Browsing by Author "Kayar, Z."

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Diamond-Alpha Hardy-Copson Type Dynamic Inequalities-I
(CRC Press, 2024) Kayar, Z.; Kaymakçalan, B.
The dual results, the delta and nabla inequalities, and their special cases, continuous and discrete inequalities, are unified into diamond alpha case and new forms of such results as well as new diamond alpha Hardy-Copson type dynamic inequalities are established by developing a new method, which does not require the Integration by Parts Formula and the Fundamental Theorem of Calculus. These theorems are standard arguments in the proofs of the similar theorems in the delta and nabla approaches but do not exist in the diamond-alpha calculus. © 2025 selection and editorial matter, Ravi P. Agarwal, Bipan Hazarika and Sanket Tikare. All rights reserved.
Diamond-Alpha Hardy-Copson Type Dynamic Inequalities-Ii
(CRC Press, 2024) Kayar, Z.; Kaymakçalan, B.
The standard diamond-alpha Hardy-Copson type dynamic inequalities, which are derived for p > 1, are complemented to the new case p < 0. The novel approach for these complements is that the directions of the standard inequalities remain same while these complements are established by using a new method. This method does not require the Integration by Parts Formula and the Fundamental Theorem of Calculus, which are the standard arguments in the proofs of the similar theorems in the delta and nabla approaches but do not exist in the diamond-alpha calculus. Since it is the first time for analyzing the case p < 0, these complementary inequalities are novel not only for the diamond alpha calculus but also for the special cases, which are delta, nabla, continuous and discrete cases. © 2025 selection and editorial matter, Ravi P. Agarwal, Bipan Hazarika and Sanket Tikare. All rights reserved.
Generalized Diamond Alpha Bennett-Leindler Dynamic Inequalities
(De Gruyter, 2023) Kayar, Z.; Kaymakçalan, B.; Pelen, N.N.
The dual results; delta and nabla inequalities and their special cases; continuous and discrete inequalities are unified into diamond alpha case and new forms of such results as well as new diamond alpha Bennett-Leindler-type dynamic inequalities are established by developing a novel method, which does not require the Integration by Parts Formula and the Fundamental Theorem of Calculus. These theorems are standard arguments in the proofs of Bennett-Leindler-type dynamic inequalities in the delta and nabla approaches but do not follow naturally in the diamond alpha calculus. © 2023 Walter de Gruyter GmbH, Berlin/Bostonl. All rights reserved.
Lyapunov-Type Inequalities for Higher-Dimensional Hamiltonian Systems on Time Scales: Anew Generalized Vector Zero Approach
(Academic Press inc Elsevier Science, 2022) Kayar, Z.; Zafer, A.
By defining a generalized zero for a vector-valued function and making use of it, we obtain new Lyapunov-type inequalities for a general linear 2n x2n Hamiltonian system z(Delta) = JH(t) z of dynamic equations on time scales. The new definition is an extension from scalar functions to valued functions with respect to a matrix. Our approach in the proofs is different in the sense that several tools such as matrix measure, exponential bound function, and Dini derivatives on time scales are employed. As a classical application, we also show how the new inequalities are useful for related boundary value problems. (C) 2022 Elsevier Inc. All rights reserved.
Matrix Measure Approach To Lyapunov-Type Inequalities for Linear Hamiltonian Systems With Impulse Effect
(Academic Press inc Elsevier Science, 2016) Kayar, Z.; Zafer, A.
We present new Lyapunov-type inequalities for Hamiltonian systems, consisting of 2n-first-order linear impulsive differential equations, by making use of matrix measure approach. The matrix measure estimates of fundamental matrices of linear impulsive systems are crucial in obtaining sharp inequalities. To illustrate usefulness of the inequalities we have derived new disconjugacy criteria for Hamiltonian systems under impulse effect and obtained new lower bound estimates for eigenvalues of impulsive eigenvalue problems. (C) 2016 Elsevier Inc. All rights reserved.
Sturm-Picone Comparison Theorems for Nonlinear Impulsive Differential Equations With Discontinuous Solutions
(Wiley, 2017) Kayar, Z.; Masiha, S. K.
The nonlinear versions of Sturm-Picone comparison theorem as well as Leighton's variational lemma and Leighton's theorem for regular and singular nonlinear impulsive differential equations with discontinuous solutions having fixedmoments of impulse actions are established. Although discontinuity of the solutions causes some difficulties, these new comparison theorems cover the old ones where impulse effects are dropped.