Abstract
In this paper, a fixed point theorem in a cone and some inequalities of the associated Green’s function are applied to obtain the existence of positive solutions of second-order three-point boundary value problem with dependence on the first-order derivative \t\t\tx″(t)+f(t,x(t),x′(t))=0,0<t<1,x(0)=0,x(1)=μx(η),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& x''(t) + f\\bigl(t, x(t), x'(t)\\bigr) =0, \\quad 0< t< 1, \\\\& x(0) =0, \\qquad x(1) =\\mu x(\\eta ), \\end{aligned}$$ \\end{document} where f: [0, 1] times [0, infty ) times R rightarrow [0, infty ) is a continuous function, mu >0, eta in (0, 1), mu eta <1. The interesting point is that the nonlinear term is dependent on the convection term.
Highlights
In recent years, there has been much attention focused on questions of solutions of twopoint, three-point, multi-point, and integral boundary value problems for nonlinear ordinary differential equations and fractional differential equations
The interesting point is that the nonlinear term is dependent on the convection term
1 Introduction In recent years, there has been much attention focused on questions of solutions of twopoint, three-point, multi-point, and integral boundary value problems for nonlinear ordinary differential equations and fractional differential equations
Summary
There has been much attention focused on questions of solutions of twopoint, three-point, multi-point, and integral boundary value problems for nonlinear ordinary differential equations and fractional differential equations. Krasnosel’skii’s fixed point theorem in a cone [18] cannot concretely solve problems whose. Via a generalization of Krasnosel’skii’s fixed point theorem in a cone [5] and some inequalities of the associated Green’s function for the associated problem, the existence of positive solutions for the second-order three-point boundary value problem is studied x (t) + f t, x(t), x (t) = 0, 0 < t < 1,. Green’s function of the following boundary value problem:. If x = x(t) satisfies the operator equation x(t) = (Tx)(t) := G(t, s)f s, x(s), x (s) ds, 0 ≤ t ≤ 1, where G(t, s) ≥ 0 is Green’s function for boundary value problem (2.3), (2.4), x = x(t) is the solution of problem (1.1), (1.2).
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