Volume 27, Issue 4

We consider 3D Navier–Stokes–Voigt equations in smooth bounded domains with homogeneous Dirichlet boundary conditions. First, we study the existence and exponential stability of strong stationary solutions to the problem. Then we show that any unstable steady state can be exponentially stabilized by using either an internal feedback control with a support large enough or a multiplicative Itô noise of sufficient intensity.

Let R be a commutative ring with 1≠0{1\neq 0}. We recall that a proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever a,b,c∈R{a,b,c\in R} and 0≠a⁢b⁢c∈I{0\not=abc\in I}, then a⁢b∈I{ab\in I} or a⁢c∈I{ac\in\sqrt{I}} or b⁢c∈I{bc\in\sqrt{I}}. In this paper, we introduce a new class of ideals that is closely related to the class of weakly 2-absorbing primary ideals. Let I⁢(R){I(R)} be the set of all ideals of R and let δ:I⁢(R)→I⁢(R){\delta:I(R)\rightarrow I(R)} be a function. Then δ is called an expansion function of ideals of R if whenever L,I,J{L,I,J} are ideals of R with J⊆I{J\subseteq I}, then L⊆δ⁢(L){L\subseteq\delta(L)} and δ⁢(J)⊆δ⁢(I){\delta(J)\subseteq\delta(I)}. Let δ be an expansion function of ideals of R . Then a proper ideal I of R (i.e., I≠R{I\not=R}) is called a weakly 2-absorbing δ-primary ideal if 0≠a⁢b⁢c∈I{0\not=abc\in I} implies a⁢b∈I{ab\in I} or a⁢c∈δ⁢(I){ac\in\delta(I)} or b⁢c∈δ⁢(I){bc\in\delta(I)}. For example, let δ:I⁢(R)→I⁢(R){\delta:I(R)\rightarrow I(R)} such that δ⁢(I)=I{\delta(I)=\sqrt{I}}. Then δ is an expansion function of ideals of R , and hence a proper ideal I of R is a weakly 2-absorbing primary ideal of R if and only if I is a weakly 2-absorbing δ-primary ideal of R . A number of results concerning weakly 2-absorbing δ-primary ideals and examples of weakly 2-absorbing δ-primary ideals are given.

For a locally integrable function f on [0,∞){[0,\infty)}, we define F⁢(t)=∫0tf⁢(u)⁢d⁢u and σα⁢(t)=∫0t(1-ut)α⁢f⁢(u)⁢d⁢uF(t)=\int_{0}^{t}f(u)\mathop{}\!du\quad\text{and}\quad\sigma_{\alpha}(t)=\int_% {0}^{t}\biggl{(}1-\frac{u}{t}\biggr{)}^{\alpha}f(u)\mathop{}\!du for t>0{t>0}. The improper integral ∫0∞f⁢(u)⁢d⁢u{\int_{0}^{\infty}f(u)\mathop{}\!du} is said to be (C,α){(C,\alpha)} integrable to L for some α>-1{\alpha>-1} if the limit limx→∞⁡σα⁢(t)=L{\lim_{x\to\infty}\sigma_{\alpha}(t)=L} exists. It is known that limt→∞⁡F⁢(t)=ℓ{\lim_{t\to\infty}F(t)=\ell} implies limt→∞⁡σα⁢(t)=ℓ{\lim_{t\to\infty}\sigma_{\alpha}(t)=\ell} for α>-1{\alpha>-1}, but the converse of this implication is not true in general. In this paper, we introduce the concept of the general control modulo of non-integer order for functions and obtain some Tauberian conditions in terms of this concept for the (C,α){(C,\alpha)} integrability method in order that the converse implication hold true. Our results extend the main theorems in [Ü. Totur and İ. Çanak, Tauberian conditions for the (C,α)(C,\alpha) integrability of functions, Positivity 21 2017, 1, 73–83].

In this paper, for 0<γ<1{0<\gamma<1} and b∈Iγ⁢(BMO){b\in I_{\gamma}(\mathrm{BMO})}, the authors give the L2⁢(ℝn){L^{2}({\mathbb{R}}^{n})} boundedness of μγ;b{\mu_{\gamma;b}}, the commutator of a fractional differential type Marcinkiewicz integral with rough variable kernel, which is an extension of some known results.

We address the (pointed) homotopy of crossed module morphisms in modified categories of interest that unify the notions of groups and various algebraic structures. We prove that the homotopy relation gives rise to an equivalence relation as well as to a groupoid structure with no restriction on either domain or co-domain of the corresponding crossed module morphisms. Furthermore, we also consider particular cases such as crossed modules in the categories of associative algebras, Leibniz algebras, Lie algebras and dialgebras of the unified homotopy definition. Finally, as one of the major objectives of this paper, we prove that the functor from simplicial objects to crossed modules in modified categories of interest preserves the homotopy as well as the homotopy equivalence.

