Nonlinear boundary value problems on time scales. (English) Zbl 0995.34016

Agarwal, Ravi P.; O’Regan, Donal

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 44, No. 4, 527-535 (2001).

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).

Reviewer: Roman Hilscher (East Lansing)

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

Keywords:

measure chain (time scale); dynamic equation; nonnegative solution; fixed-point theorem

Citations:

Zbl 0963.34020; Zbl 0995.34017

References:

[1]	Agarwal, R. P.; Bohner, M., Basic calculus on time scales and some of its applications, Results Math., 35, 3-22 (1999) · Zbl 0927.39003
[4]	Erbe, L. H.; Peterson, A., Green’s functions and comparison theorems for differential equations on measure chains, Dynamics Continuous, Discrete Impulsive Systems, 6, 121-138 (1999)
[6]	Hilgar, S., Analysis on measure chains – a unified approach to continuous and discrete calculus, Results Math., 18, 18-56 (1990)