Positive solutions of nonlocal boundary value problems involving integral conditions

Abstract

Abstract.We establish the existence of multiple positive solutions of nonlinear equations of the formwhere 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 formwhere 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 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.2000 Mathematics Subject Classification: Primary 34B18, secondary 34B10, 47H10, 47H30.

Multiple Positive Solutions of Resonant and Non-Resonant Non-Local Fourth-Order Boundary Value Problems

“…To see examples with sign-changing measures for some second-order problems, see [26,27]. Here we could give similar examples but have concentrated on simple new examples to illustrate the approach using a shift argument.…”

Existence of positive solutions for a high order fractional differential equation integral boundary value problem with changing sign nonlinearity

“…Since the nonlocal boundary value problems include the multi-point boundary value problem (A is a step function) and the Riemann integral boundary value problem (A has a continuous derivative), it has become a more general case where we study the boundary value problem with integral boundary conditions of Riemann-Stieltjes type. Many researchers have done a lot of work on this class of boundary value problems, see [12,22,25,29,30] and the references therein.…”

Positive solutions for a fractional boundary value problem

“…Signi…cant developments in fractional di¤erential equations can be …nd in the monographs of Kilbas et al [20], Miller and Ross [22], Lakshmikantham et al [21], Podlubny [23]. Ordinary di¤erential equations and fractional di¤erential equations have been studied by many authors by means of …xed point theory see [1,2,3,4,13,14,15,16,17,18,19,24,25].…”