# Vatan KarakayaYildiz Technical University · Department of Mathematical Engineering

Vatan Karakaya

Professor

## About

147

Publications

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1,496

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Citations since 2017

## Publications

Publications (147)

In this paper, firstly, we extend the nonlinear Lebesgue spaces from the setting of Hadamard spaces to the setting of p-uniformly convex metric spaces. Afterward, we establish some Δ-convergence and strong convergence theorems for a recently introduced class of generalized nonexpansive mappings in the setting of p-uniformly convex metric spaces. Fu...

Bu çalışmanın temel amacı; fiziki bilgi alanı ile metafizik bilgi alanı arasında
bulunan matematiksel bilginin yeri üzerine bir değerlendirme yapmaktır. Varlığın yaradılış hali metafiziksel bilgi olan “var olmak bakımından varlık” bilgisini
gerekli kılmaktadır. Saf güç ve imkân olarak maddenin sûret yoluyla varlığa gelmesi duyulur âlemde cisim olar...

Darbo fixed point theorem is a powerful tool which is used in many fields in mathematics. Because of this feature, many generalizations of this theorem and its relations with other subjects have been investigated. Here we introduce a generalization of an F - contraction of Darbo type mapping and define a new contraction by using both function class...

In this paper, we introduce a new type of variational inequality problem (VIP) involving nonself multivalued mappings in CAT(0) spaces. We show that this VIP problem admit a solution under suitable conditions. We also perform the convergence analysis of introduced VIP via proximal multivalued Picard-S iteration. We finish our paper with some conver...

The main purpose of this study is to investigate the knowledge and wisdom relationship
based on the definitions of God and knowledge of certain civilizations within the context
of written history. This relationship is tried to be explained under the definitions of God
and knowledge derived from eminent thinkers of Ancient Greece, Islamic and Modern...

Mevcut çalışmanın temel amacı yazılı tarih bağlamında belirli medeniyet havzalarının tanrı ve bilgi tanımlarından hareketle bilgi-hikmet ilişkisini araştırmaktır. Bu çalışmada ilk olarak Eski Yunan, İslâm ve Modern Batı medeniyet havzalarında öne çıkan düşünürler üzerinden hareketle tanrı ve bilgi tanımları verilmiştir. Daha sonra bu tanımlara daya...

The aim of this article is to define a new Jungck-Kirk type iteration method and to examine the convergence result under appropriate conditions together with other Jungck-Kirk type iteration methods in the literature. It is also to analyze whether the newly defined iteration method is stable. In addition, it has been shown through numerical example...

The aim of this article is to define a new Jungck-Kirk type iteration method and to examine the convergence result under appropriate conditions together with other Jungck-Kirk type iteration methods in the literature. It is also to analyze whether the newly defined iteration method is stable. In addition, it has been shown through numerical example...

The close relationship between chaos and dynamical systems leads to naturally consider the iteration processes that are related to dynamical systems of fixed point theory. From this natural relationship, the control of chaos that occurs in fixed point iteration dynamics will be the main focus of the article. To achieve this goal, analytical solutio...

https://www.fikriyat.com/yazarlar/sefa-saygili/2019/04/29/matematik-ve-metafizik-dusunce-iliskisi

In this paper, we define concept of approximate fixed point property of a function and a set in intuitionistic fuzzy normed space. Furthermore, we give intu-itionistic fuzzy version of some class of maps used in fixed point theory and investigate approximate fixed point property of these maps.

Matematik nesnelerin beş duyuya kapalı ve zihinsel varlıklar olduğu matematik felsefecilerinin çoğunluğu tarafından ortak kabul görmektedir. Ancak duyulara kapalı ve zihinsel bir varlık olan matematiksel bilginin duyulur dünyada bu denli etkili oluşu da yine düşünce tarihi boyunca tartışılmıştır. Bu çalışmada, duyulur cisim dünyasının nesnelerinin...

It is known that the source of knowledge, which is the subject of thinking, is the realm of concrete or intangible existence. For this reason, the issue of how knowledge is produced from existence or how existence is the subject of knowledge has been one of the main subjects of philosophy throughout the history of thought.

