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

Mirosława Zima

 Institute of Mathematics, Pedagogical University of Rzeszów, Rzeszów 35-310, Poland
Received 9 July 1999; Revised 17 December 1999

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∞.