Abdullah m Alotaibi

• PhD
• Professor (Full) at King Abdulaziz University

This work introduces τ$$ \tau $$‐Bézier–Bernstein‐integral type operators, along with local approximation results, a direct approximation theorem that leverages the Ditzian–Totik modulus of smoothness, and a quantitative Voronovskaja‐type theorem using the Ditzian–Totik modulus of continuity. Additionally, we derive the convergence rate for differe...

In this research paper, we construct a new sequence of Riemann–Liouville type fractional α$$ \alpha $$‐Bernstein–Kantorovich operators. We prove a Korovkin type approximation theorem and discuss the rate of convergence with the first order modulus of continuity of these operators. Further, we study Voronovskaja type theorem, quantitative Voronovska...

In the present article we want to study the convergence and other related properties of Schurer type Bernstein–Kantorovich–Stancu operators with Shifted knots. First we design the Bernstein–Kantorovich operators of the of Stancu type polynomials by Shifted knots of real parameters by including the Schurer positive real parameters, then obtain the c...

The aim of this article is to present modified \(\alpha \)-summation-integral type operators based on a strictly positive continuous function. The rate as per instruction of approximation in terms of modulus of continuity and Voronovskaja-type asymptotic formulas are studied. Finally, we use Maple software to show the convergence of the operators t...

In this article, we the study generalized family of positive linear operators based on two parameters, which are a broad family of many well-known linear positive operators, e.g., Baskakov-Durrmeyer, Baskakov-Szász, Szász-Beta, Lupaş-Beta, Lupaş-Szász, genuine Bernstein-Durrmeyer, and Pǎltǎnea. We first find direct estimates in terms of the second-...

In this paper, we consider a bivariate extension of blending type approximation by Lupa?-Durrmeyer type operators involving P?lya Distribution. We illustrate the convergence rate of these type operators using Peetre?s K-functional, modulus of smoothness and for functions in a Lipschitz type space.

We introduce the sequence of Stancu variant of α-Schurer-Kantorovich operators and systematically investigate some basic estimates. We also obtain the uniform convergence theorem and the order of approximation in terms of suitable modulus of continuity for our newly defined operators. Moreover, we investigate rate of convergence by means of Peetre'...

We define the summation-integral-type operators involving the ideas of Pólya–Eggenberger distribution and Bézier basis functions, and study some of their basic approximation properties. In addition, by means of the Ditzian–Totik modulus of smoothness, we study a direct theorem as well as a quantitative Voronovskaja-type theorem for our newly constr...

Two concepts—one of Darbo-type theorem and the other of Banach sequence spaces—play a very important and active role in ongoing research on existence problems. We first demonstrate the generalized Darbo-type fixed point theorems involving the concept of continuous functions. Keeping one of these theorems into our account, we study the existence of...

In the present manuscript, we consider ?-Bernstein-Durrmeyer operators involving a strictly positive continuous function. Firstly, we prove a Voronovskaja type, quantitative Voronovskaja type and Gr?ss-Voronovskaja type asymptotic formula, the rate of convergence by means of the modulus of continuity and for functions in a Lipschitz type space. Fin...

In this manuscript, we consider the Baskakov-Jain type operators involving two parameters ? and ?. Some approximation results concerning the weighted approximation are discussed. Also, we find a quantitative Voronovskaja type asymptotic theorem and Gr?ss Voronovskaya type approximation theorem for these operators. Some numerical examples to illustr...

The main purpose of this research article is to construct a Dunkl extension of $(p,q)$ ( p , q ) -variant of Szász–Beta operators of the second kind by applying a new parameter. We obtain Korovkin-type approximation theorems, local approximations, and weighted approximations. Further, we study the rate of convergence by using the modulus of continu...

The aim of this work is to give some fixed point results based on the technique of measure of noncompactness which extend the classical Darbo’s theorem. With the help of our Darbo-type theorem, we obtain the existence of solution of implicit fractional integral equations in C(I,ℓpα) (collection of all continuous functions from I=[0,a] (a>0) to ℓpα)...

In the present work, we construct a new sequence of positive linear operatorsinvolving Pólya distribution. We compute a Voronovskaja type and a Grüss–Voronovskaja type asymptotic formula as well as the rate of approximation by using the modulus of smoothness and for functions in a Lipschitz type space. Lastly, we provide some numerical results, whi...

