Browsing by Author "Kaymakcalan, Billur"

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Applications of the Novel Diamond Alpha Hardy-Copson Type Dynamic Inequalities To Half Linear Difference Equations
(Taylor & Francis Ltd, 2022) Kayar, Zeynep; Kaymakcalan, Billur
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.
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.
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.
Complements of Nabla and Delta Hardy-Copson Type Inequalities and Their Applications
(Univ Miskolc inst Math, 2025) Kayar, Zeynep; Kaymakcalan, Billur
In this paper the classical nabla and delta Hardy-Copson type inequalities, which are derived for zeta > 1, are complemented to the new case zeta < 0. These complements have exactly the same forms as the aforementioned classical inequalities except that the exponent zeta is not greater than one but it is less than zero. The obtained inequalities are not only novel but also unify the continuous and discrete cases for which the case zeta < 0 has not been considered so far either. Moreover one of the applications of Hardy-Copson type inequalities, which is to find nonoscillation criteria for the half linear differential/dynamic/difference equations, are presented by using complementary delta Hardy-Copson type inequalities.
Converses of Nabla Pachpatte-Type Dynamic Inequalities on Arbitrary Time Scales
(de Gruyter Poland Sp Z O O, 2025) Kayar, Zeynep; Kaymakcalan, Billur
Reverse Pachpatte-type inequalities are concave generalizations of the well-known Bennett-Leindler-type inequalities. We establish reverse nabla Pachpatte-type dynamic inequalities taking account of concavity. It is the first time that converses of Pachpatte-type inequalities are obtained in the nabla time scale calculus as well as for its special cases such as continuous and discrete cases and for the dual results obtained in the delta time scale calculus. Moreover, some of our results extend the related ones when concavity has been removed.
Diamond Alpha Bennett-Leindler Type Dynamic Inequalities and Their Applications
(Wiley, 2022) Kayar, Zeynep; Kaymakcalan, Billur; Pelen, Neslihan Nesliye
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.
Diamond Alpha Hardy-Copson Type Dynamic Inequalities
(Hacettepe Univ, Fac Sci, 2022) Kayar, Zeynep; Kaymakcalan, Billur
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.
Diamond-Alpha Pachpatte Type Dynamic Inequalities Via Convexity
(Springer india, 2023) Kayar, Zeynep; Kaymakcalan, Billur
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.
Hardy-Copson Type Inequalities for Nabla Time Scale Calculus
(Tubitak Scientific & Technological Research Council Turkey, 2021) Kayar, Zeynep; Kaymakcalan, Billur
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.
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.
On the Complementary Nabla Pachpatte Type Dynamic Inequalities Via Convexity
(Elsevier, 2024) Kayar, Zeynep; Kaymakcalan, Billur
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.
Pachpatte Type Inequalities and Their Nabla Unifications Via Convexity
(indian Nat Sci Acad, 2024) Kayar, Zeynep; Kaymakcalan, Billur
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.
Some Extended Nabla and Delta Hardy-Copson Type Inequalities With Applications in Oscillation Theory
(Springer Singapore Pte Ltd, 2022) Kayar, Zeynep; Kaymakcalan, Billur
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.