Extremal solutions and Green’s functions of higher order periodic boundary value problems in time scales. (English) Zbl 1056.39018

Cabada, Alberto

J. Math. Anal. Appl. 290, No. 1, 35-54 (2004).

The author develops a monotone iterative method in the presence of lower and upper solutions for the problem \[ u^{\Delta^n}(t)+\sum_{j=1}^{n-1}M_j u^{\Delta^j}(t)=f(t,u(t)), \quad t\in [a,b]=T^{\kappa^n} \]
\[ u^{\Delta^i}(a)=u^{\Delta^i}(\sigma(b)), \quad i=0,1,\dots,n-1. \] Sufficient conditions are obtained on \(f\) to guarantee the existence and approximation of a solution lying between a pair of ordered lower and upper solutions.

Reviewer: Patricia J. Y. Wong (Singapore)

MSC:

39A12	Discrete version of topics in analysis
34B27	Green’s functions for ordinary differential equations
93C70	Time-scale analysis and singular perturbations in control/observation systems
34B15	Nonlinear boundary value problems for ordinary differential equations

Keywords:

extremal solutions; Green’s function; periodic boundary value problems; time scales; Lower and upper solutions; Monotone iterative technique; Maximum principles

References:

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