Volume 53, Issue 1

In this paper, we investigate the stability of an additive-quadratic-quartic functional equation f(x+2y)+f(x− 2y)− 2f(x+y)− 2f(− x− y)− 2f(x− y)− 2f(y− x)+4f(− x)+2f(x)− f(2y)− f(− 2y)+4f(y)+4f(− y)=0$$\begin{align*}f(x+2y)& +f(x-2y)-2f(x+y)-2f(-x- y)-2f(x-y)-2f(y-x)\nonumber \\ &+4f(-x)+ 2f(x)-f(2y)-f(-2y)+4f(y)+4f(-y)=0 \end{align*}$$ by the direct method in the sense of Găvruta.

The goal of this work is to introduce and study two new types of ordered soft separation axioms, namely soft T i -ordered and strong soft T i -ordered spaces ( i = 0, 1, 2, 3, 4). These two types are formulated with respect to the ordinary points and the distinction between them is attributed to the nature of the monotone neighborhoods. We provide several examples to elucidate the relationships among these concepts and to show the relationships associate them with their parametric topological ordered spaces and p-soft T i -ordered spaces. Some open problems on the relationships between strong soft T i -ordered and soft T i -ordered spaces ( i = 2, 3, 4) are posed. Also, we prove some significant results which associate both types of the introduced ordered axioms with some notions such as finite product soft spaces, soft topological and soft hereditary properties. Furthermore, we describe the shape of increasing (decreasing) soft closed and open subsets of soft regularly ordered spaces; and demonstrate that a condition of strong soft regularly ordered is sufficient for the equivalence between p-soft T 1 -ordered and strong soft T 1 -ordered spaces. Finally, we establish a number of findings that associate soft compactness with some ordered soft separation axioms initiated in this work.

Let f be analytic in D={z:|z| < 1}{\mathbb{D}}=\{z:|z\mathrm{|\hspace{0.17em}\lt \hspace{0.17em}1\}} with f(z)=z+∑n=2∞anznf(z)=z+{\sum }_{n\mathrm{=2}}^{\infty }{a}_{n}{z}^{n}, and for α ≥ 0 and 0 < λ ≤ 1, let ℬ1(α,λ){ {\mathcal B} }_{1}(\alpha ,\lambda ) denote the subclass of Bazilevič functions satisfying |f′(z)(zf(z))1−α−1|<λ\left|f^{\prime} (z){\left(\frac{z}{f(z)}\right)}^{1-\alpha }-1\right|\lt \lambda for 0 < λ ≤ 1. We give sharp bounds for various coefficient problems when f∈ℬ1(α,λ)f\in { {\mathcal B} }_{1}(\alpha ,\lambda ), thus extending recent work in the case λ = 1.

In this study, at first we prove that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that the main result of the paper [ Proximal normal structure and nonexpansive mappings , Studia Math. 171 (2005), 283–293] immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper [ Convergence of Picard's iteration using projection algorithm for noncyclic contractions , Indag. Math. 30 (2019), no. 1, 227–239] is obtained exactly from Picard’s iteration sequence.

In this article, we study the generalized parabolic parametric Marcinkiewicz integral operators ℳΩ,h,Φ,λ(r){ {\mathcal M} }_{{\Omega },h,{\Phi },\lambda }^{(r)} related to polynomial compound curves. Under some weak conditions on the kernels, we establish appropriate estimates of these operators. By the virtue of the obtained estimates along with an extrapolation argument, we give the boundedness of the aforementioned operators from Triebel-Lizorkin spaces to L p spaces under weaker conditions on Ω and h . Our results represent significant improvements and natural extensions of what was known previously.

The structure of Jordan centralizer maps is investigated on trivial extension algebras. One may obtain some conditions under which a Jordan centralizer map on a trivial extension algebra is a centralizer map. As an application, we characterize the Jordan centralizer map on a triangular algebra.

Many mathematicians defined and studied soft separation axioms and soft continuity in soft spaces by using ordinary points of a topological space X . Also, some of them studied the same concepts by using soft points. In this paper, we introduce the concepts of soft pc−Ti{p}_{c}-{T}_{i} and soft pc−Ti⁎{p}_{c}-{T}_{i}^{\ast }, i=0,1,2i=0,1,2 by using the concept of soft pc{p}_{c}-open sets in soft topological spaces. We explore several properties of such spaces. We also investigate the relationship among these spaces and provide a counter example when it is needed.

