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Boundary value problems for second-order differential equations on unbounded domains in a Banach space. (English) Zbl 1035.34015
The author studies the following two-point boundary value problem for a second-order nonlinear differential equation in a Banach space \(X\) \[ \frac{d^2 x}{dt^2}=f \biggl(t, x(t), \frac{d x(t)}{dt}\biggr),\quad t\geq 0, \qquad x(t)=x_0, \quad \frac{d x(\infty)}{dt}=y_\infty, \] where \(x_0, y_\infty\in X\) are given vectors, and \(f: [0,\infty)\times X\times X\to X\) is a given continuous function. By virtue of the Sadovskii fixed-point theorem, the existence of solutions is investigated. Besides, the Lipschitz condition for \(f\) is not required.

34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B20 Weyl theory and its generalizations for ordinary differential equations
Full Text: DOI
[1] Dajun, G., Boundary value problems for impulsive integro-differential equation on unbounded domains in a Banach space, Appl. math. comput., 99, 1-15, (1999) · Zbl 0929.34058
[2] Lakshmikantham, V.; Leela, S., Nonlinear differential equations in abstract space, (1981), Pergamon Press Oxford · Zbl 0456.34002
[3] Deimling, K., Nonlinear functional analysis, (1985), Springer Berlin · Zbl 0559.47040
[4] Dajun, G.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press New York · Zbl 0661.47045
[5] Dajun, G.; Lakshmikantham, V.; Liu, X., Nonlinear integral equations in abstract spaces, (1996), Kluwer Academic Publishers Dordrecht · Zbl 0866.45004
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