Generalized Set-valued Variational Inclusions And Resolvent Equations,

Part I of this paper, we establish the equivalence between the general- ized nonlinear set-valued quasi-variational inclusions (i.e. 0 € N(w, y) + A(g(u), z)), the resolvent equations, and the fixed-point problem in Banach spaces, using the properties of m-accretive. This equivalence is used to develop some completely new iterative algorithms for the new class of gener- alized nonlinear set-valued quasi-variational inclusions and related optirniza- tion problems. Our results improve and generalize many known correspond- ing results in resent years. Part II of this paper, we establish the equivalence between the new set- valued quasi-variational inclusions (i.e. 0 € N(w, y) + A(z, u)) and the fixed-point problem, using the properties of maximal monotone. This equiv- alence is used to develop the Mann and Ishilcawa type perturbed iterative algorithms for the new kind of completely generalized set-valued variational inclusions. Our algorithms and results improve and generalize many known corresponding algorithms and results in resent years.