Browsing by Author "Kaymakcalan, Billur"

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Citation - WoS: 5
Citation - Scopus: 8
Applications of the Novel Diamond Alpha Hardy-Copson Type Dynamic Inequalities To Half Linear Difference Equations
(Taylor & Francis Ltd, 2022) Kaymakcalan, Billur; Kayar, Zeynep
This paper is devoted to novel diamond alpha Hardy-Copson type dynamic inequalities, which are zeta < 0 complements of the classical ones obtained fort zeta > 1, and their applications to difference equations. We obtain two kinds of diamond alpha Hardy-Copson type inequalities for zeta < 0, one of which is mixed type and established by the convex linear combinations of the related delta and nabla inequalities while the other one is new and is obtained by using time scale calculus rather than algebra. In contrast to the works existing in the literature, these complements are derived by preserving the directions of the classical inequalities. Therefore both kinds of our results unify some of the known delta and nabla Hardy-Copson type inequalities obtained for zeta < 0 into one diamond alpha Hardy-Copson type inequalities and offer new types of diamond alpha Hardy-Copson type inequalities which have the same directions as the classical ones and can be considered as complementary inequalities. Moreover the application of these inequalities in the oscillation theory of half linear difference equations provides several nonoscillation criteria for such equations.
Citation - WoS: 14
Citation - Scopus: 16
Bennett-Leindler Type Inequalities for Nabla Time Scale Calculus
(Springer Basel Ag, 2021) Kayar, Zeynep; Kaymakcalan, Billur; Pelen, Neslihan Nesliye
In this study, we generalize the converse of Hardy and Copson inequalities, which are known as Bennett and Leindler type inequalities, for nabla time scale calculus. This generalization allows us not only to unify all the related results existing in the literature for an arbitrary time scale but also to obtain new results which are analogous to the results of the delta time scale calculus.
Citation - WoS: 6
The Complementary Nabla Bennett-Leindler Type Inequalities
(Ankara Univ, Fac Sci, 2022) Kayar, Zeynep; Kaymakcalan, Billur
We aim to find the complements of the Bennett-Leindler type inequalities in nabla time scale calculus by changing the exponent from 0 < zeta < 1 to zeta > 1. Different from the literature, the directions of the new inequalities, where zeta > 1, are the same as that of the previous nabla Bennett-Leindler type inequalities obtained for 0 < zeta < 1. By these settings, we not only complement existing nabla Bennett-Leindler type inequalities but also generalize them by involving more exponents. The dual results for the delta approach and the special cases for the discrete and continuous ones are obtained as well. Some of our results are novel even in the special cases.
Citation - WoS: 22
Citation - Scopus: 26
Constantin's Inequality for Nabla and Diamond-Alpha Derivative
(Springer international Publishing Ag, 2015) Kaymakcalan, Billur; Pelen, Neslihan Nesliye; Guvenilir, Ayse Feza
Calculus for dynamic equations on time scales, which offers a unification of discrete and continuous systems, is a recently developed theory. Our aim is to investigate Constantin's inequality on time scales that is an important tool used in determining some properties of various dynamic equations such as global existence, uniqueness and stability. In this paper, Constantin's inequality is investigated in particular for nabla and diamond-alpha derivatives.
Citation - WoS: 10
Citation - Scopus: 12
Diamond Alpha Bennett-Leindler Type Dynamic Inequalities and Their Applications
(Wiley, 2022) Kaymakcalan, Billur; Pelen, Neslihan Nesliye; Kayar, Zeynep
In this paper, two kinds of dynamic Bennett-Leindler type inequalities via the diamond alpha integrals are derived. The first kind consists of eight new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together due to the fact that convex linear combinations of delta and nabla Bennett-Leindler type inequalities give diamond alpha Bennett-Leindler type inequalities. The second kind involves four new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Bennett-Leindler type inequalities. For the second type, choosing alpha=1 or alpha=0 not only yields the same results as the ones obtained for delta and nabla cases but also provides novel results for them. Therefore, both kinds of our results expand some of the known delta and nabla Bennett-Leindler type inequalities, offer new types of these inequalities, and bind and unify them into one diamond alpha Bennett-Leindler type inequalities. Moreover, an application of dynamic Bennett-Leindler type inequalities to the oscillation theory of the second-order half linear dynamic equation is developed and presented for the first time ever.
