Birkhäuser Advanced Texts Basler Lehrbücher Alessandro Fonda Playing Around Resonance An Invitation to the Search of Periodic Solutions for Second Order Ordinary Differential Equations Birkhäuser Advanced Texts Basler Lehrbücher Series editors Steven G Krantz, Washington University, St Louis, USA Shrawan Kumar, University of North Carolina at Chapel Hill, Chapel Hill, USA Jan Nekováˇr, Université Pierre et Marie Curie, Paris, France More information about this series at: http://www.springer.com/series/4842 Alessandro Fonda Playing Around Resonance An Invitation to the Search of Periodic Solutions for Second Order Ordinary Differential Equations Alessandro Fonda Dipartimento di Matematica e Geoscienze UniversitJa degli Studi di Trieste Trieste, Italy ISSN 1019-6242 ISSN 2296-4894 (electronic) BirkhRauser Advanced Texts Basler LehrbRucher ISBN 978-3-319-47089-4 ISBN 978-3-319-47090-0 (eBook) DOI 10.1007/978-3-319-47090-0 Library of Congress Control Number: 2016958441 © Springer International Publishing AG 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This book is published under the trade name Birkhäuser, www.birkhauser-science.com The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To Rodica Contents Preliminaries on Hilbert Spaces 1.1 The Hilbert Space Structure 1.2 Some Examples of Hilbert Spaces 1.3 Fundamental Properties 1.4 Subspaces 1.5 Orthogonal Subspaces 1.6 The Orthogonal Projection 1.7 Basis in a Hilbert Space 1.8 Linear Functions 1.9 Weak Convergence 1.10 Concluding Remarks 1 10 13 14 19 25 28 Operators in Hilbert Spaces 2.1 First Definitions 2.2 The Adjoint Operator 2.3 Resolvent Set and Spectrum 2.4 Selfadjoint Operators 2.5 Operators in Real Hilbert Spaces 2.6 Concluding Remarks 31 31 33 36 39 42 45 The Semilinear Problem 3.1 The Main Problem 3.2 Properties of the Differential Operator 3.3 The Linear Equation 3.3.1 The Case > 3.3.2 The Case < 3.3.3 Conclusions 3.4 The Contraction Theorem 3.5 Nonresonance: Existence and Uniqueness 3.6 Equations in Hilbert Spaces 3.7 Concluding Remarks 47 47 49 52 53 56 57 57 59 61 68 vii viii Contents The Topological Degree 4.1 The Brouwer Degree 4.2 Further Considerations on the Brouwer Degree 4.3 The Leray–Schauder Degree 4.4 Concluding Remarks 71 71 87 90 98 Nonresonance and Topological Degree 5.1 The Use of Schauder Theorem 5.2 Lower and Upper Solutions 5.3 The Continuation Principle 5.4 Asymmetric Oscillators 5.5 Nonlinear Nonresonance 5.6 Non-bilateral Conditions 5.7 The Ambrosetti–Prodi Problem 5.8 Concluding Remarks 101 101 104 108 111 112 119 129 133 Playing Around Resonance 6.1 Some Useful Inequalities 6.2 Resonance at the First Eigenvalue 6.3 Landesman–Lazer: Resonance at Higher Eigenvalues 6.4 The Lazer–Leach Condition 6.5 Landesman–Lazer Conditions: The Asymmetric Case 6.6 Lazer–Leach Conditions for the Asymmetric Oscillator 6.7 More Subtle Nonresonance Conditions 6.8 Concluding Remarks 137 137 139 141 144 145 149 151 155 The Variational Method 7.1 Definition of the Functional 7.2 Minimization 7.3 The Ekeland Principle 7.4 The Search of Saddle Points 7.5 Concluding Remarks 157 157 161 164 165 171 At Resonance, Again 8.1 Resonance at the First Eigenvalue 8.2 Subharmonic Solutions 8.3 Ahmad–Lazer–Paul: Resonance at Higher Eigenvalues 8.4 Landesman–Lazer vs Ahmad–Lazer–Paul 8.5 Periodic Nonlinearities 8.6 Concluding Remarks 173 174 176 181 184 187 190 Lusternik–Schnirelmann Theory 9.1 The Periodic Problem for Systems 9.2 An Equivalent Functional 9.3 Some Hints on Differential Equations 9.4 Lusternik–Schnirelmann Category 9.5 Multiplicity of Critical Points 193 193 194 197 199 201 Contents ix 9.6 9.7 Relative Category 206 Concluding Remarks 211 10 The Poincaré–Birkhoff Theorem 10.1 The Multiplicity Result 10.2 A Modified System 10.3 The Variational Setting 10.4 Finite Dimensional Reduction 10.5 Periodic Solutions of the Original System 10.6 The Poincaré–Birkhoff Theorem on an Annulus 10.7 Concluding Remarks 213 214 215 218 221 223 225 227 11 A Myriad of Periodic Solutions 11.1 Equations Depending on a Parameter 