Existence of asymptotically almost periodic solutions for some second-order hyperbolic integrodifferential equations (Q2227623)

The authors study the following hyperbolic integro-differential equation in a separable Banach space \(H\): \[ \frac{d^2u}{dt^2} + A^2u - \int_{-\infty}^t g(t-s) A^2u(s) ds = f(t,u),\quad t >0, \] with initial conditions \[ u(-t) = u_0(t), t \geq 0,\quad u^\prime (0) = u_1, \] where \(A \colon D(A) \subset H \to H\) is a positive bounded below self-adjoint operator and the function \(f \colon [0,\infty) \times H \to H\) is asymptotically almost periodic in the first variable uniformly in the second one. Under some suitable assumptions on the kernel function \(g \colon [0,\infty) \to [0,\infty)\) and supposing that the function \(f\) is Lipschitz in the second variable with a sufficiently small constant, the authors transform the above system to the Cauchy problem for a first-order semilinear diffential equation in a Hilbert space whose linear part generates a \(C_0\)-semigroup of contractions and a nonlinearity is Lipschitz with the same constant. The application of the Banach contraction map principle yields the existence of an asymptotically almost periodic mild solution to above problem.