Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let α > 0, and let A be an α-inverse strongly-monotone mapping of C into H. Let T be a generalized hybrid mapping of C into H. Let B andW be maximal monotone operators on H such that the domains of B andW are included in C. Let 0 < k < 1, and let g be a k-contraction of H into itself. Let V be a γ -strongly monoto...

In this work we introduce for extended real valued functions, defined on a Banach space X, the concept of K directionally Lipschitzian behavior, where K is a bounded subset of X. For different types of sets K (e.g., zero, singleton, or compact), the K directionally Lipschitzian behavior recovers well-known concepts in variational analysis (locally Lipschitzian, directionally Lipschitzian, or co...

We show that the variational inequality V I(C,A) has a unique solution for a relaxed (γ, r)-cocoercive, μ-Lipschitzian mapping A : C → H with r > γμ, where C is a nonempty closed convex subset of a Hilbert space H . From this result, it can be derived that, for example, the recent algorithms given in the references of this paper, despite their becoming more complicated, are not general as they ...

Consider the variational inequality V I(C, F ) of finding a point x∗ ∈ C satisfying the property 〈Fx∗, x − x∗〉 ≥ 0 for all x ∈ C, where C is a nonempty closed convex subset of a real Hilbert space H and F : C → H is a nonlinear mapping. If F is boundedly Lipschitzian and strongly monotone, then we prove that V I(C, F ) has a unique solution and iterative algorithms can be devised to approximate...

A d-space X = (X, %, μ) is a compact set X with respect to a quasi-metric % and a Borel measure μ such that the measure of a ball of radius r is equivalent to r, where d > 0. The paper deals with spaces B p(X;H) of Besov type where 1 < p < ∞ and s ∈ R. Here H is a bi-Lipschitzian map of the snowflaked version (X, %, μ) of X for some 0 < ε < 1, onto a fractal d/ε-set Γ = HX in some R, reducing t...

Lipschitz self mappings of metric spaces appear in many branches of mathematics. In this paper we introduce a modification of the Lipschitz condition which takes into account not only the mapping itself but also the behaviour of a finite number of its iterates. We refer to such mappings as mean Lipschitzian. The study of this new class of mappings seems potentially interesting and leads to some...

We introduce a new general system of variational inclusions in Banach spaces and propose a new iterative scheme for finding common element of the set of solutions of the variational inclusion with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mapping and the set of fixed point of nonexpansive semigroups in a uniformly convex and 2-uniformly smooth Banach space. Further...

The behavior of a minimizing point when an objective function is tilted by adding a small linear term is studied from the perspective of second-order conditions for local optimality. The classical condition of a positive-definite Hessian in smooth problems without constraints is found to have an exact counterpart much more broadly in the positivity of a certain generalized Hessian mapping. This...