Spatially almost periodic complex‐valued solutions of the Boussinesq equations

We extend several ideas developed in E. Dinaburg, Ya. G. Sinai and D. Li's papers (see Lemma 2.2 , Remark   2.1 , Lemma 2.4 , Remark   2.2 , Remark 2.3 ) and construct a unique mild solution (u,θ) of the system 1.1 satisfying 1.3 with spatially almost periodic initial data . Moreover, (u,θ) is also a spatially almost periodic complex‐valued mild solution. Assume further that (see Definition 1.6 ). Then, we overcome the difficulty in the work of Y. Giga et al. (see Introduction below) and show that the above mild solution also satisfies 1.4 for all .     

Norms and lower bounds of some matrix operators on Fibonacci weighted difference sequence space

Norm of an operator T:X→Y is the best possible value of U satisfying the inequality and lower bound for T is the value of L satisfying the inequality where ‖.‖_X and ‖.‖_Y are the norms on the spaces X and Y, respectively. The main goal of this paper is to compute norms and lower bounds for some matrix operators from the weighted sequence space ?_p(w) into a new space called as Fibonacci weighted difference sequence space. For this purpose, we firstly introduce the Fibonacci difference matrix and the space consisting of sequences whose ‐transforms are in .     

Fractional powers of the noncommutative Fourier's law by the S‐spectrum approach

Let e_?, for ? = 1,2,3, be orthogonal unit vectors in and let be a bounded open set with smooth boundary ?Ω. Denoting by a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: into the conservation of energy law, here a, b, are given functions. With the S‐spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, the fractional powers of T exist in the sense of the S‐spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory.     

The structure of the fractional powers of the noncommutative Fourier law

In the recent years, there has been a lot of interest in fractional diffusion and fractional evolution problems. The spectral theory on the S‐spectrum turned out to be an important tool to define new fractional diffusion operators stating from the Fourier law for nonhomogeneous materials. Precisely, let e_?, e_?=1,2,3 be orthogonal unit vectors in and let be a bounded open set with smooth boundary ?Ω. Denoting by a point in Ω, the heat equation is obtained replacing the Fourier law given by into the conservation of energy law. In this paper, we investigate the structure of the fractional powers of the vector operator T, with homogeneous Dirichlet boundary conditions. Recently, we have found sufficient conditions on the coefficients a, b, such that the fractional powers of T exist in the sense of the S‐spectrum approach. In this paper, we show that under a different set of conditions on the coefficients a, b, c, the fractional powers of T have a different structure.     

Controllability of second‐order Sobolev‐type impulsive delay differential systems

In this paper, we study some controllability results for second‐order Sobolev‐type impulsive integrodifferential systems with finite delay. The results are obtained by using the concepts of measure of noncompactness and Mönch's fixed point theorem. Particularly, we consider the damping term y′(·) and find a control u such that the mild solution satisfies y(T) = y_T and . Finally, an application is given to illustrate the obtained results.     

Fundamental solutions for discrete dynamical systems involving the fractional Laplacian

We prove representation results for solutions of a time‐fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example, chain processes in chemistry and radioactivity. Our focus is in the problem , where 0<β ≤ 2, 0<α ≤ 1, , (?Δ_d)^α is the discrete fractional Laplacian, and is the Caputo fractional derivative of order β. We discuss important special cases as consequences of the representations obtained.     

An Osgood type regularity criterion for the 2D Oldroyd‐B model

By means of the Littlewood‐Paley decomposition and the div‐curl Theorem by Coifman‐Lions‐Meyer‐Semmes, we prove an Osgood type regularity criterion for the 2D incompressible Oldroyd‐B model, that is, where denotes the Fourier localization operator whose spectrum is supported in the shell {|ξ|≈2^j}.     

Positive ground state solutions for fractional Kirchhoff type equations with critical growth

We study the existence of positive ground state solutions for the following fractional Kirchhoff type equation where a,b > 0 are constants, μ is a positive parameter, with and s ∈ (0,1). Under suitable assumptions on V(x), by using a monotonicity trick and a global compactness principle, we prove that the equation admits a positive ground state solution if and μ > 0 large enough.     

Existence of nontrivial solutions for fractional Schrödinger equations with critical or supercritical growth

In this paper, we study the following fractional Schrödinger equation with critical or supercritical growth where 0 < s < 1, N > 2s, λ > 0, , , ( ? Δ)^s denotes the fractional Laplacian of order s and f is a continuous superlinear but subcritical function. Under some suitable conditions, we prove that the equation has a nontrivial solution for small λ > 0 by variational methods. Our main contribution is related to the fact that we are able to deal with the case .     

Sharp oscillation and nonoscillation tests for delay dynamic equations

In this paper, we give new sufficient conditions for both oscillation and nonoscillation of the delay dynamic equation where and satisfy τ(t) ≤ σ(t) for all large t and . As an important corollary, we obtain the time scale invariant integral condition for nonoscillation: for all large t. Also, with some examples, we show that newly presented results are sharp.     