11.2 Superlinear Problems 11.3 Forced Superlinear Equations 11.4 Concluding Remarks 231 231 243 250 253 A Spaces of Continuous Functions A.1 Uniform Convergence A.2 Continuous Functions with Compact Domains A.3 Uniformly Continuous Functions A.4 The Ascoli–Arzelà Theorem A.5 The Stone–Weierstrass Theorem 255 255 257 258 259 261 B Differential Calculus in Normed Spaces B.1 The Fréchet Differential B.2 Some Computational Rules B.3 The Mean Value Theorem B.4 The Gateaux Differential B.5 Partial Differentials B.6 The Implicit Function Theorem B.7 Higher Order Differentials 265 265 267 270 272 273 276 282 C A Brief Account on Differential Forms C.1 Preliminary Definitions C.2 The External Differential C.3 Pull-Back Functions C.4 Integrating M-Differential Forms Over M-Surfaces C.5 Differentiable Manifolds C.6 Orientation C.7 The Stokes–Cartan Theorem 287 287 289 290 291 292 293 294 Bibliography 297 Index 307 Introduction This book is an introduction to the problem of the existence of solutions to some type of semilinear boundary value problems It arises from a series of courses which I have given to undergraduate and graduate students in the last few years The aim of the book is to give the possibility to any good student to reach a research level in this field, starting from the basic knowledge of mathematical analysis which is usually acquired before graduation To this aim, I will develop some tools which could be used to attack many different boundary value problems, arising from ordinary or partial differential equations However, I have chosen to deal mainly with the periodic problem for a second-order scalar ordinary differential equation One reason for this choice is that this apparently simple model already shows so many different aspects, and can be approached by such different techniques, that it seems the ideal starting point to the further understanding of more technical boundary value problems Another reason comes, of course, from its intrinsic importance in the applications So, I will be concerned with an equation of the type x00 C g.t; x/ D ; (1) where g W R R ! R is a continuous function, which is T-periodic in its first variable The main problem will be to find some conditions on the function g which guarantee the existence of T-periodic solutions of Eq (1) More generally, we will deal with the problem P/ x00 C g.t; x/ D ; x.0/ D x.T/ ; x0 0/ D x0 T/ ; where g W Œ0; T R ! R is continuous Indeed, if g.t; x/ is defined on R R, and T-periodic in its first variable, it is easy to see that any solution x.t/ of problem (P) can be extended to the whole R as a T-periodic solution of Eq (1) xi C.6 Orientation 293 Proposition C.5.1 For every x M, there are a constant continuously differentiable function x W Ix ! RN , where Ix D such that x 0/ Œ 1; 1M ; Œ 1; 1M x > and an injective if x 62 @M ; Œ0; 1 ; if x @M ; D x, and B.x; x/ \M x Ix /  M: If x @M , then x Œ 1; 1M f0g/  @M ; x Œ 1; 1M 0; 1/ \ @M D Ø : Moreover, for every u Ix , the vectors @ x @ x u/ ; : : : ; u/ @u1 @uM are linearly independent A function x with the above properties is said to be a local parametrization of M at the point x C.6 Orientation As we have seen in Proposition C.5.1, for every x M, if u Ix , we have that the set of M linearly independent vectors Ä @ x @ x u/ ; : : : ; u/ @u1 @uM is a basis for a M-dimensional space, named the tangent space to M at x u/, which we denote by T x u/ M In particular, if u D 0, we have the tangent vector space Tx M Now, if we fix u Ix , the point x u/ surely belongs to the image of many other local parametrizations Let, for instance, x0 W Ix0 ! RN be such that x u/ D x0 v/, for some v Ix0 We can then modify x0 , if necessary, so that the two bases of the tangent vector space T x u/ M D T x0 v/ M associated to x and x0 have the same orientation, i.e., the matrix transforming one basis to the other has a positive determinant We call coherent such a choice of the local parametrizations A coherent choice of the local parametrizations is always possible in a neighborhood of x