HALF-LINEAR DIFFERENTIAL EQUATIONS

Transcription

1 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

2 Preface Basic Theory Existence and uniqueness First order half-linear system and other forms of half-linear equations Half-linear trigonometric functions Half-linear Priifer transformation Half-linear Riccati transformation Existence and uniqueness theorem An alternative approach to the existence theory Sturmian theory Picone's identity Energy functional Roundabout theorem Sturmian separation and comparison theorems More on the proof of the separation theorem Disconjugacy on various types of intervals Transformation of independent variable Reciprocity principle * Sturmian theory for Mirzov's system Leighton-Wintner oscillation criterion Differences between linear and half-linear equations Wronskian identity Transformation formula Fredholm alternative Some elementary half-linear equations Equations with constant coefficients 32

3 1.4.2 Euler type half-linear differential equation Kneser type oscillation and nonoscillation criteria Notes and references 44 Methods of Oscillation Theory Variational principle Formulation of variational principle Wirtinger inequality Applications Riccati technique Preliminaries More general Riccati transformation Riccati inequality Half-linear Hartman-Wintner theorem Positive solution of generalized Riccati equation Modified Riccati inequality Applications Comparison theorems Hille-Wintner comparison theorems Leighton comparison theorems Multiplied coefficient comparison Telescoping principle Comparison theorem with respect to p Notes and references 81 Oscillation and Nonoscillation Criteria Criteria of classical type Hille-Nehari type criteria Other criteria \ Criteria by averaging technique Coles type criteria Generalized Kamenev criterion Generalized H-function averaging technique - Philos type criterion Further extensions of Hille-Nehari type criteria Q,H type criteria Hille-Nehari type weighted criteria and extensions Notes and references 120 Nonoscillatory Solutions Asymptotic of nonoscillatory solutions Integral conditions and classification of solutions The case c negative Uniqueness in M~ The case c positive Generalized Fubini's theorem and its applications 137

4 xi 4.2 Principal solution Principal solution of linear equations Mirzov's construction of principal solution Construction of Elbert and Kusano Comparison theorem for eventually minimal solutions of Riccati equations Sturmian property of the principal solution Principal solution of reciprocal equation Integrals associated with eventually minimal solution of Riccati equation Limit characterization of the principal solution Integral characterization of the principal solution Another integral characterization Oscillation criteria and (non)principal solution Half-linear differential equations and Karamata functions Existence of regularly varying solutions Existence of slowly varying solutions Notes and references Various Oscillation Problems Conjugacy and disconjugacy Lyapunov inequality Vallee-Poussin type inequality Focal point criteria Lyapunov-type focal points and conjugacy criteria Further related results Perturbation principle General idea Singular Leighton's theorem Leighton-Wintner type oscillation criterion Hille-Nehari type oscillation criterion Hille-Nehari type nonoscillation criterion Perturbed Euler equation Linearization method in half-linear oscillation theory Nonoscillation domains and (almost) periodicity Disconjugacy domain and nonoscillation domain Equations with periodic coefficients Equations with almost periodic coefficients Strongly and conditionally oscillatory equation Strong (non)oscillation criteria Oscillation constant Function sequence technique Function sequences and Riccati integral equation Modified approaches Distance between zeros of oscillatory solutions Asymptotic formula for distribution of zeros 255

5 xii Contents Quickly oscillating solution Slowly oscillating solution Half-linear Sturm-Liouville problem Basic Sturm-Liouville problem Regular problem with indefinite weight Singular Sturm-Liouville problem Singular eigenvalue problem associated with radial p-laplacian Rotation index and periodic potential Energy functional and various boundary conditions Disfocality Nonexistence of coupled points Comparison theorems of Leighton-Levin type Miscellaneous Extended Hartman-Wintner criterion Half-linear Milloux and Armellini-Tonelli-Sansone theorems Interval oscillation criteria Notes and references BVP's for Half-Linear Differential Equations Eigenvalues, existence, and nonuniqueness problems Basic boundary value problem Variational characterization of eigenvalues Existence and (non)uniqueness below the first eigenvalue Homotopic deformation along p and Leray-Schauder degree Multiplicity nonresonance results Fredholm alternative for one-dimensional p-laplacian Resonance at the first eigenvalue Resonance at higher eigenvalues Boundary value problems at resonance Ambrosetti-Prodi type result Landesman-Lazer solvability condition Fucik spectrum Notes and references Partial Differential Equations with p-laplacian Eigenvalues and comparison principle Dirichlet BVP with p-laplacian Second eigenvalue of p-laplacian Comparison and antimaximum principle for p-laplacian Fucik spectrum for p-laplacian Boundary value problems at resonance Resonance at the first eigenvalue in higher dimension Resonance at the first eigenvalue - multiplicity results

6 xiii Landesman-Lazer result in higher dimension Oscillation theory of PDE's with p-laplacian Picone's identity for equations with p-laplacian Nonexistence of positive solutions in R N Oscillation criteria Equations involving pseudolaplacian Notes and references Half-Linear Difference Equations Basic information Linear difference equations Discretization, difficulties versus eases Half-linear discrete oscillation theory Discrete roundabout theorem and Sturmian theory Methods of half-linear discrete oscillation theory Refinements of Riccati technique Discrete oscillation criteria Hille-Nehari discrete nonoscillation criteria Some discrete comparison theorems Half-linear dynamic equations on time scales Essentials on time scales, basic properties Oscillation theory of half-linear dynamic equations Notes and references Related Differential Equations and Inequalities Quasilinear differential equations Quasilinear equations with constant coefficients Existence, uniqueness and singular solutions Asymptotic of nonoscillatory solutions Sufficient and necessary conditions for oscillation Generalized Riccati transformation and applications (Half-)linearization technique Singular solutions of black hole and white hole type Curious doubly singular equations Coupled quasilinear systems Forced half-linear differential equations Two oscillatory criteria Forced super-half-linear oscillation Half-linear differential equations with deviating arguments Oscillation of equation with nonnegative second coefficient Oscillation of equation with nonpositive second coefficient Existence and asymptotic behavior of nonoscillatory solutions Higher order half-linear differential equations p-biharmonic operator Higher order half-linear eigenvalue problem 463

7 xiv Contents 9.5 Inequalities related to half-linear differential equations Inequalities of Wirtinger and Hardy type Inequalities of Opial type Notes and references 469 Bibliography 471 Index 497 Author Index 515