In a previous work the authors proved under a complex assumption on the set-valued mapping, the existence of Lipschitz solutions for second order convex sweeping processes in p-uniformly smooth and q-uniformly convex Banach spaces. In the present work we prove the same result, under a condition on the distance function to the images of the set-valued mapping. Our assumption is much simpler than...

We develop a constructive completion method in general Minkowski spaces, which successfully extends a completion procedure due to Bückner in twoand three-dimensional Euclidean spaces. We prove that this generalized Bückner completion is locally Lipschitz continuous, thus solving the problem of finding a continuous selection of the diametric completion mapping in finite dimensional normed spaces...

and Applied Analysis 3 2. Main Results In this paper, we are interested in the following classes of mappings. Definition 8. Let T : H → H be a mapping.Then T is called (a) β-strongly monotone if there exists β > 0 such that ⟨T (x) − T (y) , x − y⟩ ≥ β 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 2 , ∀x, y ∈ H, (12) (b) ξLipschitz if there exists ξ > 0 such that 󵄩󵄩󵄩󵄩T (x) − T (y) 󵄩󵄩󵄩󵄩 ≤ ξ 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 , ∀x, y ∈ H. (13) Tha...

‎this contribution mainly focuses on some aspects of lipschitz groups‎, ‎i.e.‎, ‎metrizable groups with lipschitz multiplication and inversion map‎. ‎in the main result it is proved that metric groups‎, ‎with a translation-invariant metric‎, ‎may be characterized as particular group objects in the category of metric spaces and lipschitz maps‎. ‎moreover‎, ‎up to an adjustment of the metric‎, ‎a...

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

Let $(X,d)$ be an infinite compact metric space, let $(B,parallel . parallel)$ be a unital Banach space, and take $alpha in (0,1).$ In this work, at first we define the big and little $alpha$-Lipschitz vector-valued (B-valued) operator algebras, and consider the little $alpha$-lipschitz $B$-valued operator algebra, $lip_{alpha}(X,B)$. Then we characterize its second dual space.

We prove the existence of solutions for third-order nonconvex state-dependent sweeping process with unbounded perturbations of the form: −A x 3 t ∈ N K t, ẋ t ; A ẍ t F t, x t , ẋ t , ẍ t G x t , ẋ t , ẍ t a.e. 0, T , A ẍ t ∈ K t, ẋ t , a.e. t ∈ 0, T , x 0 x0, ẋ 0 u0, ẍ 0 υ0, where T > 0, K is a nonconvex Lipschitz set-valued mapping, F is an unbounded scalarly upper semicontinuous convex set-v...

In this paper, a new concept of -proximal mapping for a proper subdifferentiable functional (which may not be convex) on a Banach space is introduced. An existence and Lipschitz continuity of the -proximal mapping are proved. By using properties of the -proximal mapping, a new class of general mixed variational inequalities is introduced and studied in Banach spaces. An existence theorem of sol...

Graph-indexed random walks (or equivalently M -Lipschitz mappings of graphs) are a generalization of standard random walk on Z. For M ∈ N, an M-Lipschitz mapping of a connected rooted graph G = (V,E) is a mapping f : V → Z such that root is mapped to zero and for every edge (u, v) ∈ E we have |f(u) − f(v)| ≤ M . We study two natural problems regarding graph-indexed random walks. 1. Computing th...

This paper is concerned with some new projection methods for solving variational inequality problems with monotone and Lipschitz-continuous mapping in Hilbert space. First, we propose the projected reflected gradient algorithm with a constant stepsize. It is similar to the projected gradient method, namely, the method requires only one projection onto the feasible set and only one value of the ...