Volume 30, Issue 2

Sufficient conditions are given for a function F ⁢ ( p ) {F(p)} to be the Laplace transform of a function f ⁢ ( t ) {f(t)} or a distribution f . No assumption on f is given a priori. It is not even assumed that f = 0 {f=0} for t < 0 {t<0} .

The generalization of Hardy uncertainty principle for the windowed Bochner–Fourier transform on the Heisenberg group is proved. We consider a Bochner measurable function Ψ : G → X {\Psi:G\to X} , where X is a completely separable Hilbert space X and Let G be a completely separable, unimodular, connected nilpotent Lie group. We establish that if ϕ ∈ C C ⁢ ( G ) {\phi\in C_{C}(G)} is a non-trivial window function and Ψ ∈ L 2 ⁢ ( G ) {\Psi\in L^{2}(G)} satisfies ∥ V ϕ ⁢ ( Ψ ) ⁢ ( g , χ ) ∥ HB ≤ c 2 ⁢ ( g ) ⁢ exp ⁡ ( - β ⁢ ∥ χ ∥ 2 ) \|V_{\phi}(\Psi)(g,\chi)\|_{\rm HB}\leq c_{2}(g)\exp(-\beta\|\chi\|^{2}) , β > 0 {\beta>0} , then Ψ = 0 {\Psi=0} almost everywhere.

The purpose of this paper is to study the controllability in the α-norm for some impulsive partial functional integrodifferential equation with infinite delay in Banach space. To do this, we give sufficient conditions ensuring the controllability by assuming that the undelayed part admits a resolvent operator in the sense of Grimmer and that the delayed part is continuous with respect to the fractional power of the generator. The results are obtained by using the Schauder fixed-point theorem.

We study the convergence of a random iterative sequence of a family of operators on infinite-dimensional Hilbert spaces, inspired by the stochastic gradient descent (SGD) algorithm in the case of the noiseless regression. We identify conditions that are strictly broader than previously known for polynomial convergence rate in various norms, and characterize the roles the randomness plays in determining the best multiplicative constants. Additionally, we prove almost sure convergence of the sequence.

In 2015, Srivastava and Singh [S. K. Srivastava and U. Singh, Trigonometric approximation of periodic functions belonging to weighted Lipschitz class W ⁢ ( L p , Ψ ⁢ ( t ) , β ) W(L^{p},\Psi(t),\beta) , Function Spaces in Analysis, Contemp. Math. 645, American Mathematical Society, Providence 2015, 283–291] determined the order of approximation of periodic functions belonging to W ⁢ ( L p , Ψ ⁢ ( u ) , β ) {W(L^{p},\Psi(u),\beta)} -class, which is a weighted version of Lip ⁢ ( ω ⁢ ( u ) , p ) {\mathrm{Lip}(\omega(u),p)} -class with weight function sin β ⁢ p ⁡ ( y 2 ) {\sin^{\beta p}(\frac{y}{2})} through matrix means of their trigonometric Fourier series. It is a well-known fact that the product summability methods are stronger than the single summability methods, and they can approximate a wider class of functions. Therefore, in this article, an effort is made to determine the degree of approximation for periodic functions from the same weighted Lipschitz class W ⁢ ( L p , Ψ ⁢ ( u ) , β ) {W(L^{p},\Psi(u),\beta)} , p ≥ 1 {p\geq 1} , using C δ . T {C^{\delta}.T} -means of their trigonometric Fourier series.

The aim of this write-up is to investigate some covering properties in neutrosophic topological spaces. We define proximate cover, almost compactness, almost countable compactness and almost Lindelöfness in connection with neutrosophic topological spaces and study some properties entangled with them.

In this paper, we are interested in studying the Cauchy problem for a weakly coupled system of two semi-linear structurally damped σ k {\sigma_{k}} -evolution equations, where σ k ≥ 1 {\sigma_{k}\geq 1} for k = 1 , 2 {k=1,2} . Our first purpose involves the proof of global (in time) existence of small data energy solutions by mixing additional L m k {L^{m_{k}}} regularity for the data, where m k ∈ [ 1 , 2 ) {m_{k}\in[1,2)} . We want to point out that in some cases of choosing suitable parameters m k {m_{k}} , with k = 1 , 2 {k=1,2} , the obtained lower bound of one exponent p or q related to power nonlinearities on the right-hand sides is really smaller than the critical exponent, the so-called modified Fujita exponent. The second aim of this paper is to discuss a blow-up result for Sobolev solutions with any different fractional values of σ k ≥ 1 {\sigma_{k}\geq 1} when m 1 = m 2 {m_{1}=m_{2}} .

