Asymptotic behavior of the solutions of a partial differential equation with piecewise constant argument

In this paper, we study the partial differential equation with piecewise constant argument of the form: x(t)(t, s) = A(t)x(t, s) + B(t, s)x([t], s) + C(t, s)x(t, [s]) + D(t, s)x([t], [s]) + f(x(t, [s])), t, s is an element of IR+ = (0, infinity), where A(t) is a k x k invertible and continuous matrix function on IR+; B(t, s), and D(t, s) are kxk continuous and bounded matrix functions on IR+ x IR+ [t] and [s] are the integral parts of [t] and [s], respectively; and f : IR+ -> IR(+ )is a continuous function. More precisely, under some conditions on the matrices A(t), B(t, s), C(t, s), and D(t, s) and the function f, we investigate the asymptotic behavior of the solutions of the above equation.