As an effective tool to unify discrete and continuous analysis, time scale calculus have been widely applied to study dynamic systems in both theoretical and practical aspects. In addition to such a classical role of unification, the dynamic equation...
In this paper, we give new sufficient conditions for boundedness and exponential stability of solutions for nonlinear Volterra integro dynamic equations from above on unbounded time scales using first Lyapunovs method. To prove this result we reduce...
In this work, by using one dynamic Gronwall–Bihari-type integral inequality on time scales, an interesting asymptotic behavior property of high-order nonlinear dynamic equations on time scales was obtained, which also generalized two classical...
This paper presents some oscillation criteria for second-order half-linear dynamic equations defined on unbounded above arbitrary time scales. These criteria offer sufficient conditions for all solutions of the equations to display oscillatory behavi...
In this paper, we make a detailed analysis of the structure of non-oscillatory solutions for second order superlinear and sublinear dynamic equations on time scales. The sufficient and necessary conditions for existence of non-oscillatory solutions a...
In this paper, we are concerned with the oscillation of solutions to a class of third-order nonlinear neutral dynamic equations on time scales. New oscillation criteria are presented by employing the Riccati transformation and integral averaging tech...
In this paper, the non-eigenvalue forms of Liouville’s formulas for delta, nabla and α -diamond matrix dynamic equations on time scales are given and proved. Meanwhile, a diamond matrix exponential function (or α -matrix e...
Under a couple of canonical and mixed canonical-noncanonical conditions, we investigate the oscillation and asymptotic behavior of solutions to a class of third-order nonlinear neutral dynamic equations with mixed deviating arguments on time scales....
In the sense of an arbitrary time scale, some new sufficient conditions on oscillation are presented in this paper for a class of nonlinear third-order delay dynamic equations involving a local fractional derivative with a super-linear neutral term....
Using higher order delta derivatives on time scales, we demonstrated a few dynamic inequalities of the Opial type in this paper. Our findings expanded upon and generalised earlier findings in the literature. Furthermore, we give the discrete and cont...
In this paper, we obtain sufficient conditions for Hyers–Ulam and Hyers–Ulam–Rassias stability of an abstract second–order nonlinear dynamic equation on bounded time scales. An illustrative example is given to show the applicability of the theoretica...
In this paper, we introduce a new metric on the space of fuzzy continuous functions on time scales by using the exponential function, e γ ( t , t 0 ) , where γ > 0 is a constant. Then, we provide some conditions to prove an ex...
This study presents novel and generalizable sufficient conditions for determining the oscillatory behavior of solutions to higher-order half-linear neutral delay dynamic equations on time scales. Utilizing the Riccati transformation technique in comb...
In this article, we establish certain time-scale-retarded dynamic inequalities that contain nonlinear retarded integral equations on various time scales. These inequalities extend and generalize some significant inequalities existing in the literatur...
This paper investigates second-order functional dynamic equations with mixed nonlinearities on an arbitrary unbounded above-time scale, T. We will use a unified time scale approach and the well-known Riccati technique to derive oscillation criteria o...
In this paper, by employing matched spaces for time scales, we introduce a δ -almost periodic function and obtain some related properties. Also the hull equation for homogeneous dynamic equation is introduced and results of the existence a...
The aim of this paper is to deduce the asymptotic and Hille-type criteria of the dynamic equations of third order on time scales. Some of the presented results concern the sufficient condition for the oscillation of all solutions of third-order dynam...
In this paper, we consider the oscillation behavior of the following second-order nonlinear dynamic equation. λ(s)Ψ1φΔ(s)y(φ(s))ΔΔ+η(s)Φ(y(τ(s)))=0,s∈[s0,∞)T. By employing generalized Riccati transformation and inequality scaling technique, we establ...
We present here more general concepts of Hausdorff derivatives (structural Nabla derivatives) on a timescale. We examine structural Nabla integration on temporal scales. Using the fixed-point theorem, we establish adequate criteria for the question o...
This work aims to develop new iterative Kneser-type criteria for determining the oscillatory behaviour of half-linear second-order advanced dynamic equations on arbitrary unbounded-above time scales T. The results extend and refine previously establi...
In this paper, we consider two universal higher order dynamic equations with several delay functions. We will establish two oscillatory criteria of the first equation and a sufficient and necessary condition for the second equation with a nonoscillat...
In this paper, the functional dynamic equation of second order is studied on an arbitrary time scale under milder restrictions without the assumed conditions in the recent literature. The Nehari, Hille, and Ohriska type oscillation criteria of the eq...
For a class of nonlinear higher-order neutral dynamic equations on a time scale, we analyze the existence and asymptotic behavior of nonoscillatory solutions on the basis of hypotheses that allow applications to equations with different integral conv...
The objective of this paper is to derive new Hille type and Ohriska type criteria for third-order nonlinear dynamic functional equations in the form of a2(ζ)φα2a1ζφα1xΔ(ζ)ΔΔ+q(ζ)φαx(...
In this paper, we introduce the concept of a n 0 -order weighted pseudo Δ n 0 δ -almost automorphic function under the matched space for time scales and we present some properties. The results are valid for q-difference...
