Over the last four decades,intensive work has been carried out in the field of theory of nonautonomous third-order ordinary and delay differential equations. However, it is only recently that the global attractivity of third-order equations has been given a serious study. During these years,new investigations were developed and results of principal importance were obtained.In particular, suitable oscillation criteria for third-order linear and nonlinear differential equations were established, with em- phasis on the oscillation of third-order nonhomogeneous differential equations with respect to oscillation and nonoscillation of third-order homogeneous differential equations.Existence and nonexistence conditions for oscillatory and nonoscillatory solutions of various types were found and asymptotic formulae for solutions of a sufficiently wide class of linear and nonlinear equations were derived.These studies notwithstanding, one would observe that oscillation and asymptotic behaviour of nonoscillatory solutions of third-order delay differential equations has rarely been studied.
In this book,an attempt is made to sum up these results.The necessity of such an attempt is felt especially as the well-known monograph of M.Gregus,Third Order Linear Differential Equations, Reidel,Dordrecht,The Netherlands,1982,devoted to related topics, reflects results which are 30 years old.
This book consists of seven chapters.In the first one,we consider the third-order linear differential equation, with constant coefficients, of the form x"+ ax"＋bx'＋cx = 0,(1)where a,b and c are constants.Eight different cases of a,bandc were considered while dealing with(1). Further, it is observed that different structures of solution spaces of(1)appears for different cases ona,bandc.In this chapter,anintroduction is given to the oscillation theory of the nonhomogeneous equation x"＋ax"+b.x'＋cx = f,(2)wherea,b,c and f are constants. An apparatus of comparison theorems is developed, which allows us to establish criteria for(2)to have oscillatory or nonoscil-[latory solutions with the help of oscillation and nonoscillation of(1. An overall idea on oscillation of third-order delay differential equations is also included in this chapter. Additionally, some basic results are incorporated, which are needed in the forthcoming chapters.Chapter 2 deals with the linear equation

Introduction 1

1 Introduction 1

1.1 Preliminaries 1

1.2 Third-Order Linear Differential Equations with Constant Coefficients 8

1.3 Introduction to Third-Order Delay Differential Equations 19

1.4 Introduction to Third-Order Canonical Differential Equations 22

1.5 Some Basic Results 29

1.6 Some Useful Results from Analysis 39

1.7 Notes 41

References41

2Behaviour of Solutions of Linear Homogeneous Differential Equations of Third Order 45

2.1 Behaviour of Solutions of x"＋a()x"+b(1)x'＋c(t)x=0 with a(t) ≥ 0, b(t)≤0 and c(t)> 0 46

2.2 Behaviour of Solutions of x"＋a(t)x"+b()x'+c(1)x=0 with a(t)≤ 0, b(t)≤0 and c(t)>0 65

2.3 Behaviour of Solutions of x"＋a(t)x"+b(t)x'+c(t)x=0 with a(t)≤ 0,b(t)≤0 and c(t)<0 73

2.4 Behaviour of Solutions of x"＋a(t)x"+b(t)x'+c()x=0 with a(t)≥ 0, b(t)≤0 and c(t)<0 83

2.5 Behaviour of Solutions of x"＋a(t)x"＋b(t)x'+c(t)x=0 with a(t)≥ 0,b(t)≥ 0 and c(t)>0 93

2.6 Behaviour of Solutions of x"＋a(t)x"+b()x'+c(t)x=0 with a(t)≤0,b(t)≥0 and c(t)<0 93

2.7 Oscillation and Nonoscillation of Third-Order Linear Differential Equations of the Form(r2(t)(r1(t)x'))'+r3(t)x=0 119

2.8 Open Problems and Discussions 143

2.9 Notes 145

References 145

3Oscillation of Solutions of Linear Nonhomogeneous Differential Equations of Third Order 147

