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Zbl 0995.34016
Agarwal, Ravi P.; O'Regan, Donal
Nonlinear boundary value problems on time scales.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 44, No.4, A, 527-535 (2001). ISSN 0362-546X

This 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) ${\bold T}$, i.e., $$y^{\Delta\Delta}(t)+f(t,y(\sigma(t)))=0, \quad t\in[a,b]\cap {\bold T},$$ 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 (${\bold T}={\bbfR}$) and difference (${\bold T}={\bbfZ}$) equations theories. The results extend the ones by {\it L.~Erbe} and {\it A.~Peterson} [Math. Comput. Modelling 32, No.~5-6, 571---585 (2000; Zbl 0963.34020)], and are also closely related to results by {\it C.~J.~Chyan, J.~Henderson} and {\it 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).
[Roman Hilscher (East Lansing)]
MSC 2000:
*34B18 Positive solutions of nonlinear boundary value problems
34B45 Boundary value problems on graphs and networks
39A99 Difference equations

Keywords: measure chain (time scale); dynamic equation; nonnegative solution; fixed-point theorem

Citations: Zbl 0963.34020; Zbl 0995.34017

Cited in: Zbl 0995.34017

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