Existence results and iterative method for a fully fourth-order nonlinear integral boundary value problem

Abstract

We consider the boundary value problem $$ \begin{array}{@{}rcl@{}} u^{\prime\prime\prime\prime}(t)&=&f(t,u(t),u^{\prime}(t),u^{\prime\prime}(t),u^{\prime\prime\prime}(t)), \quad 0<t<1,\\ u^{\prime}(0)&=&u^{\prime\prime}(0)= u^{\prime}(1)=0, u(0)={{\int}_{0}^{1}} g(s)u(s) ds, \end{array} $$ where $f: [0, 1] \times \mathbb {R}^{4} \rightarrow \mathbb {R}^{+},\ a: [0, 1] \rightarrow \mathbb {R}^{+}$ are continuous functions. For f = f(u(t)), very recently in Benaicha and Haddouchi (An. Univ. Vest Timis. Ser. Mat.-Inform. 1(54): 73–86, 2016) the existence of positive solutions was studied by employing the fixed point theory on cones. In this paper, by the method of reducing the boundary value problem to an operator equation for the right-hand sides we establish the existence, uniqueness, and positivity of solution and propose an iterative method on both continuous and discrete levels for finding the solution. We also give error analysis of the discrete approximate solution. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative method.