On the Theory of Terminal Value Problems for Ordinary Differential Equations
Aftabizadeh, A. R.
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It is well known  that the comparison principle for initial value problems of ordinary differential equations is a very useful tool in the study of qualitative and quantitative theory. Recently, attempts have been made to study the corresponding comparison principle for terminal value problems (TVP for short), [1,3,4]. Since the theory related to TVP's is much more complicated than that of initial value problems, the development of the theory corresponding to these problems has met with difficulties. For example, one of the difficulties arises because the existence of a solution of the TVP [see pdf for notation] need not imply that the TVP [see pdf for notation] has a solution. In this paper, we shall first discuss the theory of terminal differential inequalities, existence of extremal solutions of TVP's and the corresponding comparison principle. Our approach is parallel to that of initial value problems and the appropriate modifications incorporated appear as a natural setting for this theory. We shall also discuss existence of solutions of TVP in a sector extending the method of upper and lower solutions and develop monotone iterative techniques to obtain multiple solutions of TVP in a sector.