EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR NONLINEAR IMPULSIVE DIFFERENTIAL EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS

In this paper, we aim to study differential equations (x)over dot(t) = f(t, x(t)), t is an element of[0,T], t not equal t(i), i = 1,2, ..., p, with nonlocal boundary conditions Ax(0) + integral(T)(0)n(t)x(t)dt = B, and subject to impulsive conditions x(t(i)(+)) - x(t(i)) = I-i(x(t(i))), i = 1,2, ..., p, where 0 = t(0) < t(1) < ... < t(p) < t(p+1) = T, A is an element of R-nxn, n(t) is an element of R-nxn known matrices such that det N not equal 0, N = A + integral(T)(0)n(t)dt; f : [0,T] x R-n -> R-n and I-i : R-n -> R-n are given functions; Delta x(t(i)) = x(t(i)(+)) - x(t(i)(-)), where x(t(i)(+)) = lim(h -> 0+) x(t(i) + h), x(t(i)(-)) = lim(h -> 0+) x(t(i) - h) = x(t(i)) are right- and left-hand limits of x(t) at t = t(i), respectively. The Green function is constructed and the considered problem is reduced to an equivalent integral equation. The existence and uniqueness of the solutions for the given problem are analyzed using the Banach contraction principle. The Schaefer fixed point theorem is then used to prove the existence of the solutions. The continuous dependence of the solutions on the right side of the boundary conditions is also established.