Existence of solutions for a class of third-order nonlinear boundary value problems. (English) Zbl 1053.34017

Du, Zengji; Ge, Weigao; Lin, Xiaojie

J. Math. Anal. Appl. 294, No. 1, 104-112 (2004).

The considered scalar nonlinear boundary value problem has the form \[ \begin{gathered} x'''(t)+ f(t,x(t), x'(t), x''(t))= 0,\\ x(0)= 0,\quad g(x'(0), x''(0))= A,\quad h\bigl(x'(1),\;x''(1)\bigr)= B.\end{gathered} \] The functions \(f\), \(g\), \(h\) are continuous. It is assumed that \(f\) and \(h\) are increasing in their second arguments, and \(g\) is decreasing in its second argument. Additionally, the Nagumo condition is imposed. By means of lower and upper solutions, sufficient conditions for the solvability are formulated. The proof of the existence result employs the Leray-Schauder degree. The result is illustrated by an example, where the differential equation has quadratic growth in \(x\) and \(x'\).

Reviewer: Sergei A. Brykalov (Ekaterinburg)

MSC:

34B15

Nonlinear boundary value problems for ordinary differential equations

Keywords:

nonlinear boundary value problem; Nagumo condition; upper and lower solutions; Leray-Schauder degree; nonlinear boundary conditions

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