The problem of the parabolic equation with constant coefficients and time displacement is investigated under non-local and non-homogeneous boundary conditions. Under minimal conditions on the data, the uniqueness of the solution is demonstrated and an explicit analytical representation of it is obtained.
We give a new formula of the Laplace transform expressed using the delta derivative for the generalized time scales. This formula F (z) T =LT{f(t)}(z)= ∫∞ 0 ze⊖(z-1)(σ(t),0)f(t)∆t combines the theory of classical Laplace and Z transforms. By choosing the time scale to be a set of real numbers, the classical Laplace transformation is obtained in a modified form, and if the time scale is chosen to be a set of integers, the classical Z transformation is obtained. Formulas for the Laplace transform of elementary functions, as well as formulas for real and complex shifting properties, are derived. These formulas are introduced for specific functions.
Let H1 and H2 be Hilbert spaces with the unit operators I1 and I2, respectively, and Ajk be bounded operators acting from Hj into Hk (j, k = 1, 2). We consider the block operator matrices A = (Ajk) and F(z) = diag (Kˆj (z)Ij ), where Kˆ1(z) and Kˆ2(z) are scalar analytic functions. The set of all z ∈ C, such that F(z) − A is boundedly invertible is called the F-regular set of A. The complement of the F-regular set to the complex plane is called the F-spectrum (the functional spectrum) of A. It is shown that the notion of the functional spectrum enables us to investigate, from the unified point of view, various types of coupled systems, in particular, systems of integral, fractional differential, integro-differential and differential-difference equations. We derive a bound for the functional spectrum and discuss applications of the obtained bound to the stability of the considered systems.
In this paper, the trigonometric fuzzy Korovkin theorem, originally established by G. A. Anastassiou and S. G. Gal (Nonlinear Functional Analysis and Applications, 11 (2006), 385-395), is extended to the k-dimensional setting. The proof is based on a new approach that differs from the original one and does not rely on the fuzzy modulus of continuity. An illustrative example is included to confirm the applicability and validity of the proposed generalization.
The (weak) wedgeness for FK-spaces was first defined by Bennett in 1974. Then, some results of Bennett (1974) were improved by İnce (2002) and Daˇgadur (2004) for all (weak) wedge FK spaces. In this paper, the concept of wedgeness for an FDK-space X containing Φ is defined, and some fundamental characterizations related to this space and compactness of the inclusion mapping are studied. Also, some results for a summability domain X (ν) A to be (weak) ν−wedge are obtained. Moreover, necessary and sufficient conditions for some double sequence spaces are given.