The perception of mathematical objects by the human mind has been one of the fundamental questions of philosophy in general and of philosophy of mathematics in particular throughout history. Many schools of thought have put forward different opinions on how the concepts of number and geometric, which are abstaract entities, are perceived by the hum...

The studies about hybrid mappings are mainly focused for single-valued mappings in Hilbert spaces. We define a new class of multivalued mappings in CAT (κ) spaces which contains the multival-ued nonexpansive mappings, α-nonexpansive mappings and some hybrid mappings such as (α, β)-hybrid mappings and we study existence and convergence of this new c...

The purpose of this work is to investigate the behavior of function sequences under integral type transformations as a generalization of Darbo's theorem. It is also to obtain the existence of the fixed point of this transformation involving the function sequences. In addition, the study will be explained with an interesting example.

We introduce a new contractive condition and a new iterative method in n normed space setting. We employ both of these to study convergence , stability, and data dependence. The results presented here extend and improve some recent results announced in the existing literature.

In this article, we introduce a new class of contractive mappings and study analytical and computational aspects of a special case of Jungck-Khan iterative algorithm generated by this class of mappings. In particular, we improve upon strong convergence, rate of convergence and data dependence results existing in the current literature. Analytical a...

In this study, we introduce a new three step iteration process and show that the iteration process converges to the unique fixed point by two theorems under different conditions of contractive mappings on the generalized G-convex metric spaces. Also, we investigate data dependence result for this iterative process in the generalized G-convex metric...

The purpose of this work is to investigate the existence of fixed points of some mappings in fixed point theory by combining some important concepts which are F-weak contractions, multivalued mappings, integral transformations and α-admissible mappings. In fixed point theory, it is important to find fixed points of some classess under For F-weak co...

In the present paper, we show that $S^*$ iteration method can be used to approximate fixed point of almost contraction mappings. Furthermore, we prove that this iteration method is equivalent to CR iteration method and it produces a slow convergence rate compared to the CR iteration method for the class of almost contraction mappings. We also prese...

A lot of work has been done on fixed point studies using function
classes. In this work, the existence of fixed point is investigated by taking function sequences instead of function classes. This idea has been applied to the Darbo fixed point theorem and obtained some results. Also, the study was supported by an interesting example held the condit...

We study a weaker and more natural notion of stability called weak \(w^{2}\)-stability to get an insight in the corresponding results obtained by Măruşter and Măruşter (J Comput Appl Math 276:110–116, 2015) and Wang (J Comput Appl Math 285:226–229, 2015). A data dependence result for fixed points of strongly demicontractive operators is also establ...

The main purpose of this work is to extend the properties of multivalued transformations to the integral type transformations and to obtain the existence of fixed points under F-contraction. In addition, the results of this study were evaluated with some interesting example.

In this paper, we study convergence and data dependence of SP and normal-S iterative methods for the class of almost contraction mappings under some mild conditions. The validity of these theoretical results is confirmed with numerical examples. It has been observed that a special case of SP iterative method, namely normal-S iterative method, perfo...

In this study, a new three step iterative algorithm was introduced with the help of Jungck-contraction principle which is one of the remerkable generalizations of Banach-contraction principle. Also, the convergence and stability results were obtained for the pair of nonself mappings which satisfy a certain contractive condition by using this iterat...

In this paper, we introduce a new variational inequality problem(VIP) associated with nonself multivalued nonexpansive mappings in $CAT(0)$ spaces.

In this work, we prove the existence of solutions for a tripled system of integral equations using some new results of fixed point theory associated withmeasure of noncompactness. These results extend the results in some previous works. Also, the condition under which the operator admits fixed points is more general than the others in literature.

We propose a new Jungck-S iteration method for a class of quasi-contractive operators on a convex metric space and study its strong convergence, rate of convergence and stability. We also provide conditions under which convergence of this method is equivalent to Jungck-Ishikawa iteration method. Some numerical examples are provided to validate the...

