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Existence of nodal solutions for Lidstone eigenvalue problems. (English) Zbl 1136.34016
The authors study nodal solutions to the Lidstone eigenvalue problem $$(-1)^m u^{(2m)}(x)= ra(x) f(u(x)),\qquad 0< x< 1,$$ $$u^{(2i)}(0)= u^{(2i)}(1)= 0,\qquad i= 0,\dots, m-1.$$ Sufficient conditions are obtained for the existence of solutions with exactly $k-1$ simple zeros in $(0,1)$. These conditions use the eigenvalue $\lambda_k$ of the corresponding linear problem. Illustrative examples are given.

34B15Nonlinear boundary value problems for ODE
Full Text: DOI
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