Semicontinuous Set-valued Maps Research Articles

Topological Degree for Operators of Class (S)+ with Set-Valued Perturbations and Its New Applications

We investigate the topological degree for generalized monotone operators of class (S)+ with compact set-valued perturbations. It is assumed that perturbations can be represented as the composition of a continuous single-valued mapping and an upper semicontinuous set-valued mapping with aspheric values. This allows us to extend the standard degree theory for convex-valued operators to set-valued mappings whose values can have complex geometry. Several theoretical aspects concerning the definition and main properties of the topological degree for such set-valued mappings are discussed. In particular, it is shown that the introduced degree has the homotopy invariance property and can be used as a convenient tool in checking the existence of solutions to corresponding operator inclusions. To illustrate the applicability of our approach to studying models of real processes, we consider an optimal feedback control problem for the steady-state internal flow of a generalized Newtonian fluid in a 3D (or 2D) bounded domain with a Lipschitz boundary. By using the proposed topological degree method, we prove the solvability of this problem in the weak formulation.

Two Positive Solutions for Elliptic Differential Inclusions

The existence of two positive solutions for an elliptic differential inclusion is established, assuming that the nonlinear term is an upper semicontinuous set-valued mapping with compact convex values having subcritical growth. Our approach is based on variational methods for locally Lipschitz functionals. As a consequence, a multiplicity result for elliptic Dirichlet problems having discontinuous nonlinearities is pointed out.

Fixed point theorems for continuous single-valued and upper semicontinuous set-valued mappings in $p$-vector and locally $p$-convex spaces

The goal of this paper is to establish a general fixed point theorem for compact single-valued continuous mappings in Hausdorff $p$-vector spaces, and a fixed point theorem for upper semicontinuous set-valued mappings in locally $p$-convex spaces for $p\in (0, 1]$. These results not only provide a solution to Schauder conjecture in the affirmative under the setting of $p$-vector spaces for compact single-valued continuous mappings, but also show the existence of fixed points for upper semicontinuous set-valued mappings defined on $s$-convex subsets in Hausdorff locally $p$-convex spaces, which would be fundamental for nonlinear functional analysis, where $s, p \in (0, 1]$.

Fixed point theorems and applications in p-vector spaces

The goal of this paper is to develop new fixed points for quasi upper semicontinuous set-valued mappings and compact continuous (single-valued) mappings, and related applications for useful tools in nonlinear analysis by applying the best approximation approach for classes of semiclosed 1-set contractive set-valued mappings in locally p-convex and p-vector spaces for p in (0, 1]. In particular, we first develop general fixed point theorems for quasi upper semicontinuous set-valued and single-valued condensing mappings, which provide answers to the Schauder conjecture in the affirmative way under the setting of locally p-convex spaces and topological vector spaces for p in (0, 1]; then the best approximation results for quasi upper semicontinuous and 1-set contractive set-valued mappings are established, which are used as tools to establish some new fixed points for nonself quasi upper semicontinuous set-valued mappings with either inward or outward set conditions under various boundary situations. The results established in this paper unify or improve corresponding results in the existing literature for nonlinear analysis, and they would be regarded as the continuation of the related work by Yuan (Fixed Point Theory Algorithms Sci. Eng. 2022:20, 2022)–(Fixed Point Theory Algorithms Sci. Eng. 2022:26, 2022) recently.

On various entropies of set-valued maps

In this paper, the complexity of an upper semi-continuous set-valued map F on a compact metric space is considered via entropy-like invariants from various perspectives. Several entropies of topological version, including pointwise entropies hp(F) and hm(F), branch entropy hi(F) and tree entropy ht(F), are introduced and investigated. Some basic properties about them are given and the relations among them and the classical entropy htop(F) are considered. The calculation or estimation of these entropies for certain finitely-generated set-valued maps on intervals or graphs are discussed. As the counterparts of ht(F), the measure-theoretic lower and upper tree entropies h_tμ(F) and h‾tμ(F) are introduced for any Borel probability measure μ in a way resembling Hausdorff dimension. A variational inequality ht(F)≥sup⁡{h‾tμ(F)|μ∈P(X)}, where P(X) is the set of Borel probability measures on X, is obtained, and as its applications, the lower bounds of ht(F) are given for certain set-valued maps respectively.

