This paper deals with the resolvent, asymptotic stability and boundedness of the solution of time-varying Volterra integro-dynamic system on time scales in which the coefficient matrix is not necessarily stable. We generalize to a time scale some known properties about asymptotic behavior and boundedness from the continuous case. Some new results for the discrete case are obtained.

We study S-asymptotically ω-periodic mild solutions of the semilinear Volterra equation u′(t) = (a ∗ Au)(t) + f(t, u(t)), considered in a Banach space X, where A is the generator of an (exponentially) stable resolvent family. In particular, we extend recent results for semilinear fractional integro-differential equations considered in [4] and for semilinear Cauchy problems of first order given ...

The loop equation formalism is used to compute the 1/N expansion of the resolvent for the Gaussian β ensemble up to and including the term at O(N−6). This allows the moments of the eigenvalue density to be computed up to and including the 12-th power and the smoothed density to be expanded up to and including the term at O(N−6). The latter contain non-integrable singularities at the endpoints o...

Abstract We investigate off-diagonal decay properties of the generalized Stokes semigroup with bounded measurable coefficients on $$\mathrm {L}^2_{\sigma } ({\mathbb {R}}^d)$$ L σ 2 ( R <mml:...

Definition 1.1. Let X and Z be Banach spaces and suppose that A and B are two linear operators such that (i) A :D(A)→X, B :D(B)→ Z, with D(A), D(B)⊂X, (ii) D(A) ⊂ D(B) and for some ω ∈ R the resolvent set of A satisfies ρ(A) ⊃ ]ω,∞[. A one-parameter family of linear operators (Y (t))t≥0, which satisfies (1) Y (t) : → Z, with X ⊃ ⊇D(B), and for any t ≥ 0 and f ∈D(B) ∥∥Y (t)f ∥∥ ≤M exp(ωt)‖Bf ‖, ...

The periodic boundary value problem is discussed for a class of fractional evolution equations. The existence and uniqueness results of mild solutions for the associated linear fractional evolution equations are established, and the spectral radius of resolvent operator is accurately estimated. With the aid of the estimation, the existence and uniqueness results of positive mild solutions are o...

We consider the Klein-Gordon equation on a star-shaped network composed of n half-axes connected at their origins. We add a potential which is constant but different on each branch. The corresponding spatial operator is self-adjoint and we state explicit expressions for its resolvent and its resolution of the identity in terms of generalized eigenfunctions. This leads to a generalized Fourier t...