HALF-LINEAR DIFFERENTIAL EQUATIONS

ONDREJ DOSLY Department of Mathematics Masaryk University Brno, Czech Republic PAVEL REHAK Mathematical Institute Academy of Sciences of the Czech Republic Brno, Czech Republic

ELSEVIER 2005 Amsterdam - Boston - Heidelberg - London - New York - Oxford Paris - San Diego - San Francisco - Singapore - Sydney - Tokyo

Contents

Preface Basic Theory 1.1 Existence and uniqueness 1.1.1 First order half-linear system and other forms of half-linear equations 1.1.2 Half-linear trigonometric functions 1.1.3 Half-linear Priifer transformation 1.1.4 Half-linear Riccati transformation 1.1.5 Existence and uniqueness theorem 1.1.6 An alternative approach to the existence theory 1.2 Sturmian theory 1.2.1 Picone's identity 1.2.2 Energy functional 1.2.3 Roundabout theorem 1.2.4 Sturmian separation and comparison theorems 1.2.5 More on the proof of the separation theorem 1.2.6 Disconjugacy on various types of intervals 1.2.7 Transformation of independent variable 1.2.8 Reciprocity principle * 1.2.9 Sturmian theory for Mirzov's system 1.2.10 Leighton-Wintner oscillation criterion 1.3 Differences between linear and half-linear equations 1.3.1 Wronskian identity 1.3.2 Transformation formula 1.3.3 Fredholm alternative 1.4 Some elementary half-linear equations 1.4.1 Equations with constant coefficients

1 2 2 4 7 8 8 9 13 13 14 14 16 18 20 21 22 22 25 26 27 30 31 32 32

Contents

1.5

1.4.2 Euler type half-linear differential equation 1.4.3 Kneser type oscillation and nonoscillation criteria Notes and references

Methods of Oscillation Theory 2.1 Variational principle 2.1.1 Formulation of variational principle 2.1.2 Wirtinger inequality 2.1.3 Applications 2.2 Riccati technique 2.2.1 Preliminaries 2.2.2 More general Riccati transformation 2.2.3 Riccati inequality 2.2.4 Half-linear Hartman-Wintner theorem 2.2.5 Positive solution of generalized Riccati equation 2.2.6 Modified Riccati inequality 2.2.7 Applications 2.3 Comparison theorems 2.3.1 Hille-Wintner comparison theorems 2.3.2 Leighton comparison theorems 2.3.3 Multiplied coefficient comparison 2.3.4 Telescoping principle 2.3.5 Comparison theorem with respect to p 2.4 Notes and references

37 43 44 47 47 48 48 49 50 50 52 52 54 59 63 66 69 69 73 75 77 80 81

Oscillation and Nonoscillation Criteria 83 3.1 Criteria of classical type 84 3.1.1 Hille-Nehari type criteria 85 3.1.2 Other criteria \ 87 3.2 Criteria by averaging technique 91 3.2.1 Coles type criteria 91 3.2.2 Generalized Kamenev criterion 96 3.2.3 Generalized H-function averaging technique - Philos type criterion 97 3.3 Further extensions of Hille-Nehari type criteria 99 3.3.1 Q,H type criteria 100 3.3.2 Hille-Nehari type weighted criteria and extensions 113 3.4 Notes and references 120

Nonoscillatory Solutions

123

4.1

123 124 126 132 134 137

Asymptotic of nonoscillatory solutions 4.1.1 Integral conditions and classification of solutions 4.1.2 The case c negative 4.1.3 Uniqueness in M~ 4.1.4 The case c positive 4.1.5 Generalized Fubini's theorem and its applications

Contents 4.2

Principal solution 4.2.1 Principal solution of linear equations 4.2.2 Mirzov's construction of principal solution 4.2.3 Construction of Elbert and Kusano 4.2.4 Comparison theorem for eventually minimal solutions of Riccati equations 4.2.5 Sturmian property of the principal solution 4.2.6 Principal solution of reciprocal equation 4.2.7 Integrals associated with eventually minimal solution of Riccati equation 4.2.8 Limit characterization of the principal solution 4.2.9 Integral characterization of the principal solution 4.2.10 Another integral characterization 4.2.11 Oscillation criteria and (non)principal solution 4.3 Half-linear differential equations and Karamata functions 4.3.1 Existence of regularly varying solutions 4.3.2 Existence of slowly varying solutions 4.4 Notes and references

5 Various Oscillation Problems 5.1 Conjugacy and disconjugacy 5.1.1 Lyapunov inequality 5.1.2 Vallee-Poussin type inequality 5.1.3 Focal point criteria 5.1.4 Lyapunov-type focal points and conjugacy criteria 5.1.5 Further related results 5.2 Perturbation principle 5.2.1 General idea 5.2.2 Singular Leighton's theorem 5.2.3 Leighton-Wintner type oscillation criterion 5.2.4 Hille-Nehari type oscillation criterion 5.2.5 Hille-Nehari type nonoscillation criterion 5.2.6 Perturbed Euler equation 5.2.7 Linearization method in half-linear oscillation theory . . . . 5.3 Nonoscillation domains and (almost) periodicity 5.3.1 Disconjugacy domain and nonoscillation domain 5.3.2 Equations with periodic coefficients 5.3.3 Equations with almost periodic coefficients 5.4 Strongly and conditionally oscillatory equation 5.4.1 Strong (non)oscillation criteria 5.4.2 Oscillation constant 5.5 Function sequence technique 5.5.1 Function sequences and Riccati integral equation 5.5.2 Modified approaches 5.6 Distance between zeros of oscillatory solutions 5.6.1 Asymptotic formula for distribution of zeros

