Existence of Positivity of the Solutions for Higher Order Three-Point Boundary Value Problems involving p-Laplacian

Authors : Ravi SANKAR, Sreedhar NAMBURİ, Kapula RAJENDRA PRASAD

Pages : 451-459

Doi:10.31197/atnaa.845044

View : 9 | Download : 6

Publication Date : 2022-12-30

Article Type : Research Paper

Abstract :The present study focusses on the existence of positivity of the solutions to the higher order three-point boundary value problems involving $p$-Laplacian $$[\\phi_{p}insert ignore into journalissuearticles values(x^{insert ignore into journalissuearticles values(m);}insert ignore into journalissuearticles values(t););]^{insert ignore into journalissuearticles values(n);}=ginsert ignore into journalissuearticles values(t,xinsert ignore into journalissuearticles values(t););,~~t \\in [0, 1],$$ $$ \\begin{aligned} x^{insert ignore into journalissuearticles values(i);}insert ignore into journalissuearticles values(0);=0, &\\text{~for~} 0\\leq i\\leq m-2,\\\\ x^{insert ignore into journalissuearticles values(m-2);}insert ignore into journalissuearticles values(1);&-\\alpha x^{insert ignore into journalissuearticles values(m-2);}insert ignore into journalissuearticles values(\\xi);=0,\\\\ [\\phi_{p}insert ignore into journalissuearticles values(x^{insert ignore into journalissuearticles values(m);}insert ignore into journalissuearticles values(t););]^{insert ignore into journalissuearticles values(j);}_{\\text {at} ~ t=0}&=0, \\text{~for~} 0\\leq j\\leq n-2,\\\\ [\\phi_{p}insert ignore into journalissuearticles values(x^{insert ignore into journalissuearticles values(m);}insert ignore into journalissuearticles values(t););]^{insert ignore into journalissuearticles values(n-2);}_{\\text {at} ~ t=1}&-\\alpha[\\phi_{p}insert ignore into journalissuearticles values(x^{insert ignore into journalissuearticles values(m);}insert ignore into journalissuearticles values(t););]^{insert ignore into journalissuearticles values(n-2);}_{\\text {at} ~ t=\\xi}=0, \\end{aligned} $$ where $m,n\\geq 3$, $\\xi\\ininsert ignore into journalissuearticles values(0,1);$, $\\alpha\\in insert ignore into journalissuearticles values(0,\\frac{1}{\\xi});$ is a parameter. The approach used by the application of Guo--Krasnosel\`skii fixed point theorem to determine the existence of positivity of the solutions to the problem.
Keywords : three point, nonlinear, boundary value problem, p Laplacian, Green\`s function, positive solution