We define a new class of multivalued mappings and prove existence, stability of fixed point sets and convergence results for this class of multivalued hybrid mappings in CAT(κ) spaces which are more general than Hilbert spaces and CAT(0) spaces. We also show with an example that this class is not included in the class of multivalued nonexpansive ma...

In the present paper, we investigate the convergence, equivalence of convergence, rate of convergence and data dependence results using a three step iteration process for mappings satisfying certain contractive condition in hyperbolic spaces. Also we give non-trivial examples for the rate of convergence and data dependence results to show effciency...

Bu çalışmada amacımız, Türk eğitim sisteminin temel problemlerinden bir kısmını ortaya koymak ve bu problemlerin çözüm önerileri üzerine tartışma açmaktır. Ayrıca bu çalışmada, Cumhuriyet öncesi beslendiği kaynaklarla ilişkisi zayıflamış olan eğitim anlayışını Cumhuriyet sonrasındaki uygulama hataları ile birlikte irdeleyerek günümüze kadar ulaşan...

Bu iki alan ilişkisine geçmeden önce hem matematik hem de bilimsel bilgi kavramlarını kısaca tanıtmak yerinde olacaktır.
Bilimsel bilginin tarihsel süreçte ana hatlarıyla geçirdiği evreler şu şekilde sıralanabilir:
i) Bilimsel bilgi için doğrulamacı yaklaşım, yani positivizim;
ii) Karl Popper’ın temsil ettiği yanlışlamacı bilgi; yanlışlanabilir do...

Our purpose in this study is to dedect some of the key problems of Turkish educational system and to discuss possible answers to these problems. Namely, we will be focusing on the loosend connections to pre-Republic era and the misconceptions of understaning developed under the Republic era of Turkey and its repercussions today. Following the chang...

The focus of this paper is to present some concepts of intuitionistic fuzzy T-convergence in intuitionistic fuzzy 2-Banach spaces. We will modify and correct the definition of regularity of 2-norms on 2-Banach spaces, which was given by Gürdal et al. in (2009, Nonlinear Analysis: Theory, Methods & Applications, 71, 1654-1661) to guarantee uniquenes...

In this study, chaos structures of dynamics which are formed by applying Ishikawa iteration processes to function classes with chaotic behavior in discrete dynamic systems have been investigated. Later, the control mechanisms for chaotic structures have been developed according to the stable and unstable state of fixed points of this transformation...

In this paper, we investigate existence of n -tuplet coincidence point theorems in partially ordered probabilistic metric spaces. Also, we gave uniqueness of n -tuplet fixed point theorems in this space.

We study convergence, rate of convergence and data dependency of normal−S iterative method for a fixed point of a discontinuous operator T on a Banach space. We also prove some Collage type theorems for T. The main aim here is to show that there is a close relationship between the concepts of data dependency of fixed points and the collage theorems...

In this study, we introduce a new iteration scheme and prove the strong convergence result for this iteration method. We also compare the rate of convergence with the iterative scheme and the fixed point iteration scheme known as Picard-S due to Gursoy. Then we prove that this new iteration method is equivalent to convergence of the iteration schem...

In this paper, we introduce a new type of Darbo's fixed point theorem by using concept of function sequences with shifting distance property. Afterward, we investigate existence of fixed point under this the theorem. Also we are going to give interesting example held the conditions of sequences of functions

In this study, the emphasis will be placed on the importance of mathematical thinking in our lifes and our world of thought. It is to be argued that the real life involving of objects, as stated in Ibn Sina’s “Kitabu’ş Şifa Metaphysic” book, is rough and that the delicacy of the relations between these objects can only be understood by reason, in o...

The aim of this paper is to find new iterative Newton-like schemes inspired by the modified Newton iterative
algorithm and prove that these iterations are faster than the existing ones in the literature. We further investigate their behavior and finally illustrate the results by numerical examples.

In this paper by using the measure of noncompactness concept, we present new fixed point theorems for multivalued maps. In further we introduce a new class of mappings which are general than Meir-Keeler mappings. Finally, we use these results are then used to investigate the existence of weak solutions to an Evolution differential inclusion with a...

