Browsing by Author "Graef, John R."

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Asymptotic Behavior of Solutions of Volterra Integro-Differential Equations With and Without Retardation
(Rocky Mt Math Consortium, 2021) Graef, John R.; Tunc, Osman
Asymptotic stability, uniform stability, integrability, and boundedness of solutions of Volterra integro- differential equations with and without constant retardation are investigated using a new type of Lyapunov- Krasovskii functionals. An advantage of the new functionals used here is that they eliminate using Gronwall's inequality. Compared to related results in the literature, the conditions here are more general, simple, and convenient to apply. Examples to show the application of the theorems are included.
Behavior of Solutions of Nonlinear Functional Volterra Integro-Differential Equations With Multiple Delays
(Dynamic Publishers, inc, 2016) Graef, John R.; Tunc, Cemil; Sevgin, Sebaheddin
The authors consider the nonlinear functional Volterra integro-differential equation with multiple delays x'(t) = -a(t)x(t) + Sigma(n)(i=1) integral(t)(t-tau i) b(i)(t,s)f(i)(x(s))ds. They give sufficient conditions so that solutions are bounded, belong to L-1, or belong to L-2. They also prove the stability and global asymptotic stability of the zero solution. Their technique of proof involves defining appropriate Lyapunov functionals.
Continuability and Boundedness of Multi-Delay Functional Integro-Differential Equations of the Second Order
(Springer-verlag Italia Srl, 2015) Graef, John R.; Tunc, Cemil
In this paper, the authors discuss the continuability and boundedness of solutions of a second order functional integro-differential equation with multiple delays. The proof involves the construction of a Lyapunov-Krasovskii type functional.
Global Asymptotic Stability and Boundedness of Certainmulti-Delay Functional Differential Equations of Third Order
(Wiley-blackwell, 2015) Graef, John R.; Tunc, Cemil
In this paper, the authors give sufficient conditions for the boundedness and global asymptotic stability of solutions to certain nonlinear multi-delay functional differential equations of the third order. The technique of proof involves defining an appropriate Lyapunov-Krasovskii functional and applying LaSalle's invariance principle. An example is included to illustrate the results. Copyright (C) 2014 JohnWiley & Sons, Ltd.
On Unique Solutions of Integral Equations by Progressive Contractions
(Univ Tabriz, 2025) Graef, John R.; Tunc, Osman; Tunc, Cemil
The authors consider Hammerstein-type integral equations for the purpose of obtaining new results on the uniqueness of solutions on an infinite interval. The approach used in the proofs is based on the technique called progressive contractions due to T. A. Burton. Here the authors apply the Burtons method to a general Hammerstein type integral equation that also yields the existence of solutions. In most of the existing literature, investigators prove uniqueness of solutions of integral equations by applying some type of fixed point theorem which can be tedious and challenging, often patching together solutions on short intervals after making complicated translations. In this article, using the progressive contractions throughout three simple short steps, each of the three steps is an elementary contraction mapping on a short interval, we improve the technique due to T. A. Burton for a general Hammerstein type integral equation and obtain the uniqueness of solutions on an infinite interval. These are advantages of the used method to prove the uniqueness of solutions.
Razumikhin Qualitative Analyses of Volterra Integro-Fractional Delay Differential Equation With Caputo Derivatives
(Elsevier, 2021) Graef, John R.; Tunc, Cemil; Sevli, Hamdullah
A non-linear system of Volterra integro-fractional delay differential equations with Caputo fractional derivatives is considered. New sufficient conditions for uniform stability, asymptotic stability, and Mittag-Leffler stability of the zero solution of the unperturbed system, and the boundedness of all solutions of the perturbed system, are presented. The technique of proof involves the Razumikhin method with an appropriate Lyapunov function. For illustrative purposes, two examples are provided. (C) 2021 Published by Elsevier B.V.
Stability of Time-Delay Systems Via the Razumikhin Method
(Springer int Publ Ag, 2022) Graef, John R.; Tunc, Cemil; Tunc, Osman
The authors consider the time delay systems both with and without a perturbation term <(x) over dot>(t) = -Dx(t) + C integral(t)(t-h) x(s)ds + P(t, x(t)) and <(x) over dot>(t) = Dx(t) + C integral(t)(t-h) x(s)ds, where x(t) is an element of R-n is the state vector, D and C is an element of R-nxn are constant matrices, P is an element of C(R+ x R-n, R-n) and h > 0 is a constant time delay. They use the Razumikhin method to obtain some new conditions for the uniform asymptotic stability, instability, and exponential stability of the zero solution, the square integrability of the norms of all solutions of the unperturbed equation, and the boundedness of solutions of the perturbed equation. In the process, they are able to give a much simpler version of a recent result by Tian et al. (Appl Math Lett 101:106058, 2020).
Ulam-Hyers Stability of Ψ-Hilfer Volterra Integro-Differential Equations of Fractional Order Containing Multiple Variable Delays
(Mdpi, 2025) Graef, John R.; Tunc, Osman; Tunc, Cemil
The authors consider a nonlinear psi-Hilfer fractional-order Volterra integro-differential equation (psi-Hilfer FrOVIDE) that incorporates N-multiple variable time delays into the equation. By utilizing the psi-Hilfer fractional derivative, they investigate the Ulam-Hyers-Rassias and Ulam-Hyers stability of the equation by using fixed-point methods. Their results improve existing ones both with and without delays by extending them to nonlinear psi-Hilfer FrOVIDEs that incorporate N-multiple variable time delays.