Volume 8, Issue 4

In this paper we consider a plate equation with infinite memory in the presence of nonlinear feedbacks with and without delay. Under suitable condition on the weight of the delayed feedback compared with the weight of the non-delayed feedback, we use the energy method to establish an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity.

The main goal of this paper is to develop a potential theoretical approach to study the Dunkl Laplacian Δk{\Delta_{k}}, which is a standard example of differential-difference operators. Introducing the Green kernel relative to Δk{\Delta_{k}}, we prove that the Dunkl Laplacian generates a Balayage space and we investigate the associated family of harmonic measures. Therefore, by means of harmonic kernels, we give a characterization of all Δk{\Delta_{k}}-harmonic functions on a large class of open subsets U of ℝd{\mathbb{R}^{d}}. We also establish existence and uniqueness results of a solution of the corresponding Dirichlet problem.

Let A be the generator of a strongly continuous cosine family (cos⁡(t⁢A))t∈ℝ{(\cos(tA))_{t\in\mathbb{R}}} on a complex Banach space E . The paper develops an operational calculus for integral transforms and functions of A using the generalized harmonic analysis associated to certain hypergroups. It is shown that characters of hypergroups which have Laplace representations give rise to bounded operators on E . Examples include the Mellin transform and the Mehler–Fock transform. The paper uses functional calculus for the cosine family cos⁡(t⁢Δ){\cos(t\sqrt{\Delta})} which is associated with waves that travel at unit speed. The main results include an operational calculus theorem for Sturm–Liouville hypergroups with Laplace representation as well as analogues to the Kunze–Stein phenomenon in the hypergroup convolution setting.

In this paper, we discuss the existence of symmetric positive solutions for the systems of higher-order boundary value problems on time scales. Our results extend some recent work in the literature. The analysis of this paper mainly relies on the monotone iterative technique.

Recently, a characterization of frames in twisted shift-invariant spaces in L2⁢(ℝ2⁢n){L^{2}(\mathbb{R}^{2n})} has been obtained in [16]. Using this result, we prove a sampling theorem on a subspace of a twisted shift-invariant space in this paper.