By V. Lakshmikantham
Read or Download Differential and Integral Inequalities: Functional Partial, Abstract and Complex Differential Equations v. 2: Theory and Applications: Functional PDF
Best mathematics books
This can be the main complete survey of the mathematical lifetime of the mythical Paul Erd? s, some of the most flexible and prolific mathematicians of our time. For the 1st time, the entire major parts of Erd? s' examine are lined in one undertaking. due to overwhelming reaction from the mathematical group, the undertaking now occupies over 900 pages, prepared into volumes.
- Dynamical systems 08: Singularity theory II
- Trigonometric Delights (Princeton Science Library)
- Mathematical Problems and Proofs: Combinatorics, Number Theory, and Geometry
- Seminaire Bourbaki 1973-1974 exposes 436-452
- Six Degrees: The Science of a Connected Age
- M-Solid Varieties of Algebras (Advances in Mathematics)
Additional info for Differential and Integral Inequalities: Functional Partial, Abstract and Complex Differential Equations v. 2: Theory and Applications: Functional
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)
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 .