xi 140 141 142 144 147 148 149 150 154 158 164 166 170 171 183 187 189 190 190 191 193 196 200 201 201 202 205 207 210 212 215 217 217 229 230 237 238 240 243 243 250 254 255

xii

Contents

5.7

5.8

5.9

5.10

5.6.2 Quickly oscillating solution 5.6.3 Slowly oscillating solution Half-linear Sturm-Liouville problem 5.7.1 Basic Sturm-Liouville problem 5.7.2 Regular problem with indefinite weight 5.7.3 Singular Sturm-Liouville problem 5.7.4 Singular eigenvalue problem associated with radial p-Laplacian 5.7.5 Rotation index and periodic potential Energy functional and various boundary conditions 5.8.1 Disfocality 5.8.2 Nonexistence of coupled points 5.8.3 Comparison theorems of Leighton-Levin type Miscellaneous 5.9.1 Extended Hartman-Wintner criterion 5.9.2 Half-linear Milloux and Armellini-Tonelli-Sansone theorems 5.9.3 Interval oscillation criteria Notes and references

258 259 261 261 263 267 269 270 282 284 290 293 298 298 301 305 309

6 BVP's for Half-Linear Differential Equations 311 6.1 Eigenvalues, existence, and nonuniqueness problems 312 6.1.1 Basic boundary value problem 312 6.1.2 Variational characterization of eigenvalues 312 6.1.3 Existence and (non)uniqueness below the first eigenvalue . 315 6.1.4 Homotopic deformation along p and Leray-Schauder degree 318 6.1.5 Multiplicity nonresonance results 320 6.2 Fredholm alternative for one-dimensional p-Laplacian 325 6.2.1 Resonance at the first eigenvalue 325 6.2.2 Resonance at higher eigenvalues 335 6.3 Boundary value problems at resonance 337 6.3.1 Ambrosetti-Prodi type result 337 6.3.2 Landesman-Lazer solvability condition 342 6.3.3 Fucik spectrum 347 6.4 Notes and references 350 7 Partial Differential Equations with p-Laplacian 353 7.1 Eigenvalues and comparison principle 353 7.1.1 Dirichlet BVP with p-Laplacian 354 7.1.2 Second eigenvalue of p-Laplacian 357 7.1.3 Comparison and antimaximum principle for p-Laplacian . . 358 7.1.4 Fucik spectrum for p-Laplacian 361 7.2 Boundary value problems at resonance 364 7.2.1 Resonance at the first eigenvalue in higher dimension . . . 364 7.2.2 Resonance at the first eigenvalue - multiplicity results . . . 367

Contents

7.3

.7.4

xiii 7.2.3 Landesman-Lazer result in higher dimension Oscillation theory of PDE's with p-Laplacian 7.3.1 Picone's identity for equations with p-Laplacian 7.3.2 Nonexistence of positive solutions in RN 7.3.3 Oscillation criteria 7.3.4 Equations involving pseudolaplacian Notes and references

368 371 372 373 378 380 381

8

Half-Linear Difference Equations 8.1 Basic information 8.1.1 Linear difference equations 8.1.2 Discretization, difficulties versus eases 8.2 Half-linear discrete oscillation theory 8.2.1 Discrete roundabout theorem and Sturmian theory 8.2.2 Methods of half-linear discrete oscillation theory 8.2.3 Refinements of Riccati technique 8.2.4 Discrete oscillation criteria 8.2.5 Hille-Nehari discrete nonoscillation criteria 8.2.6 Some discrete comparison theorems 8.3 Half-linear dynamic equations on time scales 8.3.1 Essentials on time scales, basic properties 8.3.2 Oscillation theory of half-linear dynamic equations 8.4 Notes and references

383 383 383 384 386 386 390 391 397 401 408 412 412 414 416

9

Related Differential Equations and Inequalities 9.1 Quasilinear differential equations 9.1.1 Quasilinear equations with constant coefficients 9.1.2 Existence, uniqueness and singular solutions 9.1.3 Asymptotic of nonoscillatory solutions 9.1.4 Sufficient and necessary conditions for oscillation 9.1.5 Generalized Riccati transformation and applications . . . . 9.1.6 (Half-)linearization technique 9.1.7 Singular solutions of black hole and white hole type . . . . 9.1.8 Curious doubly singular equations 9.1.9 Coupled quasilinear systems . 9.2 Forced half-linear differential equations 9.2.1 Two oscillatory criteria 9.2.2 Forced super-half-linear oscillation 9.3 Half-linear differential equations with deviating arguments 9.3.1 Oscillation of equation with nonnegative second coefficient . 9.3.2 Oscillation of equation with nonpositive second coefficient . 9.3.3 Existence and asymptotic behavior of nonoscillatory solutions 9.4 Higher order half-linear differential equations 9.4.1 p-biharmonic operator 9.4.2 Higher order half-linear eigenvalue problem

417 417 418 421 422 424 426 428 431 434 435 438 438 440 444 444 453 457 462 462 463

xiv

Contents

9.5 Inequalities related to half-linear differential equations 9.5.1 Inequalities of Wirtinger and Hardy type 9.5.2 Inequalities of Opial type 9.6 Notes and references

466 466 468 469

Bibliography

471

Index

497

Author Index

515