Stability for a Class of Differential Set-Valued Inverse Variational Inequalities in Finite Dimensional Spaces

In this paper, we introduce and study a new class of differential set-valued inverse variational inequalities in finite dimensional spaces. By applying a result on differential inclusions involving an upper semicontinuous set-valued mapping with closed convex values, we first prove the existence of Carathéodory weak solutions for differential set-valued inverse variational inequalities. Then, by the existence result, we establish the stability for the differential set-valued inverse variational inequality problem when the constraint set and the mapping are perturbed by two different parameters. The closedness and continuity of Carathéodory weak solutions with respect to the two different parameters are obtained.

Fixed Point Theorems for Upper Semicontinuous Set-Valued Mappings in p-Vector Spaces

Fixed Point Theorems for Upper Semicontinuous Set-Valued Mappings in p-Vector Spaces

Degree for weakly upper semicontinuous perturbations of quasi-m-accretive operators.

In the paper, we provide the construction of a coincidence degree being a homotopy invariant detecting the existence of solutions of equations or inclusions of the form Ax ∈ F(x), x ∈ U, where [Formula: see text] is an m-accretive operator in a Banach space E, [Formula: see text] is a weakly upper semicontinuous set-valued map constrained to an open subset U of a closed set K ⊂ E. Two different approaches are presented. The theory is applied to show the existence of non-trivial positive solutions of some nonlinear second-order partial differential equations with discontinuities. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.

A Bilocal Problem Associated to a Fractional Differential Inclusion of Caputo-Fabrizio Type

A fractional differential inclusion defined by Caputo-Fabrizio fractional derivative with bilocal boundary conditions is studied. A nonlinear alternative of Leray-Schauder type, Bressan-Colombo selection theorem for lower semicontinuous set-valued maps with decomposable values and Covitz-Nadler set-valued contraction principle are employed in order to obtain the existence of solutions when the set-valued map that define the problem has convex or non convex values.

Set optimization problems of generalized semi-continuous set-valued maps with applications

In this paper, some properties of the lower (upper) semi-continuity for set-valued maps taking values in an abstract pre-ordered set are showed, which are then applied to study the existence of solutions for abstract set optimization problems. Moreover, some relationships between minimal solutions for abstract set-valued optimization problems with the vector and set criteria are given under mild conditions. As applications, existence results of solutions for set optimization problems are applied to obtain the existence of saddle points for the set-valued map taking values in the pre-ordered set with the set criterion and of solutions for vector optimization problems whose image space is a real vector space not necessarily endowed with a topology.

Almost mixed semi-continuous perturbation of Moreau's sweeping process

In this work, we introduce a new concept of semi-continuous set-valued mappings, called almost mixed semi-continuity, by taking maps that are upper semi-continuous with almost convex values in some points and lower semi-continuous in remaining points. We generalize earlier results obtained for both mixed semi-continuous maps and almost convex sets. We discuss the existence of solution for evolution problems driven by the so-called sweeping process subject to external forces, known as perturbation to the system, by this type of set-valued mappings. Finally, we give some topological properties of the attainable and solution sets in order to solve an optimal time problem.

Topological entropy of Markov set-valued functions

We investigate the entropy for a class of upper semi-continuous set-valued functions, called Markov set-valued functions, that are a generalization of single-valued Markov interval functions. It is known that the entropy of a Markov interval function can be found by calculating the entropy of an associated shift of finite type. In this paper, we construct a similar shift of finite type for Markov set-valued functions and use this shift space to find upper and lower bounds on the entropy of the set-valued function.

The first common fixed point theorem for commutative set-valued mappings

In this paper, we establish a common fixed point theorem for commutative set-valued convex mappings. We also prove the existence of a fixed point in a continuously expanding set under a non-convex upper semicontinuous set-valued mapping. Applying this result, we develop a common fixed point theorem for a family of non-convex set-valued upper semicontinuous mappings which provides an answer to a question of Lau and Yao.