3.1 Nonoscillatory Behaviour of Solutions of Nonhomogeneous Linear Differential Equations 148

3.2 Oscillatory Behaviour of Solutions of Nonhomogeneous Linear Differential Equations 169

3.3 Nonoscillation of Nonhomogeneous Linear Differential Equations 177

3.4 Asymptotic Behaviour of Oscillatory Solutions of Third-Order Linear Nonhomogeneous Differential Equations 184

3.5 Open Problems and Discussions 186

3.6Notes 190

References 190

4Oscillation and Nonoscillation of Homogeneous Third-Order Nonlinear Differential Equations 193

4.1 Behaviour of Solutions of x"＋b(t)x'+c(t)x°=0 and x"+b(t)x'＋c(t) f(x)= 0 197

4.2 Behaviour of Solutions of x"+a(t)x"+b(t)x'+c((1)x°=0 and x"＋a(t)x"＋b(t)x'＋c(t)f(x)=0 239

4.3 Behaviour of Solutions of x"+c(t)x"= 0 and x"＋c(t)f(x)=0 250

4.4 Behaviour of Solutions of (r2(t)(r1(t)x')')'+q(t)x'+p(t)x°=0and (r2(t)(ri1(t)x')')'+g(t)x'+p(t)f(x)=0 277

4.5 Open Problems and Discussions 300

4.6 Notes 301

References 301

5Oscillation and Nonoscillation of Nonlinear Nonhomogeneous Differential Equations of Third Order 305

5.1 Oscillatory and Nonoscillatory Behaviour of Solutions of x"+a(t)x"＋b(t)x'＋c(t)x= f(t,x,x',x") 306

5.2 Nonoscillation of Solutions of (r2(t)(ri(t)x')'+q(t)(x^)+p(t))x= f(t) 319

5.3 Open Problems and Discussions 334

5.4 Notes 334

References 334

Oscillatory and Asymptotic Behaviour of Solutions of Third-Order Delay Differential Equations 336

6.1 Asymptotic Behaviour of Solutions of Linear Delay Differential Equations of the Form x"(t)＋a(t)x"(t)＋b(t)x'(t)＋c(t))x(g(t))= 0 336

6.2 Oscillation and Asymptotic Properties of Solutions of the Linear Delay Differential Equation r7（rz（（（roo）））'+p（t）x（g（t））= 0 378

6.3 Oscilltion and Asymptotic Behaviour of Solutions of the Linear Delay Differential Equation x"(t)＋p(t)x(g())=0....400

6.4 Asymptotic Behaviour of Solutions of Nonlinear Delay-Differential Equation of the Form x"(t)＋q(t)x'(t)＋p(t)f(x(g(t)))=0 414

6.5 Oscillation of Solutions of the Nonlinear Delay Differential

Equation(r(0)(x"(t))Y)'+p(t)x'(g(t))=0and(r(t)(x"(r))')'+p(t)f(x(g(t)))=0 421

6.6 Oscillation and Asymptotic Behaviour of Solutions of (r2(t)(r1(t)x'())')'＋q(t)x'(t)＋p(t)f(x(g(t)))=0 428

6.7 Oscillation of Solutions of Third-Order Differential Equations with Distributed Deviating Arguments 435

6.8 Oscillation and Asymptotic Behaviour of Nonoscillatory Solutions of (r2(t)(rt)x'(t))')'+p(t)f(x(g(t))=0 445

6.9 Nonoscillation of Solutions of Nonlinear Nonhomogeneous

Delay Differential Equations of Third Order 448

6.10 Open Problems and Discussions 450

6.11Notes 451

References 451

7 Stability of Third-Order Differential Equations 455

7.1 Stability of Solutions of x"＋V(x,x')x"＋f(x,x')=0 456

7.2 Stability of Solutions of x"＋ψ（x，x）x"＋f（x，x'）=p(t) 465

7.3 Stability of Solutions of x"＋ψ（x，x'，x"）x"＋f（x，x'）= p（t，x，x'，x"） 474

7.4 Stability of Solutions of x"(t)=P1x"(t)+P2x"(t-t)＋g1x'(t)＋g2x'(t-t)+rix(t)+r2x(t-r) 481

7.5 Open Problems and Discussions 501

7.6 Notes 502

References 502

Index 503

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