Uniqueness of positive solutions for a class of fourth-order boundary value problems.

*(English)*Zbl 1222.34027Summary: The purpose of this paper is to investigate the existence and uniqueness of positive solutions for the following fourth-order boundary value problem:

\[ y^{(4)}(t) = f(t, y(t)),\quad t \in [0 ,1],\quad y(0) = y(1) = y'(0) = y'(1) = 0. \]

Moreover, under certain assumptions, we prove that the above boundary value problem has a unique symmetric positive solution. Finally, we present some examples and compare our results with the ones obtained in recent papers. Our analysis relies on a fixed-point theorem in partially ordered metric spaces.

\[ y^{(4)}(t) = f(t, y(t)),\quad t \in [0 ,1],\quad y(0) = y(1) = y'(0) = y'(1) = 0. \]

Moreover, under certain assumptions, we prove that the above boundary value problem has a unique symmetric positive solution. Finally, we present some examples and compare our results with the ones obtained in recent papers. Our analysis relies on a fixed-point theorem in partially ordered metric spaces.

##### MSC:

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

47N20 | Applications of operator theory to differential and integral equations |

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\textit{J. Caballero} et al., Abstr. Appl. Anal. 2011, Article ID 543035, 13 p. (2011; Zbl 1222.34027)

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##### References:

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