Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions

Abstract

We establish new existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam. We consider the widely studied boundary conditions corresponding to clamped and hinged ends and many non-local boundary conditions, with a unified approach. Our method is to show that each boundary-value problem can be written as the same type of perturbed integral equation, in the space C[0, 1], involving a linear functional α [u] but, although we seek positive solutions, the functional is not assumed to be positive for all positive u. The results are new even for the classic boundary conditions of clamped or hinged ends when α[u] = 0, because we obtain sharp results for the existence of one positive solution; for multiple solutions we seek optimal values of some of the constants that occur in the theory, which allows us to impose weaker assumptions on the nonlinear term than in previous works. Our non-local boundary conditions contain multi-point problems as special cases and, for the first time in fourth-order problems, we allow coefficients of both signs.

Multiple Positive Solutions of Resonant and Non-Resonant Non-Local Fourth-Order Boundary Value Problems

“…In particular, for ω = 0 a larger value has been found in [29] (with a different (s)) by another method.…”

“…Some kind of positivity on the functionals β i is needed in order to have positive solutions, a solution u will satisfy β i [u] ≥ 0 but we do not suppose that β i We use fixed point index theory, based on the methods developed in [26,28]. The fourth-order equation when ω = 0 with a variety of BCs, and with one BC of the non-local type that we study here, has been studied, with similar methods, in detail in [29] in the non-resonant case.…”

Multiple Positive Solutions of Fourth-Order Impulsive Differential Equations with Integral Boundary Conditions and One-Dimensional p-Laplacian

“…To identify a few, we refer the reader to 24-46 and references therein. In particular, we would like to mention some results of Zhang et al 34 , Kang et al 44 , and Webb et al 45 . In 34 , Zhang et al studied the following fourth-order boundary value problem with integral boundary conditions where λ is a positive parameter, f ∈ C 0, 1 × P, P , θ is the zero element of E, and g, h ∈ L 1 0, 1 .…”

Existence of solution for nonlinear fourth-order three-point boundary value problem

“…For Some other results on fourth-order boundary value problem, we refer the reader to the papers ( [10,11,12,13,14]). …”