## 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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### References:

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