Stability Analysis of Impulsive Functional Differential by Ivanka M. Stamova

Stability Analysis of Impulsive Functional Differential by Ivanka M. Stamova

By Ivanka M. Stamova

This e-book is dedicated to impulsive practical differential equations that are a ordinary generalization of impulsive traditional differential equations (without hold up) and of practical differential equations (without impulses). this present day, the qualitative concept of such equations is below speedy improvement. After a presentation of the elemental idea of life, specialty and continuability of suggestions, a scientific improvement of balance idea for that type of difficulties is given which makes the e-book detailed. It addresses to a large viewers resembling mathematicians, utilized researches and practitioners.

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Extra resources for Stability Analysis of Impulsive Functional Differential Equations (De Gruyter Expositions in Mathematics)

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1 will offer Lyapunov stability results. The obtained theorems are parallel to the classical theorems of Lyapunov for ordinary differential equations and show the role of delay and impulses. 2 will deal with boundedness properties for impulsive functional differential equations. By means of piecewise continuous Lyapunov functions coupled with the Razumikhin technique, sufficient conditions for equi-boundedness, uniform boundedness and uniform-ultimate boundedness of the solutions of such equations will be given.

T I t0 ; '0 /k ˇ. 36) 48 2 Lyapunov stability and boundedness (A) ¤ tk , k D 1; 2; : : : . t// is continuous at 0; x. C 0// D V . ; x. ˇ/ and C V . ; x. 16) , we have V . V . ; x. /// > V . ; x. kx. /k/ V . ; x. ˛/; and hence kx. 16) V . ; x. kx. M /º/ D M max¹M; c. 37). (B) D tj for some j 2 ¹1; 2; : : : ; k; : : :º. 14. 16) is uniformly bounded. b. // and ˇ > B. 30) be true. Let us denote Á ; k D 0; 1; 2; : : : ; : k D t0 C k c. ˛/ D Á . 16) are uniformly ultimately bounded. c. 17. t/j Ä M for some constant M > 0; p 2 C ŒRC RC ; RC .

Let D const > 0 be such that b. ˛/. k'0 kr / < b. ˛/; 1 . 3 implies that it is uniformly stable, then the solution x Á 0 is uniformly asymptotically stable. 5. 1) is uniformly asymptotically stable. Proof. 4. 4). ˛/ 1 ln b. ˛/. Let " > 0 and T Then for ' 2 PCŒŒ r; 0;  W k'0 kr < . 1) is uniformly attractive. t//; t D tk ; k D 1; 2; : : : ; where f W Œt0 ; 1/ PCŒŒ r; 0;  ! Rn ; Ik W ! 1 tk D 1. Let '1 2 PCŒŒ r; 0; . 6. 1 37 Lyapunov stability of the solutions (e) uniformly attractive, if the numbers and T in (d) are independent of t0 2 R; (f) asymptotically stable, if it is stable and attractive; (g) uniformly asymptotically stable, if it is uniformly stable and uniformly attractive; (h) unstable, if (a) does not hold.

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