Comparison Principle and Non Linear Contract in Abstract Spaces
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The theory of differential and integral equations exploits comparison and iterative techniques which do not fall under the Contractive Mapping Principle. For they make use of partial orderings and maximal solutions; concepts which have no significance in a metric space. These methods take their proper place in the theory of cones , . In this paper, Banach's Contraction Mapping Principle and comparison and iterative methods are brought together under a single roof which houses various results from the theory of differential equations , in addition to an interesting generalization of Banach's Theorem . One 'It thus led, in a natural way to a generalized Gronwell-Reid-Bellman Inequality and a discussion of nonlinear contractions of a space whose open neighborhoods are conic segments. The advantage of such a uniform principle is two-fold. First, by means of it, one is able to present many different results in one stroke while focusing more clearly on the basic ideas involved; second, one is able to isolate concepts widely used in the theory of differential equations for further application to the general theory of nonlinear analysis.