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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.

MSC:
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B20 Weyl theory and its generalizations for ordinary differential equations
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[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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