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Li and Dong Boundary Value Problems 2011, 2011:59 http://www.boundaryvalueproblems.com/content/2011/1/59 RESEARCH Open Access Multiple positive solutions for a fourth-order integral boundary value problem on time scales Yongkun Li* and Yanshou Dong * Correspondence: yklie@ynu.edu. cn Department of Mathematics, Yunnan University Kunming, Yunnan 650091 People’s Republic of China Abstract In this article, we investigate the multiplicity of positive solutions for a fourth-order system of integral boundary value problem on time scales. The existence of multiple positive solutions for the system is obtained by using the fixed point theorem of cone expansion and compression type due to Krasnosel’skill. To demonstrate the applications of our results, an example is also. | Li and Dong Boundary Value Problems 2011 2011 59 http www.boundaryvalueproblems.eom content 2011 1 59 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Multiple positive solutions for a fourth-order integral boundary value problem on time scales Yongkun Li and Yanshou Dong Correspondence yklie@ynu.edu. cn Department of Mathematics Yunnan University Kunming Yunnan 650091 People s Republic of China Springer Abstract In this article we investigate the multiplicity of positive solutions for a fourth-order system of integral boundary value problem on time scales. The existence of multiple positive solutions for the system is obtained by using the fixed point theorem of cone expansion and compression type due to Krasnosel skill. To demonstrate the applications of our results an example is also given in the article. Keywords positive solutions fixed points integral boundary conditions time scales 1 Introduction Boundary value problem BVP for ordinary differential equations arise in different areas of applied mathematics and physics and so on the existence and multiplicity of positive solutions for such problems have become an important area of investigation in recent years lots of significant results have been established by using upper and lower solution arguments fixed point indexes fixed point theorems and so on see 1-8 and the references therein . Especially the existence of positive solutions of nonlinear BVP with integral boundary conditions has been extensively studied by many authors see 9-18 and the references therein . However the corresponding results for BVP with integral boundary conditions on time scales are still very few 19-21 . In this article we discuss the multiple positive solutions for the following fourth-order system of integral BVP with a parameter on time scales x 4A t kf t x t xA t xAA t y t yA t yAA t 0 t e 0 ơ T t y 4A t rrg t x t xÁ t xÁÁ t y t yÁ t yÁÁ t 0 t e 0 ơ T t x 0 xA 0 0 y 0 yA 0 0 d1xAA 0 - x 0 xAA s A1 s As 0 .
