Applications of Nonlinear One-Dimensional Continuous, Discontinuous Integral Inequalities and Discrete Inequalities

2016 ◽  
pp. 423-534
Author(s):  
Yuming Qin
Keyword(s):  
Integral Inequalities ◽  
One Dimensional ◽  
Discrete Inequalities

scholarly journals Chebyshev-Type Integral Inequalities for Continuous Fields of Operators Concerning Khatri–Rao Products and Synchronous Properties

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 422
Author(s):  
Arnon Ploymukda ◽  
Pattrawut Chansangiam
Keyword(s):  
Radon Measure ◽  
Geometric Mean ◽  
Integral Inequalities ◽  
Counting Measure ◽  
Integrable Functions ◽  
Discrete Inequalities ◽  
Finite Space ◽  
Continuous Fields ◽  
Adjoint Operators ◽  
Locally Compact Hausdorff Space

We consider bounded continuous fields of self-adjoint operators which are parametrized by a locally compact Hausdorff space Ω equipped with a finite Radon measure μ . Under certain assumptions on synchronous Khatri–Rao property of the fields of operators, we obtain Chebyshev-type inequalities concerning Khatri–Rao products. We also establish Chebyshev-type inequalities involving Khatri–Rao products and weighted Pythagorean means under certain assumptions of synchronous monotone property of the fields of operators. The Pythagorean means considered here are three classical symmetric means: the geometric mean, the arithmetic mean, and the harmonic mean. Moreover, we derive the Chebyshev–Grüss integral inequality via oscillations when μ is a probability Radon measure. These integral inequalities can be reduced to discrete inequalities by setting Ω to be a finite space equipped with the counting measure. Our results provide analog results for matrices and integrable functions. Furthermore, our results include the results for tensor products of operators, and Khatri–Rao/Kronecker/Hadamard products of matrices, which have been not investigated in the literature.

scholarly journals Applications of one- and two-dimensional Volterra inequalities in differential equations of the hyperbolic type

2003 ◽  
Vol 2003 (53) ◽  
pp. 3373-3383
Author(s):  
Lechosław Hącia
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Hyperbolic Type ◽  
Integral Inequalities ◽  
Two Dimensional ◽  
Volterra Type ◽  
One Dimensional ◽  
Dimensional Integral ◽  
Hyperbolic Differential Equations

Some variants of one-dimensional and two-dimensional integral inequalities of the Volterra type are applied to study the behaviour properties of the solutions to various boundary value problems for partial differential equations of the hyperbolic type. Moreover, new types of integral inequalities for one and two variables, being a generalization of the Gronwall inequality, are presented and used in the theory of nonlinear hyperbolic differential equations.

scholarly journals DISCRETE POINCARE-TYPE INEQUALITIES

1998 ◽  
Vol 29 (2) ◽  
pp. 145-153
Author(s):  
WING-SUM CHEUNG
Keyword(s):  
Discrete Analogue ◽  
Integral Inequalities ◽  
Independent Variables ◽  
Type Integral ◽  
Discrete Inequalities ◽  
Poincaré Type Inequalities

In this paper some discrete analogue of Poincare-type integral inequalities involving many independent variables are established. These in turn can be used to serve as generators of other interesting discrete inequalities.

Extremum principle for a nonlinear problem in magneto-elasticity

1972 ◽  
Vol 72 (2) ◽  
pp. 315-318 ◽  
Author(s):  
N. Anderson ◽  
A. M. Arthurs ◽  
R. R. Hall
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Problem ◽  
Boundary Value ◽  
Minimum Principle ◽  
Variational Solution ◽  
Extremum Principle ◽  
Integral Inequalities ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Elastic Boundary

AbstractA minimum principle associated with a class of magneto-elastic boundary-value problems is presented. The principle depends on a result on integral inequalities which appears to be new. An accurate variational solution is obtained for an illustrative one-dimensional problem.

scholarly journals Opial and Pólya Type Inequalities Via Convexity

2018 ◽  
Vol 60 (1) ◽  
pp. 145-159 ◽  
Author(s):  
S. H. Saker ◽  
D. M. Abdou ◽  
I. Kubiaczyk
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Hölder’S Inequality ◽  
Jensen’S Inequality ◽  
Chain Rule ◽  
Integral Inequalities ◽  
Taylor Formula ◽  
Special Cases ◽  
Discrete Inequalities ◽  
Classical Integral

Abstract In this paper, we prove some new dynamic inequalities related to Opial and Pólya type inequalities on a time scale 𝕋. We will derive the integral and discrete inequalities of Pólya’s type as special cases and also derive several classical integral inequalities of Opial’s type that has been obtained in the literature as special cases. The main results will be proved by using the chain rule, Hölder’s inequality and Jensen’s inequality, Taylor formula on time scales.

scholarly journals Integral Inequalities of Chebyshev Type for Continuous Fields of Hermitian Operators Involving Tracy–Singh Products and Weighted Pythagorean Means

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1256
Author(s):  
Arnon Ploymukda ◽  
Pattrawut Chansangiam
Keyword(s):  
Hausdorff Space ◽  
Geometric Mean ◽  
Integral Inequalities ◽  
Kronecker Products ◽  
Counting Measure ◽  
Discrete Inequalities ◽  
Finite Space ◽  
Continuous Fields ◽  
Continuous Field ◽  
Locally Compact Hausdorff Space

In this paper, we establish several integral inequalities of Chebyshev type for bounded continuous fields of Hermitian operators concerning Tracy-Singh products and weighted Pythagorean means. The weighted Pythagorean means considered here are parametrization versions of three symmetric means: the arithmetic mean, the geometric mean, and the harmonic mean. Every continuous field considered here is parametrized by a locally compact Hausdorff space equipped with a finite Radon measure. Tracy-Singh product versions of the Chebyshev-Grüss inequality via oscillations are also obtained. Such integral inequalities reduce to discrete inequalities when the space is a finite space equipped with the counting measure. Moreover, our results include Chebyshev-type inequalities for tensor product of operators and Tracy-Singh/Kronecker products of matrices.

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