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Tham khảo luận văn - đề án 'van existence theorem for an implicit integral equation with discontinuous right-hand side"', luận văn - báo cáo phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | AN EXISTENCE THEOREM FOR AN IMPLICIT INTEGRAL EQUATION WITH DISCONTINUOUS RIGHT-HAND SIDE GIOVANNI ANELLO Received 8 December 2004 Revised 9 March 2005 Accepted 22 March 2005 We establish a result concerning the existence of solutions for the following implicit integral equation g u t y t x0 Ị0 f t u r dr where p is not supposed continuous with respect to the second variable. Copyright 2006 Giovanni Anello. 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 Let a 0 x0 e R and let E be a metric space. Let f 0 a X E 0 to g E R and p 0 a X R R be given functions. The aim of this paper is to establish an existence theorem for an implicit integral equation of the type g u t pit X0 f t u r dr I 1.1 0 where function p is not supposed continuous with respect to the second variable. The reason for studying 1.1 arises mainly from the paper 3 . Indeed 3 Theorem A gives the existence of solutions for 1.1 assuming among the other hypotheses that p is a Caratheodory function and that f does not depend on t e 0 a . We note that using the arguments employed in the proof of Theorem A of 3 it seems that it is not possible neither to weaken the assumption of continuity of the function p in the second variable nor to assume f dependent on t e 0 a . The purpose of the present paper goes just in this direction. Namely studying 1.1 by means of quite different arguments from that ones used in 3 we are able to suppose f dependent on t e 0 a and to remove the continuity of p in the second variable. In particular as regards to this latter our assumptions allow y t to be discontinuous at each point. The abstract framework where 1.1 is studied is that of set-valued analysis. In particular we will deduce our result by using a recent selection theorem for multifunction of two variables see 2 Theorem 2 jointly to 9 Theorem
