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Báo cáo hoa học: " Research Article Multiple Positive Solutions for Nonlinear First-Order Impulsive Dynamic Equations on Time Scales with Parameter"

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 Multiple Positive Solutions for Nonlinear First-Order Impulsive Dynamic Equations on Time Scales with Parameter | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 830247 9 pages doi 10.1155 2009 830247 Research Article Multiple Positive Solutions for Nonlinear First-Order Impulsive Dynamic Equations on Time Scales with Parameter Da-Bin Wang and Wen Guan Department of Applied Mathematics Lanzhou University of Technology Lanzhou Gansu 730050 China Correspondence should be addressed to Da-Bin Wang wangdb@lut.cn Received 13 February 2009 Accepted 14 May 2009 Recommended by Victoria Otero-Espinar By using the Leggett-Williams fixed point theorem the existence of three positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales with parameter are obtained. An example is given to illustrate the main results in this paper. Copyright 2009 D.-B. Wang and W. Guan. 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 T be a time scale that is T is a nonempty closed subset of R. Let T 0 be fixed and 0 T be points in T an interval 0 T T denoting time scales interval that is 0 T T 0 T n T. Other types of intervals are defined similarly. Some definitions concerning time scales can be found in 1-5 . In this paper we are concerned with the existence of positive solutions for the following nonlinear first-order periodic boundary value problem on time scales xA t p t x ơ t Af f x ơ t t e J 0 T T if tk k 1 2 . m x if x it Ti. x t k 1.2 m 1.1 x k x tk k x Lk . . . m . x 0 x ơ T where A 0 is a positive parameter f e CJ X 0 to 0 to Ik e C 0 to 0 to p 0 T T 0 to is right-dense continuous tk e 0 T T 0 t1 tm T and for each 2 Advances in Difference Equations k 1 2 . m x tk limh 0 x tk h and x tk limh 0-x tk h represent the right and left limits of x i at t tk. The theory of impulsive differential equations is emerging .