Class Of Multivalued Mappings Research Articles

Diametrically Regular Mappings and Browder’s Theorem Without the Endpoint Condition

In this paper, we introduce the class of diametrically regular mappings and prove that it includes the class of multi-valued mappings having endpoints. We also prove the existence of endpoints and the convergence of an iterative process for mappings of this class. Our approach provides a new idea of proving Browder’s theorem without the endpoint condition.

Optimal feedback control problem for Bingham media motion with periodic boundary conditions

We study the optimal feedback control problem for the motion of Bingham media with periodic boundary conditions in two- and three-dimensional cases. First, the considered problem is interpreted as an operator inclusion with a multivalued right-hand side. Then, the approximation-topological approach to hydrodynamic problems and the degree theory for a class of multivalued maps are used to prove the existence of solutions of this inclusion. Finally, we prove that, among the solutions of the considered problem, there exists one minimizing the given cost functional.

Optimal Feedback Control Problem for Bingham Media Motion with Periodic Boundary Conditions

We study the optimal feedback control problem for the motion of Bingham media with periodic boundary conditions in two- and three-dimensional cases. First, the considered problem is interpreted as an operator inclusion with a multivalued right-hand side. Then, the approximation-topological approach to hydrodynamic problems and the degree theory for a class of multivalued maps are used to prove the existence of solutions of this inclusion. Finally, we prove that, among the solutions of the considered problem, there exists one minimizing the given cost functional.

On the Solution of a System of Hamilton–Jacobi Equations of Special Form

The paper is concerned with the investigation of a system of first-order Hamilton–Jacobi equations. We consider a strongly coupled hierarchical system: the first equation is independent of the second, and the Hamiltonian of the second equation depends on the gradient of the solution of the first equation. The system can be solved sequentially. The solution of the first equation is understood in the sense of the theory of minimax (viscosity) solutions and can be obtained with the help of the Lax–Hopf formula. The substitution of the solution of the first equation in the second Hamilton–Jacobi equation results in a Hamilton–Jacobi equation with discontinuous Hamiltonian. This equation is solved with the use of the idea of M-solutions proposed by A. I. Subbotin, and the solution is chosen from the class of multivalued mappings. Thus, the solution of the original system of Hamilton–Jacobi equations is the direct product of a single-valued and multivalued mappings, which satisfy the first and second equations in the minimax and M-solution sense, respectively. In the case when the solution of the first equation is nondifferentiable only along one Rankine–Hugoniot line, existence and uniqueness theorems are proved. A representative formula for the solution of the system is obtained in terms of Cauchy characteristics. The properties of the solution and their dependence on the parameters of the problem are investigated.

Existence and convergence for a new multivalued hybrid mapping in CAT(κ) spaces

Most of the studies about hybrid mappings are carried out for single-valued mappings in Hilbert spaces. We define a new class of multivalued mappings in CAT (k) spaces which contains the multivalued generalization of (α, β) - hybrid mappings defined on Hilbert spaces. In this paper, we prove existence and convergence results for a new class of multivalued hybrid mappings on CAT(κ) spaces which are more general than Hilbert spaces and CAT(0) spaces.

A generalization of Reich’s fixed point theorem for multi-valued mappings

Motivated by a problem concerning multi-valued mappings posed by Reich [S. Reich, Some fixed point problems, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 57 (1974) 194-198] and a paper of Jleli and Samet [M. Jleli, B. Samet, A new generalization of the Banach contraction principle, J. Inequal. Appl. 2014:38 (2014) 1-8], we consider a new class of multi-valued mappings that satisfy a ?-contractive condition in complete metric spaces and prove some fixed point theorems. These results generalize Reich?s and Mizoguchi-Takahashi?s fixed point theorems. Some examples are given to show the usability of the obtained results.

The multi-morphisms and their properties and applications

Abstract In this paper a new class of multi-valued mappings (multi-morphisms) is defined as a version of a strongly admissible mapping, and its properties and applications are presented.

New common fixed point theorems for multivalued maps

Common fixed point theorems for a new class of multivalued maps are obtained, which generalize and extend classical fixed point theorems of Nadler and Reich and some recent Suzuki type fixed point theorems.

Convergence theorems for a finite family of multi-valued strictly pseudo-hybrid mappings in Hilbert spaces

In this paper, we introduce a new class of multi-valued mappings in a Hilbert space, called \((k,\lambda )\)-strictly pseudo-hybrid. We establish weak and strong convergence of one-step iterative method to a common fixed point of a finite family of those multi-valued mappings in a Hilbert space. Our results extend many known recent results in the literature.

An example of a piecewise linear ergodic polymorphism

We consider dynamical systems introduced by Vershik and called polymorphisms. In particular, such systems encompass the class of multivalued mappings of a closed interval onto itself which have an invariant measure. Polymorphisms arise in different areas of mathematics and mechanics, for example, in the problem of the destruction of the adiabatic invariant. We are concerned with the ergodic properties of polymorphisms. The first section deals with the main notions. In Secs. 2 and 3, we consider an example of a three-parameter family of ergodic polymorphisms formed by piecewise linear mappings.

