nadler.  Multi-valued contraction mappings. , 1969 .

[15]  S. S. Chang,et al.  Set-Valued Variational Inclusions in Banach Spaces , 2000 .

[16]  Samir Adly,et al.  Perturbed algorithms and sensitivity analysis for a general class of variational inclusions , 1996 .

[17]  Nan-jing Huang,et al.  Generalized nonlinear variational inclusions with noncompact valued mappings , 1996 .

[18]  Byung-Soo Lee,et al.  Generalized Set-Valued Variational Inclusions in Banach Spaces , 2000 .

[19]  G. Cohen.  Auxiliary problem principle extended to variational inequalities , 1988 .