Volume 25, Issue 3

In this paper, we study the existence of mild solutions for a class of semilinear fractional order integro-differential inclusions with infinite delay in Banach spaces. Sufficient conditions for the existence of solutions are derived by using a nonlinear alternative of Leray–Schauder type for multivalued maps due to Martelli. An example is given to illustrate the theory.

In this paper, we show that the generalized Kato spectrum of a bounded operator in a Banach space is invariant under perturbation by commuting quasi-nilpotent operators, and the Kato spectrum is stable under additive commuting nilpotent perturbations. Our results are used to give an equivalent definition of the generalized Kato spectrum.

In the present work the Cauchy problem for an abstract evolution equation with a Lipschitz-continuous operator is considered, where the main operator represents the sum of positive definite self-adjoint operators. The fourth-order accuracy decomposition scheme is constructed for an approximate solution of the problem. The theorem on the error estimate of an approximate solution is proved. Numerical calculations for different model problems are carried out using the constructed scheme. The obtained numerical results confirm the theoretical conclusions.

The aim of this paper is to establish the relation between the two classes of functions BV*⁢(q;α){\mathrm{BV}_{*}(q;\alpha)} and Λ*⁢BVp{\Lambda_{*}\mathrm{BV}^{p}}. It uses a new shorter proof and extends the results of Gogatishvili, Goginava and Tephnadze.

In the paper, we investigate the relation between the properties of functions and their Fourier–Haar coefficients. We show that for some classes of functions Fourier–Haar coefficients have constant signs and order of magnitude. In 1964, Golubov proved in [B. I. Golubov, On Fourier series of continuous functions with respect to a Haar system (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 28 1964, 1271–1296] that if f⁢(x)∈C⁢(0,1){f(x)\in C(0,1)}, then its Fourier–Haar coefficients have constant signs when f⁢(x){f(x)} is a nonincreasing function on [0,1]{[0,1]}, and in some cases those coefficients have a certain order of magnitude. In the present paper, we continue to investigate the properties of functions which follow from the behavior of their Fourier–Haar coefficients.

The study of oscillation theory for fractional differential equations has been initiated by Grace et al. [5]. In this paper we establish some new criteria for the oscillation of fractional differential equations with the Caputo derivative of the form DarC⁢x⁢(t)=e⁢(t)+f⁢(t,x⁢(t)),t>0,a>1{{}^{C}D_{a}^{r}x(t)=e(t)+f(t,x(t)),t>0,a>1}, where r=α+n-1,α∈(0,1){r=\alpha+n-1,\alpha\in(0,1)}, and n≥1{n\geq 1} is a natural number. We also present the conditions under which all solutions of this equation are asymptotic to tn-1{t^{n-1}} as t→∞{t\to\infty}.

An optimal control problem for the vibration equation of an elastic plate is considered when the control function is included in the coefficient of the highest order derivative and the right-hand side of the equation. The solvability of the initial boundary value problem is shown, the theorem on the existence of an optimal control is proved and a necessary condition of optimality in the form of an integral equation is obtained.

Let 𝔓⁢(ℝn){\mathfrak{P}({\mathbb{R}^{n}})} be the power set of ℝn{\mathbb{R}^{n}} and let φ:𝔓⁢(ℝn)→[0,∞]{\varphi:\mathfrak{P}({\mathbb{R}^{n}})\rightarrow[0,\infty]} be a set function. In this paper, the authors introduce a class of generalized Hausdorff capacities Hφ{H_{\varphi}} with respect to φ. Some basic properties of Hφ{H_{\varphi}} including the strong subadditivity are obtained. An equivalent variant of Hφ{H_{\varphi}} defined via dyadic cubes is also introduced and proved to be Choquet capacity. The authors then prove the boundedness of some maximal operators, such as the Hardy–Littlewood maximal operator, on Lebesgue spaces with respect to Hφ{H_{\varphi}}. As an application, the predual spaces of weighted Morrey spaces are described via these capacities.

The present paper deals with the bivariate (p,q){(p,q)}-Baskakov–Kantorovich operators and their approximation properties. First we construct the operators and obtain some auxiliary results such as calculations of moments and central moments, etc. Our main results consist of uniform convergence of the operators via the Korovkin theorem and rate of convergence in terms of modulus of continuity.

New exact solutions are obtained for unsteady magnetohydrodynamic (MHD) flows of a generalized second-grade fluid near a uniform accelerating plate. The generalized second-grade fluid saturates the porous space. A fractional derivative is used in the governing equation. Analytical expressions for the velocity and shear stress fields are obtained by using the Laplace transform technique for fractional calculus. The obtained solutions are expressed in the series form in terms of Fox H -functions. Similar solutions for an ordinary second-grade fluid passing through a porous space are also derived. Moreover, several graphs are constructed for the pertinent parameters to analyze the characteristics of the velocity and shear stress field.