This study is focused on finding the solution to the initial value problem for the fractional elliptic equations driven by the Wiener process. First, with some strong conditions on the input data, we establish the regularity of the solution. With relaxed conditions of the input data function, our forward problem is ill-posed in the sense of Hadamard. For this ill-posed problem, the truncation method is used to construct a regularized solution. Under prior assumptions for the exact solution, the convergence rate is obtained.

In this paper, we generalize the notions of the Köthe–Toeplitz duals of sequence spaces by introducing the concepts of bicomplex α-dual, bicomplex β-dual and bicomplex γ-dual, and also we compute them for some bicomplex sequence spaces l p ⁢ ( 𝔹 ⁢ ℂ ) {l_{p}(\mathbb{BC})} for 1 ≤ p ≤ ∞ {1\leq p\leq\infty} , c 0 ⁢ ( 𝔹 ⁢ ℂ ) {c_{0}(\mathbb{BC})} and c ⁢ ( 𝔹 ⁢ ℂ ) {c(\mathbb{BC})} . Furthermore, we define a concept of bicomplex multiplier space as the bicomplex version of multiplier space of two sequence spaces and support this definition with examples.

Goebel’s coincidence theorem, a remarkably simple extension of Banach’s contraction principle widely applied in analysis, has found significant utility in the theory of differential and integral equations. Over the past fifty years, researchers have endeavored to generalize the definition of a metric space, thereby extending the scope of Goebel’s coincidence theorem to diverse settings. This survey paper overviews valuable insights into Goebel’s coincidence theorem’s historical background, its relevance in the field of fixed point theory, and its practical implications in solving problems related to differential and integral equations.

In this paper, we study boundary controllability for the linear extension problem of a wave equation with space-dependent coefficients and having an internal degeneracy. For this purpose, we mainly focus on the well-posedness and the boundary null controllability of a relaxed version of the original problem, namely, to some degenerate transmission problem. The key ingredient is to derive direct and inverse inequalities for the associated homogeneous degenerate adjoint problem. By these inequalities, we deduce that the transmission problem has a unique solution by transposition and this solution is null controllable. Moreover, we give an explicit formula of the controllability time.

We describe a framework for random pairwise comparisons matrices, inspired by selected constructions related to the so called inconsistency reduction of pairwise comparisons (PC) matrices. In order to build up structures on random pairwise comparisons matrices, the set up for (deterministic) PC matrices for non-reciprocal PC matrices is completed. Basic concepts such as inconsistency indices and geometric mean method are extended to random pairwise comparisons matrices and completed by new notions which seem useful to us. Two procedures for (random) inconsistency reduction are sketched, based on well-known existing objects, and a fiber bundle-like decomposition of random pairwise comparisons is proposed.

In this article we introduce the notion of statistical convergence difference sequences of bi-complex numbers. Some properties of these sequence spaces like BC-module, Banach BC-module, BC-balanced set, BC-convex set, solidness, and symetricity are studied.

In this work we prove some new Bernstein-type inequalities for the polynomials with restricted zeros. Our results strengthen some recently proved Erdős–Lax and Turán-type inequalities by Kumar and others. We further extend the obtained results to the polar derivative of a polynomial.

The goal of the present article is to offer several new retarded nonlinear inequalities of Gronwall, Bellman, and Pachpatte kind for a class of integral and integro-differential equations. These inequalities generalize and provide new formulations of some well-known results in the mathematical framework of integral and differential inequalities that have been derived currently as well as in earlier times. These results can be utilized to investigate diverse aspects, both qualitative and quantitative, of a class of aforementioned equations. We propose a few applications to ensure the effectiveness of these inequalities.

The paper is designed to highlight the presence of an optimal solution to a differential system comprising fractional order derivatives, precisely the ψ-Caputo derivative, under the prescribed initial constraints. This is attained through a new variant of cyclic (noncyclic) condensing operators incorporating a pair of maps. The altered notion facilitates in evolving best proximity point (pair) results as well as coupled best proximity point results with supportive corollaries.

Recently, the authors Guzman et al. (2018) introduced a new simple well-behaved definition of the fractional derivative called non-conformable fractional derivative. In this paper we study the exponential stability of non-conformable fractional-order systems by using the Lyapunov function and Gronwall inequality. These inequalities can be used as handy tools to research stability problems of nonlinear systems. Sufficient conditions for exponential stability are given using the Lyapunov theory. Further, deals with the state feedback stabilization problems for a family of nonlinear systems satisfying a Lipschitz continuity condition. The stability of the controller is proved by means of the new Lyapunov stability theorem given in this paper. A numerical example is given to illustrate the efficiency of the obtained result.