##
**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.

\[ y^{(2i)}(0)= y^{(2i)}(1)= 0,\quad 0\leq i\leq m-1,\tag{2} \] 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.

Reviewer: I.Rachůnková (Olomouc)

### MSC:

34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

### Keywords:

Lidstone boundary value problem; multiple positive solutions; Green function; concave solutions; Leggett-Williams fixed point theorem
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\textit{J. M. Davis} et al., J. Math. Anal. Appl. 237, No. 2, 710--720 (1999; Zbl 0935.34020)

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### References:

[1] | Agarwal, R. P.; Wong, P. J.Y., Lidstone polynomials and boundary value problems, Comput. Math. Appl., 17, 1397-1421 (1989) · Zbl 0682.65049 |

[2] | D. Anderson, Multiple positive solutions for a three-point boundary value problem, Math. Comput. Modelling, in press.; D. Anderson, Multiple positive solutions for a three-point boundary value problem, Math. Comput. Modelling, in press. · Zbl 0906.34014 |

[3] | Avery, R., Existence of multiple positive solutions to a conjugate boundary value problem, MSR Hotline, 2 (1998) · Zbl 0960.34503 |

[4] | Coddington, E.; Levinson, N., Theory of Ordinary Differential Equations (1955), McGraw-Hill: McGraw-Hill New York · Zbl 0064.33002 |

[5] | J. M. Davis, Differentiability of solutions of differential equations with respect to nonlinear Lidstone boundary conditions, Nonlinear Differential Equations, in press.; J. M. Davis, Differentiability of solutions of differential equations with respect to nonlinear Lidstone boundary conditions, Nonlinear Differential Equations, in press. |

[6] | J. M. Davis, Differentiation of solutions of Lidstone boundary value problems with respect to the boundary data, Math. Comput. Modelling, in press.; J. M. Davis, Differentiation of solutions of Lidstone boundary value problems with respect to the boundary data, Math. Comput. Modelling, in press. · Zbl 0991.34016 |

[7] | J. M. Davis, P. W. Eloe, and, J. Henderson, Comparison of eigenvalues for discrete Lidstone boundary value problems, Dynam. Systems Appl, in press.; J. M. Davis, P. W. Eloe, and, J. Henderson, Comparison of eigenvalues for discrete Lidstone boundary value problems, Dynam. Systems Appl, in press. · Zbl 0941.39009 |

[8] | J. M. Davis, and, J. Henderson, Triple positive symmetric solutions for a Lidstone boundary value problem, Differential Equations Dynam. Systems, in press.; J. M. Davis, and, J. Henderson, Triple positive symmetric solutions for a Lidstone boundary value problem, Differential Equations Dynam. Systems, in press. · Zbl 0981.34014 |

[9] | Davis, J. M.; Henderson, J., Uniqueness implies existence for fourth order Lidstone boundary value problems, Panamer. Math. J., 8, 23-35 (1998) · Zbl 0960.34011 |

[10] | Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag: Springer-Verlag New York · Zbl 0559.47040 |

[11] | P. W. Eloe, J. Henderson, and, B. Thompson, Extremal points for impulsive Lidstone boundary value problems, Math. Comput. Modelling, in press.; P. W. Eloe, J. Henderson, and, B. Thompson, Extremal points for impulsive Lidstone boundary value problems, Math. Comput. Modelling, in press. · Zbl 0963.34022 |

[12] | Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press San Diego · Zbl 0661.47045 |

[13] | J. Henderson, and, H. B. Thompson, Existence of multiple solutions for some \(n\); J. Henderson, and, H. B. Thompson, Existence of multiple solutions for some \(n\) · Zbl 1013.34017 |

[14] | J. Henderson, and, H. B. Thompson, Multiple symmetric solutions for a second order boundary value problem, preprint.; J. Henderson, and, H. B. Thompson, Multiple symmetric solutions for a second order boundary value problem, preprint. · Zbl 0949.34016 |

[15] | T. M. Lamar, Analysis of a \(2n\); T. M. Lamar, Analysis of a \(2n\) |

[16] | T. M. Lamar, Comparison theory for eigenvalue problems with Lidstone boundary conditions, preprint.; T. M. Lamar, Comparison theory for eigenvalue problems with Lidstone boundary conditions, preprint. · Zbl 1332.34041 |

[17] | Leggett, R.; Williams, L., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28, 673-688 (1979) · Zbl 0421.47033 |

[18] | Wong, P. J.Y., Triple positive solutions of conjugate boundary value problems, Comput. Math. Appl., 36, 19-35 (1998) · Zbl 0936.34018 |

[19] | P. J. Y. Wong, and, R. P. Agarwal, Multiple solutions of difference equations with Lidstone conditions, preprint.; P. J. Y. Wong, and, R. P. Agarwal, Multiple solutions of difference equations with Lidstone conditions, preprint. · Zbl 0973.39001 |

[20] | P. J. Y. Wong, and, R. P. Agarwal, Results and estimates on multiple solutions of Lidstone boundary value problems, preprint.; P. J. Y. Wong, and, R. P. Agarwal, Results and estimates on multiple solutions of Lidstone boundary value problems, preprint. · Zbl 0966.34017 |

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