In the present article we study the approximation properties of Phillips operators by q-Dunkl generalization. We construct the operators in a new q-Dunkl form and obtain the approximation properties in weighted function space. We give the rate of convergence in terms of Lipschitz class by initiate the modulus of continuity and finally, we present s...

In this work, we construct the genuine Durrmeyer–Stancu type operators depending on parameter α in [0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[0,1]$\end{docume...

We construct the Baskakov–Kantorovich operators based on shape parameter \(\alpha\) by linking with Stancu operators to approximate functions over unbounded intervals. We establish local approximation results with the help of suitable modulus of continuity, \({\mathcal {K}}\)-functional and Lipschitz-type space. Further, we obtain the weighted appr...

In the present paper, we intend to make an approach to introduce and study the applications of fractional-order difference operators by generating Orlicz almost null and almost convergent sequence spaces. We also show that aforesaid spaces are linearly isomorphic and BK-spaces. Further, we investigate inclusion relations between newly formed sequen...

In this work, we present a Durrmeyer-type operator having the basis functions in summation and integration due to Aral (Math Commun 24:119–131, 2019) and Pǎltǎnea (Carpath J Math 24(3):378–385, 2008) that preserve the constant functions. We compute the rate of convergence of these operators in a weighted space and also computed a quantitative Voron...

The aim of this paper is to define two sorts of convergence in measure, that is, outer and inner statistical convergence, for double sequences of fuzzy-valued measurable functions and demonstrate that both kinds of convergence are equivalent in a finite measurable set. We also define the notion of statistical convergence in measure for double seque...

Abstract The main purpose of this article is to introduce a new generalization of q-Phillips operators generated by Dunkl exponential function. We establish some approximation results for these operators. We also determine the order of approximation, and the rate of convergence in terms of the modulus of continuity of order one and two. Moreover, w...

Abstract The main purpose of this paper is to introduce a generalized class of Dunkl type Szász operators via post quantum calculus on the interval [12,∞) $[ \frac{1}{2},\infty )$. This type of modification allows a better estimation of the error on [12,∞) $[ \frac{1}{2},\infty ) $ rather than [0,∞) $[ 0,\infty )$. We establish Korovkin type result...

Abstract We apply the concept of measure of noncompactness to study the existence of solution of second order differential equations with initial conditions in the sequence space n(ϕ) $n(\phi)$.

The “oldest quartic” functional equation 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...

The object of this paper to construct Dunkl type Szász operators via post-quantum calculus. We obtain some approximation results for these new operators and compute convergence of the operators by using the modulus of continuity. Furthermore, we obtain the rate of convergence of these operators for functions belonging to the Lipschitz class. We als...

The purpose of this paper is to introduce the notion of weighted almost convergence of a sequence and prove that this sequence endowed with the sup-norm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddside...

The present paper deals with the construction of Baskakov Durrmeyer operators, which preserve the linear functions, in (p,q) -calculus. More precisely, using (p,q) -Gamma function we introduce genuine mixed type Baskakov Durrmeyer operators having Baskakov and Szász basis functions. After construction of the operators and calculations of their mome...

The aim of this paper is to introduce a new generalization of Bleimann-Butzer-Hahn operators by using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document...

In the present paper, we construct a new sequence of Bernstein Kantorovich operators depending on a parameter α. The uniform convergence of the operators, rate of convergence in local and global senses in terms of first and second order modulus of continuity are studied. Some graphs and numerical results presenting the advantages of our constructio...

The study of infinite matrices is important in the theory of summability and in approximation. In particular, Toeplitz matrices or regular matrices and almost regular matrices have been very useful in this context. In this paper, we propose to use a more general matrix method to obtain necessary and sufficient conditions to sum the conjugate derive...

In the present paper, we introduce a new modification of Szász-Mirakyan operators based on (p,q)-integers and investigate their approximation properties. We obtain weighted approximation and Voronovskaya-type theorem for new operators.

We define the notions of pointwise and uniform statistical convergence of double sequences of fuzzy valued functions and obtain relationships between these two kinds of convergence. We further introduce the notion of equi-statistical convergence of double sequences of fuzzy valued functions and show that uniformly statistically convergent double se...

The purpose of this paper is to introduce a modification of q-Dunkl generalization of exponential functions. These types of operators enable better error estimation on the interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{...

In this article, we use the technique based upon measures of noncompactness in conjunction with a Darbo-type fixed point theorem with a view to studying the existence of solutions of infinite systems of second-order differential equations in the Banach sequence space ℓp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepack...

