# Research Article Convex Solutions of a Nonlinear Integral Equation of Urysohn Type"

## Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Convex Solutions of a Nonlinear Integral Equation of Urysohn Type | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 917614 13 pages doi 2009 917614 Research Article Convex Solutions of a Nonlinear Integral Equation of Urysohn Type Tiberiu Trif Faculty of Mathematics and Computer Science Babeậ-Bolyai University Str. Kogalniceanu Nr. 1 400084 Cluj-Napoca Romania Correspondence should be addressed to Tiberiu Trif ttrif@ Received 4 August 2009 Accepted 25 September 2009 Recommended by Donal O Regan We study the solvability of a nonlinear integral equation of Urysohn type. Using the technique of measures of noncompactness we prove that under certain assumptions this equation possesses solutions that are convex of order p for each p e -1 0 . r with r -1 being a given integer. A concrete application of the results obtained is presented. Copyright 2009 Tiberiu Trif. 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. 1. Introduction Existence of solutions of differential and integral equations is subject of numerous investigations see . the monographs 1-3 or 4 . Moreover a lot of work in this domain is devoted to the existence of solutions in certain special classes of functions . positive functions or monotone functions . We merely mention here the result obtained by Caballero et al. 5 concerning the existence of nondecreasing solutions to the integral equation of Urysohn type x t a t u t x t v t s x s ds t e 0 T 0 where T is a positive constant. In the special case when u t x x2 or even u t x x the authors proved in 5 that if a is positive and nondecreasing v is positive and nondecreasing in the first variable when the other two variables are kept fixed and they satisfy some additional assumptions then there exists at least one positive nondecreasing solution x 0 T R to . A similar existence result but .

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