KKM-type theorems for best proximity points

Let us consider two nonempty subsets A , B of a normed linear space X , and let us denote by 2 B the set of all subsets of B . We introduce a new class of multivalued mappings { T : A → 2 B } , called R-KKM mappings, which extends the notion of KKM mappings. First, we discuss some sufficient conditions for which the set ∩ { T ( x ) : x ∈ A } is nonempty. Using this nonempty intersection theorem, we attempt to prove a extended version of the Fan–Browder multivalued fixed point theorem, in a normed linear space setting, by providing an existence of a best proximity point.

Fixed point theorems for two new classes of multivalued mappings

The notions of weak*-nonexpansive mapping and generalized R -multivalued mapping are introduced. Fixed points, common fixed points and some properties of the set of fixed points (common fixed points) of these maps are studied.

Exponentiating a Bundle of Linear Operators

This paper deals with the concept of exponentiability for a special class of multivalued maps. To be more precise, we discuss the exponentiability of a multivalued map F: X⇉X expressible in the form F(x) = {Ax:A ∈ Ξ}, with Ξ denoting a collection of linear continuous operators defined on a Banach space X. Among other results, we prove that, under suitable assumptions on Ξ, the Painleve–Kuratowski limit $${\left[ {\exp F} \right]}{\left( x \right)} = {\mathop {\lim }\limits_{N \to \infty } }{\left( {I + \frac{1}{N}F} \right)}^{N} {\left( x \right)}$$ exists for all x ∈ X, and it admits the representation [expF](x) = {eAx:A ∈ clco(Ξ)}. The operation of exponentiation has therefore a convexification effect on Ξ. By exploiting the above-mentioned representation formula, we derive general properties for the semigroup {SF(t)}t⩾0 defined by $$S_{F} {\left( t \right)}x = {\mathop {\lim }\limits_{N \to \infty } }{\left( {I + \frac{t}{N}F} \right)}^{N} {\left( x \right)}.$$

On the Cone of Bounded Lower Semicontinuous Functions

We prove that the cone of bounded lower semicontinuous functions defined on a Tychonoff space X is algebraically and structurally isomorphic and isometric to a convex cone contained in the cone of all bounded lower semicontinuous functions defined on the Stone-Cech compactification βX if and only if the space X is normal. We apply this theorem to the study of relationship between a class of multivalued maps and sublinear operators. Using these results, we obtain new proofs of theorems about continuous selections.

Some fixed point theorems for multivalued maps in ordered Banach spaces and applications

The existence of maximal and minimal fixed points for various set‐valued operators is discussed. This paper presents some new fixed point theorems in ordered Banach spaces. A necessary and sufficient condition for the existence of the fixed point to a class of multivalued maps has been obtained. The uniqueness of the positive fixed point has been discussed. The results extend and improve the corresponding results. As an application, we utilize the results to study the existence and uniqueness of positive fixed points for a class of convex operators. In the end, we give a simple application to certain integral equations.

Mixed Semicontinuous Mappings and Their Applications to Differential Inclusions

A class of multivalued mappings, called by us mixed semicontinuous mappings, which are upper semicontinuous in some points and lower semicontinuous in remaining points is considered. A general selection theorem for mixed semicontinuous maps is proved. Then applications to differential inclusions with mixed semicontinuous right-hand sides are presented. Note that our applications generalize earlier existence results obtained both on the whole Euclidean space R n and on a compact proximate retract of R n .

Homotopy and epi maps

The notion of a 0 - (L, T) epi map is introduced for a large class of multivalued maps. Our results lead to existence principles for the operator inclusion \(Lx \in Gx\).

Continuous selections of multivalued maps with non-convex non-closed decomposable values

A class of multivalued maps with non-convex non-closed decomposable values is distinguished, and theorems are proved on the existence of continuous selections for such maps. This class contains multivalued maps whose values are extreme points of continuous multivalued maps with closed convex decomposable values in a Banach space of Bochner-integrable functions. The proofs are based on the Baire category theorem. It is known that the set of extreme points of a closed convex set is in general not closed. Hence the results or paper answer the question of the existence of continuous selections for multivalued maps with non-convex non-closed values.

The eclipsing concept to approximate a multi-valued mapping

One of the uses of the derivative of a nonlinear function F is to define first-order approximations of F by special functions, namely affine functions. Our aim is to extend this concept of first order approximations to the case when F is multi-valued; how should the concept of affinity be generalized for such functions? We first show why a mere copy of the single-valued case is hardly appropriate; a more sophisticated definition is needed for affinity. Restricting our study to convex compact valued F we consider various possibilities. We conclude by exhibiting a class of multi-valued mappings, which we call eclipsing, and which seem to impose themselves as natural generalizations of affine mappings. They correspond to linearizing the support function of Fwhich thus offers to be the most reasonable thing to do.

New results in the theory of multivalued mappings. I. Topological characteristics and solvability of operator relations

The first part of the survey describes the fundamental results in the theory of multivalued mappings, obtained during the period 1981–1986. The survey contains a description of fixed-point and coincidence theorems for various classes of multivalued mappings, generalizing a series of classical results. Results on the theory of the topological degree of multivalued mappings are also given.