A sequence of real numbers $\{x_{n}\}_{n\in \mathbb{N}}$ is said to be $\alpha \beta$-statistically convergent of order $\gamma$ (where $0<\gamma\leq 1$) to a real number $x$ \cite{a} if for every $\delta>0,$ $$\underset{n\rightarrow \infty} {\lim} \frac{1}{(\beta_{n} - \alpha_{n} + 1)^\gamma}~ |\{k \in [\alpha_n,\beta_n] : |x_{k}-x|\geq \delta \}|...

In this paper, we propose a modified version of Picard type iterative algorithm for finding a fixed point of a nonexpansive mapping defined on a closed convex subset of a Hilbert space. We prove the strong convergence of the sequence generated by the proposed algorithm to a fixed point of a nonexpensive map, such fixed point is also a solution of a...

The studies about hybrid mappings are mainly focused on single-valued mappings. Now, we give definition of multivalued generalization of generalized hybrid mappings which is defined in CAT (0) spaces and also studied on CAT (κ) spaces. This new definition is general than multivalued nonexpansive mappings, multivalued hybrid mappings and multivalued...

The objective of the present work is to analyze stability in the sense of Hyers-Ulam and Hyers-Ulam-Rassias for nonlinear Volterra Fredholm integro-differential equation by using fixed point approach.

We introduce a different class of s-type operators by using the generalized weighted mean sequence space c 0 (u, v), then it is shown that this new class of operators is a quasi-Banach operator ideal. Moreover, their injectivity and surjectivity are investigated according to sort of s-number. Finally, we proof that it is a closed operator ideal und...

In this paper, we introduce a three step iteration method and show that this method can be used to approximate fixed point of weak contraction mappings. Furthermore, we prove that this iteration method is equivalent to Mann iterative scheme and converges faster than Picard-S iterative scheme for the class of weak contraction mappings. We also prese...

In this paper, some results on the existence of n-tuplet fixed points for multivalued contraction mappings are proved via measure of noncompactness. As an application, the existence of solutions for a system of integral inclusions is studied.

We study convergence and data dependence of iterates presented in (B. Xu and M. A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002), 444-453.) for hemicontractive operators in the framework of a real Banach space. Some nontrivial numerical examples in a Banach space setting whic...

The main purpose of this work is to determine of fine spectra and subspectra such as approximate point spectrum, defect spectrum and compression spectrum of the difference operator with periodic coefficients over the sequence spaces ℓ1..

In this paper, we show that the iteration method defined in "V. Karakaya, Y. Atalan, K. Dogan and NH. Bouzara, Some Fixed Point Results for a New Three Steps Reration Process in Banach Spaces, Fixed Point Theory" converges to solution of the functional Volterra-Fredholm integral equation with deviating argument in a Banach space. Furthermore, a dat...

Most of the studies about hybrid mappings are carried out for single-valued mappings in Hilbert spaces. We define a new class of multivalued mappings in CAT (k) spaces which contains the multivalued generalization of (α, β)-hybrid mappings defined on Hilbert spaces. In this paper, we prove existence and convergence results for a new class of multiv...

In this study, we introduce a new iterative processes to approximate common fixed points of an infinite family of quasi-nonexpansive mappings and obtain a strongly convergent iterative sequence to the common fixed points of these mappings in a uniformly convex Banach space. Also we prove that this process to approximate zeros of an infinite family...

In this paper, we introduce a new iteration method and show that this iteration method can be used to approximate fixed point of almost contraction mappings. Furthermore, we prove that the new iteration method is equivalent to both Mann iteration method and Picard-Mann hybrid iteration method and also converges faster than Picard-Mann hybrid iterat...

In this study, we give definition of some multivalued hybrid mappings which are general than many mappings in the existing literature, then we give some existence and convergence results for these mappings in CAT({\kappa})-spaces

In this work, using measure of noncompactness some result on the existence of coupled and tripled fixed point for multivalued set contraction mapping are investigated. As application, the existence of solution, for a system of integral inclusion is studied.

In this paper we combine the concept of measure of noncompactness with the concept of C-class functions to obtain an extention of the Darbo fixed point theorem. Then, we study the existence of coupled fixed point for mappings of C-class. Finally, we applicate the obtained results to investigate the solvability of a system of a nonlinear integral eq...

