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).


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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