Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations

Infante, G. and Webb, J.R.L. (2006) Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations. Proceedings of the Edinburgh Mathematical Society, 49(3), pp. 637-656. (doi: 10.1017/S0013091505000532)

Abstract

Motivated by some non-local boundary-value problems (BVPs) that arise in heat-flow problems, we establish new results for the existence of non-zero solutions of integral equations of the form [FORMULA] where G is a compact set in Rn. Here α[u] is a positive functional and f is positive, while k and γ may change sign, so positive solutions need not exist. We prove the existence of multiple non-zero solutions of the BVPs under suitable conditions. We show that solutions of the BVPs lose positivity as a parameter decreases. For a certain parameter range not all solutions can be positive, but for one of the boundary conditions we consider we show that there are positive solutions for certain types of nonlinearity. We also prove a uniqueness result.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Webb, Professor Jeffrey
• Authors: Infante, G., and Webb, J.R.L.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Proceedings of the Edinburgh Mathematical Society
• ISSN: 0013-0915
• ISSN (Online): 1464-3839
• Published Online: 25 January 2007