nonlinear analysis & differential equations, an introduction - schmitt & thompson

Nonlinear Analysis and Di erential Equations An Introduction Klaus Schmitt Department of Mathematics University of Utah Russell C Thompson Department of Mathematics and Statistics Utah State University August 14, 2000 ii Copyright c 1998 by K Schmitt and R Thompson iii Preface The subject of Di erential Equations is a well established part of mathematics and its systematic development goes back to the early days of the development of Calculus Many recent advances in mathematics, paralleled by a renewed and ourishing interaction between mathematics, the sciences, and engineering, have again shown that many phenomena in the applied sciences, modelled by di erential equations will yield some mathematical explanation of these phenomena (at least in some approximate sense) The intent of this set of notes is to present several of the important existence theorems for solutions of various types of problems associated with di erential equations and provide qualitative and quantitative descriptions of solutions At the same time, we develop methods of analysis which may be applied to carry out the above and which have applications in many other areas of mathematics, as well As methods and theories are developed, we shall also pay particular attention to illustrate how these ndings may be used and shall throughout consider examples from areas where the theory may be applied As di erential equations are equations which involve functions and their derivatives as unknowns, we shall adopt throughout the view that di erential equations are equations in spaces of functions We therefore shall, as we progress, develop existence theories for equations de ned in various types of function spaces, which usually will be function spaces which are in some sense natural for the given problem iv Table of Contents I Nonlinear Analysis Chapter I Analysis In Banach Spaces Introduction Banach Spaces Di erentiability, Taylor's Theorem Some Special Mappings Inverse Function Theorems The Dugundji Extension Theorem Exercises Chapter II The Method of Lyapunov-Schmidt 27 Introduction De nition Properties of the Brouwer Degree Completely Continuous Perturbations Exercises Introduction Continuation Principle A Globalization of the Implicit Function Theorem The Theorem of Krein-Rutman Global Bifurcation Exercises Chapter IV Global Solution Theorems 3 11 20 22 25 Introduction 27 Splitting Equations 27 Bifurcation at a Simple Eigenvalue 30 Chapter III Degree Theory 33 33 33 38 42 47 49 49 49 52 54 57 61 II Ordinary Di erential Equations 63 Chapter V Existence and Uniqueness Theorems 65 v vi TABLE OF CONTENTS Introduction The Picard-Lindelof Theorem The Cauchy-Peano Theorem Extension Theorems Dependence upon Initial Conditions Di erential Inequalities Uniqueness Theorems Exercises Chapter VI Linear Ordinary Di erential Equations Introduction Preliminaries Constant Coe cient Systems Floquet Theory Exercises Chapter VII Periodic Solutions Introduction Preliminaries Perturbations of Nonresonant Equations Resonant Equations Exercises Chapter VIII Stability Theory Introduction Stability Concepts Stability of Linear Equations Stability of Nonlinear Equations Lyapunov Stability Exercises Chapter IX Invariant Sets Introduction Orbits and Flows Invariant Sets Limit Sets Two Dimensional Systems Exercises Chapter X Hopf Bifurcation 65 66 67 69 72 74 78 79 81 81 81 83 85 88 91 91 91 92 94 100 103 103 103 105 108 110 118 121 121 122 123 125 126 128 129 Introduction 129 A Hopf Bifurcation Theorem 129 TABLE OF CONTENTS vii Chapter XI Sturm-Liouville