Monotone and convex positive solutions for fourth-order multi-point boundary value problems

Abstract

The existence results of multiple monotone and convex positive solutions for some fourth-order multi-point boundary value problems are established. The nonlinearities in the problems studied depend on all order derivatives. The analysis relies on a fixed point theorem in a cone. The explicit expressions and properties of associated Green's functions are also given. MSC: 34B10; 34B15.

A survey on stationary problems, Green's functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions

“…Zhao et al [239] investigated the second-order BVP with four-point BC. The existence results of multiple monotone and convex positive solutions for some fourth-order multipoint BVP was established by Liu, Weiguo and Chunfang in [118]. Ma [122] studied the existence and nonexistence of positive solutions of nonlinear periodic BVP and Green's function for corresponding linear problem.…”

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

“…Higherorder formulae can be used to determine a more accurate numerical solution with the same mesh as for the original solution. Specifically, the global error can be estimated by: e HO = ║Y p -Y q ║ ∞ (10) where Y p is the original discrete solution of order p and Y q is the more accurate discrete solution of order q > p. In [14] symmetric formulae are used, q = p + 2.…”

“…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. In this paper we use two-point osculatory interpolation; essentially this is a generalization of interpolation using Taylor polynomials.…”

A shifted Legendre spectral method for fractional-order multi-point boundary value problems

“…Prompted by the application of multi-point boundary value problems to applied mathematics and physics, these problems have provoked a great deal of attention by many authors (see, for instance, [28][29][30][31][32][33][34] and references therein). In pursuit of this, we use the shifted Legendre tau method to solve numerically the following FDE:…”