Well-Posedness for Fractional Cauchy Problems Involving Discrete Convolution Operators

This work focused on establishing sufficient conditions to guarantee the well-posedness of the following nonlinear fractional semidiscrete model: {D-t(beta) u(n, t) = Bu(n, t) + f(n - ct, u(n, t)), n is an element of Z, t > 0, u(n, 0) = phi (n), n is an element of Z, under the assumptions that beta is an element of (0, 1], c > 0 some constant, and B is a discrete convolution operator with kernel b is an element of l(1) (Z), which is the infinitesimal generator of the Markovian C-0-semigroup and suitable nonlinearity f. We present results concerning the existence and uniqueness of solutions, as well as establishing a comparison principle of solutions according to the respective initial values.