Positive solutions of nonlocal boundary value problems involving integral conditions

Webb, J.R.L. and Infante, G. (2008) Positive solutions of nonlocal boundary value problems involving integral conditions. No DEA - Nonlinear Differential Equations and Applications, 15(1-2), pp. 45-67. (doi: 10.1007/s00030-007-4067-7)

Abstract

We establish the existence of multiple positive solutions of nonlinear equations of the form -u ''(t) = g(t)f(t, u(t)), t epsilon (0,1) where g, f are non-negative functions, subject to various nonlocal boundary conditions. The common feature is that each can be written as an integral equation, in the space C[0, 1], of the form u(t) = gamma(t)alpha[u] + integral(1)(0) k(t, s)g(s)f(s, u(s))ds where alpha[u] is a linear functional given by a Stieltjes integral but is not assumed to be positive for all positive u. Our new results cover many nonlocal boundary conditions previously studied on a case by case basis for particular positive functionals only, for example, many m-point BVPs are special cases. Even for positive functionals our methods give improvements on previous work. Also we allow weaker assumptions on the nonlinear term than were previously imposed.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Infante, Dr Gennaro and Webb, Professor Jeffrey
• Authors: Webb, J.R.L., and Infante, G.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: No DEA - Nonlinear Differential Equations and Applications
• Journal Abbr.: NoDEA
• Publisher: Birkhaeuser Verlag AG
• ISSN: 1021-9722
• ISSN (Online): 1420-9004