A class of three-point boundary-value problems for second-order impulsive integro-differential equations in Banach spaces

In this paper, we consider the following boundary-value problems for second-order three-point nonlinear impulsive integrodifferential equation of mixed type in a real Banach space x ''(t) + f(t, x(t), x'(t), (Ax)(t), (Bx)(t)) = theta, t is an element of J, t not equal t(k), Delta x vertical bar(t=tk) = I-k(x(t(k))), Delta x'vertical bar(t=tk) = (I) over bar (k) (x(t(k)), x'(t(k))), k = 1, 2, ..., m, x(0) = theta, x(1) = rho x(eta), where theta is the zero element of E, (Ax)(t) = integral(t)(0) g(t,s)x(s)ds, (Bx)(t) = integral(1)(0)h(t, s)x(s)ds, g is an element of C[D, R+], D = {(t, s) is an element of J x J : t >= s}, h is an element of C[J x J, R], and Delta x vertical bar(t=tk) denotes the jump of x(t) at t = t(k), Delta x'vertical bar(t=tk) denotes the jump of x'(t) at t = t(k). Some new results are obtained for the existence and multiplicity of positive solutions of the above problems by using the fixed-point index theory and fixed-point theorem in the cone of strict set contraction operators. Meanwhile, an example is worked out to demonstrate the main results. (C) 2007 Elsevier Ltd. All rights reserved.