We have compared rate of convergence among various iterative methods. Also, we have established an equivalency result between convergence of two recently introduced iterative methods and we have proven a data dependence result for one of them. Mathematics Subject Classification: 47H06, 54H25.

In this work, we define new sequence spaces by using the matrix obtained by product of factorable matrix and generalized difference matrix of order m. Afterward, we investigate topological structure which are completeness, AK-property, AD-property. Also, we compute the α-, β- and γ - duals, and obtain bases for these sequence spaces. Finally we giv...

In this paper, we present a new iteration for set contraction mappings in a Banach space. In further, we introduce the concept of set stability for these class of mappings. Finally, we present some results on the existence of fixed points for power set contraction mappings, these results include in special case many results existing in literature l...

The purpose of this paper is to prove new common fixed point theorems for commuting mappings. We also deduce new classes of k-set contraction mappings and guarantee the existence of their fixed points. As applications, we establish an integral version of these results. Finally, we also introduce and study the resolvability of a new type of integral...

In this paper, we introduce a new iteration method and show that this iteration method can be used to approximate fixed point of almost contraction mappings. Furthermore, we prove that the new iteration method is equivalent to both Mann iteration method and Picard-Mann hybrid iteration method and also converges faster than Picard-Mann hybrid iterat...

In the study by Baliarsingh and Dutta [Internat. J.Anal., Vol.2014(2014), Article ID 786437], the authors computed the spectrum and the fine spectrum of the product operator G (u, v; Delta) over the sequence space l(1). The product operator G (u, v; Delta) over l(1) is defined by (G (u, v; Delta) x)(k) = Sigma(k)(i=0)u(k)v(i) (x(i)-x(i-1)) with x(k...

In this paper, we generalized the Darbo theorem to Frechet spaces. In further, we use this theorem to prove an alternative theorem for set contraction mappings in Frechet spaces. Finally, we study the solvability of a nonlinear integral equation.

In this work, we investigate the ideal of all bounded linear operators between any arbitrary Banach spaces whose sequence of approximation numbers belong to the generalized modular spaces of Cesaro type defined by weighted means. Also, we show that the completeness of obtained operator ideals.

In this paper, we define concept of approximate fixed point property of a
function and a set in intuitionistic fuzzy normed space. Furthermore, we give
intuitionistic fuzzy version of some class of maps used in fixed point theory
and investigate approximate fixed point property of these maps.

In this paper, we continue the theme of analytical and numerical treatment of Jungck type iterative schemes. In particular, we focus on a special case of Jungck-Khan iterative scheme introduced in [Analytical and numerical treatment of Jungck-type iterative schemes, Appl. Math. Comput. 231 (2014) 521–535] to get an insight in the strong convergence...

In this work, we prove the existence of solutions for a tripled system of
integral equations using some new results of fixed point theory associated with
measure of noncompactness. These results extend some previous works in the
literature, since the condition under which the operator admits fixed points is
more general than the others in literatur...

Our aim in this paper is to present results of existence of fixed points for
continuous operators in Banach spaces using measure of noncompactness under an
integral condition. This results are generalization of results given by A.
Aghajania and M. Aliaskaria which are generalization of Darbo's fixed point
theorem. As application we use these result...

In this paper, we study the stability of a new iterative scheme that we had
introduced. Moreover we compare its rate of convergence with Picard-Mann
iterative scheme. Finally, we apply this iterative process for the resolution
of delay equation.

In this study, we introduce a new iterative processes to approximate common
fixed points of an infinite family of quasi-nonexpansive mappings and obtain a
strongly convergent iterative sequence to the common fixed points of these
mappings in a uniformly convex Banach space. Also we prove that this process to
approximate zeros of an infinite family...

The concept of αβ-statistical convergence was introduced and studied by
Aktu˘glu (Korovkin type approximation theorems proved via αβ-statistical convergence,
J Comput Appl Math 259:174–181, 2014). In this work, we generalize the
concept of αβ-statistical convergence and introduce the concept of weighted αβ-
statistical convergence of order γ , weig...