Boundary Value Problems Introduction Linear Boundary Value Problems Completeness of Eigenfunctions Exercises Bibliography Index 135 135 135 138 141 143 144 viii TABLE OF CONTENTS Part I Nonlinear Analysis 130 in a neighborhood of there is a curve of eigenvalues and eigenvectors fu (0; )a( ) = ( )a( ) d a( ) 6= 0; ( ) = i; Re d j 6= 0: (2) Then there exist postive numbers and and a C function (u; ; ) : (? ; ) ! C2 R R; where C2 is the space of periodic C Rn ? valued functions, such that (u(s); (s); (s)) solves the equation du + f (u; ) = 0; (3) d nontrivially, i.e u(s) 6= 0; s 6= and (0) = 1; (0) = ; u(0) = 0: (4) Furthermore, if (u1 ; ) is a nontrivial solution of (1) of period , with j ? 1j < ; j ? 0j < ; ju1 (t)j < ; t 0; ]; then there exists s (? ; ) such that = (s); = (s) and u1 ( t) = u(s)( + ); = t 0; ]; 0; ): Proof We note that u(t) will be a solution of (1) of period ; whenever u( ); = t is a solution of period of (3) We thus let X = C2 and Y = ; respectively, continuous ? periodic functions C2 be Banach spaces of C endowed with the usual norms and de ne an operator F :X R R !Y (5) (u; ; ) 7! u0 + f (u; ); = dd : Then F belongs to class C and we seek nontrivial solutions of the equation F (u; ; ) = 0; (6) with values of close to 1, close to ; and u 6= 0: We note that F (0; ; ) = 0; for all R; R: Thus the claim is that the value (1; ) of the two dimensional parameter ( ; ) is a bifurcation value Theorem II.1 tells us that u0 + fu (0; )u cannot be a linear homeomorphism of X onto Y: This is guaranteed by the assumptions, since the functions i i = Re(e a( )); = Im(e a( )) are ? periodic solutions of u0 + fu (0; )u = 0; (7) A HOPF BIFURCATION THEOREM 131 and they span the the kernel of Fu (0; 1; 0); kerFu (0; 1; ) = f ; = ? 00 g: It follows from the theory of linear di erential equations that the image, imFu (0; 1; 0); is closed in Y and that imFu (0; 1; ) = ff Y : hf; i i = 0; i = 0; 1g; where h ; i denotes the L2 inner product and f ; g forms a basis for kerf?u0 + T fu (0; )ug; the di erential equation adjoint operator of u0 + fu (0; )u: In fact = and h i ; j i = ij ; the Kronecker delta Thus Fu (0; 1; ) is a linear Fredholm mapping from X to Y having a two dimensional kernel as well as a two dimensional cokernel We now write, as in the beginning of Chapter II, X = V W Y = Z T: We let U be a neighborhood of (0; 1; 0; 0) in V R R R and de ne G on U as follows (s ; G(v; ; ; s) = sFF (0;( ;0 + v); + v);; s 6= 0: )( ) s = u We now want to solve the equation G(v; ; ; s) = 0; for v; ; in terms of s in a neighborhood of R: We note that G is C and G(v; ; ; 0) = ( + v)0 + fu(0; )( + v): Hence G(0; ; ; 0) = 00 + fu (0; ) : Thus, in order to be able to apply the implicit function theorem, we need to di erentiate the map (v; ; ) 7! G(v; ; ; s) evaluate the result at (0; 1; ; 0) and show that this derivative is a linear homeomorphism of V R R onto Y: Computing the Taylor expansion, we obtain G(v; ; ; s) = G(0; 1; 0; s) + G (0; 1; 0; s)( ? 1) G (0; 1; ; s)( ? ) + Gv (0; 1; 0; s)v + ; (8) and evaluating at s = we get Gv; ; (0; 1; 0; 0)(v; ? 1; ? ) = fv (0; ) ( ? 1) +fv (0; ) ( ? ) (9) +(v0 + fv (0; )v): 132 Note that the mapping v 7! v0 + fv (0; )v is a linear homeomorphism of V onto T: Thus we need to show that the map ( ; ) 7! fv (0; ) ( ? 1) + fv (0; ) ( ? ) only belongs to T if = and = and for all Z there exists a unique ( ; ) such that fv (0; ) ( ? 1) + fv (0; ) ( ? ) = : By the characterization of T; we have that fv (0; ) ( ? 1) + fv (0; ) ( ? ) T if and only if < fv (0; ) ; i > ( ? 1)+ < fv (0; ) ; i > ( ? ) = 0; (10) i = 1; 2: Since fv (0; ) = ? 