In this paper, we unify and extend the linear quadratic pursuit–evasion games to dynamic equations on time scales. A mixed strategy for a single pursuer and evader is studied in two settings. In the open-loop setting, the corresponding controls...
By using Bochner transform, Stepanov almost periodic functions inherit some basic properties directly from almost periodic functions. Recently, this old work was extended to time scales. However, we show that Bochner transform is not valid on time sc...
In this paper, we examine the oscillatory behavior of solutions to a class of half-linear third-order dynamic equations with deviating arguments α2(η)ϕδ2α1ηϕδ1uΔ(η)ΔΔ+p(η)ϕ&delt...
This paper aims to improve Hille oscillation criteria for the third-order functional dynamic equation p2(ξ)ϕγ2p1ξϕγ1yΔ(ξ)ΔΔ+a(ξ)ϕγy(g(ξ))=0, on an above-unbounded time scale T. The obt...
The aim of this study is to refine the known Riccati transformation technique to provide new oscillation criteria for solutions to second-order dynamic equations over time. It is important to note that the convergence or divergence of some improper i...
In this paper, the sharp Hille-type oscillation criteria are proposed for a class of second-order nonlinear functional dynamic equations on an arbitrary time scale, by using the technique of Riccati transformation and integral averaging method. The o...
This study aims to establish novel iterative oscillation criteria for second-order half-linear advanced dynamic equations in non-canonical form. The results extend and enhance recently established criteria for this type of equation by various authors...
This paper is dedicated to exploring the existence, uniqueness and Ulam stability analysis applied to a specific class of mathematical equations known as nonlinear impulsive Volterra Fredholm integro-dynamic adjoint equations within finite time scale...
In this work, we develop enhanced Hille-type oscillation conditions for arbitrary-time, second-order quasilinear functional dynamic equations. These findings extend and improve previous research that has been published in the literature. Some example...
In contrast to previous research on periodic averaging principles for various types of impulsive stochastic differential equations (ISDEs), we establish an averaging principle without periodic assumptions of coefficients and impulses for impulsive st...
In this study, novel criteria are derived to ensure the oscillation of solutions in nonlinear advanced noncanonical dynamic equations. The obtained results are reminiscent of the criteria proposed by Hille and Ohriska for canonical dynamic equations....
In this work, we deduce a new criterion that guarantees the oscillation of solutions to linear Sturm–Liouville delay noncanonical dynamic equations; these results emulate the criteria of the Hille and Ohriska types for canonical dynamic equatio...
The objective of the research article is two-fold. Firstly, we present a fixed point result in the context of triple controlled metric type spaces with a distinctive contractive condition involving the controlled functions. Secondly, we consider an i...
We are studying first order differential inclusions with periodic boundary conditions where the Stieltjes derivative with respect to a left-continuous non-decreasing function replaces the classical derivative. The involved set-valued mapping is not a...
As a powerful tool that can be used to solve both continuous and discrete equations, the Lie symmetry analysis of dynamical systems on a time scale is investigated. Applying the method to the Burgers equation and Euler equation, we get the symmetry o...
We prove some new dynamic inequalities of the Gronwall–Bellman–Pachpatte type on time scales. Our results can be used in analyses as useful tools for some types of partial dynamic equations on time scales and in their applications in envi...
The coupled cubic-quintic nonlinear Schrödinger (CQNLS) equation is a universal mathematical model describing many physical situations, such as nonlinear optics and Bose–Einstein condensate. In this paper, in order to simplify the process...
In order to merge continuous and discrete analyses, a number of dynamic derivative equations have been put out in the process of developing a time-scale calculus. The investigations that incorporated combined dynamic derivatives have led to the propo...
In this paper, based on the concept of complete-closed time scales attached with shift direction under non-translational shifts (or S-CCTS for short), as a first attempt, we develop the concepts of S-equipotentially almost automorphic sequences, disc...
In this article, with the help of Leibniz integral rule on time scales, we prove some new dynamic inequalities of Gronwall–Bellman–Pachpatte-type on time scales. These inequalities can be used as handy tools to study the qualitative and q...
The present article aims to study the oscillatory properties of a class of second-order dynamic equations on time scales. We consider during this study the noncanonical case, which did not receive much attention compared to the canonical dynamic equa...
As a new usage of Leibniz integral rule on time scales, we proved some new extensions of dynamic Gronwall–Pachpatte-type inequalities on time scales. Our results extend some existing results in the literature. Some integral and discrete inequal...
Some possible correspondences between the Scale Relativity Theory and the Space–Time Theory can be established. Since both the multifractal Schrödinger equation from the Scale Relativity Theory and the General Relativity equations for a gr...
We established several novel inequalities of Gronwall–Pachpatte type on time scales. Our results can be used as handy tools to study the qualitative and quantitative properties of the solutions of the initial boundary value problem for a partia...
We show theoretically that the mean turbulent dynamics can be described by a kinetic theory representation with a single free relaxation time that depends on space and time. A proper kinetic equation is constructed from the Klimontovich-type kinetic...
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