
Rifat ÇolakFırat University · Department of Mathematics
Rifat Çolak
Prof. Dr.
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81
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Introduction
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March 1978 - April 2018
Publications
Publications (81)
The applications of a Fibonacci sequence in mathematics extend far beyond their initial discovery and theoretical significance. The Fibonacci sequence proves to be a versatile tool with real-world implications and the practical utility of manifests in various fields, including optimization algorithms, computer science and finance. In this research...
In this research paper, we introduce some concepts of λf-density in connection with modulus functions under certain conditions. Furthermore, we establish some relations between the sets of λf-statistically convergent and λf-statistically bounded sequences.
In this paper, we establish relations between the sets of strongly Cesàro summable sequences of complex numbers for modulus functions f and g satisfying various conditions. Furthermore, for some special modulus functions, we obtain relations between the sets of strongly Cesàro summable and statistically convergent sequences of complex numbers.
In this study we establish the relations between the sets of difference sequences which are statistically convergent in connection with modulus functions.
The expansion of new convergence methods accelerated the development of the theory of sequence spaces. One of the most important types is called statistical convergence. This concept has been a major topic of study in recent years. In this research paper, we introduce the concept of λ−Wijsman weak statistical convergence and λ−Wijsman weak summabil...
This research paper focuses on defining the relationships between the sets of strongly lacunary summable and lacunary statistically convergent sequences of complex numbers by using different modulus functions f and g under certain conditions and different orders α, β ∈ (0, 1] such that α ≤ β. Furthermore, for some special modulus functions, we esta...
In the present paper, the notion of discrete weighted mean method of summability isextended the concept of statistical convergence. We also give the notion of statistical (M,P_{λ})-summability and [M,P_{λ}]_{q}-summability. We introduced some properties of this modes of convergence
In this study we first established the relations between f-density and g-density of a subset of the set of positive integers for any modulus functions f and g. Using the obtained facts we establish the relationship between the sets Sf and Sg of statistically convergent and BSf and BSg of statistically bounded sequences which defined by modulus func...
In this study we introduce df-statistical convergence and df-strong Cesàro summability with respect to a modulus for a sequence in a metric space. Furthermore we give the relations between the set of df-statistically convergent sequences and the set of df-strongly Cesàro summable sequences with respect to a modulus. Besides this we give the relatio...
In this study we introduce the concepts of Wijsman asymptotically deferred statistical equivalence of order α and Wijsman strong deferred Cesàro asymptotically equivalence of order α for set sequences.
In the present paper, we introduce and study d-statistical convergence of order α d-statistical boundedness of order α and d-strong p-Cesàro summability of order α for sequences in a metric space. Furthermore, we investigate the relations between the sets of sequences which are d-statistically convergent of order α, between the sets of sequences wh...
In this paper we introduce some new sequence spaces by using a sequence of moduli F = (fk) , give some topological properties and inclusion relations related to these sequence spaces. We also give the β-dual of [ĉ, F,p]∞(Δm).
In this study, using the generalized difference operator Δim and a sequence λ = (λn) which is a non-decreasing sequence of positive numbers tending to ∞ such that λn+1 ≤ λn+1, λ1 = 1, we introduce the concepts of λ(Δim)–statistical convergence of order α (α ∈ (0, 1]) and strong λ(Δim)−Cesàro summablility of order α (α > 0). We establish some connec...
In this paper, we introduce and study \(\lambda _{d} -\)statistical convergence, \(\lambda _{d}-\)statistical boundedness and strong \(\left( V,\lambda \right) _{d}-\)summability of sequences in metric spaces. Furthermore we establish some relations between the sets of \(\lambda _{d} \)-statistically convergent sequences, between the sets of \(\lam...
In this paper the concepts of Δm v -statistical convergence of order α and strong (p, Δm )-Ces`aro summability of order α are introduced for sequences of complex (or real) numbers. Some relations between the Δm v -statistical convergence of order α and strong (p, Δm v )-Ces`aro summability of order α are given. Also some relations between the space...