00 = ; we may write equation (10) as two equations in the two unknowns ? and ? 0; < fv (0; ) ; > ( ? ) = (11) ( ? 1)+ < fv (0; ) ; > ( ? ) = 0; which has only the trivial solution if and only if < fv (0; ) ; >6= 0: Computing this latter expression, one obtains < fv (0; ) ; >= Re (0); which by hypothesis is not zero The uniqueness assertion we leave as an exercise For much further discussion of Hopf bifurcation we refer to 12] The following example of the classical Van der Pol oscillator from nonlinear electrical circuit theory (see 12]) will serve to illustrate the applicability of the theorem Example Consider the nonlinear oscillator x00 + x ? (1 ? x2 )x0 = 0: (12) This equation has for for certain small values of nontrivial periodic solutions with periods close to : A HOPF BIFURCATION THEOREM 133 We rst transform (12) into a system by setting = x x0 u0 + ?1 ? + u= u1 u2 and obtain u2 u2 = 0 (13) We hence obtain that f (u; ) = ?1 + u20u ? and fu (0; ) = ?1 ; ? whose eigenvalues satisfy ( + ) + = 0: d Letting = 0; we get (0) = i; and computing d = we obtain + + = 0; or = ? = ? ; for = 0: Thus by Theorem II.1 there exists > +2 and continuous functions (s); (s); s (? ; ) such that (0) = 0; (0) = and (12) has for s 6= a nontrivial solution x(s) with period (s): This solution is unique up to phase shift 134 Chapter XI Sturm-Liouville Boundary Value Problems Introduction In this chapter we shall study a very classical problem in the theory of ordinary di erential equations, namely linear second order di erential equations which are parameter dependent and are subject to boundary conditions While the existence of eigenvalues (parameter values for which nontrivial solutions exist) and eigenfunctions (corresponding nontrivial solutions) follows easily from the abstract Riesz spectral theory for compact linear operators, it is instructive to deduce the same conclusions using some of the results we have developed up to now for ordinary di erential equations While the theory presented below is for some rather speci c cases, much more general problems and various other cases may be considered and similar theorems may be established We refer to the books 4], 5] and 14] where the subject is studied in some more detail Linear Boundary Value Problems Let I = a; b] be a compact interval and let p; q; r C (I; R); with p; r positive on I: Consider the linear di erential equation (p(t)x0 )0 + ( r(t) + q(t))x = 0; t I; (1) where is a complex parameter We seek parameter values (eigenvalues) for which (1) has nontrivial solutions (eigensolutions or eigenfunctions) when it is subject to the set of boundary conditions x(a) cos ? p(a)x0 (a) sin = (2) x(b) cos ? p(b)x0 (b) sin = 0; where and are given constants and (without loss in generality, < ; 0< ) Such a boundary value problem is called a Sturm-Liouville boundary value problem 135 136 We note that (2) is equivalent to the requirement c1 x(a) + c2 x0 (a) = 0; jc1 j + jc2 j 6= c3 x(b) + c4 x0 (b) = 0; jc3 j + jc4 j 6= 0: We have the following lemma (3) Lemma Every eigenvalue of equation (1) subject to the boundary conditions (2) is real Proof Let the di erential operator L be de ned by L(x) = (px0 )0 + qx: Then, if is an eigenvalue L(x) = ? rx; for some nontrivial x which satis es the boundary conditions Hence also L(x) = ? rx = ? rx: Therefore xL(x) ? xL(x) = ?( ? )rxx: Hence Zb a (xL(x) ? xL(x)) dt = ?