Lp-continuous extreme selectors of multifunctions with decomposable values: Existence theorems

The existence theorems of L _p -continuous selectors that values are extreme points are proved for a class of multivalued maps. Applications to multivalued maps appearing in multivalued differential equations are presented.Supported in part by RFFI Grant 93-011-264.     

Multivalued linear projections

A multivalued linear projection operator P defined on linear space X is a multivalued linear operator which is idempotent and has invariant domain. We show that a multivalued projection can be characterised in terms of a pair of subspaces and then establish that the class of multivalued linear projections is closed under taking adjoints and closures. We apply the characterisations of the adjoint and completion of a projection together with the closed graph and closed range theorems to give criteria for the continuity of a projection defined on a normed linear space. A new proof of the theorem on closed sums of closed subspaces in a Banach space (cf. Mennicken and Sagraloff [9, 10]) follows as a simple corollary. We then show that the topological decomposition of a space may be expressed in terms of multivalued projections. The paper is concluded with an application to multivalued semi-Fredholm relations with generalised inverses.     

On almost periodic multivalued maps

We consider multivalued maps with compact images in a complete metric spaceU. Conditions on the set of selections of a multivalued mapF are obtained which are necessary and sufficient for the multivalued mapF to be Stepanov almost periodic. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 82–90, July, 2000.     

On the problem of stochastic differential inclusions

In this paper we study a stochastic differential equation with multivalued maximally monotone drift operator. Under certain assumptions on the growth of the multivalued operator we prove a theorem on the existence and uniqueness of the solution of such an equation.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 54–59, 1987.     

Nonlinear second-order multivalued boundary value problems

In this paper we study nonlinear second-order differential inclusions involving the ordinary vectorp-Laplacian, a multivalued maximal monotone operator and nonlinear multivalued boundary conditions. Our framework is general and unifying and incorporates gradient systems, evolutionary variational inequalities and the classical boundary value problems, namely the Dirichlet, the Neumann and the periodic problems. Using notions and techniques from the nonlinear operator theory and from multivalued analysis, we obtain solutions for both the ‘convex’ and ‘nonconvex’ problems. Finally, we present the cases of special interest, which fit into our framework, illustrating the generality of our results.     

Multivalued fractals in <Emphasis Type="Italic">b</Emphasis>-metric spaces

Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems theory in several topics of applied sciences. It is known that examples of fractals and multivalued fractals are coming from fixed point theory for single-valued and multivalued operators, via the so-called fractal and multi-fractal operators. On the other hand, the most common setting for the study of fractals and multi-fractals is the case of operators on complete or compact metric spaces. The purpose of this paper is to extend the study of fractal operator theory for multivalued operators on complete b-metric spaces.     

A multivalued version of Sharkovskiĭ’s theorem holds with at most two exceptions

A multivalued version of Sharkovskiĭ’s theorem is formulated for M-maps on linear continua and, more generally, for triangular M-maps on a Cartesian product of linear continua. This improves the main result of [AP1] in the sense that our multivalued analogue holds with at most two exceptions. A further specification requires some additional restrictions. For instance, 3- orbits of m-maps imply the existence of k-orbits for all k ? \mathbbNk \in {\mathbb{N}} , except possibly for k ?k \in {4, 6}. It is also shown that, on every connected linearly ordered topological space, an M-map with orbits of all periods can always be constructed. This demonstrates that Baldwin’s classification of linear continua in terms of admissible periods [Ba] is useless for multivalued maps.     

General proximal point algorithm involving η-maximal accretiveness framework in Banach spaces

First, based on η-maximal accretiveness, a generalization to Rockafellar’s theorem (1976) in the context of approximating a solution to a general inclusion problem involving a multivalued η-maximal accretive mapping using the proximal point algorithm in a q-uniformly smooth Banach space setting is considered. Then an application to a minimization problem of a functional is examined. The general framework for η-maximal accretiveness generalizes the general theory of multivalued maximal monotone mappings.      

Inclusion problems via subsolution–supersolution method with applications to hemivariational inequalities

Abstract

Existence and location of solutions to a Dirichlet problem driven by (p, q)-Laplacian and containing a (convection) multivalued term fully depending on the solution and its gradient are established through the method of subsolution–supersolution. This result extends preceding works, in particular improving the growth condition for the lower order terms and allowing multivalued nonlinearities. A criterion for the existence of positive solutions with a priori estimates is obtained. Finally, an application to hemivariational inequalities is given.     

σ-Extensions of discrete multivalued groups

The paper is devoted to the theory of singly generated multivalued groups. We construct new classes of such groups and find criteria that allow to check whether such a group is a coset group. For this purpose, we introduce the construction of a σ-extension of a singly generated multivalued group with Hermitian generator. This construction is based on the relation of the theory of such groups with the theory of symmetric graphs. The main result of the paper is as follows: the suggested construction of σ-extensions of singly generated bicoset multivalued groups with Hermitian generators is equivariant with respect to morphisms of graph-theoretic nature. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 225–242.     