In this paper we introduce q-Szász-Mirakjan-Kantorovich operators generated by a Dunkl generalization of the exponential function and we propose two different modifications of the q-Szász-Mirakjan-Kantorovich operators which preserve some test functions. We obtain some approximation results with the help of the well-known Korovkin theorem and the w...

In this paper, we derive some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the sequence space \(\ell_{p}(r,s,t;B^{(m)})\) which is related to \(\ell_{p}\) spaces. By applying the Hausdorff measure of noncompactness, we obtain the necessary and sufficient conditions for su...

In this paper, we are introducing pertinent Euler–Lagrange–Jensen type k-quintic functional equations and investigate the ‘Ulam stability’ of these new k-quintic functional mappings f:XY, where X is a real normed linear space and Y a real complete normed linear space. We also solve the Ulam stability problem for Euler–Lagrange–Jensen alternative k-...

The “oldest quartic” functional equation was introduced and solved by the author of this paper (see: Glas. Mat. Ser. III 34 (54) (1999), no. 2, 243-252) which is of the form: f(x + 2y) + f(x − 2y) = 4[f(x + y) + f(x − y)] − 6f(x)+ 24f(y). Interesting results have been achieved by S.A. Mohiuddine et al., since 2009. In this paper, we are introducing...

In 1940 S. M. Ulam proposed at the University of Wisconsin the problem: "Give conditions in order for a linear mapping near an approximately linear mapping to exist". In 1982-2013, the second author solved the above Ulam problem for a variety of quadratic mappings. Interesting stability results have been achieved by S. A. Mohiuddine et al., since 2...

In 1940 S. M. Ulam proposed at the University of Wisconsin the problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist”. In 1982-2013, the second author solved the above Ulam problem for a variety of quadratic mappings. Interesting stability results have been achieved by S. A. Mohiuddine et al., since 2...

We propose a Kantorovich variant of ( p , q ) -analogue of Szász-Mirakjan operators. We establish the moments of the operators with the help of a recurrence relation that we have derived and then prove the basic convergence theorem. Next, the local approximation and weighted approximation properties of these new operators in terms of modulus of c...

The “oldest cubic” functional equation was introduced and solved by the second author of this paper (see: Glas. Mat. Ser. III 36(56) (2001), no. 1, 63-72). which is of the form: f(x + 2y) = 3f(x + y) + f(x - y) - 3f(x) + 6f(y) For further research in various normed spaces, we are introducing new cubic functional equations, and establish fundamental...

H. Aktuğlu and H. Gezer [Central European J. Math. 7 (2009), 558-567] introduced the concepts of lacunary equistatistical convergence, lacunary statistical pointwise convergence and lacunary statistical uniform convergence for sequences of functions. In this paper, we apply the notion of lacunary equistatistical convergence to prove a Korovkin type...

The boundedness character of positive solutions of the difference equation in the title, where and , , , is studied here.

In this paper, we prove some tripled fixed point theorems for Meir-Keleer condensing operator in a Banach space by using L-functions. We apply these results to establish the existence of solutions for a system of functional integral equations of Volterra type.

The concepts of lambda-equi-statistical convergence, lambda-statistical pointwise convergence and lambda-statistical uniform convergence for sequences of functions were introduced recently by Srivastava, Mursaleen and Khan [Math. Comput. Modelling, 55 (2012) 2040-2051]. In this paper, we apply the notion of lambda-equi-statistical convergence to pr...

We introduce some new generalized difference sequence spaces by means of ideal convergence, infinite matrix, and a sequence of modulus functions over -normed spaces. We also make an effort to study several properties relevant to topological, algebraic, and inclusion relations between these spaces.

The aim of this special issue is to focus on recent developments and achievements in the theory of function spaces, sequences spaces and their geometry, and compact operators and their applications in various fields of applied mathematics, engineering, and other sciences. The theory of sequence spaces is powerful tool for obtaining positive results...

The aim of this special issue is to focus on the latest developments and achievements of the theory of compact operators on function spaces and their applications in differential , functional, and integral equations. The concept of the compactness plays a fundamental role in creating the basis of several investigations conducted in nonlinear analys...

The aim of this paper is to introduce some generalized spaces of double sequences with the help of the Musielak-Orlicz function M=(Mjk) and four-dimensional bounded-regular (shortly, RH-regular) matrices A=(anmjk) over n-normed spaces. Some topological properties and inclusion relations between these spaces are investigated. MSC: 40A05, 40D25.

