On the existence of periodic solution of some quasilinear differential equations with impulses

As for ordinary differential equations, one of the problems that especially attract the attention of many mathematicians is the problem on the existence of periodic solutions of the differential equation systems with impulses. In this paper, we study the periodic solutions of the equation system under of the form. | Vietnam Journal of Mechanics , VAST , Vol. 26 , 2004, No. 1 (55 - 64) ON THE EXISTENCE OF PERIODIC SOLUTION OF SOME QUASILINEAR DIFFERENTIAL EQUATIONS WITH IMPULSES LE LUONG TAI Thai Nguyen University 1 Introduction As for ordinary differential equations, one of the problems that especially attract the attention of many mathematicians is the problem on the existence of periodic solutions of the differential equation systems with impulses. In this paper , we study the periodic solutions of the equation system under of the form: + F(t , x), t-::/= Ti, = Bi(x)x + ci(x), ± = A(x, t) x () ~xlt=r; () Bi+p(x) = Bi(x), Ci+p = ci(x), p - integers, i = 1, 2, . , () where A(t, x), Bi(x) are n x n-matrices, F(t, x), ci(x) are n-vector, the elements of which are continuous with respect to its variables (piecewise continuous with the first kind discontinuities in t at t = Ti) and T -periodic with respect to t. In the case of ordinary differential equations, this problem was discussed by A. G. Kartsatos [1], S. Saito and M. Yamamoto [2]. Here this result will be extended to the case of ordinary differential equations with impulses. Together with the system () and (), we consider the homogeneous linear differential equation system with impulses ± = Ao(t)x, () t-::/= Ti, ~x ft=r; =Bf x , Bf+P =Bf , i = 1, 2, . , () where A 0 (t) is a continuous T-periodic int matrix; Bf are the constant matrices. The following symbols are introduced. We denote by I ·II a norm of a vector or matrix, · and by the symbol 11 · llo a supremum norm of vector or matrix function respectively. For example, for the vector function