Existence and Iteration of Monotone Positive Solution for a Fourth-Order Nonlinear Boundary Value Problem

Authors : Djourdem Habib, Slimane Benaicha, Noureddine Bouteraa

Pages : 205-211

Doi:10.33401/fujma.418934

View : 18 | Download : 11

Publication Date : 2018-12-25

Article Type : Research

Abstract :This paper is concerned with the following fourth-order three-point boundary value problem BVP \[ u^{\left(4\right)}\left(t\right)=f\left(t,u\left(t\right)\right),\quad t\in\left[0,1\right], \] \[ u'\left(0\right)=u''\left(0\right)=u\left(1\right)=0,\;u'''\left(\eta\right)+\alpha u\left(0\right)=0, \] where $f\in C\left(\left[0,1\right]\times\left[0,+\infty\right),\left[0,+\infty\right)\right)$ , $\alpha\in\left[0,6\right)$ and $\eta\in\left[\frac{2}{3},1\right)$. Although corresponding Green\textquoteright s function is sign-changing, we still obtain the existence of monotone positive solution under some suitable conditions on $f$ by applying iterative method. An example is also given to illustrate the main results.
Keywords : Boundary value problem, Green’s function, Positive solution, Iterative method, Sign-changing