Periodic boundary value problems of first and second order differential equations

Recently the method of upper and lower solutions coupled with the Lyapunov-Schmidt method and monotone iterative techniques has been employed fruitfully to prove theoretical as well as constructive existence results relative to periodic boundary value problems, among others, of first and second order differential equations namely, u'=f(t,u), u(0)=u(2;), and ¸-u“=f(t,u), u(0)=u(2;), u'(0)=u'(2;). These considerations crucially depend upon lower and upper solutions α, β satisfying the relations α•β, α(0)•α(2;), β(0)≥β(2;), and α'(0)≥α'(2;), β'(0)•β'(2;), in addition to other assumptions. The problem of proving the existence results when some or all of the foregoing relations are violated is an interesting and important question. In this paper, we discuss some known results and raise some open questions.