Volume 30, Issue 1

The well-posed problem for a nonlocal boundary value problem is considered for a singular system of linear generalized ordinary differential equations. The considered system is analogous to the system corresponding to the n -order ordinary differential equation. Along with the standard well-posed question, there is considered the so-called conditionally well-posed one. The effective sufficient conditions are established for both problems.

We study products of Toeplitz operators with angular symbols on the Bergman space over the upper half-plane. We establish necessary and sufficient conditions for the product of two such Toeplitz operators to give rise to a Toeplitz operator, especially in case one of the symbols is absolutely continuous or with bounded variation. Our conditions make appeal to Volterra and Fredholm integral equations, and to Duhamel–Mikusiński convolution products as well as to functional equations involving Stieltjes integrals. We illustrate our results by concrete examples showing that there are many angular symbols satisfying our natural conditions and ensuring Toeplitzness of such Toeplitz products.

Let 𝔄 {\mathfrak{A}} be a triangular ring and let p n ⁢ ( U 1 , U 2 , … , U n ) {p_{n}(U_{1},U_{2},\dots,U_{n})} denote the ( n - 1 ) {(n-1)} th commutator of elements U 1 , U 2 , … , U n ∈ 𝔄 {U_{1},U_{2},\dots,U_{n}\in\mathfrak{A}} . Suppose that ℕ {\mathbb{N}} is the set of nonnegative integers and 𝔏 = { ξ r } r ∈ ℕ {\mathfrak{L}=\{{\xi_{r}}\}_{r\in\mathbb{N}}} is a sequence of additive mappings on 𝔄 {\mathfrak{A}} such that ξ 0 = i ⁢ d 𝔄 {\xi_{0}=id_{\mathfrak{A}}} , the identity mapping on 𝔄 {\mathfrak{A}} , and for each r ∈ ℕ {r\in\mathbb{N}} , ξ r ⁢ ( p n ⁢ ( U 1 , U 2 , … , U n ) ) = ∑ i 1 + i 2 + ⋯ + i n = r p n ⁢ ( ξ i 1 ⁢ ( U 1 ) , ξ i 2 ⁢ ( U 2 ) , … , ξ i n ⁢ ( U n ) ) {\xi_{r}(p_{n}(U_{1},U_{2},\dots,U_{n}))=\sum_{i_{1}+i_{2}+\cdots+i_{n}=r}p_{n% }(\xi_{i_{1}}(U_{1}),\xi_{i_{2}}(U_{2}),\dots,\xi_{i_{n}}(U_{n}))} for all U 1 , U 2 , … , U n ∈ 𝔄 {U_{1},U_{2},\dots,U_{n}\in\mathfrak{A}} with U 1 ⁢ U 2 ⁢ ⋯ ⁢ U n = 0 {U_{1}U_{2}\cdots U_{n}=0} . In this paper, it is shown that under certain conditions 𝔏 = { ξ r } r ∈ ℕ {\mathfrak{L}=\{{\xi_{r}}\}_{r\in\mathbb{N}}} has the standard form, that is, there exist a higher derivation { d r } r ∈ ℕ {\{{d_{r}}\}_{r\in\mathbb{N}}} on 𝔄 {\mathfrak{A}} and a family { h r } r ∈ ℕ {\{{\mathrm{h}_{r}}\}_{r\in\mathbb{N}}} of additive mappings h r : 𝔄 → 𝒵 ⁢ ( 𝔄 ) {h_{r}:\mathfrak{A}\rightarrow\mathcal{Z}(\mathfrak{A})} satisfying h r ⁢ ( p n ⁢ ( U 1 , U 2 , … , U n ) ) = 0 {h_{r}(p_{n}(U_{1},U_{2},\dots,U_{n}))=0} for all U 1 , U 2 , … , U n ∈ 𝔄 {U_{1},U_{2},\dots,U_{n}\in\mathfrak{A}} with U 1 ⁢ U 2 ⁢ ⋯ ⁢ U n = 0 {U_{1}U_{2}\cdots U_{n}=0} such that for each r ∈ ℕ {r\in\mathbb{N}} , ξ r ⁢ ( U ) = d r ⁢ ( U ) + h r ⁢ ( U ) {\xi_{r}(U)=d_{r}(U)+h_{r}(U)} for all U ∈ 𝔄 {U\in\mathfrak{A}} .

The classifying map of the integral Krichever–Hoehn formal group law is presented as a quotient map of the Lazard ring by some explicit ideal.

In this paper, we introduce and investigate a new concept that we call demicompact elements in Banach algebras as a generalization of demicompact linear operators acting on Banach spaces. Our concept is used to construct a new class of Fredholm perturbations with respect to a given Banach subalgebra B , that contains an inessential ideal k B {k_{B}} and the set of left Fredholm perturbations introduced in [A. Ben Ali and N. Moalla, Fredholm perturbation theory and some essential spectra in Banach algebra with respect to subalgebra, Indag. Math. (N. S.) 28 2017, 2, 276–286]. We show that it is a two-sided closed ideal of B playing an important role in the characterization of the Weyl spectrum of elements affiliated with B .

