Open Journal of Mathematical Analysis (OMA)

This paper concerns with the global solutions and general decay to an initial-boundary value problem of the dispersive wave equation with memory and source terms.

This work concerns the study of the controllability for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces. We give sufficient conditions that ensure the controllability of the system by supposing that its undelayed part admits a resolvent operator in the sense of Grimmer, and by making use of the measure of noncompactness and the Mönch fixed-point Theorem. As a result, we obtain a generalization of the work of K. Balachandran and R. Sakthivel (Journal of Mathematical Analysis and Applications, 255, 447-457, (2001)) and a host of important results in the literature, without assuming the compactness of the resolvent operator. An example is given for illustration.

The integrals \(\int_{-\infty}^\infty t_+^{\lambda-1} \phi(t)dt\) and \(\int_0^t(t-s)^{\lambda -1}b(s)ds\) are considered, \(\lambda\neq 0,-1,-2…\), where \(\phi\in C^\infty_0(\mathbb{R})\) and \(0\le b(s)\in L^2_{loc}(\mathbb{R})\). These integrals are defined in this paper for \(\lambda\le 0\), \(\lambda\neq 0,-1,-2,…\), although they diverge classically for \(\lambda\le 0\). Integral equations and inequalities are considered with the kernel \((t-s)^{\lambda -1}_+\).

In the present paper, we study a differential inequality involving certain differential operator. As a special case of our main result, we obtained certain differential inequalities implying sufficient conditions for meromorphic starlike and meromorphic convex functions of certain order.

The problem discussed is the Navier-Stokes problem (NSP) in \(\mathbb{R}^3\). Uniqueness of its solution is proved in a suitable space \(X\). No smallness assumptions are used in the proof. Existence of the solution in \(X\) is proved for \(t\in [0,T]\), where \(T>0\) is sufficiently small. Existence of the solution in \(X\) is proved for \(t\in [0,\infty)\) if some a priori estimate of the solution holds.

In this paper, we characterize the centre of dense irreducible subalgebras of compact elementary operators that are spectrally bounded. We show that the centre is a unital, irreducible and commutative \(C^{*}\)-subalgebra. Furthermore, the supports from the centre are orthogonal and the intersection of a nonzero ideal with the centre is non-zero.

By applying Opoola differential operator, in this article, two new subclasses \(\mathcal{M}_{\mathcal{H},\sigma}^{\mu,\beta}(m,\psi,k,\tau)\) and \(\mathcal{M}_{\mathcal{H},\sigma}^{\mu,\beta}(m,\xi,k,\tau)\) of bi-univalent functions class \(\mathcal{H}\) defined in \(\bigtriangledown\) are introduced and investigated. The estimates on the coefficients \(|l_2|\) and \(|l_3|\) for functions of the classes are also obtained.

Here we study the existence of solutions of a nonlocal two-point, with parameters, boundary value problem of a first order nonlinear differential equation. The maximal and minimal solutions will be proved. The continuous dependence of the unique solution on the parameters of the nonlocal condition will be proved. The anti-periodic boundary value problem will be considered as an application.

This paper presents an existence theorem of the solutions for a boundary value problem of fractional order differential equations with integral boundary conditions, by using measure of noncompactness combined with Mönch fixed point theorem. An example is furnished to illustrate the validity of our outcomes.

Recently, asymptotic estimates for the unimodular Fourier multipliers \(e^{i\mu(D)}\) have been studied for the function \(\alpha\)-modulation space. In this paper, using the almost orthogonality of projections and some techniques on oscillating integrals, we obtain asymptotic estimates for the unimodular Fourier multiplier \(e^{it(I-\Delta)^{\frac{\beta}{2}}}\) on the \(\alpha\)-modulation space. For an application, we give the asymptotic estimate of the solution for the Klein-Gordon equation with initial data in a \(\alpha\)-modulation space. We also obtain a quantitative form about the solution to the nonlinear Klein-Gordon equation.