The boundedness character of positive solutions of the next max-type system of difference equations with min{A, p, q}> 0, is characterized.

We define some classes of double entire and analytic sequences by means of Orlicz functions. We study some relevant algebraic and topological properties. Further some inclusion relations among the classes are also examined.

We introduce some vector-valued sequence spaces defined by a Musielak-Orlicz function and the concepts of lacunary convergence and strong (íµí°-convergence, where íµí° (íµí±Ž íµí±–íµí±˜) is an infinite matrix of complex numbers. We also make an effort to study some topological properties and some inclusion relations between these spaces.

The aim of this paper is to introduce some interval valued double difference sequence spaces by means of Musielak-Orlicz function M = ( M i j ) . We also determine some topological properties and inclusion relations between these double difference sequence spaces.

The idea of [λ, μ] -almost convergence (briefly, F[λ, μ] -convergence) has been recently introduced and studied by Mohiuddine and Alotaibi (2014). In this paper first we define a norm on F[λ, μ] such that it is a Banach space and then we define and characterize those four-dimensional matrices which transform F[λ, μ]-convergence of double sequences...

We introduce the notion of weighted A-statistical convergence of a sequence, where A represents the nonnegative regular matrix. We also prove the Korovkin approximation theorem by using the notion of weighted A-statistical convergence. Further, we give a rate of weighted A-statistical convergence and apply the classical Bernstein polynomial to cons...

Here we give natural explanations for some recent relations for the solutions of the following difference equation Z(n+2) = A(n+1)(3)/Z(n)(2)-2Z(n+1)(2), n epsilon N-0, and extend them for the case of complex initial values. We also present some other properties related to solutions of the equation, as well as some properties of the solutions of an...

In this paper, the Trefftz collocation method is applied to solve the inverse Cauchy problem of anisotropic elasticity, wherein both tractions as well as displacements are prescribed at a small part of the boundary of an arbitrary simply/multiply connected anisotropic elastic domain. The Stroh formalism is used to construct the Trefftz basis functi...

The purpose of this paper is to introduce new classes of generalized seminormed difference sequence spaces defined by a Musielak-Orlicz function. We also study some topological properties and prove some inclusion relations between resulting sequence spaces.

In the present paper we introduce double sequence space m 2 ( M , A , ϕ , p , ∥ ⋅ , … , ⋅ ∥ ) defined by a sequence of Orlicz functions over n -normed space. We examine some of its topological properties and establish some inclusion relations. MSC: 40A05, 46A45.

The aim of this paper is to establish some stability results concerning the Cauchy functional equation f (x + y) = f (x) + f (y) in the framework of intuitionistic fuzzy normed spaces.

A simple and efficient method for modeling piezoelectric composite and porous materials to solve direct and inverse 2D problems is presented in this paper. The method is based on discretizing the problem domain into arbitrary polygonal-shaped regions that resemble the physical shapes of grains in piezoelectric polycrystalline materials, and utilizi...

Some sufficient conditions which guaranty the attractivity of all positive solutions of the next system of difference equations xn=max{a,y n-1p∏;j=2kxn-jqj}, yn=max{a,xn-1∏j=2kyn-jqj}, nεN0,where a,p>0, and qj,jε{2,⋯,k}, are nonnegative numbers, are given. The boundedness character of positive solutions of the system is also studied.

The aim of this paper is to define the notions of ideal convergence, I-bounded for double sequences in setting of locally solid Riesz spaces and study some results related to these notions. We also define the notion of I*-convergence for double sequences in locally solid Riesz spaces and establish its relationship with ideal convergence.

The aim of this paper is to introduce some new double difference sequence spaces with the help of the Musielak-Orlicz function ℱ = ( F j k ) and four-dimensional bounded-regular (shortly, RH -regular) matrices A = ( a n m j k ) . We also make an effort to study some topological properties and inclusion relations between these double dif...

The idea of I-convergence of real sequences was introduced by Kostyrko et al., (2000/01) and also independently by Nuray and Ruckle (2000). In this paper, we introduce the concepts of (Δ (m) , I)-statistical convergence of order α and strong (Δ p (m) , I)-Cesàro summability of order α of real sequences and investigated their relationship.

