Positive Solutions of Nonlocal Boundary Value Problems: A Unified Approach

Abstract

We give a unified approach for studying the existence of multiple positive solutions of nonlinear differential equations of the form −u (t) = g(t)f (t, u(t)), for almost every t ∈ (0, 1),where g, f are non-negative functions, subject to various nonlocal boundary conditions. We study these problems via new results for a perturbed integral equation, in the space C[0, 1], of the form β[u] are linear functionals given by Stieltjes integrals but are not assumed to be positive for all positive u. This means we actually cover many more differential equations than the simple equation written above. Previous results have studied positive functionals only, but even for positive functionals our methods give improvements on previous work. The well-known m-point boundary value problems are special cases and we obtain sharp conditions on the coefficients, which allows some of them to have opposite signs. We also use some optimal assumptions on the nonlinear term.

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.…”

“…The idea used in [26,28] is to consider solution of the non-local problem as perturbations from the non-local problem and to seek fixed points of the following operator…”

“…Some kind of positivity on the functionals β i is needed in order to have positive solutions, a solution u will satisfy β i [u] ≥ 0 but we do not suppose that β i We use fixed point index theory, based on the methods developed in [26,28]. The fourth-order equation when ω = 0 with a variety of BCs, and with one BC of the non-local type that we study here, has been studied, with similar methods, in detail in [29] in the non-resonant case.…”

Smoothness of solutions with respect to multi-strip integral boundary conditions for nth order ordinary differential equations

“…Subsequent to that paper, Ahmad and Ntouyas have put forth a couple of additional papers devoted to solutions of boundary value problems involving multi-strip integral boundary conditions for both fractional differential equations and fractional differential inclusions; see [11,12]. It can also be pointed out that, under suitable measures, the boundary conditions can be considered in the form of Stieltjes integrals; readers can find of interest the papers, [13][14][15] and [16][17][18].…”

On two systems of non-resonant nonlocal boundary value problems

“…Nowadays, the problem of the existence of solutions for various types of nonlocal BVPs is the subject of many papers. For such problems and comments on their importance, we refer the reader to [10], [11], [14], [18], [19], [23] and the references therein.…”