Method of lower and upper solutions for a generalized boundary value problem. (English) Zbl 1090.34520

Rudolf, Boris

Arch. Math., Brno 36, Suppl., 595-602 (2000).

The author considers the scalar regular boundary value problem \[ x''=f(t,x,x'), \tag{1} \]
\[ x(a)=\int _a^bx(t)\,dg_1(t)+k_1x'(a), \tag{2} \]
\[ x(b)=\int _a^bx(t)\,dg_2(t)-k_2x'(a), \tag{3} \] where \(f \in C^0([a,b] \times \mathbb R^2)\), \(g_i\) are nondecreasing with bounded variation, \(1 \geq g_i(b)-g_i(a)\) and \(k_i \geq 0\), \(i=1,2\). By the metod of lower and upper solutions for problem (1)–(3) and by the topological degree, sufficient conditions for the solvability of problem (1)–(3) are given.

Reviewer: Svatoslav Staněk (Olomouc)

MSC:

34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations

Keywords:

boundary value problem; existence; lower and upper solutions; Leray-Schauder degree