Journal of Inequalities and Applications
Volume 6 (2000), Issue 3, Pages 359-371
doi:10.1155/S1025583401000212
On positive solutions of functional-differential equations in banach spaces
Institute of Mathematics, Pedagogical University of Rzeszów, Rzeszów 35-310, Poland
Received 9 July 1999; Revised 17 December 1999
Copyright © 2000 Mirosława Zima. 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 this paper, we deal with two point boundary value problem (BVP) for the functional-differential equation of second order
x″(t)+kx′(t)+f(t,x(h1(t)),x(h2(t)))=0,ax(−1)−bx′(−1)=0,cx(1)+dx′(1)=0,
where the function f takes values in a cone K of a Banach space E. For h1(t)=t and h2(t)=−t we obtain the BVP with reflection of the argument. Applying fixed point theorem on strict set-contraction from G. Li, Proc. Amer. Math. Soc. 97 (1986), 277–280, we prove the existence of positive solution in the space C([−1,1],E). Some inequalities involving f and the respective Green’s function are used. We also give the application of our existence results to the infinite system of functional–differential equations in the case E=l∞.