Upper and Lower Solutions Method for Fourth-Order Periodic Boundary Value Problems

Abstract

Abstract. The purpose of this paper is to prove the existence of a solution of the following periodic boundary value problemin the presence of an upper solution β and a lower solution α with β ≤ α, where f (t, u, v) satisfies one side Lipschitz condition.

Existence and uniqueness results for fourth-order four-point BVP arising in bridge design in the presence of reverse ordered upper and lower solutions

“…For the existence of solutions, method of UL solutions is extensively used to develop MI technique on the second order BVP [16,17,30,33,35,36,37,38,39]. There are also several research articles available on higher order two-point BVP with MI technique [7,8,26,32,34,42]. To the best of our knowledge only few works are there on fourth-order four-point BVP with monotone iterative technique [12,25,31,43].…”

On higher order fully periodic boundary value problems

“…Higher order periodic boundary value problems have been studied by several authors in the last decades, using different types of arguments and techniques, as it can be seen in [1][2][3] for variational methods, in [4][5][6][7][8][9][10][11][12][13][14][15][16][17], for first and higher order equations and in [18][19][20] for a linear or quasi-linear nth order periodic problem. A fully nonlinear differential equation of higher order as in (1) was studied in some works, such as, for instance, [21], for f a bounded and periodic function verifying different assumptions for n even or odd.…”