Two positive solutions for nonlinear fourth-order elastic beam equations

The aim of this paper is to study the existence of at least two non-trivial solutions to a boundary value problem for fourth-order elastic beam equations given by u (4) + Au00 + Bu = λ f(x, u) in [0, 1], u(0) = u(1) = 0, u 00(0) = u 00(1) = 0, under suitable conditions on the nonlinear term on the right hand side. Our approach is based on variational methods, and in particular, on an abstract two critical points theorem given for differentiable functionals defined on a real Banach space

Positive Solutions of a Singular Positone and Semipositone Boundary Value Problems for Fourth-Order Difference Equations

This paper studies the boundary value problems for the fourth-order nonlinear singular difference equations Δ4u(i−2)=λα(i)f(i,u(i)), i∈[2,T+2], u(0)=u(1)=0, u(T+3)=u(T+4)=0. We show the existence of positive solutions for positone and semipositone type. The nonlinear term may be singular. Two examples are also given to illustrate the main results. The arguments are based upon fixed point theorems in a cone