204Lecture142006

204Lecture142006 - Economics 204 Lecture 14Thursday, August...

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Economics 204 Lecture 14–Thursday, August 17, 2006 Revised 8/17/06, Revisions indicated by ** Diferential Equations Existence and Uniqueness oF Solutions De±nition 1 A diferential equation is an equation of the form y 0 ( t )= F ( y ( t ) ,t ) where F : R n × R R n .An initial value problem is a diFer- ential equation combined with an initial condition y ( t 0 y 0 A solution of the initial value problem is a diFerentiable function y :( a, b ) R n such that t 0 ( a, b ), y ( t 0 y 0 and, for all t ( a, b ), dy dt = F ( y ( t ) ). Theorem 2 Consider the initial value problem y 0 ( t F ( y ( t ) ) ,y ( t 0 y 0 Let U be an open set in R n × R containing ( y 0 0 ) . Suppose F : U R n is continuous. Then the initial value problem has a solution. IF, in addition, F is Lipschitz in y on U , i.e. there is a constant K such that For all ( y, t ) , ) U , | F ( ) F ) |≤ K | y ˆ y | then there is an interval ( a, b ) containing t 0 such that the solution is unique on ( a, b ) . 1
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Proof: We consider only the case in which F is Lipschitz. **Since U is open, we may choose r> 0 such that R = { ( y, t ): | y y 0 |≤ r, | t t 0 r }⊆ U Since F is continuous, we may fnd M R and 0 such that | F ( ) M For all ( ) R . Given the Lipschitz condition, we may assume that | F ( ) F ) K | y ˆ y | For all ( ) , ) R Let δ =m in 1 2 K , r M We claim the initial value problem has a unique solution on ( t 0 δ, t 0 + δ ). Let C be the space oF continuous Functions From [ t 0 0 + δ ] to R n , endowed with the sup norm k f k =sup {| f ( t ) | : t [ t 0 0 + δ ] } Let S = { z C :( z ( s ) ,s ) R For all s [ t 0 0 + δ ] } S is a closed subset oF the complete metric space C ,so S is a complete metric space. Consider the Function I : S C defned by I ( z )( t )= y 0 + Z t t 0 F ( z ( s ) ) ds I ( z ) is defned and continuous because F is bounded on R .Ob - serve that iF ( z ( s ) ) R For all s [ t 0 0 + δ ], then | I ( z )( t ) y 0 | = ± ± ± ± ± Z t t 0 F ( z ( s ) ) ds ± ± ± ± ± ≤| t t 0 | max {| F ( y, s ) | ) R } δM r 2
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so ( I ( z )( t ) ,t ) R for all t [ t 0 δ, t 0 + δ ]. Thus, I : S S .
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204Lecture142006 - Economics 204 Lecture 14Thursday, August...

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