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Multiple positive solutions for nonlinear dynamical systems on a measure chain. (English) Zbl 1045.39007

Authors’ summary: We consider the following dynamical system on a measure chain: \[ u_1^{\Delta\Delta}(t)+f_1(t,u_1(\sigma(t)),u_2(\sigma(t)))=0, \quad t\in [a,b], \]
\[ u_2^{\Delta\Delta}(t)+f_2(t,u_1(\sigma(t)),u_2(\sigma(t)))=0, \quad t\in [a,b], \] with the Sturm-Liouville boundary value conditions \[ \alpha u_i(a)-\beta u_i^{\Delta}(a)=0, \gamma u_i(\sigma(b))+\delta u_i^\Delta(\sigma(t))=0, \;\text{ for}\;i=1,2. \] Some results are obtained for the existence of three positive solutions of the above problem by using Leggett-Williams fixed point theorem.

MSC:

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
34B24 Sturm-Liouville theory
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