Generalized nonlinear quasi-variational inclusions in Banach spaces

This paper introduces a new class of generalized nonlinear quasi-variational inclusions involving generalized mm-accretive mappings in pp-uniformly smooth real Banach spaces. By using the resolvent operator technique for generalized mm-accretive mappings due to Huang et al. [N.J. Huang, Y.P. Fang, C.X. Deng, Nonlinear variational inclusions involving generalized mm-accretive mappings, in: Proceedings of the Bellman Continuum: International Workshop on Uncertain Systems and Soft Computing, Beijing, China, July, 24–27, 2002, pp. 323–327] and Nadler Theorem [S.B. Nadler Jr., Multivalued contraction mappings, Pacific J. Math. 30 (1969) 475–488], we construct an iterative algorithm for solving generalized nonlinear quasi-variational inclusions with strongly accretive and relaxed accretive mappings in pp-uniformly smooth real Banach spaces. Then we prove the existence of solutions for our inclusions without compactness assumption and convergence of the iterative sequences generated by the algorithm in pp-uniformly smooth real Banach spaces. Some special cases are also discussed.