Citation - WoS: 2
Citation - Scopus: 6
Diamond Alpha Hardy-Copson Type Dynamic Inequalities
(Hacettepe Univ, Fac Sci, 2022) Kaymakcalan, Billur; Kayar, Zeynep
In this paper two kinds of dynamic Hardy-Copson type inequalities are derived via diamond alpha integrals. The first kind consists of twelve new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together. The second kind involves another twelve new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Hardy-Copson type inequalities. Our approach is quite new due to the fact that it uses time scale calculus rather than algebra. Therefore both kinds of our results unify some of the known delta and nabla Hardy-Copson type inequalities into one diamond alpha Hardy-Copson type inequalities and offer new Hardy-Copson type inequalities even for the special cases.
Citation - WoS: 2
Citation - Scopus: 1
Diamond-Alpha Pachpatte Type Dynamic Inequalities Via Convexity
(Springer india, 2023) Kaymakcalan, Billur; Kayar, Zeynep
Diamond-alpha Pachpatte type dynamic inequalities, which are convex generalizations of diamond-alpha Hardy-Copson type inequalities, are established to harmonize and bind foregoing related results in the delta and nabla calculi. A noteworthy contribution of the paper is that new diamond-alpha dynamic inequalities as well as their delta and nabla versions are derived by making use of convexity.
Citation - WoS: 14
Citation - Scopus: 17
Discrete Gruss Type Inequality on Fractional Calculus
(Springeropen, 2015) Asliyuce, Serkan; Guvenilir, Ayse Feza; Kaymakcalan, Billur; Akin, Elvan
We give a discrete Gruss type inequality on fractional calculus.
Citation - WoS: 7
Citation - Scopus: 7
Generalization of Mitrinovic-Pecaric Inequalities on Time Scales
(Rocky Mt Math Consortium, 2021) Akin, Elvan; Kaymakcalan, Billur; El-Deeb, Ahmed A.
We prove some new inequalities of Mitrinovic-Pecaric inequalities for convex functions on an arbitrary time scale using delta integrals. These inequalities extend and improve some known dynamic inequalities in the literature. The main results will be proved by using Holder and Jensen inequalities and a simple consequence of Keller's and Poetzsche's chain rules on time scales.
Citation - WoS: 6
Citation - Scopus: 14
Generalized Diamond-Α Dynamic Opial Inequalities
(Springer, 2012) Lesaja, Goran; Tas, Kenan; Atasever, Nuriye; Kaymakcalan, Billur
We establish some new dynamic Opial-type diamond alpha inequalities in time scales. Our results in special cases yield some of the recent results on Opial's inequality and also provide new estimates on inequalities of this type. Also, we introduce an example to illustrate our result.
Citation - WoS: 13
Citation - Scopus: 15
Hardy-Copson Type Inequalities for Nabla Time Scale Calculus
(Tubitak Scientific & Technological Research Council Turkey, 2021) Kaymakcalan, Billur; Kayar, Zeynep
This paper is devoted to the nabla unification of the discrete and continuous Hardy?Copson type inequalities. Some of the obtained inequalities are nabla counterparts of their delta versions while the others are new even for the discrete, continuous, and delta cases. Moreover, these dynamic inequalities not only generalize and unify the related ones in the literature but also improve them in the special cases.
Citation - WoS: 4
Citation - Scopus: 5
Nabla Discrete Fractional Gruss Type Inequality
(Springeropen, 2014) Peterson, Allan C.; Tas, Kenan; Guvenilir, A. Feza; Kaymakcalan, Billur
Properties of the discrete fractional calculus in the sense of a backward difference are introduced and developed. Here, we prove a more general version of the Gruss type inequality for the nabla fractional case. An example of our main result is given.
Citation - WoS: 4
Citation - Scopus: 5
Necessary and Sufficient Condition for Existence of Periodic Solutions of Predator-Prey Dynamic Systems With Beddington-Deangelis Functional Response
(Springeropen, 2016) Guvenilir, A. Feza; Kaymakcalan, Billur; Pelen, Neslihan Nesliye
We consider two-dimensional predator-prey systems with Beddington-DeAngelis-type functional response on periodic time scales. For this special case, we try to find the necessary and sufficient conditions for the considered system when it has at least one w-periodic solution. This study is mainly based on continuation theorem in coincidence degree theory and will also give beneficial results for continuous and discrete cases. Especially, for the continuous case, by using the study of Cui and Takeuchi (J. Math. Anal. Appl. 317: 464-474, 2006), to obtain the globally attractive w-periodic solution of the given system, an inequality is given as a necessary and sufficient condition. Additionally, for the continuous case in this study, the open problem given in the discussion part of the study of Fan and Kuang (J. Math. Anal. Appl. 295: 15-39, 2004) is solved.
