Two Positive Solutions for a Fourth-Order Three-Point BVP with Sign-Changing Green`s Function

Authors : Habib DJOURDEM, Slimane BENAİCHA, Noureddine BOUTERAA

Pages : 60-68

Doi:10.33434/cams.452839

View : 43 | Download : 10

Publication Date : 2019-03-22

Article Type : Research Paper

Abstract :This paper concerns the fourth-order three-point boundary value problem insert ignore into journalissuearticles values(BVP); \[ u^{\leftinsert ignore into journalissuearticles values(4\right);}\leftinsert ignore into journalissuearticles values(t\right);=f\leftinsert ignore into journalissuearticles values(t,u\leftinsert ignore into journalissuearticles values(t\right);\right);,\quad t\in\left[0,1\right], \] \[ u`\leftinsert ignore into journalissuearticles values(0\right);=u``\leftinsert ignore into journalissuearticles values(0\right);=u\leftinsert ignore into journalissuearticles values(1\right);=0,\;\alpha u``\leftinsert ignore into journalissuearticles values(1\right);-u```\leftinsert ignore into journalissuearticles values(\eta\right);=0, \] where $f\in C\leftinsert ignore into journalissuearticles values(\left[0,1\right]\times\left[0,+\infty\right);,\left[0,+\infty\right);\right);$, $\alpha\in\left[0,1\right);$ and $\eta\in\left[\frac{2\alpha+10}{15-2\alpha},1\right);$. Although the corresponding Green\textquoteright s function is sign-changing, we still obtain the existence of at least two positive and decreasing solutions under some suitable conditions on $f$ by applying the two-fixed-point theorem due to Avery and Henderson. An example is also given to illustrate the main results.
Keywords : two positive solutions, Completely continuous, fourth order boundary value problem, Green extquoteright s function, two positive solutions

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