In this article, the Mesh less Local Petrov-Galerkin (MLPG) Mixed Collocation Method is developed to solve the Cauchy inverse problems of Steady-State Heat Transfer In the MLPG mixed collocation method, the mixed scheme is applied to independently interpolate temperature as well as heat flux using the same meshless basis functions The balance and c...

Kuhn-Tucker points play a fundamental role in the analysis and the numerical solution of monotone inclusion problems, providing in particular both primal and dual solutions. We propose a class of strongly convergent algorithms for constructing the best approximation to a reference point from the set of Kuhn-Tucker points of a general Hilbertian com...

In the first part of the paper, following the works of Pehlivan et al. (2004), we study the set of all A-statistical cluster points of sequences in m-dimensional spaces and make certain investigations on the set of all A-statistical cluster points of sequences in m-dimensional spaces. In the second part of the paper, we apply this notion to study a...

In this expository article, a variety of computational methods, such as Collocation, Finite Volume, Finite Element, Boundary Element, MLPG (Meshless Local Petrov Galerkin), Trefftz methods, and Method of Fundamental Solutions, etc., which are often used in isolated ways in contemporary literature are presented in a unified way, and are illustrated...

The purpose of this paper is to generalize the concept of almost convergence for double sequence through the notion of de la Vallée- Poussin mean for double sequences. We also define and characterize the generalized regularly almost conservative and almost coercive four-dimensional matrices. Further, we characterize the infinite matrices which tran...

In this paper we define the notion of statistical convergence and statistical Cauchy in a random paranormed space. We establish some relations between them and obtain a subsequential characterization for statistical convergence in random paranormed space.

Some sufficient conditions which guaranty the attractivity of all positive solutions of the next system of difference equations where a, p > 0, and q_j, j\in{2, . . .,k}, are nonnegative numbers, are given. The boundedness character of positive solutions of the system is also studied.

We propose a new class of primal-dual Fejer monotone algorithms for solving systems of com- posite monotone inclusions. Our construction is inspired by a framework used by Eckstein and Svaiter for the basic problem of finding a zero of the sum of two monotone operators. At each iteration, points in the graph of the monotone operators present in the...

In this paper we study the notion of statistical -summability, which is a generalization of statistical A-summability. We study here many other related concepts and its relations with statistical convergence and λ-statistical convergence and provide some interesting examples.

The notion of statistical convergence was defined by Fast (Colloq. Math. 2:241-244, 1951) and over the years was further studied by many authors in different setups. In this paper, we define and study statistical τ-convergence, statistically τ-Cauchy and -convergence through de la Vallée-Poussin mean in a locally solid Riesz space. MSC: 40A35,...

In this paper, we introduce an iterative process which converges strongly to a common solution of a finite family of variational inequality problems for monotone mappings with Bregman distance function. Our convergence theorem is applied to the convex minimization problem. Our theorems extend and unify most of the results that have been proved for...

In this work, we introduce the notion of intuitionistic generalized fuzzy metric space by using the idea of intuitionistic fuzzy set due to Atanassov. We determine some coupled coincidence point results for compatibility of two mappings, that is, F:X×X→X and g:X→X, in the framework of intuitionistic generalized fuzzy metric spaces endowed with part...

In this study, using the notion of ( V , λ ) -summability and λ -statistical convergence, we introduce the concepts of strong ( V , λ , p ) -summability and λ -statistical convergence of order α of real-valued functions which are measurable (in the Lebesgue sense) in the interval ( 1 , ∞ ) . Also some relations between λ -statistical convergence of...

Spingarn's method of partial inverses has found many applications in nonlinear analysis and in optimization. We show that it can be employed to solve composite monotone inclusions in duality, thus opening a new range of applications for the partial inverse formalism. The versatility of the resulting primal-dual splitting algorithm is illustrated th...

We prove a Korovkin type approximation theorem for a function of two variables by using the notion of statistical A-summability. We also construct an example by Meyer-König and Zeller operators to show that our result is stronger than those previously proved by other authors.

We prove that all positive solutions of the next system of difference equationsxn+1=A+ynpxn-1q,yn+1=A+xnpyn-1q,n∈N0,where parameters A,pA,p and q are positive numbers, are bounded if one of the following conditions is satisfied p2<4qp2<4q, or 2q⩽p<1+q,q∈(0,1), and that the system has positive unbounded solutions when p2⩾4q>4p2⩾4q>4, or p⩾1+qp⩾1+q,...

