Dynamic inequalities on time scales

264 Pages·2014·1.17 MB·English

by  Ravi Agarwal

Ravi Agarwal · Donal O'Regan
Samir Saker
Dynamic
Inequalities
On Time
Scales
Dynamic Inequalities On Time Scales

Ravi Agarwal•Donal O’Regan•Samir Saker
Dynamic Inequalities
On Time Scales
123
RaviAgarwal DonalO’Regan
DepartmentofMathematics SchoolofMathematics,Statistics,
TexasA&MUniversity–Kingsville andAppliedMathematics
Kingsville,TX,USA NationalUniversityofIreland
Galway,Ireland
SamirSaker
DepartmentofMathematics
MansouraUniversity
Mansoura,Egypt
ISBN978-3-319-11001-1 ISBN978-3-319-11002-8(eBook)
DOI10.1007/978-3-319-11002-8
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014951536
MathematicsSubjectClassification:34K11,34C10,34K20
(cid:2)c SpringerInternationalPublishingSwitzerland2014
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Ravi P Agarwal: To Sadhna, Sheba, and Danah
Donal O’Regan: To Alice, Aoife, Lorna, Daniel, and Niamh
Samir H Saker: To Mona, Meran, Maryam, Menah, and Ahmed

Preface
——————————————————————————————
All analysts spend half their time hunting through the literature for in-
equalities which they want to use and cannot prove.
G.H. Hardy.
——————————————————————————————
The study of dynamic inequalities on time scales has received a lot of
attention in the literature and has become a major field in pure and ap-
plied mathematics. This book is devoted to some fundamental dynamic
inequalities on time scales such as Young’s inequality, Jensen’s inequality,
H¨older’sinequality,Minkowski’sinequality,Steffensen’sinequality,Cˇebyˇsev’s
inequality, Opial’s inequality, Lyapunov’s inequality, Halanay’s inequality,
and Wirtinger’s inequality.
The book on the subject of time scale, i.e., measure chain, by Bohner
and Peterson [51] summarizes and organizes much of time scale calculus.
The three most popular examples of calculus on time scales are differential
calculus, difference calculus, and quantum calculus (see Kac and Cheung
[89]), i.e, when T = R, T = N, and T = qN0 = {qt : t ∈ N } where
0
q >1. There are applications of dynamic equations and inequalities on time
scales to quantum mechanics, electrical engineering, neural networks, heat
transfer, combinatorics, and population dynamics. A cover story article in
New Scientist [141] discusses several possible applications. In population
dynamics the dynamic equations can be used to model insect populations
that are continuous while in season, die out in say winter, while their eggs
are incubating or dormant, and then hatch in a new season, giving rise to a
nonoverlapping population.
Thisbookpresentsavarietyofintegralinequalities. Weassumethereader
hasagoodbackground intime scalecalculus. Thebookconsistsofsixchap-
ters. In Chap.1 we present preliminaries and basic concepts of time scale
calculus, and in Chap.2 we discuss and prove dynamic inequalities on time
scales such as Young’s inequality, Jensen’s inequality, Holder’s inequality,
Minkowski’s inequality, Steffensen’s inequality, Hermite–Hadamard inequal-
ity, and Cˇebyˇsv’s inequality. Opial type inequalities on time scales and their
vii
viii Preface
extensions with weighted functions will be discussed in Chap.3. In Chap.4
we present some inequalities of Lyapunov type for some dynamic equations,
and in Chap.5 we employ the shift operators δ± to construct delay dynamic
inequalities on time scales and use them to derive Halanay type inequalities
for dynamic equations on time scales. Using Halanay’s inequalities and the
propertiesofexponentialfunctionontimescales,weestablishnewconditions
that lead to stability for nonlinear dynamic equations. Finally in Chap.6 we
discuss Wirtinger-type inequalities on time scales and their extensions.
We wish to express our thanks to our families and friends.
Kingsville, TX, USA Ravi Agarwal
Galway, Ireland Donal O’Regan
Mansoura, Egypt Samir H. Saker
Contents
1 Preliminaries 1
1.1 Delta Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Nabla Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Diamond-α Calculus . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Taylor Monomials and Series . . . . . . . . . . . . . . . . . . 15
2 Basic Inequalities 23
2.1 Young Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Jensen Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3 H¨older Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 47
2.4 Minkowski Inequalities . . . . . . . . . . . . . . . . . . . . . . 56
2.5 Steffensen Inequalities . . . . . . . . . . . . . . . . . . . . . . 62
2.6 Hermite–Hadamard Inequalities . . . . . . . . . . . . . . . . . 69
2.7 Cˇebyˇsev Inequalities . . . . . . . . . . . . . . . . . . . . . . . 80
3 Opial Inequalities 93
3.1 Opial Type Inequalities I . . . . . . . . . . . . . . . . . . . . 94
3.2 Opial Type Inequalities II . . . . . . . . . . . . . . . . . . . . 106
3.3 Opial Type Inequalities III . . . . . . . . . . . . . . . . . . . 121
3.4 Higher Order Opial Type Inequalities . . . . . . . . . . . . . 135
3.5 Diamond-α Opial Inequalities . . . . . . . . . . . . . . . . . . 165
4 Lyapunov Inequalities 175
4.1 Second Order Linear Equations . . . . . . . . . . . . . . . . . 176
4.2 Second Order Half-Linear Equation . . . . . . . . . . . . . . . 188
4.3 Second Order Equations with Damping Terms. . . . . . . . . 198
4.4 Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . 209
5 Halanay Inequalities 215
5.1 Shift Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 216
5.2 Delay Functions Generated by Shift Operators . . . . . . . . 217
ix