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2、NKk ),dk.6GX u5, eZ#(J. 31,ILyapunov( 6GX(I)5.kI0 ILyapunovVg9).13!,LX(I)I XX?1 #n.3d:,14! X(I)u5eZ(J. 31,LILyapunov!CLyapunov 1 a ?Kn(,|XX5 6G X(I)5eZ(J. By,312! 0 (h0,eh)5.13!|ICLyapunov I C?n.3nA,Ln,X(I) )X)LX)X5.14!3 n:,X(I)u5eZ(J. A5. 31n,LCLyapunov5 RazumikhinE|(, Razumikhin.CLyapunov .| X(I)u
3、5(J. 9?OKf X3:?.? f5?y(Jk?5. c:6G,X,(h0,h),ILyapunov ,CLyapunov,RazumikhinE|. a: O175.21 2 a Stability Analysis for Impulsive Integro-diff erential Systems with Variable Times Zhao Yan School of Mathematical Sciences, Shandong Normal University Jinan, Shandong, 250014, P. R. China ABSTRACT In this p
4、aper, we consider the stability properties about the following impulsive integro- diff erential equations with variable times x0= f(t, x,Tx), t , k(x), 4x = Ik(x),t = k(x), x(t+ 0) = x0, k = 1,2,3, , (I) where f(t, x,Tx) = F(t, x) + R(t, x,Tx),Tx = R t t0 L(t, s, x(s)ds,L C(R2 + Rn,Rn). Every solu-
5、tion of it meet each hypersurface in turn exactly once. Impulsive integro-diff erential system have extensive practical background and application in the natural sciences,as the systems extensively occur in the mathematical modeling of cir- cuit simulation in physics and neuronal networks in biology
6、.The research of impulsive integro- diff erential system has aroused experts interest and attention,and various interesting results have been obtained in the past years112,19 ,especially for impulsive integro-diff erential system with fi xed times110.For example,article1,3,4studied boundedness of so
7、lutions of this system and got some direct results.However, impulsive integro-diff erential equations with variable times as an extension of systems with fi xed times have more application.But the theory of impulsive integro-diff erential systems with variable times is relatively less developed due
8、to the diffi culties created by the phenomena of beating and bifurcation etc.Up to now, the results about impul- sive systems with variable times are really few1012 .In the fi ndings appeared,article 10 gave one existence result of solutions,and article 11 gave several criteria for asymptotic stabil
9、ity of this system from which the impulsive eff ects on the stability are displayed in the result ob- tained.However,the study of stability of this system is in underway phase,and there are many problems which are not solved.Therefore,we have a large number of work to do.In this paper we study the p
10、roperties of stability of impulsive integro-diff erential system with variable times,and we get some new results. In chapter one,using the method of cone-valued Lyapunov functions and comparison prin- 3 a ciple,we discuss the property of impulsive integro-diff erential system with variable times.we
11、fi rstly give the defi nition of the cone,and furthermore we introduce the conception of cone-valued Lyapunov functions and its derivative along the solution of system (I).In section 3 of this chap- ter,we established a new comparison principle by comparing with a scalar diff erential system on cone
12、.Then base on the comparison theorem,we get several stability criteria in terms of two measures for system (I) in section 4. In chapter two,by employing the method of cone-valued Lyapunov functions,variational Lyapunov functions and comparison principle,several stability criteria for the impulsive p
13、er- turbed integro-diff erential systems (I) are established through the corresponding unperturbed diff erential systems and the comparison systems.In section 2 of this chapter,we introduce the defi nition of (h0,eh)stability.In section 3,the cone-valued variational comparison principle of system (I) is obtained by cone-valued variational Lyapunov functions.In corollaries,by taking the relevant function of comparison theorem,the sol
