Nonlinear boundary value problems on time scales.

*(English)*Zbl 0995.34016This paper contains several results on the existence of one or two nonnegative solutions to the nonlinear second-order dynamic equation on a measure chain (time scale) \({\mathbf T}\), i.e.,
\[
y^{\Delta\Delta}(t)+f(t,y(\sigma(t)))=0, \quad t\in[a,b]\cap {\mathbf T},
\]
subject to the boundary conditions
\[
y(a)=0, \quad y^\Delta(\sigma(b))=0.
\]
The theory of dynamic equations on measure chains unifies and extends the differential (\({\mathbf T}={\mathbb{R}}\)) and difference (\({\mathbf T}={\mathbb{Z}}\)) equations theories. The results extend the ones by L. Erbe and A. Peterson [Math. Comput. Modelling 32, No. 5-6, 571—585 (2000; Zbl 0963.34020)], and are also closely related to results by C. J. Chyan, J. Henderson and H. C. Lo [Tamkang J. Math. 30, No. 3, 231-240 (1999; Zbl 0995.34017)]. The proofs are based on an application of a nonlinear alternative of Leray-Schauder-type or Krasnoselskii fixed-point theorems. The paper will be useful for researchers interested in nonnegative solutions to differential or difference equations, and/or in dynamic equations on measure chains (time scales).

Reviewer: Roman Hilscher (East Lansing)

##### MSC:

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

34B45 | Boundary value problems on graphs and networks for ordinary differential equations |

39A99 | Difference equations |

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\textit{R. P. Agarwal} and \textit{D. O'Regan}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 44, No. 4, 527--535 (2001; Zbl 0995.34016)

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##### References:

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