Existence of positive solutions to n-point nonhomogeneous boundary value problem

Abstract

In this paper, we are concerned with the existence of positive solutions to a n-point nonhomogeneous boundary value problem. By using the Krasnoselskii's fixed point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of positive solution is established for the n-point nonhomogeneous boundary value problem.

The Shooting Method and Nonhomogeneous Multipoint BVPs of Second-Order ODE

“…(2) With the exception of a few, for example, [1,3], most of the results on the existence of solutions to multipoint boundary value problems are based upon fixed-point theorems of Krasnoselskii's type; see Krasnoselskii [2], Guo and Lakshmikantham [19], and recent articles of Ma [12,14,20], Liu [18], and Sun et al [13]. Other methods in nonlinear functional analysis such as Leray-Schauder continuation theorem, nonlinear alternative of Leray-Schauder fixed point theorem, and coincidence degree theory have also been used, see Mawhin [21,22].…”

“…Analogous results (Theorems 3.3 and 3.4) are then formulated and extended to nonlinear equations of the more general form y (t) + F t, y(t) = 0, t ∈ (0,1), (1.3) where the nonlinear term may not be in a separable format. In both [12,13], the Neumann boundary condition…”

“…Define For the nonhomogeneous problem, Ma [14] has the following result for the superlinear case. In a recent paper by Sun et al [13], Theorem 1.2 was extended to the multipoint Neumann problem (1.4) and (1.7). The authors also stated an analogue for the sublinear case …”

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Recommended by Kanishka PereraIn a recent paper, Sun et al (2007) studied the existence of positive solutions of nonhomogeneous multipoint boundary value problems for a second-order differential equation. It is the purpose of this paper to show that the shooting method approach proposed in the recent paper by the first author can be extended to treat this more general problem.

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Positive Solutions for Third Order Boundary Value Problems with p-Laplacian

“…(1.4) Under some assumptions, it was shown that there exists λ * > 0 such that BVP (1.3), (1.4) has at least one positive solution for 0 < λ < λ * and has no positive solution for λ > λ * . Later on, in [6,10,11,[13][14][15][16], this result was generalized to other problems with a parameter λ in the BCs. Here, we mention that, in all of the above work, neither the uniqueness of solutions nor the dependence of solutions on the parameter λ is studied, and only papers [10,11,16] studied the multiplicity of solutions.…”