We would like to have such a coherent choice globally, for all the local 294 C A Brief Account on Differential Forms parametrizations of M However, this is not always possible, as the example of the Möbius ribbon shows Whenever it is possible to make a coherent choice of all the local parametrizations of M, we say that M is orientable From now on, we will assume that M is orientable, and that the coherent choice of the local parametrizations has been made In such a situation, we say that M has been oriented Once M has been oriented, let us explain how its boundary @M inherits an orientation itself Take x @M, and let x W Ix ! RN be a local parametrization, with x 0/ D x Recall that, in this case, Ix D Œ 1; 1M Œ0; 1 Being @M a M 1/-manifold, the tangent vector space Tx @M has dimension M 1, and it is a subspace of Tx M, which has dimension M There are then two directions in Tx M which are orthogonal to Tx @M, one opposite to the other We denote by x/ the one which can be obtained as @@v 0/ D d 0/v, for some v D v1 ; : : : ; vM / with vM < (So, x/ is the so called outer normal at the point x.) We now choose a basis Œv 1/ x/; : : : ; v M 1/ x/ in Tx @M such that Œ x/; v 1/ x/; : : : ; v M 1/ x/ is a basis of Tx M, with the same orientation of the one which has been already chosen in this space Proceeding in such a way for every x, it is possible to see that @M comes out to be oriented, and we say that @M is given the induced orientation from M C.7 The Stokes–Cartan Theorem We assume now that M, besides being oriented, is compact For a M-differential form !, defined on an open set U which contains M, we want to define the real number Z !; M the integral of ! over M In the case when !jM (the restriction of ! to the set M) is equal to zero outside the image of one single local parametrization W I ! RN , we simply set Z Z M !D !: In general, we have seen in Proposition C.5.1 that M can be covered by the balls B.x; x / Being M a compact set, there is a finite subcovering: M D B.x1 ; x1 / [ [ B.xm ; xm / : C.7 The Stokes–Cartan Theorem 295 Let f ; : : : ; m g be a partition of unity associated with such a covering It is constructed as follows: let f W R ! R be the function defined as u2 exp 0; f u/ D ; if juj < ; if juj ; and set  k x/ Df xk jj jjx à : k Then, for every x V; one has C1 -smooth functions k x/ x/ D C C n x/ > 0, and we can define the k x/ x/ C ::: C n x/ : We have the following properties (i) Ä k x/ Ä , (ii) x B.xk ; xk / P H) m (iii) x M H) kD1 k x/ k x/ D 0, D Since each of the k !jM is equal to zero outside the image of one single local parametrization, we can define Z M !D m Z X kD1 M k !: It can be proved that such a definition does not depend neither on the (coherent) choice of the local parametrizations, nor from any particular choice of the partition of unity We are now ready to state the conclusive Stokes–Cartan Theorem Theorem C.7.1 If ! is a M-differential form of class C1 , defined on an open set U, and M is an oriented compact M C 1/-manifold contained in U, then Z Z d! D M @M ! 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compact operator, 102 completely continuous function, 90 complexified Hilbert space, 42 operator, 43 continuation principle, 109 contractible set, 199 contraction, 57 critical point, 159, 196 critical value, 159 deformation, 199 deformation lemma, 202 degree additivity, 71, 90 Brouwer, 71 coincidence, 98 excision, 72, 90 existence, 72, 90 homotopy invariance, 71, 90 Leray–Schauder, 90 normalization, 71, 90 Rouché’s property, 72, 90 densely defined, 34 derivative, 270 differentiable, 265, 266 differentiable manifold, 292 differential, 265 differential form, 287 direct sum of subspaces, 11 directional derivative, 265 distance, double resonance, 155 eigenvalue, 36 Ekeland Principle, 164 excision of the degree, 72, 90 external differential, 289 © Springer International Publishing AG 2016 A Fonda, Playing Around Resonance, BirkhRauser Advanced Texts Basler LehrbRucher, DOI 10.1007/978-3-319-47090-0 307 308 external product, 288 fixed point, 