Differential and Integral Inequalities: Functional Partial, by V. Lakshmikantham

By V. Lakshmikantham

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5) 'I'hus, we obtain for small h <<0 which, in turn, implies that D-x(t,) b D-y(t,). 9. Xi,) G f(t1 ? +, yields x(tl), Yt,). 5). T h u s the set 2 is empty, and the result follows. 1. 3) are replaced by D-JC(t) , f(4Y ( Q Yi). 1. 1). 1). 1) can be bracketed between its under and over functions. 2. 1). Let f~ C [ J x R x V , R] and f ( t , x, +) be nondecreasing in +, for each ( t , x). 1) defined on [ t o ,a), Yt, Then, Y(t)< 3 < do < Z t O . 1. 1). 2. 1) defined on [t, , t, a).

Go. 1). 1. 1) is generalized exponentially asymptotically stable. Suppose further that p ( t ) is continuously differentiable on J . Then, there exists a functional V ( t ,4)satisfying the following properties: (I") V E C[J x C, , R,], and V is Lipschitzian in 4 with the function K(t); (2") I/ ,$ I10 < V(t,$1 < K ( t ) II 4110 t 6 J , 4E c, ; (3") D+V(t,4) < -p'(t) v(t,+), t E J , 4 E C, . 9 Proof. 1), it follows that V ( t , $ )verifies the property (2"). Moreover, D+V(t,4 ) = lim SUP h-O+ h-"Qt + h, Xt+h(4 4)) - V(4 +)I This proves (3").

9. 9) results from the fact that uniformly for t 2 to , lim "(to , do , ~ ) ( t= ) r(t, ,&)(t) <+O T h e following theorem provides an estimate for the difference between a solution and an approximate solution of a functional differential system. 5. Let g E C [ J x R+ x Y , , R+]and g ( t , u, u) be nondecreasing in a for each ( t , u). Assume that f t C [ J x Rn x Yn,R"], and, for t E 1,x,y E R", and 4,$I E %PL, 411 < g(tt I/ " llf(t? 10) 1cI 11). 1 1) x' = f ( t , "0 "1 defined for t 3 to . +, at t == t o , existing for t (7, we have 2 t o .

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