scholarly journals On some new double dynamic inequalities associated with Leibniz integral rule on time scales

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
A. A. El-Deeb ◽  
Saima Rashid
Keyword(s):  
Time Scales ◽  
Initial Value Problems ◽  
Initial Value ◽  
Retarded Time ◽  
Special Cases ◽  
New Inequalities

AbstractIn 2020, El-Deeb et al. proved several dynamic inequalities. It is our aim in this paper to give the retarded time scales case of these inequalities. We also give a new proof and formula of Leibniz integral rule on time scales. Beside that, we also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as special cases. Furthermore, we study boundedness of some delay initial value problems by applying our results as application.

scholarly journals Intrinsic Discontinuities in Solutions of Evolution Equations Involving Fractional Caputo–Fabrizio and Atangana–Baleanu Operators

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2023
Author(s):  
Christopher Nicholas Angstmann ◽  
Byron Alexander Jacobs ◽  
Bruce Ian Henry ◽  
Zhuang Xu
Keyword(s):  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Evolution Equations ◽  
Integral Transform ◽  
Initial Value ◽  
Intrinsic Nature ◽  
Recent Interest ◽  
Special Cases ◽  
Singular Kernels

There has been considerable recent interest in certain integral transform operators with non-singular kernels and their ability to be considered as fractional derivatives. Two such operators are the Caputo–Fabrizio operator and the Atangana–Baleanu operator. Here we present solutions to simple initial value problems involving these two operators and show that, apart from some special cases, the solutions have an intrinsic discontinuity at the origin. The intrinsic nature of the discontinuity in the solution raises concerns about using such operators in modelling. Solutions to initial value problems involving the traditional Caputo operator, which has a singularity inits kernel, do not have these intrinsic discontinuities.

scholarly journals Delta-Nabla Type Maximum Principles for Second-Order Dynamic Equations on Time Scales and Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-28
Author(s):  
Jiang Zhu ◽  
Dongmei Liu
Keyword(s):  
Time Scales ◽  
Initial Value Problems ◽  
Second Order ◽  
Dynamic Equations ◽  
Maximum Principles ◽  
Uniqueness Theorems ◽  
Initial Value ◽  
Approximation Theorems ◽  
Construction Techniques ◽  
Linear And Nonlinear

Some delta-nabla type maximum principles for second-order dynamic equations on time scales are proved. By using these maximum principles, the uniqueness theorems of the solutions, the approximation theorems of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear and nonlinear initial value problems and boundary value problems on time scales are proved, the oscillation of second-order mixed delat-nabla differential equations is discussed and, some maximum principles for second order mixed forward and backward difference dynamic system are proved.

scholarly journals New Inequalities of Opial's Type on Time Scales and Some of Their Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Samir H. Saker
Keyword(s):  
Time Scales ◽  
Second Order ◽  
Dynamic Equations ◽  
Linear Dynamic ◽  
Special Cases ◽  
Discrete Inequalities ◽  
Zeros Of Solutions ◽  
Yield Conditions ◽  
New Inequalities

We will prove some new dynamic inequalities of Opial's type on time scales. The results not only extend some results in the literature but also improve some of them. Some continuous and discrete inequalities are derived from the main results as special cases. The results will be applied on second-order half-linear dynamic equations on time scales to prove several results related to the spacing between consecutive zeros of solutions and the spacing between zeros of a solution and/or its derivative. The results also yield conditions for disfocality of these equations.

scholarly journals Some Dynamic Inequalities via Diamond Integrals for Function of Several Variables

2021 ◽  
Vol 5 (4) ◽  
pp. 207
Author(s):  
Muhammad Bilal ◽  
Khuram Ali Khan ◽  
Hijaz Ahmad ◽  
Ammara Nosheen ◽  
Khalid Mahmood Awan ◽  
...  
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Jensen’S Inequality ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Several Variables ◽  
Time Scale Calculus ◽  
Special Cases ◽  
Function Of Several Variables ◽  
New Inequalities

In this paper, Jensen’s inequality and Fubini’s Theorem are extended for the function of several variables via diamond integrals of time scale calculus. These extensions are used to generalize Hardy-type inequalities with general kernels via diamond integrals for the function of several variables. Some Hardy Hilbert and Polya Knop type inequalities are also discussed as special cases. Classical and new inequalities are deduced from the main results using special kernels and particular time scales.

scholarly journals On Some New Weighted Steffensen-Type Inequalities on Time Scales

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2670
Author(s):  
Ahmed A. El-Deeb ◽  
Omar Bazighifan ◽  
Jan Awrejcewicz
Keyword(s):  
Time Scales ◽  
Main Idea ◽  
Special Cases ◽  
New Inequalities

The motivation of this paper is to explore some new inequalities of Steffensen-type which were demonstrated by Pečarić and Kalamir in 2014. The main idea is to investigate a class of certain inequalities by employing diamond-α dynamic integral on time scales. In addition, to obtain some new inequalities as special cases, we also extend our results to continuous and discrete calculations.

scholarly journals Existence-uniqueness of solutions for fuzzy nabla initial value problems on time scales

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
R. Leelavathi ◽  
G. Suresh Kumar ◽  
M. S. N. Murty ◽  
R. V. N. Srinivasa Rao
Keyword(s):  
Time Scales ◽  
Initial Value Problems ◽  
Initial Value ◽  
Uniqueness Of Solutions

Extensions of Dynamic Inequalities of Hardy’s Type on Time Scales

2015 ◽  
Vol 65 (5) ◽  
Author(s):  
S. H. Saker ◽  
Donal O’Regan
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Hölder Inequality ◽  
Chain Rule ◽  
Simple Consequence ◽  
Hardy Type ◽  
Special Cases ◽  
Discrete Inequalities ◽  
New Inequalities

AbstractIn this paper using some algebraic inequalities, Hölder inequality and a simple consequence of Keller’s chain rule we prove some new inequalities of Hardy type on a time scale T. These inequalities as special cases contain some integral and discrete inequalities when T = ℝ and T = ℕ.

scholarly journals The Taylor series method of order $p$ and Adams-Bashforth method on time scales

2021 ◽  
Author(s):  
Svetlin Georgiev ◽  
Inci Erhan
Keyword(s):  
Time Scales ◽  
Taylor Series ◽  
Initial Value Problems ◽  
Arbitrary Order ◽  
Trapezoidal Rule ◽  
Dynamic Equations ◽  
Series Method ◽  
Initial Value ◽  
Nonlinear Dynamic Equations ◽  
Taylor Series Method

A recent study on the Taylor series method of second order and the trapezoidal rule for dynamic equations on time scales has been continued by introducing a derivation of the Taylor series method of arbitrary order $p$ on time scales. The error and convergence analysis of the method is also obtained. The 2 step Adams-Bashforth method for dynamic equations on time scales is concluded and applied to examples of initial value problems for nonlinear dynamic equations. Numerical results are presented and discussed.

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