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Nonlinear boundary value problems on time scales. (English) Zbl 0995.34016
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) \({\mathbf T}\), 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).

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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