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TOPOLOGICAL METHODS

IN

NONLINEAR ANALYSIS



Vol. 15, No. 1           March 2000



TABLE OF CONTENTS

Title and Author(s) Page
item A Result on the Singular Perturbation Theory for Differential Inclusions in Banach Spaces
A. Andreini, M. Kamenskii and P. Nistri
ABSTRACT. We provide conditions which ensure that the solution set of the Cauchy problem for a singularly perturbed system of differential inclusions in infinite dimensional Banach spaces is upper semicontinuous with respect to the parameter \varepsilon\ge0 of the perturbation. The main tools are represented by suitable introduced measures of noncompactness and the topological degree theory in locally convex spaces.
1
item Total and Local Topological Indices for Maps of Hilbert and Banach Manifolds
Yu. E. Gliklikh
ABSTRACT. Total and local topological indices are constructed for various types of continuous maps of infinite-dimensional manifolds and ANR's from a broad class. In particular the construction covers locally compact maps with compact sets of fixed points (e.g. maps having a certain finite iteration compact or having compact attractor or being asymptotically compact etc.); condensing maps (k-set contraction) with respect to Kuratowski's or Hausdorff's measure of non-compactness on Finsler manifolds; maps, continuous with respect to the topology of weak convergence, etc.

The characteristic point is that all conditions are formulated in internal terms and the index is in fact internal while the construction is produced through transition to the enveloping space. Examples of applications are given.