In this paper, we introduce and examine the concepts of I2(u)-convergence
of order (α,β) and
uniformly strong p-Cesàro summability of
order (α,β), of double
sequences of complex (or real) numbers, where α,β are real numbers such that
0
In this paper, we introduce the concept Ŝαλ – statistical convergence of order α. Also some relations between Ŝαλ – statistical convergence of order α and strong ŵαp(λ) – summability of order α are given. Furthermore some relations between the spaces ŵα(p)[λ, M] – and Ŝαλ – are examined. © 2014, Eduem - Editora da Universidade Estadual de Maringa....
The idea of difference sequences of real (or complex) numbers was introduced by Kizmaz [8]. In this paper, using the difference operator and a lacunary sequence, we introduce and examine the class of sequence bvθ (Δ, F). We study some of its properties like solidity, symmetricity, etc.
In this paper, we introduce the concept Ŝαλ - statistical convergence of order α. Also some relations between Ŝαλ -statistical convergence of order α and strong ŵβp(μ) -summability of order β are given. Furthermore some relations between the spaces ŵα(p) [λ,f] and Ŝαλ are examined.
We intend to make a new approach and introduce the concepts of statistical convergence of order and strongly -Cesàro summability of order for double sequences of complex or real numbers. Also, some relations between the statistical convergence of order and strong -Cesàro summability of order are given.
In this study, we introduce the sets
$\left[ V,\lambda ,p\right] _{\Updelta }^{{\mathcal{F}}},\left[ C,1,p\right] _{\Updelta }^{{\mathcal{F}}}$
and examine their relations with the classes of
$ S_{\lambda }\left( \Updelta ,{\mathcal{F}}\right)$
and
$ S_{\mu }\left( \Updelta ,{\mathcal{F}}\right)$
of sequences for the sequences
$\left( \lamb...
The sequences defined in Example 3 and Example 4 do not serve our purpose for any λ = (λn). Because this sequences are just the sequences x=(xk)=(k) and x=(xk)=1 respectively and any term of these sequences can not be 0. In this short not we give Example 3* and Example 4* to show that the inclusions given in Theorem 2.4 and Theorem 2.9 are strict f...
The sequences defined in Example 3 and Example 4 do not serve our purpose for any lambda = (lambda(n)). Because this sequences are just the sequences x = (x(k)) = (k) and x = (x(k)) = (1) respectively and any term of these sequences can not be 0. In this short not we give Example 3* and Example 4* to show that the inclusions given in Theorem 2.4 an...
In the existing literature the relations between the set of λ−statistical convergent sequences S λ and the set of statistical convergent sequences S, and the set of strongly (V,λ)−summable sequences [V, λ] and the set of statistical convergent sequences already have been examined. In this paper, we determine the relations between the sets S λ and S...
In this paper, we introduce the concept of lambda-statistical convergence of order alpha. Also some relations between the lambda-statistical convergence of order alpha and strong (V, lambda)-summability of order alpha are given.
In this article the concepts of statistical convergence of order α and strongly p-Cesàro summability of order α are introduced for sequences of complex (or real) numbers. Also some relations between the statistical convergence of order α and strongly p-Cesàro summability of order α are given.
In this paper we define the sequence space $${w}_{{\mathcal{F}}}\left(f,p,\Updelta\right)$$ which is called the space of strongly $$\Updelta p$$-Cesàro summable sequences with modulus f. Furthermore the fuzzy Δ-statistically pre-Cauchy sequence is defined and the necessary and sufficient conditions are given
for a sequence of fuzzy numbers to be fu...
The idea of difference sequences of real (or complex) numbers was generalized by Et and Çolak [Et M, Çolak R. On some generalized difference sequence spaces. Soochow J Math 1995; 21(4): 377–86; Çolak R, Et M. On some generalized difference sequence spaces and related matrix transformations. Hokkaido Math J 1997; 26(3): 483–92]. In this paper, using...
A four-dimensional matrix transformation is said to be regular if it maps every bounded-convergent double sequence into a convergent sequence with the same limit. Firstly, Robison [G.M. Robison, Divergent double sequences and series, Trans. Amer. Math. Soc. 28 (1926) 50–73] presented the necessary and sufficient conditions for regular matrix transf...
The purpose is to introduce the concepts of almost lacunary statistical convergence and strongly almost lacunary convergence of generalized difference sequences of fuzzy numbers. We give some relations related to these concepts.