( ? ) Zb a rxxdt: Integrating the latter expression and using the fact that both x and x satisfy the boundary conditions we obtain the value for this expression and hence = : We next let u(t; ) = u(t) be the solution of (1) which satis es the (initial) conditions u(a) = sin ; p(a)u0 (a) = cos ; then u 6= and satis es the rst set of boundary conditions We introduce the following transformation (Prufer transformation) p u = u2 + p2 (u0 )2 ; = arctan pu0 : Then and are solutions of the di erential equations = ? ( r + q) ? sin cos p and = cos2 + ( r + q) sin2 : p (4) (5) LINEAR BOUNDARY VALUE PROBLEMS 137 Further (a) = : (Note that the second equation depends upon only, hence, once is known, may be determined by integrating a linear equation and hence u is determined We have the following lemma describing the dependence of upon : Lemma Let be the solution of (5) such that (a) = : Then satis es the following conditions: (b; ) is a continuous strictly increasing function of ; lim !1 (b; ) = 1; lim !?1 (b; ) = 0: Proof The rst part follows immediately from the discussion in Sections V.5 and V.6 To prove the other parts of the lemma, we nd it convenient to make the change of independent variable s= Zt d p( ) ; a which transforms equation (1) to d x00 + p( r + q)x = 0; = ds : (6) We now apply the Prufer transformation to (6) and use the comparison theorems in Section V.6 to deduce the remaining parts of the lemma Using the above lemma we obtain the existence of eigenvalues, namely we have the following theorem Theorem The boundary value problem (1)-(2) has an unbounded in nite sequence of eigenvalues < < < with nlim n = 1: !1 The eigenspace associated with each eigenvalue is one dimensional and the eigenfunctions associated with k have precisely k simple zeros in (a; b): Proof The equation + k = (b; ) has a unique solution k ; for k = 0; 1; : This set f k g1 has the desired k=0 properties We also have the following lemma 138 Lemma Let ui ; i = j; k; j 6= k be eigenfunctions of the boundary value prob- lem (1)-(2) corresponding to the eigenvalues j and k : Then uj and uk are orthogonal with respect to the weight function r; i.e huj ; uk i = Zb a ruj uk = 0: (7) In what is to follow we denote by fui g1 the set of eigenfunctions whose i=0 existence is guaranteed by Theorem with ui an eigenfunction corresponding to i ; i = 0; 1; which has been normalized so that Zb a ru2 = 1: i (8) We also consider the nonhomogeneous boundary value problem (p(t)x0 )0 + ( r(t) + q(t))x = rh; t I; (9) where h L2 (a; b) is a given function, the equation being subject to the boundary conditions (2) and solutions being interpreted in the Caratheodory sense We have the following result Lemma For = k equation (9) has a solution subject to the boundary conditions (2) if and only if Zb a ruk h = 0: If this is the case, and w is a particular solution of (9)-(2), then any other solution has the form w + cuk ; where c is an arbitrary constant Proof Let v be a solution of (p(t)x0 )0 + ( k r(t) + q(t))x = 0; which is linearly independent of uk ; then c (uk v0 ? u0k v) = p ; where c is a nonzero constant One veri es that ! v(t) Z t ru h + u Z b rvh w(t) = c a k k t R is a solution of (9)-(2) (for = k ), whenever ab ruk h = holds Completeness of Eigenfunctions We note that it follows from Lemma that (9)-(2) has a solution for every R b k ; k = 0; 1; 2; if and only if a ruk h = 0; for k = 0; 1; 2; : Hence, since fui g1 forms an orthonormal system for the Hilbert space L2 (a; b) (i.e L2 (a; b) r i=0 with weight function r de ning the inner product), fuig1 will be a complete i=0 COMPLETENESS OF EIGENFUNCTIONS R 139 orthonormal system, once we can show that ab uk h = 0; for k = 0; 1; 2; ; implies that h = (see 23]) The aim of this section is to prove completeness The following lemma will be needed in this discussion Lemma If 6= k ; k = 0; 1; (9) has a solution subject