Scopus İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8651
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Article Citation - WoS: 1Citation - Scopus: 1Pachpatte Type Inequalities and Their Nabla Unifications Via Convexity(indian Nat Sci Acad, 2024) Kaymakcalan, Billur; Kayar, ZeynepNabla 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.Article Citation - WoS: 1Citation - Scopus: 2On the Complementary Nabla Pachpatte Type Dynamic Inequalities Via Convexity(Elsevier, 2024) Kaymakcalan, Billur; Kayar, ZeynepPachpatte 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.Article Citation - WoS: 2Citation - Scopus: 6Diamond Alpha Hardy-Copson Type Dynamic Inequalities(Hacettepe Univ, Fac Sci, 2022) Kaymakcalan, Billur; Kayar, ZeynepIn 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.Article Citation - WoS: 6Citation - Scopus: 9Applications of the Novel Diamond Alpha Hardy-Copson Type Dynamic Inequalities To Half Linear Difference Equations(Taylor & Francis Ltd, 2022) Kaymakcalan, Billur; Kayar, ZeynepThis 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.Article Citation - WoS: 9Citation - Scopus: 12Some Extended Nabla and Delta Hardy-Copson Type Inequalities With Applications in Oscillation Theory(Springer Singapore Pte Ltd, 2022) Kaymakcalan, Billur; Kayar, ZeynepWe 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.Article Citation - WoS: 10Citation - Scopus: 12Diamond Alpha Bennett-Leindler Type Dynamic Inequalities and Their Applications(Wiley, 2022) Kaymakcalan, Billur; Pelen, Neslihan Nesliye; Kayar, ZeynepIn 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.