A Class of Nonlinear Singular Difference Inequality in Engineering

2014 ◽  
Vol 577 ◽  
pp. 824-827
Author(s):  
Wu Sheng Wang ◽  
Zong Yi Hou
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
Jensen Inequality ◽  
Fractional Difference ◽  
Difference Inequality ◽  
Amplification Method ◽  
Weakly Singular ◽  
Change Of Variable ◽  
Explicit Bounds ◽  
The Mean

In this paper, we discuss a class of new nonlinear weakly singular difference inequality. Using change of variable, discrete Jensen inequality, amplification method, the mean-value theorem for integrals and Gamma function, explicit bounds for the unknown functions in the inequality is given clearly. The derived results can be applied in the study of fractional difference equations in Engineering.

Estimation of Unknown Function in a Class of Singular Difference Inequality in Engineering

2014 ◽  
Vol 945-949 ◽  
pp. 2467-2470
Author(s):  
Chun Miao Huang ◽  
Wu Sheng Wang
Keyword(s):  
Beta Function ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Jensen Inequality ◽  
Fractional Difference ◽  
Difference Inequality ◽  
Amplification Method ◽  
Change Of Variable ◽  
Explicit Bounds ◽  
The Mean

In this paper, we discuss a class of new weakly singular difference inequality, which is solved using change of variable, discrete Jensen inequality, Beta function, the mean-value theorem for integrals and amplification method, and explicit bounds for the unknown functions is given clearly. The derived results can be applied in the study of fractional difference equations in Engineering.

A Class of Singular Volterra-Fredholm Type Difference Inequality in Engineering

2014 ◽  
Vol 571-572 ◽  
pp. 132-138
Author(s):  
Wu Sheng Wang ◽  
Chun Miao Huang
Keyword(s):  
Beta Function ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Jensen Inequality ◽  
Fredholm Type ◽  
Difference Inequality ◽  
Amplification Method ◽  
Change Of Variable ◽  
Explicit Bounds ◽  
The Mean

In this paper, we discuss a class of new weakly singular Volterra-Fredholm difference inequality, which is solved using change of variable, discrete Jensen inequality, Beta function, the mean-value theorem for integrals and amplification method, and explicit bounds for the unknown functions is given clearly. The derived results can be applied in the study of fractional difference equations in engineering.

scholarly journals A Class of Iterative Nonlinear Difference Inequality with Weakly Singularity

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Chunmiao Huang ◽  
Wu-Sheng Wang ◽  
Xiaoliang Zhou
Keyword(s):  
Gamma Function ◽  
Mean Value ◽  
Hölder Inequality ◽  
Mean Value Theorem ◽  
Jensen Inequality ◽  
Difference Inequality ◽  
Amplification Method ◽  
Weakly Singular ◽  
Change Of Variable ◽  
The Mean

We discuss a class of new nonlinear weakly singular difference inequality, which is solved by change of variable, discrete Hölder inequality, discrete Jensen inequality, the mean-value theorem for integrals and amplification method, and Gamma function. Explicit bound for the unknown function is given clearly. Moreover, an example is presented to show the usefulness of our results.

A Class of Nonlinear Difference Inequality with Weakly Singularity in Engineering

2014 ◽  
Vol 945-949 ◽  
pp. 2426-2429
Author(s):  
Zong Yi Hou ◽  
Wu Sheng Wang
Keyword(s):  
Gamma Function ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Difference Inequality ◽  
Amplification Method ◽  
Weakly Singular ◽  
Change Of Variable ◽  
Explicit Bounds ◽  
The Mean

In this paper, we discuss a class of new nonlinear weakly singular difference inequality, which is solved by change of variable, the mean-value theorem for integrals and amplification method, Gamma function, and explicit bounds for the unknown functions is given clearly.

A Class of Volterra-Fredholm Difference Inequality with Weakly Singularity in Engineering

2014 ◽  
Vol 571-572 ◽  
pp. 139-142 ◽  
Author(s):  
Wu Sheng Wang ◽  
Zong Yi Hou
Keyword(s):  
Schwarz Inequality ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Jensen Inequality ◽  
Difference Inequality ◽  
Amplification Method ◽  
Function Change ◽  
Change Of Variable ◽  
Explicit Bounds ◽  
The Mean

In this paper, we discuss a class of new Volterra-Fredholm weakly singular difference inequality. The explicit bounds for the unknown functions are given clearly by discrete Jensen inequality, Cauchy-Schwarz inequality, Gamma function, change of variable, the mean-value theorem for integrals and amplification method. The derived results can be applied in the study of fractional difference equations in engineering.

