Posner, E. Xin, X.
Partially Ordered Sets
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Discrete Mathematics - Sets
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On derivations of partially ordered sets : Mathematica Slovaca
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In this paper we investigate the existence of periodic solutions to delay differential equations in partially ordered metric spaces. More precisely, we consider the following equations:. A solution of 1. It is well known from the theory of delay differential equations see, for example, [ 1 ] that 1. This type of problems was already investigated for delay or ordinary differential equations in real-valued spaces by Drici et al.
The papers cited therein provide additional reading on this topic.
The aim of this paper is to extend the fixed point results of contraction mappings in vectorial partially ordered sets by using vectorial norms introduced by Agarwal [ 12 , 13 ] which will allow us to investigate the existence of periodic solutions of vectorial delay differential equations. For more on fixed point theory, the reader may consult the books [ 14 , 15 ]. To be able to prove such results, we will need some monotonicity result of the flow. The main difficulty encountered comes from the fact that we are not working on a classical normed vector space.
In fact, consider the example of the evolution of a burning zone.
The velocity of its evolution is not the same if the burning area is narrow or if it is wide. In this case we cannot study the trajectory of one point without taking into account the others. Therefore we need to consider mappings defined on the subsets of R n which have nice properties with respect to a metric distance other than the regular norm. The first version of the classical Banach contraction principle in partially ordered metric spaces was given by Ran and Reurings [ 16 ]. Their result was used in applications to linear and nonlinear matrix equations.
Theorem 2. Assume that one of the following conditions holds :. Then f has a fixed point. Clearly, the vector linear space C [ a , b ] , R n is partially ordered.
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We have. Obviously, we have. Clearly, one may define a norm on C [ a , b ] , R n to make it a normed vector space. But it seems that a vector-valued norm will be useful in our applications to delay differential equations.
Then we have. This concept is not new.