17
item Some Topological Properties of a Nonconvex Integral Inclusion
A. Cernea
ABSTRACT. We consider a conconvex parametrized integral inclusion and we prove that the solution set is a retract of Banach space.
33
item A Coincidence Theory Involving Fredholm Operators of Nonnegative Index
D. Gabor and W. Kryszewski
ABSTRACT. We construct a homotopy invariant appropriate for studying the existence of coincidence points of Fredholm operators of nonnegative index and multivalued admissible maps. Cohomotopy methods are used as a more suitable tool than homological ones. Both finite and infinite dimensional cases are investigated.
43
item A Remark to the Schauder Fixed Point Theorem
V. Seda
ABSTRACT. In the paper some sufficient conditions are established in order that a continuous map have a fixed point. The results are related to those obtained by R. D. Nussbaum in [18], L. Gorniewicz and D. Rozploch-Nowakowska in [12], S. Szufla in [21] and D. Bugajewski in [6].
61
item Use Finite Family of Multivalued Maps for Constructing Stable Absorption Operator
S. V. Grigorieva and V. N. Ushakov
ABSTRACT. The differential game of pursuit-evasion over a fixed time segment is considered. the problem of construction of the stable absorption operator of control system is investigated. The attainability sets is appointed with the help of the stable absorption operator. The partition of the conjugate space on the finite regions of convexity of Hamiltonian is used for constructing stable absorption operator.
75
item Discontinuous Mayer Control Problem Under State-Constraints
S. Plaskacz and M. Quincampoix
ABSTRACT. This paper deals with Mayer's problem for control systems with state constraints and, possibly, discontinuous terminal cost. The main result of this paper consists in the characterization of the value function as the unique solution to an Hamilton-Jacobi equation. The above characterization extends results already obtained in the case of regular cost functions and under some controlability assumptions on the boundary of the set of constraints.
91
item Solutions of Implicit Evolution Inclusions with Pseudo-Monotone Mappings
W. M. Bian
ABSTRACT. Existence results are given for the implicit evolution inclusions
(Bx(t))'+A(t, x(t))\ni f(t)$ and $(Bx(t))'+A(t, x(t))-G(t, x(t))\ni f(t)
with B a bounded linear operator, A(t, . ) a bounded, coercive and pseudo-monotone set-valued mapping and G a set-valued mapping of non-monotone type. Continuity of the solution set of first inclusion with respect to f is also obtained which is used to solve the second inclusion.
101
item Eigenvalue Stability for Multivalued Operators
P. Lavilledieu and A. Seeger		 115
item Higher-Order Necessary Optimality Conditions for Extremum Problems in Topological Vector Spaces
L. Mikolajczyk and M. Studniarski
ABSTRACT. We present a higher-order extension of the well-known theorem of Ben-Tal and Zowe on second-order necessary optimality conditions in topological vector spaces. We also examine the connection between this extension and the results of Furukawa and Yoshinaga which are stated in terms of higher-order variational sets and Neustadt derivatives.
129
item On a Controllability Problem for Systems Governed by Semilinear Functional Differential Inclusions in Banach Spaces
V. Obukhovskii and P. Rubbioni
ABSTRACT. For a Banach space E, a given pair (\overline p, \overline x)\in[0,a]\times E, and control system governed by a semilinear functional differential includion of the form
x'(t)\in Ax(t)+F(t,x(t),Tx)
the existence of a mild trajectory of x(t) satisfying the condition x(\overline p)=\overline x is considered. Using topological methods we develop an unified approach to the cases when a multivalued nonlinearity F is Caratheodory upper semicontinuous or almost lower semicontinuous and an abstract extension operator T allows to deal with variable and infinite delay. For the Caratheodory case, the compactness of the solutions set and, as a corollary, an optimization result are obtained.
141
item Infinite Products of Resolvents of Accretive Operators
S. Reich and A. J. Zaslavski
ABSTRACT. We study the space Mm of all m-accretive operators on a Banach space X endowed with an appropriate complete metrizable uniformity and the space \overline{\Cal M}{}^*_m which is the closure in Mm of all those operators which have a zero. We show that for a generic operator in Mm all infinite products of its resolvents become eventually close to each other and that a generic operator in \overline{\Cal M}{}^*_m has a unique zero and all the infinite products of its resolvents converge uniformly on bounded subsets of X to this zero.
153
item A Few Properties of the Kobayashi Distance and Their Applications
J. Kapeluszny and T. Kuczumow
ABSTRACT. Let N be a norming set in a Banach space X. In this paper we prove the lower semicontinuity with respect to the topology \sigma(X,N) of the Kobayashi distance in a bounded, relatively compact in \sigma(X,N), convex and open subset of a Banach space. We apply this result to the Denjoy-Wolff type theorem.
169
item A Short Proof of the Converse to the Contraction Principle and Some Related Results
J. Jachymski
ABSTRACT. We simplify a proof of Bessaga's theorem given in the monograph of Deimling. Moreover, our argument let us also obtain the following result.

Let F be a selfmap of an arbitrary set \Omega and \alpha\in (0,1). Then F is an \alpha-similarity with respect to some complete metric d for \Omega (that is, d(Fx,Fy)=\alpha d(x,y) for all x,y\in\Omega) if and only if F is injective and F has a unique fixed point.

Finally we present that the converse to the Contraction Principle for bounded spaces is independent of the Axiom of Choice.

179
item An Example Concerning Equivariant Deformations
M. Izydorek and A. Vidal
ABSTRACT. We give an example of Z2-space X with a property that the identity map \text{\rm{id}}_X:X\to X as well as its restriction to the fixed point set of the group action \text{\rm{id}}^{Z_2}:X^{Z_2}\to X^{Z_2} are deformable to fixed point free maps whereas there is no fixed point free map in the equivariant homotopy class of the identity [\text{\rm{id}}_X]_{ Z_2}.
187
item Coincidence and Fixed Point Theorems with Applications
Q. H. Ansari, A. Idzik and J.-Ch. Yao
ABSTRACT. In this paper, we first establish a coincidence theorem under the noncompact settings. Then we derive some fixed point theorems for a family of functions. We apply our fixed point theorem to study nonempty intersection problems for sets with convex sections and obtain a social equilibrium existence theorem. We also introduce a concept of a quasi-variational inequalities and prove an existence result for a solution to such a system.
191



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