scholarly journals CERTAIN DEVELOPMENTS ON SOLUTIONS OF DYNAMIC INEQUALITIES IN THE PREMISE OF TIME SCALES

Fractals ◽  
2021 ◽  
pp. 2240024
Author(s):  
ZAREEN A. KHAN ◽  
KADIR KAYNAK
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Nonlinear Dynamic ◽  
Dynamic Equations ◽  
Unknown Parameters ◽  
Independent Variable ◽  
Research Plan ◽  
Scale Scheme

On the grounds of few perceived Gronwall inequalities, we inspect some new relevant nonlinear dynamic inequalities on time scales related to one independent variable. To guarantee the evenness of information division and with the conversation of peculiar cases, it is indicated that these inequalities catch dynamic, delay, Volterra–Fredholm, discrete, fractional, etc. The inequalities suggested right here to other specific bounds on unknown parameters may be handled as powerful equipment for determining the features of various dynamic equations on time scales. It is trusted that this research plan will open new possibilities in in-depth examination of time scale scheme structure of varying nature.

scholarly journals Solvability for a class of integral inequalities with maxima on the theory of time scales and their applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zareen A. Khan
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Unknown Function ◽  
Nonlinear Dynamic ◽  
Time Interval ◽  
Specific Class ◽  
Initial Condition ◽  
Basic Technique ◽  
Arbitrary Time ◽  
Independent Variable

Abstract This paper is committed to introducing some Bihari type inequalities for scalar functions of one independent variable under an initial condition associated with an arbitrary time scale $\mathbb{T}$ T . The integrals involve the maximum of an unknown function over a past time interval. We not only solve some new estimated bounds of a specific class of retarded and nonlinear dynamic inequalities but also derive and unify continuous inequalities along with the corresponding discrete analogs of some known results with ‘maxima’ on time scales. We illustrate some applications of the considered inequalities to represent the advantages of our work. The main results will be proved by utilizing some examination procedures and the basic technique of Keller’s chain rule on time scales.

scholarly journals Li–Yorke Chaos in Hybrid Systems on a Time Scale

2015 ◽  
Vol 25 (14) ◽  
pp. 1540024 ◽  
Author(s):  
Marat Akhmet ◽  
Mehmet Onur Fen
Keyword(s):  
Differential Equations ◽  
Hybrid Systems ◽  
Time Scales ◽  
Time Scale ◽  
Impulsive Differential Equations ◽  
Reduction Technique ◽  
Duffing Equation ◽  
Dynamic Equations ◽  
Definition Of ◽  
First Time

By using the reduction technique to impulsive differential equations [Akhmet & Turan, 2006], we rigorously prove the presence of chaos in dynamic equations on time scales (DETS). The results of the present study are based on the Li–Yorke definition of chaos. This is the first time in the literature that chaos is obtained for DETS. An illustrative example is presented by means of a Duffing equation on a time scale.

scholarly journals Improved Oscillation Results for Functional Nonlinear Dynamic Equations of Second Order

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1897
Author(s):  
Taher S. Hassan ◽  
Yuangong Sun ◽  
Amir Abdel Menaem
Keyword(s):  
Dynamic Equation ◽  
Time Scale ◽  
Nonlinear Dynamic ◽  
Recent Literature ◽  
Second Order ◽  
Dynamic Equations ◽  
Oscillation Criteria ◽  
Arbitrary Time ◽  
Nonlinear Dynamic Equations ◽  
Type Oscillation

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 equation are investigated. The presented results confirm that the study of the equation in this formula is superior to other previous studies. Some examples are addressed to demonstrate the finding.

scholarly journals Solvability and Stability of Impulsive Set Dynamic Equations on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
Shihuang Hong ◽  
Jing Gao ◽  
Yingzi Peng
Keyword(s):  
Differential Equations ◽  
Time Scales ◽  
Difference Equations ◽  
Time Scale ◽  
Dynamic Equations ◽  
Stability Of Solutions ◽  
Real Numbers ◽  
Generalized Derivative ◽  
Special Cases ◽  
Existence And Stability

A class of new nonlinear impulsive set dynamic equations is considered based on a new generalized derivative of set-valued functions developed on time scales in this paper. Some novel criteria are established for the existence and stability of solutions of such model. The approaches generalize and incorporate as special cases many known results for set (or fuzzy) differential equations and difference equations when the time scale is the set of the real numbers or the integers, respectively. Finally, some examples show the applicability of our results.

scholarly journals Oscillatory and Asymptotic Properties on a Class of Third Nonlinear Dynamic Equations with Damping Term on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-17
Author(s):  
Qinghua Feng ◽  
Huizeng Qin
Keyword(s):  
Time Scales ◽  
Nonlinear Dynamic ◽  
Asymptotic Properties ◽  
The Other ◽  
Dynamic Equations ◽  
Damping Term ◽  
Third Order ◽  
Discrete Analysis ◽  
Other Hand ◽  
Nonlinear Dynamic Equations

We establish some new oscillatory and asymptotic criteria for a class of third-order nonlinear dynamic equations with damping term on time scales. The established results on one hand extend some known results in the literature on the other hand unify continuous and discrete analysis. For illustrating the validity of the established results, we also present some applications for them.

scholarly journals Kamenev-Type Oscillation Criteria of Second-Order Nonlinear Dynamic Equations on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Yang-Cong Qiu ◽  
Qi-Ru Wang
Keyword(s):  
Time Scales ◽  
Nonlinear Dynamic ◽  
Second Order ◽  
Dynamic Equations ◽  
Riccati Technique ◽  
Oscillation Criteria ◽  
Function Classes ◽  
Nonlinear Dynamic Equations ◽  
Generalized Riccati Technique ◽  
Type Oscillation

Using functions from some function classes and a generalized Riccati technique, we establish Kamenev-type oscillation criteria for second-order nonlinear dynamic equations on time scales of the form(p(t)ψ(x(t))k∘xΔ(t))Δ+f(t,x(σ(t)))=0. Two examples are included to show the significance of the results.

scholarly journals Oscillation Criteria for Some New Generalized Emden-Fowler Dynamic Equations on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Haidong Liu ◽  
Puchen Liu
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Analytical Techniques ◽  
Dynamic Equations ◽  
Oscillation Criteria ◽  
Special Cases

By means of novel analytical techniques, we have established several new oscillation criteria for the generalized Emden-Fowler dynamic equation on a time scale𝕋, that is,(r(t)|ZΔ(t)|α-1ZΔ(t))Δ+f(t,x(δ(t)))=0, with respect to the case∫t0∞r-1/α(s)Δs=∞and the case∫t0∞r-1/α(s)Δs<∞, whereZ(t)=x(t)+p(t)x(τ(t)),  αis a constant,|f(t,u)|⩾q(t)|uβ|,βis a constant satisfyingα⩾β>0, andr,p, andqare real valued right-dense continuous nonnegative functions defined on𝕋. Noting the parameter valueαprobably unequal toβ, our equation factually includes the existing models as special cases; our results are more general and have wider adaptive range than others' work in the literature.

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