We give some results concerning the existence of
tripled fixed points for a class of condensing operators in Banach spaces. Further, as an application, we study the existence of solutions for a general
system of nonlinear integral equations.

We introduce a new class of sequences named as mαΔr,ϕ,p and, for this space, we study some inclusion relations, topological properties, and geometrical properties such as order continuous, the Fatou property, and the Banach-Saks property of type p.

We put forward a new general iterative process. We prove a convergence result as well as a stability result regarding this new iterative process for weak contraction operators.

In this work, we classify and calculate spectra such as point spectrum, continuous spectrum and residual spectrum over sequences spaces
ℒ∞, cℒ∞, c and c0 according to a new matrix operator W which is obtained by matrix product.

The aim of this study is to conduct quantitative and qualitative analysis of biological and dosimetric materials which contain organic and inorganic materials and to make the determination by using the spectral theorem Beer-Lambert law. Beer-Lambert law is a system of linear equations for the spectral theory. It is possible to solve linear equation...

In this paper, wedefine a new generalized difference sequence space C-(p) Delta(m)(lambda) and consider it equipped with the Luxemburg norm under which it is a Banach space and we show that the space C-(p) Delta(m)(lambda) possess Banach Saks property of type p, uniform opial property and property (H), where p = (p(n)) is a bounded sequence of posi...

We consider the recently introduced notion of -statistical convergence (Das, Savas and Ghosal, Appl. Math. Lett., 24(9) (2011), 1509–1514, Savas and Das, Appl. Math. Lett. 24(6) (2011), 826–830) in probabilistic normed spaces and in the following (Şençimen and Pehlivan (2008 vol. 26, 2008 vol. 87, 2009)) we introduce
the notions like strong -statis...

We introduce a new sequence space which is defined by the operator W=(w nk ) on the sequence space ℓ(p). We define a modular functional on this space and investigate structure of this space equipped with Luxemburg norm. Also we study some geometric properties which are called Kadec-Klee, k-NUC, and uniform Opial properties and prove that this new s...

For an arbitrary n positive integer, we investigate the existence of n-tuplet coincidence points in intuitionistic fuzzy normed space. Results of the paper are more general than those of the coupled and the tripled fixed point works in intuitionistic fuzzy normed space.

Banach-Saks type is calculated for two types of Banach sequence spaces and Gurariǐ modulus of convexity is estimated from
above for the spaces of one type among them.

We introduce a new iteration method what is called Picard-S iteration. We
show that the Picard-S iteration method can be used to approximate fixed point
of contraction mappings. Also, we show that our new iteration method is
equivalent and converges faster than CR iteration method for the aforementioned
class of mappings. Furthermore, by providing...

We show that iterative scheme due to Karahan and \"Ozdemir (2013) can be used
to approximate fixed point of contraction mappings. Furthermore, we prove that
CR iterative scheme converges faster than the iterative scheme due to Karahan
and \"Ozdemir (2013) for the class contraction mappings. Finally, we prove a
data dependence result for contraction...

The purpose of this paper is to introduce a new Kirk type iterative algorithm called Kirk multistep iteration and to study its convergence. We also prove some theorems related to the stability results for the Kirk multistep and Kirk-SP iterative processes by employing certain contractive-like operators. Our results generalize and unify some other r...

We introduce Kirk-multistep-SP and Kirk-S iterative algorithms
and we prove some convergence and stability results for these iterative
algorithms. Since these iterative algorithms are more general than some
other iterative algorithms in the existing literature, our results
generalize and unify some other results in the literature.

In this work, our purpose is to introduce I−convergence of sequences of functions in intuitionistic fuzzy normed space by combining the I−convergence, the sequences of functions and the intuitionistic fuzzy normed spaces, and to investigate relations among concepts such as I−convergence, statistical convergence and the usual convergence of sequence...

## Projects

Projects (5)

The main aim of this project is to model some physics and real world problems with classes of differential equations and integral equations, then study their solutions using some concepts of fixed point theory such as iterative algorithms, Darbo fixed point theorem under some topological and functional concepts.