scholarly journals Analysis of discrete fractional operators

2015 ◽  
Vol 9 (1) ◽  
pp. 139-149 ◽  
Author(s):  
Ferhan Atici ◽  
Meltem Uyanik
Keyword(s):  
Fractional Calculus ◽  
Difference Operator ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Fractional Operators ◽  
Discrete Fractional Calculus ◽  
Fractional Difference ◽  
The Mean ◽  
Discrete Domain

In this paper, we introduce two new monotonicity concepts for a nonnegative or nonpositive valued function defined on a discrete domain. We give examples to illustrate connections between these new monotonicity concepts and the traditional ones. We then prove some monotonicity criteria based on the sign of the fractional difference operator of a function f, ??f with 0 < ? < 1. As an application, we state and prove the mean value theorem on discrete fractional calculus.

Several Estimations of a Class of Integral Inequality in Engineering

2014 ◽  
Vol 962-965 ◽  
pp. 2748-2751
Author(s):  
Wu Sheng Wang ◽  
Ji Ting Huang
Keyword(s):  
Singular Integral ◽  
Integral Inequality ◽  
Analysis Techniques ◽  
Amplification Method ◽  
Weakly Singular ◽  
Change Of Variable ◽  
Explicit Bounds

In this paper, we discuss a class of new nonlinear weakly singular integral inequality. Under different assumptions, the inequality is solved by analysis techniques, such as: change of variable, amplification method, and three explicit bounds for the unknown functions are given clearly.

scholarly journals Difference monotonicity analysis on discrete fractional operators with discrete generalized Mittag-Leffler kernels

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Pshtiwan Othman Mohammed ◽  
Faraidun Kadir Hamasalh ◽  
Thabet Abdeljawad
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
Fractional Operators ◽  
Fractional Difference ◽  
Monotonicity Properties ◽  
The Mean ◽  
Monotonicity Analysis

AbstractIn this paper, we present the monotonicity analysis for the nabla fractional differences with discrete generalized Mittag-Leffler kernels $( {}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y )(\eta )$ ( a − 1 A B R ∇ δ , γ y ) ( η ) of order $0<\delta <0.5$ 0 < δ < 0.5 , $\beta =1$ β = 1 , $0<\gamma \leq 1$ 0 < γ ≤ 1 starting at $a-1$ a − 1 . If $({}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y ) ( \eta )\geq 0$ ( a − 1 A B R ∇ δ , γ y ) ( η ) ≥ 0 , then we deduce that $y(\eta )$ y ( η ) is $\delta ^{2}\gamma $ δ 2 γ -increasing. That is, $y(\eta +1)\geq \delta ^{2} \gamma y(\eta )$ y ( η + 1 ) ≥ δ 2 γ y ( η ) for each $\eta \in \mathcal{N}_{a}:=\{a,a+1,\ldots\}$ η ∈ N a : = { a , a + 1 , … } . Conversely, if $y(\eta )$ y ( η ) is increasing with $y(a)\geq 0$ y ( a ) ≥ 0 , then we deduce that $({}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y )(\eta ) \geq 0$ ( a − 1 A B R ∇ δ , γ y ) ( η ) ≥ 0 . Furthermore, the monotonicity properties of the Caputo and right fractional differences are concluded to. Finally, we find a fractional difference version of the mean value theorem as an application of our results. One can see that our results cover some existing results in the literature.

scholarly journals On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1303
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Faraidun Kadir Hamasalh
Keyword(s):  
Time Scale ◽  
Initial Value Problems ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Discrete Fractional Calculus ◽  
Initial Value ◽  
Forward Difference ◽  
The Mean ◽  
Monotonicity Analysis ◽  
Monotonicity Results

Monotonicity analysis of delta fractional sums and differences of order υ∈(0,1] on the time scale hZ are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional h-difference and delta Caputo fractional h-differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that y(z) is υ-increasing on Ma+υh,h, where the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and then, we can show that y(z) is υ-increasing on Ma+υh,h, where the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) for each z∈Ma+h,h. Conversely, if y(a+υh) is greater or equal to zero and y(z) is increasing on Ma+υh,h, we show that the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) on Ma,h. Furthermore, we consider some related results for strictly increasing, decreasing, and strictly decreasing cases. Finally, the fractional forward difference initial value problems and their solutions are investigated to test the mean value theorem on the time scale hZ utilizing the monotonicity results.

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