From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability

We consider a generalized equation governed by a strongly monotone and Lipschitz single-valued mapping and a maximally monotone set-valued mapping in a Hilbert space. We are interested in the sensitivity of solutions w.r.t. perturbations of both mappings. We demonstrate that the directional differentiability of the solution map can be verified by using the directional differentiability of the single-valued operator and of the resolvent of the set-valued mapping. The result is applied to quasi-generalized equations in which we have an additional dependence of the solution within the set-valued part of the equation.

Strong convergence theorems for strongly monotone mappings in Banach spaces

Let $E$ be a uniformly smooth and uniformly convex real Banach space and $E^*$ be its dual space. Suppose $A : E\rightarrow E^*$ is bounded, strongly monotone and satisfies the range condition such that $A^{-1}(0)\neq \emptyset$. Inspired by Alber \cite{b1}, we introduce Lyapunov functions and use the new geometric properties of Banach spaces to show the strong convergence of an iterative algorithm to the solution of $Ax=0$.

A Method of approximation for a zero of the sum of maximally monotone mappings in Hilbert spaces

Our purpose of this study is to construct an algorithm for finding a zero of the sum of two maximally monotone mappings in Hilbert spaces and discus its convergence. The assumption that one of the mappings is α-inverse strongly monotone is dispensed with. In addition, we give some applications to the minimization problem. Our method of proof is of independent interest. Finally, a numerical example which supports our main result is presented. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.