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Triple positive solutions and dependence on higher order derivatives. (English) Zbl 0935.34020

The authors consider the Lidstone boundary value problem \[ y^{(2m)}(t)= f(y(t),\dots, y^{(2j)}(t),\dots,y^{(2(m- 1))}(t)),\quad 0\leq t\leq 1,\tag{1} \]
\[ y^{(2i)}(0)= y^{(2i)}(1)= 0,\quad 0\leq i\leq m-1,\tag{2} \] where \((-1)^mf> 0\) is continuous.
They impose growth conditions on \(f\) and make use of inequalities involving an associated Green function which will ensure the existence of at least three positive symmetric concave solutions to (1), (2). In constrat to earlier works devoted to problems of this type, \(f\) may depend on higher-order derivatives. The proofs are based on the Leggett-Williams fixed point theorem which deals with fixed points of a cone-preserving operator defined on an ordered Banach space. A crucial feature in the construction of an appropriate cone is an inequality that gives a lower bound on positive concave functions as a function of their norm. The validity of an analogous assertion for \(f\) depending explicitly on \(t\) is claimed in the final remark.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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References:

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