By using the projection methods, we suggest and analyze the iterative schemes for finding the approximation solvability of a system of general variational inequalities involving different nonlinear operators in the framework of Hilbert spaces. Moreover, such solutions are also fixed points of a Lipschitz mapping. Some interesting cases and examples of applying the main results are discussed and...

This paper is concerned with an optimal control problem governed by a semilinear, nonsmooth operator differential equation. The nonlinearity is locally Lipschitz-continuous and directionally differentiable, but not Gâteaux-differentiable. Two types of necessary optimality conditions are derived, the first one by means of regularization, the second one by using the directional differentiability ...

We give a simple quantitative condition, involving the “mapping content” of Azzam–Schul, which implies that Lipschitz map from Euclidean space to metric must be close factoring through tree. Using results Azzam–Schul and present authors, this gives checkable conditions for have large piece its domain on it behaves like an orthogonal projection. The proof involves new lower bounds continuity sta...

In this paper, we present a family of conjugate gradient projection methods for solving large-scale nonlinear equations. At each iteration, it needs low storage and the subproblem can be easily solved. Compared with the existing solution methods for solving the problem, its global convergence is established without the restriction of the Lipschitz continuity on the underlying mapping. Prelimina...

Various definitions of directional derivatives in topological vector spaces are compared. Directional derivatives in the sense of G~teaux, Fr6chet, and Hadamard are singled out from the general framework of cr-directional differentiability. It is pointed out that, in the case of finite-dimensional spaces and locally Lipschitz mappings, all these concepts of directional differentiability are equ...

In this paper, for a Lipschitz pseudocontractive mapping T, we study the strong convergence of the iterative scheme generated by ( ) (( ) ) , 1 1 1 n n n n n n n Tx u x x β + β − α + α − = + when { } { } n n α β , satisfy ( ) ; 0 lim i = α ∞ → n n ( ) ; ii 1 ∞ = α ∑ ∞ = n n ( ) . 0 lim iii = β ∞ → n n JING HAN and YISHENG SONG 148

in this paper, we consider a class of time-dependent neutral stochastic evolution equations with the infinite delay and a fractional brownian motion in a hilbert space. we establish the existence and uniqueness of mild solutions for these equations under non-lipschitz conditions with lipschitz conditions being considered as a special case. an example is provided to illustrate the theory

‎in this work we prove malliavin differentiability for the solution to an sde with locally lipschitz and semi-monotone drift‎. ‎to prove this formula‎, ‎we construct a sequence of sdes with globally lipschitz drifts and show that the $p$-moments of their malliavin derivatives are uniformly bounded‎.

Let X be a normed linear space. We investigate properties of vector functions F : [a, b] → X of bounded convexity. In particular, we prove that such functions coincide with the delta-convex mappings admitting a Lipschitz control function, and that convexity K aF is equal to the variation of F ′ + on [a, b). As an application, we give a simple alternative proof of an unpublished result of the fi...

Let (X,‖ · ‖) be an infinite-dimensional Banach space with the unit ball B and the unit sphere S. Since the works of Nowak [11], Benyamini and Sternfeld [1], and Lin and Sternfeld [10], it is known that S is a Lipschitzian retract of B. It means that there exists a mapping (retraction) R : B → S satisfying Rx = x for all x ∈ S and also being Lipschitzian. If R satisfies the Lipschitz condition ...