Citation - WoS: 3
A New Transform Method in Nabla Discrete Fractional Calculus
(Springer, 2012) Jarad, Fahd; Kaymakcalan, Billur; Tas, Kenan
Starting from the definition of the Sumudu transform on a general nabla time scale, we define the generalized nabla discrete Sumudu transform. We obtain the nabla discrete Sumudu transform of Taylor monomials, fractional sums, and differences. We apply this transform to solve some fractional difference equations with initial value problems. MSC: 44A15, 44A55.
Citation - WoS: 8
Citation - Scopus: 11
Novel Diamond Alpha Bennett-Leindler Type Dynamic Inequalities and Their Applications
(Springernature, 2022) Kayar, Zeynep; Kaymakcalan, Billur
For the exponent zeta > 1, the diamond alpha Bennett-Leindler type inequalities are established by developing two methods, one of which is based on the convex linear combinations of the related delta and nabla inequalities, while the other one is new and is implemented by using time scale calculus rather than algebra. These inequalities can be considered as the complementary to the classical ones obtained for 0 < zeta < 1. Since both methods provide different diamond alpha Bennett-Leindler type inequalities, we can obtain various diamond alpha unifications of the known delta and nabla BennettLeindler type inequalities. Moreover, the second method offers new Bennett-Leindler type inequalities even for the special cases such as delta and nabla ones. Moreover, an application of dynamic Bennett-Leindler type inequalities to the oscillation theory of the second-order half linear dynamic equation is developed and presented for the first time ever.
Citation - WoS: 14
Citation - Scopus: 25
On Coupled Fixed Point Theorems on Partially Ordered G-Metric Spaces
(Springeropen, 2012) Karapinar, Erdal; Kaymakcalan, Billur; Tas, Kenan
In this manuscript, we extend, generalize and enrich some recent coupled fixed point theorems in the framework of partially ordered G-metric spaces in a way that is essentially more natural.
Citation - WoS: 1
Citation - Scopus: 2
On the Complementary Nabla Pachpatte Type Dynamic Inequalities Via Convexity
(Elsevier, 2024) Kaymakcalan, Billur; Kayar, Zeynep
Pachpatte type inequalities are convex generalizations of the well-known Hardy-Copson type inequalities. As Hardy-Copson type inequalities and convexity have numerous applications in pure and applied mathematics, combining these concepts will lead to more significant applications that can be used to develop certain branches of mathematics such as fuctional analysis, operator theory, optimization and ordinary/partial differential equations. We extend classical nabla Pachpatte type dynamic inequalities by changing the interval of the exponent delta from delta > 1 to delta < 0. Our results not only complement the classical nabla Pachpatte type inequalities but also generalize complementary nabla Hardy-Copson type inequalities. As the case of delta < 0 has not been previously examined, these complementary inequalities represent a novelty in the nabla time scale calculus, specialized cases in continuous and discrete scenarios, and in the dual outcomes derived in the delta time scale calculus.
Citation - WoS: 1
Citation - Scopus: 1
Pachpatte Type Inequalities and Their Nabla Unifications Via Convexity
(indian Nat Sci Acad, 2024) Kaymakcalan, Billur; Kayar, Zeynep
Nabla unifications of the discrete and continuous Pachpatte type inequalities, which are convex generalizations of Hardy-Copson type inequalities, are established. These unifications also yield dual results, namely delta Pachpatte type inequalities. Some of the dual results and some discrete and continuous versions of nabla Pachpatte type inequalities have appeared in the literature for the first time.
Citation - WoS: 9
Citation - Scopus: 12
Some Extended Nabla and Delta Hardy-Copson Type Inequalities With Applications in Oscillation Theory
(Springer Singapore Pte Ltd, 2022) Kaymakcalan, Billur; Kayar, Zeynep
We extend classical nabla and delta Hardy-Copson type inequalities from zeta > 1 to 0 < zeta < 1 and also use these novel inequalities to find necessary and sufficient condition for the nonoscillation of the related half linear dynamic equations. Since ordinary differential equations and difference equations are special cases of dynamic equations, our results cover these equations as well. Moreover, the obtained inequalities are not only novel but also unify the continuous and discrete cases for which the case 0 < zeta < 1 has not been considered so far.
Citation - WoS: 1
Some Results on Predator-Prey Dynamic Systems With Beddington-Deangelis Type Functional Response on Time Scale Calculus
(Dynamic Publishers, inc, 2017) Kaymakçalan, Billur; Pelen, Neslihan Nesliye; Guvenilir, Ayse Feza; Kaymakcalan, Billur; Matematik
We consider two dimensional predator-prey system with Beddington-DeAngelis type functional response on time scales. For this special case, we try to find under which conditions the system is permanent and globally attractive. This study gives beneficial results for continuous and discrete cases and also for solving open problems related to the dynamical properties of the systems which include the species that have unusual life cycle.