We present upper and lower estimates of the error of approximation of periodic functions by Fejér means in the Lebesgue spaces L2πp{L}_{2{\pi }}^{p}. The estimates are given in terms of a K -functional for 1≤p≤∞1\le p\le \infty and in terms of the first modulus of continuity in the case 1<p<∞1\lt p\lt \infty . We pay attention to the involved constants.

The existence of a.e. monotonic solutions for functional quadratic Hammerstein integral equations with the perturbation term is discussed in Orlicz spaces. We utilize the strategy of measure of noncompactness related to the Darbo fixed point principle. As an application, we discuss the presence of solution of the initial value problem with nonlocal conditions.

Our main interest in this article is to introduce and study the class of θ -generalized demimetric mappings in Hadamard spaces. Also, a Halpern-type proximal point algorithm comprising this class of mappings and resolvents of monotone operators is proposed, and we prove that it converges strongly to a fixed point of a θ -generalized demimetric mapping and a common zero of a finite family of monotone operators in a Hadamard space. Furthermore, we apply the obtained results to solve a finite family of convex minimization problems, variational inequality problems and convex feasibility problems in Hadamard spaces.

We examine how implicit functions on ILB-Fréchet spaces can be obtained without metric or norm estimates which are classically assumed. We obtain implicit functions defined on a domain D which is not necessarily open, but which contains the unit open ball of a Banach space. The corresponding inverse function theorem is obtained, and we finish with an open question on the adequate (generalized) notion of differentiation, needed for the corresponding version of the Fröbenius theorem.

Riemann-Liouville fractional differential equations with a constant delay and impulses are studied in this article. The following two cases are considered: the case when the lower limit of the fractional derivative is fixed on the whole interval of consideration and the case when the lower limit of the fractional derivative is changed at any point of impulse. The initial conditions as well as impulsive conditions are defined in an appropriate way for both cases. The explicit solutions are obtained and applied to the study of finite time stability.

In this note, we will revisit the special atom space introduced in the early 1980s by Geraldo De Souza and Richard O’Neil. In their introductory work and in later additions, the space was mostly studied on the real line. Interesting properties and connections to spaces such as Orlicz, Lipschitz, Lebesgue, and Lorentz spaces made these spaces ripe for exploration in higher dimensions. In this article, we extend this definition to the plane and space and show that almost all the interesting properties such as their Banach structure, Hölder’s-type inequalities, and duality are preserved. In particular, dual spaces of special atom spaces are natural extension of Lipschitz and generalized Lipschitz spaces of functions in higher dimensions. We make the point that this extension could allow for the study of a wide range of problems including a connection that leads to what seems to be a new definition of Haar functions, Haar wavelets, and wavelets on the plane and on the space.

The purpose of this article is to study the method of approximation for zeros of the sum of a finite family of maximally monotone mappings and prove strong convergence of the proposed approximation method under suitable conditions. The method of proof is of independent interest. In addition, we give some applications to the minimization problems and provide a numerical example which supports our main result. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

In this article, we present a sufficient condition about the length of the interval for the existence and uniqueness of mild solutions to a fractional boundary value problem with Sturm-Liouville boundary conditions when the data function is of Lipschitzian type. Moreover, we present an application of our result to the eigenvalues problem and its connection with a Lyapunov-type inequality.

In this article, we prove the generalized Hyers-Ulam stability for the following additive-quartic functional equation:f(x+3y)+f(x−3y)+f(x+2y)+f(x−2y)+22f(x)+24f(y)=13[f(x+y)+f(x−y)]+12f(2y),f(x+3y)+f(x-3y)+f(x+2y)+f(x-2y)+22f(x)+24f(y)=13{[}f(x+y)+f(x-y)]+12f(2y),where f maps from an additive group to a complete non-Archimedean normed space.

In this manuscript, a numerical approach for the stronger concept of exact controllability ( total controllability ) is provided. The proposed control problem is a nonlinear fractional differential equation of order α∈(1,2]\alpha \in (1,2] with non-instantaneous impulses in finite-dimensional spaces. Furthermore, the numerical controllability of an integro-differential equation is briefly discussed. The tool for studying includes the Laplace transform, the Mittag-Leffler matrix function and the iterative scheme. Finally, a few numerical illustrations are provided through MATLAB graphs.

In this work, we introduce two new inertial-type algorithms for solving variational inequality problems (VIPs) with monotone and Lipschitz continuous mappings in real Hilbert spaces. The first algorithm requires the computation of only one projection onto the feasible set per iteration while the second algorithm needs the computation of only one projection onto a half-space, and prior knowledge of the Lipschitz constant of the monotone mapping is not required in proving the strong convergence theorems for the two algorithms. Under some mild assumptions, we prove strong convergence results for the proposed algorithms to a solution of a VIP. Finally, we provide some numerical experiments to illustrate the efficiency and advantages of the proposed algorithms.