In this paper, we examine the validity of bicomplex versions of some crucial inequalities with respect to the hyperbolic-valued norm | ⋅ | 𝕜 {|\cdot|_{\Bbbk}} and we discuss some topological and geometric concepts such as completeness, convexity, strict convexity and uniform convexity in the bicomplex setting with respect to the hyperbolic-valued norm ∥ ⋅ ∥ 𝔻 , ⋅ {\|\cdot\|_{\mathbb{D},\cdot}} by defining the concept of 𝔻 {\mathbb{D}} -normed Banach bicomplex A -module and constructing 𝔻 {\mathbb{D}} -normed Banach bicomplex 𝔹 ⁢ ℂ {\mathbb{BC}} -modules l p 𝕜 ⁢ ( 𝔹 ⁢ ℂ ) {l_{p}^{\Bbbk}(\mathbb{BC})} .

We prove the existence and uniqueness of a weak solution to a Dirichlet boundary value problem for a class of nonlinear degenerate elliptic equations in the setting of weighted Sobolev spaces. Our proof is based on the weighted Sobolev spaces theory and the Browder–Minty theorem. First, we transform the problem into an equivalent operator equation; second, we utilized the Browder–Minty theorem to prove the existence and uniqueness of a weak solution to the considered problem.

This paper deals with the problem of the global uniform stability of nonlinear time-varying systems in the presence of perturbations. The main novelty relies on the fact that the proposed approach for stability analysis allows for the computation of the bounds which characterize the asymptotic convergence of solutions to a small ball centered at the origin. Therefore, we generalize some results which have already be announced in the literature. Furthermore, we provide a numerical example to validate the effectiveness of our main result.

In this paper, we solve the Fekete–Szegő problem of a strongly α-close-to-convex functions associated with generalized fractional operator.

If V 1 , … , V n {V_{1},\dots,V_{n}} are translations of a Vitali set by rational numbers, then we prove that ⋃ i = 1 n V i {\bigcup_{i=1}^{n}V_{i}} contains no measurable subset of positive measure. This provides a decomposition of ℝ {{\mathbb{R}}} as a countable union of disjoint sets, any finite union of which has Lebesgue inner measure zero. As a consequence, we present a function δ : ℝ → ( 0 , + ∞ ) {\delta:{\mathbb{R}}\rightarrow(0,+\infty)} for which there is no measurable function f : ℝ → ℝ {f:{\mathbb{R}}\rightarrow{\mathbb{R}}} satisfying 0 < f ≤ δ {0<f\leq\delta} , on any measurable set of positive Lebesgue measure.

In this paper, we study, on the interval I = [ a , b ] {I=[a,b]} , the problem of the solvability and unique solvability of systems of higher-order differential equations with argument deviation u i ( m i ) ( t ) = p i ( t ) u i + 1 ( τ i ( t ) ) + q i ( t )   ( i = 1 , … , n ) u_{i}^{(m_{i})}(t)=p_{i}(t)u_{i+1}(\tau_{i}(t))+q_{i}(t)\quad(i=1,\ldots,n) and u i ( m i ) ( t ) = f i ( t , u i + 1 ( τ i ( t ) ) ) + q 0 ⁢ i ( t )   ( i = 1 , … , n ) , u_{i}^{(m_{i})}(t)=f_{i}\bigl{(}t,u_{i+1}(\tau_{i}(t))\bigr{)}+q_{0i}(t)\quad(% i=1,\ldots,n), under the periodic boundary conditions u i ( j ) ( b ) - u i ( j ) ( a ) = c i ⁢ j   ( i = 1 , … , n , j = 0 , … , m i - 1 ) , u_{i}^{(j)}(b)-u_{i}^{(j)}(a)=c_{ij}\quad(i=1,\ldots,n,\,j=0,\ldots,m_{i}-1), where u n + 1 = u 1 {u_{n+1}=u_{1}} , n ≥ 2 {n\geq 2} , m i ≥ 1 {m_{i}\geq 1} , p i ∈ L ∞ ⁢ ( I ; R ) {p_{i}\in L_{\infty}(I;R)} , q 0 ⁢ i ∈ L ⁢ ( I ; R ) {q_{0i}\in L(I;R)} , f i : I × R → R {f_{i}:I\times R\to R} are Carathéodory class functions, and τ i : I → I {\tau_{i}:I\to I} are measurable functions. The optimal conditions are obtained, which guarantee the unique solvability of the linear problem and take into account the effect of argument deviation. Based on these results, the optimal conditions of the solvability and unique solvability are proved for the nonlinear problem.

Consider ℜ {\Re} as a prime ring which is non-commutative in structure with a suitable characteristic. Here, 𝒵 ⁢ ( ℜ ) {\mathcal{Z}(\Re)} is the center of ℜ {\Re} and 𝒬 {\mathcal{Q}} is the Utumi ring of quotients where 𝒞 {\mathcal{C}} is the extended centroid of ℜ {\Re} . Suppose 𝒫 {\mathcal{P}} to be a Lie ideal of ℜ {\Re} which is non-central. Let 𝒦 {\mathcal{K}} be a generalized derivation of ℜ {\Re} related with derivation μ of ℜ {\Re} . If 𝒦 {\mathcal{K}} satisfies certain typical algebraic identities, then we prove that 𝒦 {\mathcal{K}} is either the identity map or the zero map or the scalar map and further information is also drawn on the associated scalar unless ℜ {\Re} embeds in M 2 ⁢ ( 𝒞 ) {M_{2}(\mathcal{C})} , a matrix ring of order 2 × 2 {2\times 2} over 𝒞 {\mathcal{C}} .

In this paper, we study the existence and attractivity of solutions for an implicit differential equation with composite fractional derivative. By means of Schauder’s fixed point theorem, sufficient conditions for the main results are investigated. An example is presented to illustrate our theory