## 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

### Citations:

Zbl 0963.34020; Zbl 0995.34017
Full Text:

### References:

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