Higher order multi-point boundary value problems

Abstract

Key words Solutions, boundary value problems, lower and upper solutions, Nagumo condition, fixed point theorem MSC (2010) 34B15, 34B18We consider the boundary value problem u, u , . . . , u (n −1) = 0, t ∈ (0, 1),where n ≥ 2 and m ≥ 1 are integers, tj ∈ [0, 1] for j = 1, . . . , m, and f and gi , i = 0, . . . , n − 1, are continuous. We obtain sufficient conditions for the existence of a solution of the above problem based on the existence of lower and upper solutions. Explicit conditions are also found for the existence of a solution of the problem. The differential equation has dependence on all lower order derivatives of the unknown function, and the boundary conditions cover many multi-point boundary conditions studied in the literature. Schauder's fixed point theorem and appropriate Nagumo conditions are employed in the analysis. Examples are given to illustrate the results.

Existence of Solutions for a Higher Order Multi-Point Boundary Value Problem

“…The following lemma is an analogue to [4,Lemma 3.1], and can be proved in the same way. We omit the details here.…”

“…Motivated partly by the recent papers [4,5,11,14,18], here we study the general BVP (1.1), (1.2) and obtain conditions for the existence of one and two solutions. As applications, we present results for a special case of BVP (1.1), (1.2), i.e., the BVP consisting of Eq.…”

Solution of Nonlinear High Order Multi-Point Boundary Value Problems By Semi-Analytic Technique

“…Next, the software determines a more accurate solution P 2(n+1)+1 by increasing n number of fit order for derivative of approximate with derivative of exact. Then, an estimate of the norm of the global error, e MPRE , is given by: (9) We apply this algorithm for example 1, as follows: Apply equation (9) by the following (for more details see Table (3)):…”

“…Wang et al [8] studied the existence of nontrivial solutions for nonlinear higher order MPBVP on time scales with all derivatives. Graef et al [9] obtained sufficient conditions for the existence of a solution of the higher order MPBVP based on the existence of lower and upper solutions. Liu et al [10] established the existence results of multiple monotone and convex positive solutions for some fourth-order MPBVPs.…”

Higher Order $${\phi}$$ -Laplacian BVP with Generalized Sturm–Liouville Boundary Conditions

“…On the other hand, the general Sturm-Liouville boundary conditions considered here are not covered by several other multi-point boundary value problems of higher order such as those considered, for example, in [3,4] for n-th order problems, or in [2] for second order ones.…”