Existence of Extremal Solutions and Comparison Results for Delay Differential Equations in Abstract Cones
Laughn, R. L.
Vatsala, A. S.
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The study of Cauchy problem for ordinary differential equations in a Banach space has been extensive [2,4,10]. It is of interest to look at the corresponding problem for delay differential equations since such equations occur in many physical problems. Existence of solutions of such equations are considered in [6,8,9] using monotonicity conditions and dissipative conditions.In this paper our objective is to prove the existence of extremal solutions for the delay differential equation [see pdf for notation] (1.1) relative to a cone k of the Banach space E. For this purpose, we begin by proving an existence result under a simple set of conditions without assuming uniform continuity on f we then develop needed theory of differential inequalities and utilize this to show the existence of extremal solutions for (1.1) Several useful comparison theorems are then proved including a flow invariance result. Our results generalize some of the recent results for equations without delay [5,7].