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Báo cáo hóa học: " POSITIVE SOLUTIONS OF SOME THREE-POINT BOUNDARY VALUE PROBLEMS VIA FIXED POINT INDEX FOR WEAKLY INWARD A-PROPER MAPS"

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: POSITIVE SOLUTIONS OF SOME THREE-POINT BOUNDARY VALUE PROBLEMS VIA FIXED POINT INDEX FOR WEAKLY INWARD A-PROPER MAPS | POSITIVE SOLUTIONS OF SOME THREE-POINT BOUNDARY VALUE PROBLEMS VIA FIXED POINT INDEX FOR WEAKLY INWARD A-PROPER MAPS GENNARO INFANTE Received 23 November 2003 and in revised form 2 November 2004 We use the theory of fixed point index for weakly inward A-proper maps to establish the existence of positive solutions of some second-order three-point boundary value problems in which the highest-order derivative occurs nonlinearly. 1. Introduction In the present paper we discuss the existence of positive solutions of the nonlinear three-point boundary value problem BVP -u t f t u ú u t G 0 1 1.1 with the nonlocal boundary conditions BCs u 0 0 au n u 1 0 n 1 an 1 1.2 in which the second derivative may occur nonlinearly. Positive solutions for the case f t u u u g t h u have been studied by Ma 15 and Webb 20 21 when f t u u u h t u by He and Ge 5 and also by Lan 11 . The case f t u u u g t h u u has been studied by Feng 4 . The results in 4 15 are obtained by means of Krasnosel skii s theorem 8 the ones in 5 use Leggett and Williams theorem 14 and the results in 11 20 21 are achieved via the classical fixed point index for compact maps see for example 1 . Lafferriere and Petryshyn 9 and Cremins 2 studied existence of positive solutions of the so-called Picard boundary value problem -u t f t u u u 1.3 with BCs u 0 u 1 0 1.4 Copyright 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005 2 2005 177-184 DOI 10.1155 FPTA.2005.177 178 Positive solutions of some BVPs by means of fixed point index theory for A-proper maps. A key restriction in 2 9 is that f must take positive values. Lan and Webb 13 improved the results of 2 9 by allowing f to possibly take some negative values. Here we will exploit Lan and Webb s theory 12 of fixed point index for weakly inward A-proper maps to prove new results on the existence of positive solutions of the BVP 1.1 - 1.2 . We mention that with very little change this technique may be applied to a variety of BCs e.g. other