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 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.

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 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.

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 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 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.

scholarly journals On a sum analogous to Dedekind sum and its mean square value formula

2002 ◽  
Vol 32 (1) ◽  
pp. 47-55 ◽  
Author(s):  
Zhang Wenpeng
Keyword(s):  
Asymptotic Property ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Mean Square ◽  
Dedekind Sum ◽  
The Mean

The main purpose of this paper is using the mean value theorem of DirichletL-functions to study the asymptotic property of a sum analogous to Dedekind sum, and give an interesting mean square value formula.

The mean-value theorem for elliptic operators on stratified sets

2007 ◽  
Vol 81 (3-4) ◽  
pp. 365-372
Author(s):  
S. N. Oshchepkova ◽  
O. M. Penkin
Keyword(s):  
Mean Value ◽  
Elliptic Operators ◽  
Mean Value Theorem ◽  
The Mean
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