Topological methods in existence theorems
by
N. M. Benkafadar
University of Constantine, Algeria

In the note one builds a topological characteristic for a class of pairs of single-valued maps defined in Hausdorff topological spaces. This topological characteristic is more subtle than the classical Brouwer-Hopf degree that formed the basis of the Leray-Schauder degree theory for completely continuous vector fields. For this purpose one defines a class of quintuples called (n, G)-compatible, which contains naturally other classes elaborated before by different authors. Applying this concept, one obtains existence therorems for some systems of equations of a general form. The construction is based on categorical aspects. Let us give more notions on this topic.

A pair (X, A) of spaces is called Hausdorff if X is a Hausdorff topological space and A subset or equal X. A Hausdorff pair of the type (X, \emptyset) is identified with the Hausdorff space X. A continuous single-valued map f : (X, A) --> (Y, B) defined between Hausdorff pairs is a continuous single-valued map f : X --> Y such that f(A) subset or equal B. The composition of two single-valued maps acting on Hausdorff pairs is defined as the classical composition of maps. In this way one obtains a category Top⁽²⁾ [5]. A quintuple

\Pi = [(X, X\A), (Y, Y\B), (Z, ~ Z   ), f, g]

in the category Top⁽²⁾, expresses the following concept: f:(Z, [Z\tilde]) --> (X, X\A) and g:(Z, [Z\tilde]) --> (Y, Y\B) are two morphisms in Top⁽²⁾. Let H be the singular homology functor with coefficients in an abelian group G, from the category Top⁽²⁾of Hausdorff pairs and continuous maps to the category GAG of graded abelian groups and homomorphisms of degree zero [6]. For instance, for each quintuple

\Pi = [(X, X\A), (Y, Y\B), (Z, ~ Z   ), f, g]

in Top⁽²⁾ one could associate in GAG a quintuple:

H*(\Pi) = [H*(X, X\A; G), H*(Y, Y\B; G), H*(Z, ~ Z   ; G), H*(f), H*(g)] =[{Hn(X, X\A;G)}, {Hn(Y, Y\B;G)}, {Hn(Z, ~ Z   ;G)}, {Hn(f)}, {Hn(g)}]n >= 0

Imposing some conditions on the rank n >= 0 on H_*(\Pi) one defines the notion of (n, G)-compatible quintuple \Pi and one associates to every quintuple of this type a topological characteristic denoted by \zeta(\Pi), thereafter one studies its properties. The main result resides in the fact that using \zeta(\Pi) one states some existence theorems for a class of systems of equations of the form:

ì í î f(z) in A g(z) in B.

This class of systems of equations is vast. Indeed, it is not difficult to check that if the appropriate objects and morphisms are chosen in the category Top⁽²⁾, the problem of the existence of solutions for the system of equations, above defined, becomes equivalent to a fixed point problem whose generalization is the coincidence point problem. Furthermore, several results of the fixed-point theory for single-valued maps can be carried over to the case of multi-valued mappings.

References

[1] Benkafadar N.M. Local degree for some class of single-valued maps, Algebra and Number Theory, 2000, V 1, N. 1, 22-29.

[2] Benkafadar N., Gel 'man B. D. On a local degree of one class of multivalued vector fields in infinite-dimensional Banach spaces, Abstract and Applied Analysis, 1996, V. 1, N. 4, 381-396.

[3] Borisovitch Y. G. Modern approach to the theory of topological characteristics of non-linear operators. II, Lecture Notes in Mathematics, 1990, N 1453, 21-50.

[4] Borisovitch Y. G. Modern approach to the theory of topological characteristics of non-linear operators. I, Lecture Notes in Mathematics, 1988, N 1334, 199-220.

[5] Dold A. Lectures on Algebraic Topology, Springer-Verlag , 1972.

[6] Spanier E. H. Algebraic Topology, McGraw-Hill, 1966.

Date received: June 16, 2001