to the boundary conditions (2) for every h L2 (a; b): Proof For = k ; k = 0; 1; we let u be a nontrivial solution of (1) which satis es the rst boundary condition of (2) and let v be a nontrivial solution of (1) which satis es the second boundary condition of (2) Then c uv0 ? u0 v = p with c a nonzero constant De ne the Green's function s t G(t; s) = v(t))u((s);; a s b: c v(s u t) t Then w(t) = Zb a (10) G(t; s)r(s)h(s)ds is the unique solution of (9) - (2) We have the following corollary Corollary Lemma de nes a continuous mapping G : L2 (a; b) ! C a; b]; by Further (11) h 7! G(h) = w: hGh; wi = hh; Gwi: Proof We merely need to examine the de nition of G(t; s) as given by equation (10) Let us now let S = fw L2 (a; b) : hui ; hi = 0; i = 0; 1; 2; g: (12) Using the de nition of G we obtain the lemma Lemma G : S ! S: We note that S is a linear manifold in L2 (a; b) which is weakly closed, i.e if fxn g S is a sequence sucht that hxn ; hi ! hx; hi; 8h L2 (a; b); then x S 140 Lemma If S 6= f0g; then there exists x S such that hG(x); xi 6= 0: Proof If hG(x); xi = for all x S; then, since S is a linear manifold, we have for all x; y S and R = hG(x + y); x + yi = hG(y); xi; in particular, choosing x = G(y) we obtain a contradiction, since for y 6= G(y) 6= 0: 10 Lemma If S 6= f0g; then there exists x S nf0g and 6= such that G(x) = x: Proof Since there exists x S such that hG(x); xi 6= we set = inf fhG(x); xi : x S; kxk = 1; if hG(x); xi 0; 8x S g supfhG(x); xi : x S; kxk = 1; if hG(x); xi 0; for some x S g: We easily see that there exists x0 S; kx0 k = such that hG(x0 ); x0 i = 6= 0: If S is one dimensional, then G(x0 ) = x0 : If S has dimension greater than 1, x+ then there exists 6= y S such that hy; x0 i = 0: Letting z = p01+ y we nd that hG(z ); z i has an extremum at = and thus obtain that hG(x0 ); yi = 0; for any y S with hy; x0 i = 0: Hence since hG(x0 ) ? x0 ; x0 i = it follows that hG(x0 ); G(x0 ) ? x0 i = and thus hG(x0 ) ? x0 ; G(x0 ) ? x0 i = 0; proving that is an eigenvalue Combining the above results we obtain the following completeness theorem 11 Theorem The set of eigenfunctions fuig1 forms acomplete orthonormal sysi=0 tem for the Hilbert space L2 (a; b): r Proof Following the above reasoning, we merely need to show that S = f0g: If this is not the case, we obtain, by Lemma 10, a nonzero element h S and a nonzero number such that G(h) = h: On the other hand w = G(h) satis es the boundary conditions (2) and solves (9); hence h satis es the boundary conditions and solves the equation (p(t)h0 )0 + ( r(t) + q(t))h = r h; t I; (13) i.e ? = j for some j: Hence h = cuj for some nonzero constant c; contradicting that h S: EXERCISES 141 Exercises Find the set of eigenvalues and eigenfunctions for the boundary value problem x00 + x = x(0) = = x0 (0): Supply the details for the proof of Lemma Prove Lemma 4 Prove that the Green's function given by (10) is continuous on the square ( a; b]2 and that @G@tt;s) is continuous for t 6= s: Discuss the behavior of this derivative as t ! s: Provide the details of the proof of Corollary Also prove that G : L2 (a; b) ! L2 (a; b) is a compact mapping Let G(t; s) be de ned by equation (10) Show that G(t; s) = X ui(t)ui (s) i=0 ? i ; where the convergence is in the L2 norm Replace the boundary conditions (2) by the periodic boundary conditions x(a) = x(b); x0 (a) = x0 (b): Prove that the existence and completeness part of the above theory may be established provided the functions satisfy p(a) = p(b); q(a) = q(b); r(a) = r(b): Apply the previous exercise to the problem x00 + x = 0; x(0) = x(2 ) x0 (0) = x0 (2 ): Let the di erential operator