Extremal solutions and Green's functions of higher order periodic boundary value problems in time scales

In this paper we develop the monotone method in the presence of lower and upper solutions for the problem uΔn(t)+∑lower limit j=1, upper limit n−1 MjuΔj(t)=f<fen><cp type="lpar" STYLE="S">t,u(t)<cp type="rpar" STYLE="S"></fen>, t∈[a,b],uΔi(a)=uΔi<fen><cp type="lpar" STYLE="S">σ(b)<cp type="rpar" STYLE="S"></fen>, i=0,…,n−1. Here <f>f :[a,b]×R→R</f> is such that <f>f(· ,x)</f> is rd-continuous in <f>I</f> for every <f>x∈R</f> and <f>f(t,·)</f> is continuous in <f>R</f> uniformly at <f>t∈I</f>, <f>Mj∈R</f> are given constants and <f>[a,b]=Tκn</f> for an arbitrary bounded time scale <f>T</f>. We obtain sufficient conditions in <f>f</f> to guarantee the existence and approximation of solutions lying between a pair of ordered lower and upper solutions <f>α</f> and <f>β</f>. To this end, given <f>M>0</f>, we study some maximum principles related with operators Tn±[M]u(t)≡uΔn(t)+∑lower limit j=1, upper limit n−1 MjuΔj(t)±Mu(t), in the space of periodic functions. [Copyright &y& Elsevier]