Volume 28, Issue 4

Four Ulam type stability concepts for non-instantaneous impulsive fractional differential equations with state dependent delay are introduced. Two different approaches to the interpretation of solutions are investigated. We study the case of an unchangeable lower bound of the Caputo fractional derivative and the case of a lower bound coinciding with the point of jump for the solution. In both cases we obtain sufficient conditions for Ulam type stability. An example is also provided to illustrate both approaches.

Let ℛ{\mathscr{R}} be a prime ring, 𝒬r{\mathscr{Q}_{r}} the right Martindale quotient ring of ℛ{\mathscr{R}} and 𝒞{\mathscr{C}} the extended centroid of ℛ{\mathscr{R}}. In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m -potent commutators of skew derivations involving multilinear polynomials, i.e., ([δ⁢(f⁢(x1,…,xn)),f⁢(x1,…,xn)])m=[δ⁢(f⁢(x1,…,xn)),f⁢(x1,…,xn)],\big{(}[\delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})]\big{)}^{m}=[% \delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})], where 1<m∈ℤ+{1<m\in\mathbb{Z}^{+}}, f⁢(x1,x2,…,xn){f(x_{1},x_{2},\ldots,x_{n})} is a non-central multilinear polynomial over 𝒞{\mathscr{C}} and δ is a skew derivation of ℛ{\mathscr{R}}.

This paper relates to a recent trend in complex differential equations which studies solutions in a given domain. The classical settings in complex equations were widely studied for meromorphic solutions in the complex plane. For functions in the complex plane, we have a lot of results of general nature, in particular, the classical value distributions theory describing numbers of a -points. Many of these results do not work for functions in a given domain. A recent principle of derivatives permits us to study the numbers of Ahlfors simple islands for functions in a given domain; the islands play, to some extend, a role similar to that of the numbers of simple a -points. In this paper, we consider a large class of higher order differential equations admitting meromorphic solutions in a given domain. Applying the principle of derivatives, we get the upper bounds for the numbers of Ahlfors simple islands of similar solutions.

We investigate the solvability of three-dimensional dynamical mixed boundary value problems of electro-magneto-elasticity theory for homogeneous anisotropic bodies with interior cracks. Using the Laplace transform technique, the potential method, and the theory of pseudodifferential equations, we prove the existence and uniqueness theorems and analyze asymptotic properties of solutions near the crack edges and near the lines where the different boundary conditions collide.

Let (ℌ,μ,α){({\mathfrak{H}},\mu,\alpha)} be a regular Hom-algebra of arbitrary dimension and over an arbitrary base field 𝔽{{\mathbb{F}}}. A basis ℬ={ei}i∈I{{\mathcal{B}}=\{e_{i}\}_{i\in I}} of ℌ{{\mathfrak{H}}} is called multiplicative if for any i,j∈I{i,j\in I}, we have that μ⁢(ei,ej)∈𝔽⁢ek{\mu(e_{i},e_{j})\in{\mathbb{F}}e_{k}} and α⁢(ei)∈𝔽⁢ep{\alpha(e_{i})\in{\mathbb{F}}e_{p}} for some k,p∈I{k,p\in I}. We show that if ℌ{{\mathfrak{H}}} admits a multiplicative basis, then it decomposes as the direct sum ℌ=⊕rℑr{{\mathfrak{H}}=\bigoplus_{r}{{\mathfrak{I}}}_{r}} of well-described ideals admitting each one a multiplicative basis. Also, the minimality of ℌ{{\mathfrak{H}}} is characterized in terms of the multiplicative basis and it is shown that, in case ℬ{{\mathcal{B}}}, in addition, it is a basis of division, then the above direct sum is composed by means of the family of its minimal ideals, each one admitting a multiplicative basis of division.

In this paper, we introduce an interesting kind of convergence for a double sequence called the uniform convergence at a point. We give an example and demonstrate a Korovkin-type approximation theorem for a double sequence of functions using the uniform convergence at a point. Then we show that our result is stronger than the Korovkin theorem given by Volkov and present several graphs. Finally, in the last section, we compute the rate of convergence.

It is shown that the cardinality of the continuum is not real-valued measurable if and only if there exists no nonzero σ-finite diffused measure μ on the real line such that all Vitali sets (respectively all Bernstein sets) are μ-measurable.

We contribute to the study of Rota–Baxter operators on types of algebras other than associative and Lie algebras. If A is an algebra of a certain type and R is a Rota–Baxter operator on A , one can define a new multiplication on A by means of R and the previous multiplication and ask under what circumstances the new algebra is of the same type as A . Our first main result deals with such a situation in the case of BiHom-Lie algebras. Our second main result is a BiHom analogue of Aguiar’s theorem that shows how to obtain a pre-Lie algebra from a Rota–Baxter operator of weight zero on a Lie algebra. The BiHom analogue does not work for BiHom-Lie algebras, but for a new concept we introduce here, called left BiHom-Lie algebra, at which we arrived by defining first the BiHom version of Leibniz algebras.

In this paper, we find the condition on a function ω and a weight v which ensures the equivalency of norms of the Riesz potential and the fractional maximal function in generalized weighted Morrey spaces ℳp,ω⁢(ℝn,v){{\mathcal{M}}_{p,\omega}({\mathbb{R}}^{n},v)} and generalized weighted central Morrey spaces ℳ˙p,ω⁢(ℝn,v){\dot{\mathcal{M}}_{p,\omega}({\mathbb{R}}^{n},v)}, when v belongs to the Muckenhoupt A∞{A_{\infty}}-class.

We study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation (p⁢(t)⁢|x′|α⁢sgn⁡x′)′+q⁢(t)⁢|x|α⁢sgn⁡x=0,(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0, where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions ∫a∞d⁢sp⁢(s)1/α=∞ or ∫a∞d⁢sp⁢(s)1/α<∞.\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty. The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

In this paper, we establish weighted higher order exponential type inequalities in the geodesic space (X,d,μ){({X,d,\mu})} by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given.

The subject of the paper is suggested by G. Janelidze and motivated by his earlier result giving a positive answer to the question posed by S. MacLane whether the Galois theory of homogeneous linear ordinary differential equations over a differential field (which is Kolchin–Ritt theory and an algebraic version of Picard–Vessiot theory) can be obtained as a particular case of G. Janelidze’s Galois theory in categories. One ground category in the Galois structure involved in this theory is dual to the category of commutative rings with unit, and another one is dual to the category of commutative differential rings with unit. In the present paper, we apply the general categorical construction, the particular case of which gives this Galois structure, by replacing “commutative rings with unit” by algebras from any variety V\mathscr{V} of universal algebras satisfying the amalgamation property and a certain condition (of the syntactical nature) for elements of amalgamated free products which was introduced earlier, and replacing “commutative differential rings with unit” by V\mathscr{V}-algebras equipped with additional unary operations which satisfy some special identities to construct a new Galois structure. It is proved that this Galois structure is admissible. Moreover, normal extensions with respect to it are characterized in the case where V\mathscr{V} is any of the following varieties: abelian groups, loops and quasigroups.