Nonlinear boundary value problems on time scales (Q5937342)

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., NEWLINE\[NEWLINE y^{\Delta\Delta}(t)+f(t,y(\sigma(t)))=0, \quad t\in[a,b]\cap {\mathbf T}, NEWLINE\]NEWLINE subject to the boundary conditions NEWLINE\[NEWLINE y(a)=0, \quad y^\Delta(\sigma(b))=0. NEWLINE\]NEWLINE 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 \textit{L.~Erbe} and \textit{A.~Peterson} [Math. Comput. Modelling 32, No.~5-6, 571---585 (2000; Zbl 0963.34020)], and are also closely related to results by \textit{C.~J.~Chyan, J.~Henderson} and \textit{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).