Journal of Inequalities and Applications
Volume 6 (2000), Issue 2, Pages 199-226
doi:10.1155/S1025583401000121
Nonlinear systems of differential inequalities and solvability of certain boundary value problems
1Department of Mathematics, Palacký University, 779 00 OLOMOUC, Tomkova 40, Czech Republic
2Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, 115 67 PRAHA 1, Czech Republic
Received 25 July 1999; Revised 10 October 1999
Copyright © 2000 Irena Rachůnková and Milan Tvrdý. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In the paper we present some new existence results for nonlinear second order generalized periodic boundary value problems of the form
u″=f(t,u,u′),
u(a)=u(b),
u′(a)=w(u′(b)).These results are based on the method of lower and upper functions defined as solutions of the system of differential inequalities associated with the problem and their relation to the Leray–Schauder topological degree of the corresponding operator. Our main goal consists in a fairly general definition of these functions as couples from 𝔸ℂ[a,b]×𝔹𝕍[a,b]. Some conditions ensuring their existence are indicated, as well.