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

Authors : Djourdem HABİB, Slimane BENAİCHA, Noureddine BOUTERAA

Pages : 205-211

Doi:10.33401/fujma.418934

View : 15 | Download : 8

Publication Date : 2018-12-25

Article Type : Research Paper

Abstract :This paper is concerned with the following fourth-order three-point boundary value problem BVP \[ u^{\leftinsert ignore into journalissuearticles values(4\right);}\leftinsert ignore into journalissuearticles values(t\right);=f\leftinsert ignore into journalissuearticles values(t,u\leftinsert ignore into journalissuearticles values(t\right);\right);,\quad t\in\left[0,1\right], \] \[ u`\leftinsert ignore into journalissuearticles values(0\right);=u``\leftinsert ignore into journalissuearticles values(0\right);=u\leftinsert ignore into journalissuearticles values(1\right);=0,\;u```\leftinsert ignore into journalissuearticles values(\eta\right);+\alpha u\leftinsert ignore into journalissuearticles values(0\right);=0, \] where $f\in C\leftinsert ignore into journalissuearticles values(\left[0,1\right]\times\left[0,+\infty\right);,\left[0,+\infty\right);\right);$ , $\alpha\in\left[0,6\right);$ 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, Greens function, Positive solution, Iterative method, Sign changing