The Borel types of some classical small subsets of the real line are considered. In particular, under Martin’s axiom it is shown that there are at least 𝐜+{{\mathbf{c}}^{+}} pairwise incomparable Borel types of generalized Luzin sets (resp. of generalized Sierpiński sets), where 𝐜{{\mathbf{c}}} stands for the cardinality of the continuum.

This paper is devoted to the analysis of the boundary value problem ∂t⁡u-Δ⁢u=f{\partial_{t}u-\Delta u=f}, with an N -dimensional space variable, subject to a Dirichlet–Robin type boundary condition on the lateral boundary of the domain. The problem is settled in a noncylindrical domain of the form Q={(t,x1)∈ℝ2:0<t<T,φ1(t)<x1<φ2(t)}×∏i=1N-1]0,bi[Q=\{(t,x_{1})\in\mathbb{R}^{2}:0<t<T,\varphi_{1}(t)<x_{1}<\varphi_{2}(t)\}% \times\prod_{i=1}^{N-1}{]0,b_{i}[}, where φ1{\varphi_{1}} and φ2{\varphi_{2}} are smooth functions. One of the main issues of the paper is that the domain can possibly be non-regular; for instance, the significant case when φ1⁢(0)=φ2⁢(0){\varphi_{1}(0)=\varphi_{2}(0)} is allowed. We prove well-posedness results for the problem in a number of different settings and under natural assumptions on the coefficients and on the geometrical properties of the domain. This work is an extension of the one-dimensional case studied in [4].

In this paper, by using a weighted identity for functions defined on an open invex subset of the set of real numbers, by using the Hölder integral inequality and by using the notion of h -preinvexity, we present weighted integral inequalities of Hermite–Hadamard-type for functions whose derivatives in absolute value raised to certain powers are h -preinvex functions. Some new Hermite–Hadamard-type integral inequalities are obtained when h is super-additive. Inequalities of Hermite–Hadamard-type for s -preinvex functions are given as well as a special case of our results.

Let K⁢〈X〉=K⁢〈X1,…,Xn〉{K\langle X\rangle=K\langle X_{1},\ldots,X_{n}\rangle} be the free algebra generated by X={X1,…,Xn}{X=\{X_{1},\ldots,X_{n}\}} over a field K . It is shown that, with respect to any weighted ℕ{\mathbb{N}}-gradation attached to K⁢〈X〉{K\langle X\rangle}, minimal homogeneous generating sets for finitely generated graded two-sided ideals of K⁢〈X〉{K\langle X\rangle} can be algorithmically computed, and that if an ungraded two-sided ideal I of K⁢〈X〉{K\langle X\rangle} has a finite Gröbner basis 𝒢{{\mathcal{G}}} with respect to a graded monomial ordering on K⁢〈X〉{K\langle X\rangle}, then a minimal standard basis for I can be computed via computing a minimal homogeneous generating set of the associated graded ideal 〈𝐋𝐇⁢(I)〉{\langle\mathbf{LH}(I)\rangle}.

In this paper, we use the theory of simplicial groups to generalize the Schur multiplier of a pair of groups (G,N){(G,N)} to its Baer invariant, 𝒱⁢M⁢(G,N){\mathcal{V}M(G,N)}, with respect to an arbitrary variety 𝒱{\mathcal{V}}. Moreover, among other things we present some behaviors of Baer invariants of a pair of groups with respect to the free product and the direct limit. Finally, we prove that the nilpotent multiplier of a pair of groups does commute with the free product of finite groups of mutually coprime orders.

In this paper, using the concept of ideal convergence, which extends the idea of ordinary convergence and statistical convergence, we are concerned with the I -uniform convergence and the I -pointwise convergence of sequences of functions defined on a set of real numbers D . We present the Arzelà–Ascoli theorem by means of ideal convergence and also the relationship between I -equicontinuity and I -continuity for a family of functions.

In this paper, we obtain sufficiency conditions for generalized β-absolute convergence (0<β≤2{0<\beta\leq 2}) of single and multiple Fourier series of functions of the class Λ-BV(p(n)↑∞,φ,[-π,π]){\Lambda\text{-}\mathrm{BV}(p(n)\uparrow\infty,\varphi,[-\pi,\pi])} and the class (Λ1,Λ2,…,ΛN)-BV(p(n)↑∞,φ,[-π,π]N){(\Lambda^{1},\Lambda^{2},\dots,\Lambda^{N})\text{-}\mathrm{BV}(p(n)\uparrow% \infty,\varphi,[-\pi,\pi]^{N})}, respectively.