Here we study the following system of difference equations \begin{align} x_n&=f^{-1}\bigg(\frac{c_nf(x_{n-2k})}{a_n+b_n\prod_{i=1}^kg(y_{n-(2i-1)})f(x_{n-2i})}\bigg),\nonumber\\ y_n&=g^{-1}\bigg(\frac{\gamma_n g(y_{n-2k})}{\alpha_n+\beta_n \prod_{i=1}^kf(x_{n-(2i-1)})g(y_{n-2i})}\bigg),\nonumber \end{align} $n\in\mathbb{N}_0,$ where $f$ and $g$ are...

In this paper, a novel Mesh less Local Petrov-Galerkin (MLPG) Mixed Collocation Method is developed for solving the inverse Cauchy problem of linear elasticity, wherein both the tractions as well as displacements are prescribed/measured at a small portion of the boundary of an elastic body. The elastic body may be isotropic/anisotropic and simply c...

The aim of this article is to introduce new hybrid iterative schemes, namely Jungck-Kirk-SP and Jungck-Kirk-CR iterative schemes, and prove convergence and stability results for these iterative schemes using certain quasi-contractive operators. Numerical examples showing the comparison of convergence rate and applications of newly introduced iterat...

In this paper we use the idea of logarithmic density to define the concept of logarithmic statistical convergence. We find the relations of logarithmic statistical convergence with statistical convergence, statistical summability ( H , 1 ) introduced by Móricz (Analysis 24:127-145, 2004) and [ H , 1 ] q -summability. We also give subsequence charac...

Suppose that a homogeneous linear differential equation has entire coefficients of finite order and a fundamental set of solutions each having zeros with finite exponent of convergence. Upper bounds are given for the number of zeros of these solutions in small discs in a neighbourhood of infinity.

In this paper we define the λ(u)-statistical convergence that generalizes, in a certain sense, the notion of λ-statistical convergence. We find some relations with sets of sequences which are related to the notion of strong convergence. MSC: 40A05; 40H05.

Korovkin type approximation theorems are useful tools to check whether a given sequence of positive linear operators on of all continuous functions on the real interval is an approximation process. That is, these theorems exhibit a variety of test functions, which assure that the approximation property holds on the whole space if it holds for...

In this paper, we first introduce the concepts of Bregman nonexpansive retract and Bregman one-local retract and then use these concepts to establish the existence of common fixed points for Banach operator pairs in the framework of reflexive Banach spaces. No compactness assumption is imposed either on C or on T, where C is a closed and convex sub...

We use the notion of (λ,μ)-statistical convergence to prove the Korovkin-type approximation theorem for functions of two variables by using the test functions 1, x, y, x 2 +y 2 . Furthermore, we define a new type of summability method via (λ,μ)-statistical convergence and use it to prove a Korovkin-type theorem. Moreover, we obtain the order of (λ,...

We introduce some double sequences spaces involving the notions of invariant mean (or σ -mean) and σ -convergence for double sequences while the idea of σ -convergence for double sequences was introduced by Çakan et al. 2006, by using the notion of invariant mean. We determine here some inclusion relations and topological results for these new doub...

We define the notions of double statistically convergent and double lacunary statistically convergent sequences in locally solid Riesz space and establish some inclusion relations between them. We also prove an extension of a decomposition theorem in this setup. Further, we introduce the concepts of double θ -summable and double statistically lacun...

Recently, Alghamdi and Mursaleen (2013) used the Hankel matrix to determine the necessary and suffcient condition to find the sum of the Walsh-Fourier series. In this paper, we propose to use the Hankel matrix as well as any general nonnegative regular matrix to obtain the necessary and sufficient conditions to sum the derived Fourier series and co...

We prove a Korovkin type approximation theorem for a function of two variables by using the notion of statistical summability (íµí°, 1, 1). We also study the rate of statistical summability (íµí°, 1, 1) of positive linear operators. Finally we construct an example to show that our result is stronger than those previously proved for Pringsheim's con...

The purpose of this paper is to define some new types of summability methods for double sequences involving the ideas of de la Vallée-Poussin mean in the framework of probabilistic normed spaces and establish some interesting results.

We define and study the concept of non-Archimedean intuitionistic fuzzy normed space by using the idea of t-norm and t-conorm. Furthermore, by using the non-Archimedean intuitionistic fuzzy normed space, we investigate the stability of various functional equations. That is, we determine some stability results concerning the Cauchy, Jensen and its P...