In this paper, by using the contraction mapping principle, we establish the existence and uniqueness of solutions for the fourth order $p$-Laplacian beam equations with Riemann-Stieltjes boundary conditions. Finally, some examples are presented to illustrate the main results.

In this paper, by using the contraction mapping principle, we establish the existence and uniqueness of solutions for the fourth order $p$-Laplacian beam equations $(\Phi_{p}(u’‘(t)))’‘=f(t, u(t), u’(t), u’‘(t))$ with integral boundary conditions $u’‘(0)=u’‘(1)=0,u(0)-\alpha u’(0)= \int_{0}^{1}g_{1}(s)u(s)ds, u(1)+\beta u’(1)= \int_{0}^{1}g_{2}(s)u(s)ds$. The monotony of iterations is also considered. As an extension, we obtain the uniqueness and iterative solutions of the fourth order $p$-Laplacian integral boundary value problems with fully nonlinear term $(\Phi_{p}(u’‘(t)))’‘=f(t, u(t), u’(t), u’‘(t), u”’(t))$. At last, some examples are presented to illustrate the main results.