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Nonnegative solutions to superlinear problems of generalized Gelfand type. (English) Zbl 0831.34029

The author considers the existence of nonnegative solutions of the boundary value problems \(y'' + \mu q(t) g(t,y) = 0\), \(\mu \geq 0\), a) \(y(0) = a \geq 0\), \(y(T) = b \geq a\) or b) \(y(0) = a \geq 0\), \(y\) bounded on \([0, \infty)\) or c) \(y(0) = a \geq 0\), \(\lim y (t)\) for \(t \to \infty\) exists. The main assumptions are the following: \(q(t) > 0\) continuous on \((0,T)\), \(g(t,y)\) continuous, nonnegative on \([0,T] \times [a, \infty)\) or on \([0, \infty) \times [a, \infty)\), respectively, \(f : [0, \infty) \to [0, \infty)\) continuous nondecreasing, \(f(u) > 0\) for \(u > a\) and \(g(t,u) \leq f(u)\). The proofs are made via fixed point theorem (nonlinear alternative).

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
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