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Multiple solutions of a boundary value problem on an unbounded domain. (English) Zbl 1108.34024
This paper is concerned with the existence of multiple solutions of the following class of boundary value problems on an unbounded domain
\[ x''(t)-a(t)x(t)+f(t,x(t))=0, \;\;\;t\in[0,\infty), \]
\[ x(0)=x_0,\;\;x(t) \text{ bounded on } [0,\infty), \] with \(x_0\in\mathbb R,\;f:[0,\infty)\times(-\infty,\infty)\to(-\infty,\infty)\) is a continuous function and \(a:[0,\infty)\to(0,\infty)\) is a continuous function such that \(a(t)\geq\alpha^2>0,\;t\geq0,\) where \(\alpha\) is a real constant.
The proofs are based on the method of lower and upper solutions and the theory of topological degree on compact domains. Then, they extend solutions to the unbounded domain with sequential arguments. An example is given. For quantitative contributions on boundary value problems on unbounded domains compare [R. P. Agarwal and D. O’Regan, Infinite interval problems for differential, difference and integral equations. Dordrecht: Kluwer Academic Publishers (2001; Zbl 0988.34002)].

MSC:
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems
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