57 flux, 202 Fourier series in L2 a; b/, 16 in a Hilbert space, 14 Fréchet-differentiable, 265 Fuˇcík spectrum, 112 functional, 23, 158 Fundamental Theorem, 198 Gateaux-differentiable, 272 generated subspace, geometrically distinct, 188, 194 gradient, 159 Gram–Schmidt orthonormalization, 26 graph, 31 Hamiltonian system, 69 higher order differential, 285 Hilbert space, complexified, 42 homotopy invariance of the degree, 71, 90 image, 31 Implicit Function Theorem, 277 induced orientation, 294 integral in Hilbert space, 197 of a differential form, 291, 294 inverse operator, 36 isochronous center, 53, 111 isomorphic Hilbert spaces, 20 Landesman–Lazer condition, 139–141, 143, 146, 149, 176, 184, 186, 187 Lazer–Leach condition, 144, 150 Leray–Schauder degree, 90 linear function, 19 bounded, 21 continuous, 21 linear manifold, linear oscillator, 52 lower solution, 104 Lusternik–Schnirelmann category, 200 M-surface, 291 Index manifold, 292 Mawhin’s Lemma, 83 maximum point, 161 Mean Value Theorem, 271 Minimax Theorem, 166 minimum point, 161 mollifiers, 65 monotone function, 61 operator, 40 mountain pass point, 169 Mountain Pass Theorem, 169 negative part, 111 Nemytskii operator, 48 Neumann series, 37 nonlinear resonance, xii norm in a Hilbert space, of a bounded linear function, 21 normalization of the degree, 71, 90 null-space, 31 operator, 31 adjoint, 33 anti-selfadjoint, 45 closed, 32 compact, 102 complexified , 43 densely defined, 34 graph, 31 image, 31 inverse, 36 monotone, 40 null-space, 31 selfadjoint, 39 square root, 63 orientable manifold, 294 oriented manifold, 294 orthogonal family, projection, 13 sets, 10 vectors, orthonormal, 14 oscillator asymmetric, 111 linear, 52 Palais–Smale condition, 169, 196 parallelogram identity, Index Parseval identity, 14 partial differential, 273 partition of unity, 295 pendulum equation, 107, 190 periodic problem, 48 Poincaré map, 151 Poincaré–Bohl Theorem, 99 Poincaré–Miranda Theorem, 99 positive part, 111 projection, 13 pull-back, 290 regular value, 76 relative category, 206 resolvent function, 55 resolvent set, 36 resonance, xii, 54, 137, 139, 141, 145, 155, 173, 174, 181, 190 rotation number, 225 Rouché’s property, 72, 90 saddle point, 166, 170 Saddle Point Theorem, 170 scalar product in L2 a; b/, in W 1;2 a; b/, in W 2;2 a; b/, in a Hilbert space, Schauder Fixed Point Theorem, 97 Schwarz inequality, second differential, 282 selfadjoint operator, 39 separable Hilbert space, 14 spectrum, 36 square root of an operator, 63 subharmonic solution, 176 subspace, sum of subspaces, superposition of deformations, 199 symplectic matrix, 69, 213 309 Banach–Steinhaus, 24 Brouwer, 98 Dancer, 150 Ding–Zanolin, 251 Dini, 262 Dolph, 59 Drabek–Invernizzi, 113 Ekeland, 164 Fabry, 155 Fabry–Fonda, 155 Fabry–Habets, 119 Fabry–Mawhin–Nkashama, 129 Fejer, 264 Fonda–Lazer, 176 Fonda–Mawhin, 61 Fonda–Ureña, 214 Hammerstein, 106 Jacobowitz–Hartman, 244 Jiang, 190 Knobloch, 104 Landesman–Lazer, 141 Lazer–Leach, 144 Lazer–McKenna, 231 Leray, 102 Leray–Schauder, 109 Mawhin–Ward, 128 Mawhin–Willem, 187 Poincaré–Birkhoff, 226 Poincaré–Bohl, 99 Poincaré–Miranda, 99 Pythagoras, Rabinowitz, 170 Riesz, 23 Schauder, 97 Schwartz, 203 Schwarz, 284 Stokes–Cartan, 295 Stone–Weierstrass, 262 torus, 194 twice differentiable, 282 upper solution, 104 tangent space, 293 Theorems: Ahmad–Lazer–Paul, 181 Ambrosetti–Rabinowitz, 169 Ascoli–Arzelà, 259 Banach, 57 weak convergence, 25 weak derivative, weakly lower semicontinuous, 159 winding number, 89 Wirtinger inequality, 138 ... 108 111 112 119 129 133 Playing Around Resonance 6.1 Some Useful Inequalities 6.2 Resonance at the First... More information about this series at: http://www.springer.com/series/4842 Alessandro Fonda Playing Around Resonance An Invitation to the Search of Periodic Solutions for Second Order Ordinary Differential... complicated Even if it is not clear what resonance (or perhaps nonlinear resonance) would mean in the general case, one can expect that a phenomenon similar to linear resonance may appear in situations