L be given by L(x) = (tx0 )0 + m x; < t < 1: t and consider the eigenvalue problem L(x) = ? tx: 142 In this case the hypotheses imposed earlier are not applicable and other types of boundary conditions than those given by (3) must be sought in order that a development parrallel to that given in Section may be made Establish such a theory for this singular problem Extend this to more general singular problems Bibliography 1] R Adams, Sobolev Spaces, Academic Press, New York, 1975 2] J Bebernes and D Eberly, Mathematical Problems from Combustion Theory, Springer Verlag, New York, 1989 3] G Birkhoff and G Rota, Ordinary Di erential Equations, Blaisdell, Waltham, 1969 4] E Coddington and N Levinson, Theory of Ordinary Di erential Equations, McGraw Hill, New York, 1955 5] R Cole, Theory of Ordinary Di erential Equations, Appleton-CenturyCrofts, New York, 1968 6] W Coppel, Stability and Asymptotic Behavior of Di erential Equations, Heath, Boston, 1965 7] H Dang and K Schmitt, Existence and uniqueness theorems for nonlinear boundary value problems, Rocky Mtn J Math., 24 (1994), pp 77{91 8] J Dieudonne, Foundations of Modern Analysis, Academic Press, New York, 1964 9] J Dugundji, Topology, Allyn and Bacon, Boston, 1966 10] N Dunford and J Schwartz, Linear Operators: Part I, Wiley, New York, 1964 11] D Gilbarg and N Trudinger, Elliptic Partial Di erential Equations of Second Order, Sringer, Berlin, 1983 12] J Guckenheimer and P Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer Verlag, New York,, 1983 13] W Hahn, Stability of Motion, Springer, Berlin, 1967 14] P Hartman, Ordinary Di erential Equations, Birkhauser, Boston, 1982 143 144 BIBLIOGRAPHY 15] E Heinz, An elementary analytic theory of the degree of mapping in ndimensional Euclidean space, J Math Mech., (1959), pp 231{247 16] E Kamke, Zur Theorie der Systeme gewohnlicher Di erentialgleichungen II, Acat Math, 58 (1932), pp 57{85 17] M A Krasnosels'kii, Positive Solutions of Operator Equations, Noordho , Groningen, 1964 18] M Krein and M Rutman, Linear operators leaving invariant a cone in a Banach space, AMS Translations, 10 (1962) 19] E Kreyszig, Introductory Functional Analysis, Wiley, New York, 1978 20] M Nagumo, A theory of degree of mapping based on in nitesimal analysis, Amer J Math., 73 (1951), pp 485{496 21] C Olech, A simple proof of a result of Z Opial, Ann Polon Math., (1960), pp 61{63 22] H L Royden, Real Analysis, 3rd ed., Macmillan Publishing Co., New York, 1988 23] W Rudin, Real and Complex Analysis, McGraw Hill, New York, 1966 24] M Schechter, Principles of Functional Analysis, Academic Press, New York, 1971 25] J Schwartz, Nonlinear Functional Analysis, Gordon and Breach, New York, 1969 26] R E Showalter, Hilbert Space Methods for partial Di erential Equations, Pitman, London, 1977 27] H Siegberg, Some historical remarks concerning degree theory, Amer Math Monthly, 88 (1981), pp 125{139 28] A Taylor, Introduction to Functional Analysis, Wiley and Sons, New York, 1972 29] G Whyburn, Analytic Topolgy, Amer Math Soc., Providence, 1942 ... properties and refer the reader for more detailed discussions to standard texts on analysis and functional analysis (e.g 8]) 4.1 Completely continuous mappings Let E and X be Banach spaces and let be an. .. 144 viii TABLE OF CONTENTS Part I Nonlinear Analysis Chapter I Analysis In Banach Spaces Introduction This chapter is devoted to developing some tools from Banach space valued function theory... natural for the given problem iv Table of Contents I Nonlinear Analysis Chapter I Analysis In Banach Spaces Introduction Banach Spaces Di erentiability, Taylor''s Theorem