scholarly journals Approximation Properties of Solutions of a Mean Value-Type Functional Inequality, II

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1299
Author(s):  
Soon-Mo Jung ◽  
Ki-Suk Lee ◽  
Michael Th. Rassias ◽  
Sung-Mo Yang
Keyword(s):  
Functional Equation ◽  
Type Equation ◽  
Functional Inequality ◽  
Unit Element ◽  
Interesting Property ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Ulam Stability ◽  
Approximation Properties ◽  
The Mean

Let X be a commutative normed algebra with a unit element e (or a normed field of characteristic different from 2), where the associated norm is sub-multiplicative. We prove the generalized Hyers-Ulam stability of a mean value-type functional equation, f(x)−g(y)=(x−y)h(sx+ty), where f,g,h:X→X are functions. The above mean value-type equation plays an important role in the mean value theorem and has an interesting property that characterizes the polynomials of degree at most one. We also prove the Hyers-Ulam stability of that functional equation under some additional conditions.

A simpler proof of Levinson's theorem

1985 ◽  
Vol 97 (3) ◽  
pp. 385-395 ◽  
Author(s):  
J. B. Conrey ◽  
A. Ghosh
Keyword(s):  
Functional Equation ◽  
Critical Line ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Levinson’S Theorem ◽  
The Mean ◽  
Image Position

In this paper we present a proof of the mean-value theorem required by Levinson to show that at least one-third of the zeros of ζ(s) are on the critical line. As in Levinson [3], letwhere Χ(s)=ζ(s)/ζ(1−s) is the usual factor from the functional equation, and letwhereandwhere

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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