Lower and upper solutions method for a problem of an elastic beam whose one end is simply supported and the other end is sliding clamped

Abstract

In this paper we develop the lower and upper solutions method for the fourth-order boundary value problem of the form    y (4) (x) + (k1 + k2)y ′′ (x) + k1k2y(x) = f (x, y(x)), x ∈ (0, 1), y(0) = y ′ (1) = y ′′ (0) = y ′′′ (1) = 0, which models a statically elastic beam with one of its ends simply supported and the other end clamped by sliding clamps, where k1 < k2 < 0 are the real constants and f : [0, 1] × R → R is a continuous function. The proof of the main result is based on the Schauder fixed point theorem.