Existence and iteration for fourth order $p$-Laplacian beam equations with integral boundary conditions

Abstract

In this paper, we consider the fourth order $p$-Laplacian beam equation $(\Phi_{p}(u''(t)))''=f(t, u(t),u''(t))$ with integral boundary conditions $u''(0)=u''(1)=0,u(0)-\alpha u'(0)= \int_{0}^{1}g_{1}(s)u(s)ds, u(1)+\beta u'(1)= \int_{0}^{1}g_{2}(s)u(s)ds$. By using the contraction mapping principle, we establish the existence and uniqueness of solutions for the problem. The monotony of iterations is also considered. At last, some examples are presented to illustrate the main results.