This paper is concerned with a system governed by nonsingular delay differential equations. We study the β -Ulam-type stability of the mentioned system. The investigations are carried out over compact and unbounded intervals. Before proceeding to the main results, we convert the system into an equivalent integral equation and then establish an existence theorem for the addressed system. To justify the application of the reported results, an example along with graphical representation is illustrated at the end of the paper.

The combined systems of integral equations have become of great importance in various fields of sciences such as electromagnetic and nuclear physics. New classes of the merged type of Urysohn Volterra-Chandrasekhar quadratic integral equations are proposed in this paper. This proposed system involves fractional Urysohn Volterra kernels and also Chandrasekhar kernels. The solvability of a coupled system of integral equations of Urysohn Volterra-Chandrasekhar mixed type is studied. To realize the existence of a solution of those mixed systems, we use the Perov’s fixed point combined with the Leray-Schauder fixed point approach in generalized Banach algebra spaces.

Given Hilbert space operators T,S∈B(ℋ)T,S\in B( {\mathcal H} ), let Δ\text{&#x0394;} and δ∈B(B(ℋ))\delta \in B(B( {\mathcal H} )) denote the elementary operators ΔT,S(X)=(LTRS−I)(X)=TXS−X{\text{&#x0394;}}_{T,S}(X)=({L}_{T}{R}_{S}-I)(X)=TXS-X and δT,S(X)=(LT−RS)(X)=TX−XS{\delta }_{T,S}(X)=({L}_{T}-{R}_{S})(X)=TX-XS. Let d=Δd=\text{&#x0394;} or δ\delta . Assuming T commutes with S∗{S}^{\ast }, and choosing X to be the positive operator S∗nSn{S}^{\ast n}{S}^{n} for some positive integer n , this paper exploits properties of elementary operators to study the structure of n -quasi [m,d]{[}m,d]-operators dT,Sm(X)=0{d}_{T,S}^{m}(X)=0 to bring together, and improve upon, extant results for a number of classes of operators, such as n -quasi left m -invertible operators, n -quasi m -isometric operators, n -quasi m -self-adjoint operators and n -quasi (m,C)(m,C) symmetric operators (for some conjugation C of ℋ {\mathcal H} ). It is proved that Sn{S}^{n} is the perturbation by a nilpotent of the direct sum of an operator S1n=S|Sn(ℋ)¯n{S}_{1}^{n}={\left(S{|}_{\overline{{S}^{n}( {\mathcal H} )}}\right)}^{n} satisfying dT1,S1m(I1)=0{d}_{{T}_{1},{S}_{1}}^{m}({I}_{1})=0, T1=TSn(ℋ)¯{{T}_{1}=T}_{\overline{{S}^{n}( {\mathcal H} )}}, with the 0 operator; if S is also left invertible, then Sn{S}^{n} is similar to an operator B such that dB∗,Bm(I)=0{d}_{{B}^{\ast },B}^{m}(I)=0. For power bounded S and T such that ST∗−T∗S=0S{T}^{\ast }-{T}^{\ast }S=0 and ΔT,S(S∗nSn)=0{\text{&#x0394;}}_{T,S}({S}^{\ast n}{S}^{n})=0, S is polaroid (i.e., isolated points of the spectrum are poles). The product property, and the perturbation by a commuting nilpotent property, of operators T,ST,S satisfying dT,Sm(I)=0{d}_{T,S}^{m}(I)=0, given certain commutativity properties, transfers to operators satisfying S∗ndT,Sm(I)Sn=0{S}^{\ast n}{d}_{T,S}^{m}(I){S}^{n}=0.

In this paper, we study the Hyers-Ulam stability of a nonautonomous semilinear reaction-diffusion equation. More precisely, we consider a nonautonomous parabolic equation with a diffusion given by the fractional Laplacian. We see that such a stability is a consequence of a Gronwall-type inequality.

In this work, we study Hasimoto surfaces for the second and third classes of curve evolution corresponding to a Frenet frame in Minkowski 3-space. Later, we derive two formulas for the differentials of the second and third Hasimoto-like transformations associated with the repulsive-type nonlinear Schrödinger equation.