In this paper, the necessary and sufficient conditions are found for the boundedness of the Riesz potential Iα{I_{\alpha}} in the local Morrey–Lorentz spaces Mp,q;λloc⁢(ℝn){M_{p,q;{\lambda}}^{\mathrm{loc}}({\mathbb{R}^{n}})}. This result is applied to the boundedness of particular operators such as the fractional maximal operator, fractional Marcinkiewicz operator and fractional powers of some analytic semigroups on the local Morrey–Lorentz spaces Mp,q;λloc⁢(ℝn){M_{p,q;{\lambda}}^{\mathrm{loc}}({\mathbb{R}^{n}})}.

For certain families of topologies, the existence of a common Sierpiński–Zygmund function (of a common Sierpiński–Zygmund function in the strong sense) is established. In this connection, the notion of a Sierpiński–Zygmund space (of a Sierpiński–Zygmund space in the strong sense) is introduced and examined. The behavior of such spaces under some standard topological operations is considered.

A new scale of weighted grand mixed norm Lebesgue spaces Lw1p1,φ1⁢(Lw2p2,φ2){L_{w_{1}}^{p_{1},\varphi_{1}}(L_{w_{2}}^{p_{2},\varphi_{2}})} is introduced and the boundedness criteria for multiple singular operators are established.

The “oldest quartic” functional equation f⁢(x+2⁢y)+f⁢(x-2⁢y)=4⁢[f⁢(x+y)+f⁢(x-y)]-6⁢f⁢(x)+24⁢f⁢(y)f(x+2y)+f(x-2y)=4[f(x+y)+f(x-y)]-6f(x)+24f(y) was introduced and solved by the second author of this paper (see J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glas. Mat. Ser. III 34(54) 1999, 2, 243–252). Similarly, an interesting “quintic” functional equation was introduced and investigated by I. G. Cho, D. Kang and H. Koh, Stability problems of quintic mappings in quasi-β-normed spaces, J. Inequal. Appl. 2010 2010, Article ID 368981, in the following form: 2⁢f⁢(2⁢x+y)+2⁢f⁢(2⁢x-y)+f⁢(x+2⁢y)+f⁢(x-2⁢y)=20⁢[f⁢(x+y)+f⁢(x-y)]+90⁢f⁢(x).2f(2x+y)+2f(2x-y)+f(x+2y)+f(x-2y)=20[f(x+y)+f(x-y)]+90f(x). In this paper, we generalize this “Cho–Kang–Koh equation” by introducing pertinent Euler–Lagrange k -quintic functional equations, and investigate the “Ulam stability” of these new k -quintic functional mappings.

In this work, the solvability of a generally nonlocal problem is investigated for a third order linear ordinary differential equation with variable principal coefficient. A novel adjoint problem and Green’s functional are constructed for a completely nonhomogeneous problem. Several illustrative applications for the theoretical results are provided.

Nonlinear functional partial differential equations with initial conditions are considered on the cone. The weak convergence of a sequence of successive approximations is proved. The proof is given by the duality principle.

For the nonlinear controlled functional differential equations with several constant delays, the local variation formulas of solutions are proved, in which the effects of the discontinuous initial condition and perturbations of delays and the initial moment are detected.

The purpose of this work is to investigate the existence of solutions for various Neumann boundary value problems associated to the Laplacian-type operators. The main results are obtained using the extension of Mawhin’s continuation theorem.

We establish off-diagonal extrapolation on mixed variable Lebesgue spaces. As its applications, we obtain the boundedness for strong fractional maximal operators. The vector-valued analogies are also considered. Additionally, the Littlewood–Paley characterization for mixed variable Lebesgue spaces is also established with the help of weighted norm inequalities and extrapolation.

Let 𝒜=𝒜p{\mathcal{A}=\mathcal{A}_{p}} be the mod⁢p{\mathrm{mod}\,p} Steenrod algebra, where p is a fixed prime and let 𝒜′{\mathcal{A}^{\prime}} denote the Bockstein-free part of 𝒜{\mathcal{A}} at odd primes. Being a connected graded Hopf algebra, 𝒜{\mathcal{A}} has the canonical conjugation χ. Using this map, we introduce a relationship between the X - and Z -bases of 𝒜′{\mathcal{A}^{\prime}}. We show that these bases restrict to give bases to the well-known sub-Hopf algebras 𝒜⁢(n-1){\mathcal{A}(n-1)}, n≥1{n\geq 1}, of 𝒜′{\mathcal{A}^{\prime}}.

In this note, using some conditions on the weight function K , we investigate the inner functions as multipliers in weighted Dirichlet spaces, and we also discuss zero sets.