On a fixed point index method for the analysis of the asymptotic behavior and boundary value problems of infinite dimensional dynamical systems and processes

Abstract

The Waiewski principle [25] plays an important role in the study of ordinary differential equations. Its applicability is largely due to the fact that in a finite dimensional euclidean space the unit sphere is not a retract of the closed unit ball. Since this is no longer true in infinite dimensional Banach spaces the direct extension of Waiewski’s principle to processes or semidynamical systems of infinite dimensional Banach spaces has a very limited applicability. Since in finite dimensional spaces the fact that the unit sphere is not a retract of the closed unit ball is equivalent to the fact that every continuous mapping of the unit closed convex ball has a fixed point, the main idea of this work is to develop a method based on fixed point index properties instead of retraction properties. Our fixed point formulation Corollary 1 is essentially equivalent in finite dimension to the Waiewski theorem. Although in infinite dimension the Waiewski theorem is no longer applicable, Theorems 1 and 2 are applicable since fixed point index methods have proved to be very useful in the solution of problems related to differential equations either in finite or infinite dimensional spaces. After the Waiewski paper several papers were written applying the Waiewski principle to the asymptotic behaviour of ordinary differential equations, Olech [ 191, Pliss [22], Mikolajska [ 161, Onuchic [20], Ize [ 121, and others. Kaplan, Lasota, and Yorke [ 131 applied the Waiewski method to the boundary value problem and Conley [4] also applied the Waiewski method to a boundary value problem for diffusion equations in biology. Since our approach uses Waiewski basic ideas in connection with fixed point index theory it should also give good results even in finite dimensions and it can also be applied to boundary value problems in Hilbert spaces. 162 0022-0396184 $3.00