In this paper, the notion of an operator γ\gamma on a supra topological space (X,μ)(X,\mu ) is studied and then utilized to analyze supra γ\gamma -open sets. The notions of μγ{\mu }_{\gamma }- g .closed sets on the subspace are introduced and investigated. Furthermore, some new μγ{\mu }_{\gamma }-separation axioms are formulated and the relationships between them are shown. Moreover, some characterizations of the new functions via operator γ\gamma on μ\mu are presented and investigated. Finally, we give some properties of S(γ,β){S}_{(\gamma ,\beta )}-closed graph and strongly S(γ,β){S}_{(\gamma ,\beta )}-closed graph.

In this article, a new problem that is called system of split mixed equilibrium problems is introduced. This problem is more general than many other equilibrium problems such as problems of system of equilibrium, system of split equilibrium, split mixed equilibrium, and system of split variational inequality. A new iterative algorithm is proposed, and it is shown that it satisfies the weak convergence conditions for nonexpansive mappings in real Hilbert spaces. Also, an application to system of split variational inequality problems and a numeric example are given to show the efficiency of the results. Finally, we compare its rate of convergence other algorithms and show that the proposed method converges faster.

Let G be a group with identity e , R be a G -graded commutative ring with a nonzero unity 1 and M be a G -graded R -module. In this article, we introduce and study the concept of almost graded multiplication modules as a generalization of graded multiplication modules; a graded R -module M is said to be almost graded multiplication if whenever a∈h(R)a\in h(R) satisfies AnnR(aM)=AnnR(M){\text{Ann}}_{R}(aM)={\text{Ann}}_{R}(M), then (0:Ma)={0}(0{:}_{M}a)=\{0\}. Also, we introduce and study the concept of almost graded comultiplication modules as a generalization of graded comultiplication modules; a graded R -module M is said to be almost graded comultiplication if whenever a∈h(R)a\in h(R) satisfies AnnR(aM)=AnnR(M){\text{Ann}}_{R}(aM)={\text{Ann}}_{R}(M), then aM=MaM=M. We investigate several properties of these classes of graded modules.

In this article, we propose an inertial extrapolation-type algorithm for solving split system of minimization problems: finding a common minimizer point of a finite family of proper, lower semicontinuous convex functions and whose image under a linear transformation is also common minimizer point of another finite family of proper, lower semicontinuous convex functions. The strong convergence theorem is given in such a way that the step sizes of our algorithm are selected without the need for any prior information about the operator norm. The results obtained in this article improve and extend many recent ones in the literature. Finally, we give one numerical example to demonstrate the efficiency and implementation of our proposed algorithm.

Generalizing the case of a normal operator in a complex Hilbert space, we give a straightforward proof of the non-hypercyclicity of a scalar type spectral operator A in a complex Banach space as well as of the collection {etA}t≥0{\{{e}^{tA}\}}_{t\ge 0} of its exponentials, which, under a certain condition on the spectrum of the operator A , coincides with the C0{C}_{0}-semigroup generated by A . The spectrum of A lying on the imaginary axis, we also show that non-hypercyclic is the strongly continuous group {etA}t∈ℝ{\{{e}^{tA}\}}_{t\in {\mathbb{R}}} of bounded linear operators generated by A . From the general results, we infer that, in the complex Hilbert space L2(ℝ){L}_{2}({\mathbb{R}}), the anti-self-adjoint differentiation operator A≔ddxA:= \tfrac{\text{d}}{\text{d}x} with the domain D(A)≔W21(ℝ)D(A):= {W}_{2}^{1}({\mathbb{R}}) is non-hypercyclic and so is the left-translation strongly continuous unitary operator group generated by A .

The exponential spline function is presented to find the numerical solution of third-order singularly perturbed boundary value problems. Convergence analysis of the method is briefly discussed, and it is shown to be sixth order convergence. To validate the applicability of the method, some model problems are solved for different values of the perturbation parameter, and the numerical results are presented both in tables and graphs. Furthermore, the present method gives more accurate solution than some methods existing in the literature.

Existence of mild solution for noninstantaneous impulsive fractional order integro-differential equations with local and nonlocal conditions in Banach space is established in this paper. Existence results with local and nonlocal conditions are obtained through operator semigroup theory using generalized Banach contraction theorem and Krasnoselskii’s fixed point theorem, respectively. Finally, illustrations are added to validate derived results.

Properties of linear regression of order statistics and their functions are usually utilized for the characterization of distributions. In this paper, based on such statistics, the concept of Pearson covariance and the pseudo-covariance measure of dependence is used to characterize the exponential, Pearson and Pareto distributions.