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Monotone iterative technique and positive solutions of lidstone boundary value problems. (English) Zbl 1049.34028

Consider the boundary value problem \[ (-1)^n w^{(2n)} (t) = f(t,w(t)), \quad 0 \leq t \leq 1,\quad w^{(2i)} (0) = w^{(2i)} (1) = 0, \quad 0 \leq i \leq n-1, \tag{\(*\)} \] under the conditions (i) \(f:[0,1] \times [0,+\infty) \rightarrow [0,+\infty)\) is continuous. (ii) \(f(t,.)\) is nondecreasing for any \(t \in [0,1].\) (iii) \(f(t,w)=f(1-t,w)\) for all \(t\in [0,1], w\in [0,\infty).\) A solution of \((*)\) is said to be symmetric, if \(w(t)=w(1-t)\). By using the monotone iterative technique, the author proves the existence of \(N\) symmetric positive solutions of \((*)\).

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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