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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) 𝐓, i.e.,

y ΔΔ (t)+f(t,y(σ(t)))=0,t[a,b]𝐓,

subject to the boundary conditions

y(a)=0,y Δ (σ(b))=0·

The theory of dynamic equations on measure chains unifies and extends the differential (𝐓=) and difference (𝐓=) 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:
34B18Positive solutions of nonlinear boundary value problems for ODE
34B45Boundary value problems for ODE on graphs and networks
39A99Difference equations