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 Generalized Bihari Type Integral Inequalities and the Corresponding Integral Equations | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 409809 20 pages doi 2009 409809 Research Article Generalized Bihari Type Integral Inequalities and the Corresponding Integral Equations László Horvath Department of Mathematics University of Pannonia Egyetem u. 10 8200 Veszprem Hungary Correspondence should be addressed to Laszlo Horvath lhorvath@ Received 2 February 2009 Accepted 23 June 2009 Recommended by Alberto Cabada We study some special nonlinear integral inequalities and the corresponding integral equations in measure spaces. They are significant generalizations of Bihari type integral inequalities and Volterra and Fredholm type integral equations. The kernels of the integral operators are determined by concave functions. Explicit upper bounds are given for the solutions of the integral inequalities. The integral equations are investigated with regard to the existence of a minimal and a maximal solution extension of the solutions and the generation of the solutions by successive approximations. Copyright 2009 Laszlo Horvath. 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 and the Main Results In this paper we study integral inequalities of the form y x f x g x J q o y dy x e X and the corresponding integral equations y x f x g x q o ydy x e X JS x where A1 X A ỳ is a measure space A2 S is a function from X into A such that the following properties hold 2 Journal of Inequalities and Applications A2 p S x TO for every x e X A2 if x2 e S x1 then S x2 c S x1 A2 xi x2 e X2 x2 e S x1 is p7--measurable A3 q is a function from 0 to into 0 to with the following conditions A3 q is concave A2 limt TO q t t 0 A4 the functions f and g belong to Lioc X p X 0 to p is -integrable over S x Yx e . It may be noted that .
