First and Second Order Convex Sweeping Processes in Reflexive Smooth Banach Spaces

Abstract

In this paper we establish new characterizations of the normal cone of closed convex sets in reflexive smooth Banach spaces and then we use those results to prove the existence of solutions for first order convex sweeping processes and their variants in reflexive smooth Banach spaces. The case of second order convex sweeping processes is also studied.

Noncompact Perturbation of Sweeping Process with Delay in Banach Spaces

“…Therefore, our work extends many problems from Hilbert spaces setting to Banach spaces setting. Moreover, our work improves many results in the literature concerning the existence of solutions for some evolution inclusions governed by sweeping process in Banach spaces, for example, [14,15]. In addition our technique allows us to discuss some sweeping process problems with noncompact perturbation in Banach spaces.…”

“…For other results concerning (2) with or without perturbation in Hilbert spaces, we refer to [8][9][10][11][12][13]. Bounkhel and Al-Yusof [14] initiated the extension of (1) from the Hilbert setting to the Banach spaces setting. In fact, Bounkhel [8] considered (2) when is a multifunction from to the family of nonempty closed convex subsets of and satisfyes a condition similar to Condition ( 1 ) in the statement of our result (Theorem 18) and…”

“…Lemma 14 (see[14, Proposition 2.2]). Let Ω be an open subset in a normal vector space , a Banach space, and a continuous set-valued mapping defined on Ω and with nonempty compact convex values in .…”

State-Dependent Delayed Sweeping Process With a Noncompact Perturbation in Banach Spaces

“…Since then, important improvements have been developed by weaken assumptions in order to obtain the most general result of existence for sweeping processes (see for example [5,9,10,16,22,23]) in Hilbert spaces setting and [2,7,8,18] in Banach spaces setting. We observe that all these papers except [2,5] were without perturbation (like [7]) or with compact valued perturbation.…”

Generalized Projections on Closed Nonconvex Sets in Uniformly Convex and Uniformly Smooth Banach Spaces

“…In 2005, Li [3] extended and studied this concept from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces. This concept has been used successfully in many applications such as variational inequalities, minimization principles, and differential inclusions (see [1,2,[4][5][6][7] and the references therein). The main result in [1][2][3] is the existence property of the operator for closed convex sets in reflexive Banach spaces (resp., in uniformly convex and uniformly smooth Banach spaces) in [3] (resp., in [1]).…”