The idea of dierence sequence spaces was introduced by Kizmaz (8) and this concept was generalized by Et and Colak (6). In this paper we define the space '( m,f,p,q,s) on a seminormed complex linear space by using modulus function and we give various properties and some inclusion relations on this space. Furthermore we study some of its properties,...
The difference sequence space m(Æ,p,Δ(r)), which is a generalization of the space m(Æ) introduced and studied by Sargent (1960), was defined by Çolak and Et (2005). In this paper we establish some geometric inequalities for this space.
The purpose of this paper is to introduce the concepts of lacunary statistical convergence and lacunary strongly convergence of generalized difference sequences of fuzzy numbers. We give some relations related to these concepts. We show that lacunary Δm-statistical convergence and lacunary strongly Δ(p)m-convergence are equivalent for Δm-bounded se...
In this paper the concept of strongly Δλp2-Cesàro summability of a sequence of fuzzy numbers is introduced. Also some inclusion relations between the set of strongly Δλp2-Cesàro convergent and Δλ2-statistically convergent sequences of fuzzy numbers are given.
The purpose of this paper is to introduce the spaces of sequences that are strongly almost $(w, \lambda)-$ summable with respect to an Orlicz function. We give some relations
related to these sequence spaces. It is also shown if a sequence is strongly $(w, \lambda)-$ summable with respect to an Orlicz function, then it is $\bar{S}_{\lambda}-$ stati...
We introduce the double sequence space ℓ 2 ∞ (p) and show that ℓ 2 ∞ (p) is a complete paranormed space. Furthermore we give its α-, β- and γ-duals and examine its perfectness and Köthe–Toeplitz reflexivity.
The idea of dierence sequence sets, X()= {x=(xk): x2X}, where X='1, c and c0 was introduced by Kizmaz (4), and then this subject has been studied and generalized by various mathematicians. In this study, we define a new sequence space denoted by m(,p )( r ) and give some properties of this sequence space. The obtained results generalize some known...
We define the sequence spaces ℓ ∞ (Δ r m ), c(Δ r m ) and c 0 (Δ r m ) (m∈ℕ and r∈ℝ), where for instance ℓ ∞ (Δ r m )={x=(x k ):(k r Δ m x k )∈ℓ ∞ }, give some topological properties, inclusion relations of these spaces and compute their continuous and α, β and γ-duals. Furthermore, we characterize some matrix classes related to these sequence spac...
We introduce and examine some properties of the double sequence spaces c 2 P (p) and c 2 PB (p).
The definition of the pα-, pβ- and pγ-duals of a sequence space was defined by Et [Internat. J. Math. Math. Sci. 24 (2000) 785–791]. In this paper we compute pα- and N-duals of the sequence spaces Δmv(X) for X=ℓ∞, c and c0, and compute β- and γ-duals of the sequence spaces Δmv(X) for X=ℓ∞, c and c0.
A sequence Θ = (kr) of positive integers is called lacunary if k0 = 0, 0 < kr < kr+1 and hr = kr – kr-1 → ∞ as r → ∞. The intervals determined by Θ are denoted by Ir = (kr-1, kr]. Let ω be the set of all sequences of complex numbers and f be a modulus function. Then we define NΘ(Δm, f) = {x є ω: lim 1/hr Σ f(|Δm xk -l|)=0 for some l} r kєIr NΘ0(Δm,...
We introduce a general sequence space X v , where X is any sequence space and establish some inclusion relations, topological results. Furthermore we give α- and β-duals of sequence spaces [ℓ(p)] v , [c 0 (p)] v , [ℓ ∞ (p)] v and [c(p)] v together with α-duals of sequence spaces ℓ(p), c 0 (p), ℓ ∞ (p) and c(p). The perfectness of sequence spaces [ℓ...
We give γ-duals of the sequence spaces ℓ ∞ (p), c(p), c 0 (p) and ℓ(p), and we examine the perfectness, normality and monotonicity the sequence spaces.
We introduce a general sequence space Δ V (X)={x=(x k ): (v k x k -v k+1 x k+1 )∈X}, where X is any sequence space. We establish some inclusion relations, topological results, in general case and we characterize the continuous, α-, β- and γ-duals of Δ V (X) for various sequence spaces X. The results of this paper, in a particular case, include the...