Stationary solutions for a superlinear Dirac equation

K. Guerlebeck In this paper, we consider the following nonlinear Dirac equation By applying the variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou, we prove the existence of nontrivial and ground state solutions for the aforementioned system under conditions weaker than those in Zhang et al. (Journal of Mathematical Physics, 2013). John Wiley & Sons, Ltd.     

A multiperiod model for estimating the cost of deposit insurance under the Basel III minimum capital requirement

Under the Basel III regime, a commercial bank is considered adequately capitalized if it maintains a ratio of capital to total risk-weighted assets or capital adequacy ratio (CAR) of at least 8%. We model a commercial bank that complies with Basel III's minimum capital requirement on an interval $[0,T]$ for $T>0$. The bank model is achieved via a specific rate of capital influx that fixes the bank's CAR at the minimum prescribed level of 8%. On the basis of this capital influx rate, we derive models for the bank's asset portfolio and capital dynamics required for maintaining the CAR at the minimum prescribed level. For the aforementioned bank, we further study a deposit insurance (DI) pricing problem with a coverage horizon equal to $T$ years. More specifically, we employ a multiperiod DI pricing model to approximate the cost of DI for the bank on the interval $[0,T]$, where the constant (minimum) CAR is maintained. We study the behaviours of the models leading to the constant (minimum) CAR, and the behaviour of the DI premium estimate by means of numerical simulations. In the simulation study pertaining to the DI premium estimate specifically, we determine the effects of changes in the bank's initial leverage level (deposit-to-asset ratio), the DI coverage horizon, and the volatility of the asset portfolio on the DI premium estimate.     

More on a hyperbolic‐cotangent class of difference equations

We present a natural method for solving the difference equation where , parameter a, and initial values x_?j, , , are real or complex numbers. As a concrete example, the case k = 1, l = 2 is studied in detail, and several interesting formulas for solutions and objects used in the analysis of the equation are given. A useful remark on solvability of difference equations on complex domains is presented and used here.     

Semiclassical asymptotic behavior of ground state for the two‐component Hartree system

We consider the semiclassical asymptotic behaviors of ground state solution for the following two‐component Hartree system: which is originated from the study on cold atoms of boson and fermion system with long‐range interaction. Under the assumption by detailed compactness analysis, we prove that there is a β₀>0 such that if β<β₀, the system has a ground state solution. For this solution, the energy estimates and the decay rates are presented, and the asymptotic profiles as ε→0 are displayed in details for β<0 and β>0, respectively. Furthermore, we show that for β<0, the phase separation phenomenon may occur.     

Existence of positive solutions to fractional elliptic problems with Hardy potential and critical growth

In this work, we study the following critical problem involving the fractional Laplacian: where s ∈ (0,1), N > 2s, , and is the fractional critical exponent, 0 < μ < Λ_N,s, the sharp constant of the Hardy‐Sobolev inequality. For suitable assumptions on g(x) and K(x), we consider the existence and multiplicity of positive solutions depending on the value of p. Moreover, we obtain an existence result for the problem when λ = 0.     

On positive solutions of fractional differential equations with change of sign

We study the existence and uniqueness of positive solutions of fractional differential equations with change of sign where 1 < α ≤ 2, is continuous and does not vanish identically on any subinterval of [0,1].     

Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces

Let C_Γ be the Cauchy integral operator on a Lipschitz curve Γ. In this article, the authors show that the commutator [b,C_Γ] is bounded (resp, compact) on the Morrey space for any (or some) p ∈ (1,∞) and λ ∈ (0,1) if and only if (resp, ). As an application, a factorization of the classical Hardy space in terms of C_Γ and its adjoint operator is obtained.     

On concentration of solution to a Schrödinger logarithmic equation with deepening potential well

In this work, we prove the existence of positive solution for the following class of problems where λ>0 and is a potential satisfying some conditions. Using the variational method developed by Szulkin for functionals, which are the sum of a C¹ functional with a convex lower semicontinuous functional, we prove that for each large enough λ>0, there exists a positive solution for the problem, and that, as λ→+∞, such solutions converge to a positive solution of the limit problem defined on the domain Ω=int(V^?1({0})).     

On a parabolic‐elliptic chemotaxis system with periodic asymptotic behavior

We study a parabolic‐elliptic chemotactic PDEs system, which describes the evolution of a biological population “u” and a chemical substance “v” in a bounded domain . We consider a growth term of logistic type in the equation of “u” in the form μu(1 ? u + f(t,x)). The function “f,” describing the resources of the systems, presents a periodic asymptotic behavior in the sense where f ^? is independent of x and periodic in time. We study the global existence of solutions and its asymptotic behavior. Under suitable assumptions on the initial data and f ^?, if the constant chemotactic sensitivity χ satisfies we obtain that the solution of the system converges to a homogeneous in space and periodic in time function.     

Weakly coupled systems of semilinear elastic waves with different damping mechanisms in 3D

We consider the following Cauchy problem for weakly coupled systems of semilinear damped elastic waves with a power source nonlinearity in three dimensions: where with b² > a² > 0 and θ ∈ [0,1]. Our interests are some qualitative properties of solutions to the corresponding linear model with vanishing right‐hand side and the influence of the value of θ on the exponents p₁,p₂,p₃ in to get results for the global (in time) existence of small data solutions.