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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) ${\bold T}$, 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).

34B18Positive solutions of nonlinear boundary value problems for ODE
34B45Boundary value problems for ODE on graphs and networks
39A99Difference equations
Full Text: DOI
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