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Positive solutions of nonlocal boundary value problems involving integral conditions

  • Jeff R. L. Webb
  • Gennaro Infante
Article

Abstract.

We establish the existence of multiple positive solutions of nonlinear equations of the form
$$-u^{\prime\prime}(t) = g(t)f(t, u(t)), t \in (0, 1),$$
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
$$u(t) = \gamma(t)\alpha [u] + \int \nolimits^1_0 k(t, s)g(s)f(s, u(s))ds$$
where α[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 non-local 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.

2000 Mathematics Subject Classification:

Primary 34B18 secondary 34B10, 47H10, 47H30 

Keywords:

Nonlocal boundary conditions fixed point index positive solution 

Copyright information

© Birkhaueser 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GlasgowGlasgowUK
  2. 2.Dipartimento di MatematicaUniversità della CalabriaCosenzaItaly

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