An Minimax Theorem for Bi-Lower-Semicontinuous Set-Valued Mappings

In this article, an approximate minimax theorem for bi-lower-semicontinuous set-valued mapping was proved, relationships between semicontinuous set-valued mappings are discussed and the existence of approximate maxmin was given. The minimax theorem in this article is the first minimax theorem that doesn’t require the set-valued mappings to be continuous.

Constant step stochastic approximations involving differential inclusions: stability, long-run convergence and applications

ABSTRACTWe consider a Markov chain whose kernel is indexed by a scaling parameter , referred to as the step size. The aim is to analyze the behaviour of the Markov chain in the doubly asymptotic regime where then . First, under mild assumptions on the so-called drift of the Markov chain, we show that the interpolated process converges narrowly to the solutions of a Differential Inclusion (DI) involving an upper semicontinuous set-valued map with closed and convex values. Second, we provide verifiable conditions which ensure the stability of the iterates. Third, by putting the above results together, we establish the long run convergence of the iterates as , to the Birkhoff center of the DI. The ergodic behaviour of the iterates is also provided. Application examples are investigated. We apply our findings to (1) the problem of nonconvex proximal stochastic optimization and (2) a fluid model of parallel queues.

Trees and generalised inverse limits on intervals

We consider inverse limits of sequences of upper semicontinuous set-valued functions fi+1:Ii+1→2Ii (where Ii=[0,1] for each i∈N), for which the graph of each bonding function is an arc. We show that any finite tree can be obtained as such an inverse limit, and one for which each bonding function is one of two specified functions. In addition, we discuss trees of height ω+1 that can be obtained as the inverse limit of such a sequence.

On positive embeddings of C(K) spaces into C(S,X) lattices

Let K and S be compact Hausdorff spaces and X a Banach lattice with λ+(X)>1, whereλ+(X)=inf⁡{max⁡{‖x−y‖,‖x+y‖}:‖x‖=1,‖y‖=1 and x,y>0}. We prove that if T is a positive isomorphism from C(K) into C(S,X) then for some natural number m there exists an upper semicontinuous set-valued mapping ψ from S to K such that⋃s∈Sψ(s)=K, and each set ψ(s) has at most m elements. This result is an extension of a Plebanek theorem, the case X=R. We also show that if ‖T‖‖T−1‖<λ+(X), then m can be taken equal to 1. This result is optimal for the classical spaces X=ℓp, with 1≤p<∞.

A Ky Fan Minimax Inequality for Quasiequilibria on Finite-Dimensional Spaces

Several results concerning existence of solutions of a quasiequilibrium problem defined on a finite-dimensional space are established. The proof of the first result is based on a Michael selection theorem for lower semicontinuous set-valued maps which holds in finite-dimensional spaces. Furthermore, this result allows one to locate the position of a solution. Sufficient conditions, which are easier to verify, may be obtained by imposing restrictions either on the domain or on the bifunction. These facts make it possible to yield various existence results which reduce to the well-known Ky Fan minimax inequality when the constraint map is constant and the quasiequilibrium problem coincides with an equilibrium problem. Lastly, a comparison with other results from the literature is discussed.

K-semistratifiable spaces and expansions of set-valued mappings

&lt;p&gt;In this paper, the concept of k-upper semi-continuous set-valued mappings is introduced. Using this concept, we give characterizations of k-semistratifiable and k-MCM spaces, which answers a question posed by Xie and Yan.&lt;/p&gt;

On inverse limits with set-valued functions on graphs, dimensionally stepwise spaces and ANRs

A space X is a dimensionally stepwise space if dim⁡X<∞ and for any 1≤m≤dim⁡X there is an open set Um of X such that dim⁡Um=m. In this paper, for given inverse sequence {Xi,fi,i+1}i=1∞ of compacta with upper semi-continuous set-valued functions, we introduce new indexes I˜({Xi,fi,i+1}) and W˜({Xi,fi,i+1}), and by use of the indexes we investigate topological structures of inverse limits of graphs with upper semi-continuous set-valued functions. Especially, we prove the following theorems.