Approximation to a set-valued mapping,I: A proposal

Given a multivalued mappingF, we address the problem of finding another multivalued mappingS that agrees locally withF. We limit ourselves to the case whenF(x) is convex compact for eachx. We propose anS obtained by linearizing the support function ofF(x) and give conditions under which thisS approximatesF in the sense of Hausdorff distance. We show equality betweenS and other proposals obtained from tangent cones to the graph ofF. Finally, we apply these results to the approximate subdifferential of a convex function.     

Existence results for hemivariational inequalities with measures

We prove existence results for multivalued quasilinear elliptic problems of hemivariational inequality type with measure data right-hand sides. In case of L ¹-data, we study existence and enclosure behaviors of solutions by an appropriate sub-supersolution approach. The proofs of our results are based on general existence theory for multivalued pseudomonotone operators, and approximation-, truncation-, and special test function techniques.     

On multivalued evolution equations in Hilbert spaces

The Cauchy problemdu/dt+Au+B(t,u)∋0,u(0)=u ₀ is studied in a separable Hilbert space setting, whenA is a multivalued maximal monotone operator, andB is a multivalued operator which is measurable with respect to the time variable and upper semi-continuous with respect to the space variable. Under some boundedness conditions onB, an existence theorem is proved, with the extra assumption, in the infinite dimensional case thatA is the subdifferential of a proper lower semi-continuous inf-compact convex function. A theorem of dependence upon the initial condition is also given.     

Multivalued differential equations in Banach space. An application to control theory

This paper is concerned with the study of existence theorems for multivalued differential systems in infinite-dimensional Banach space: the method used is based on techniques (extension theorem for linear operator, compactly convergent sequences) developed earlier by the authors for multivalued differential systems defined inn-dimensional vector spaces. As an application, the authors consider a distributed-parameter control problem arising in mathematical physics, more specifically, in the study of heat transfer in solids.This work was performed under the auspices of the National Research Council of Italy (CNR).     

Continuous Approximations of Multivalued Mappings and Fixed Points

In the present paper, we prove a fixed-point theorem for completely continuous multivalued mappings defined on a bounded convex closed subset X of the Hilbert space H which satisfies the tangential condition , where T _X (x) is the cone tangent to the set X at a point x. The proof of this theorem is based on the method of single-valued approximations to multivalued mappings. In this paper, we consider a simple approach for constructing single-valued approximations to multivalued mappings. This approach allows us not only to simplify the proofs of already-known theorems, but also to obtain new statements needed to prove the main theorem in this paper.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 212–222.Original Russian Text Copyright © 2005 by B. D. Gel’man.     

Differentiation of Multivalued Mappings. T-Derivative

We consider several approaches to the differentiation of multivalued mappings and introduce a new definition of derivative (T-derivative), which generalizes the Hukuhara derivative.     

Newton and other continuation methods for multivalued inclusions

Viability theory provides an efficient framework for looking for zeros of multivalued equations: 0 F(x). The main idea is to consider solutions of a suitable differential inclusion, viable in graph ofF. The choice of the differential inclusion is guided necessarily by the fact that any solution should converge or go through a zero of the multivalued equation. We investigate here a new understanding of the well-known Newton's method, generalizing it to set-valued equations and set up a class of algorithms which involve generalization of some homotopic path algorithms.     

Convergence of Ishikawa Iterates for a Multi-Valued Mapping with a Fixed Point

Existence of fixed points of multivalued mappings that satisfy a certain contractive condition was proved by N. Mizoguchi and W. Takahashi. An alternative proof of this theorem was given by Peter Z. Daffer and H. Kaneko. In the present paper, we give a simple proof of that theorem. Also, we define Mann and Ishikawa iterates for a multivalued map T with a fixed point p and prove that these iterates converge to a fixed point q of T under certain conditions. This fixed point q may be different from p. To illustrate this phenomenon, an example is given. This work is supported by the U. G. C. Grant No. U4/4997/97-98. G. V. R. Babu thanks the University Grants Commission, India for the financial support.     

Homogeneous Deformations of Discrete Groups

A linear deformation of a discrete group G is a special deformation of the group algebra of G resulting in the group algebra of a multivalued group. Basic results on linear deformations are presented. Bibliography: 5 titles.     

Tensor invariants of natural mechanical systems on compact surfaces and the corresponding integrals

The classification problem for tensor invariants of geodesic flows on compact surfaces is considered. Such invariants include first integrals, multivalued integrals, and symmetry fields. The problem is solved for tensor invariants of arbitrary degrees. It is shown that the invariants under consideration can vanish at some point only in the presence of two-valued symmetry fields and multivalued integrals. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 417–430, September, 1999.