Sufficient conditions for functions to form Riesz bases in L 2 and applications to nonlinear boundary-value problems

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1 Electronic Journal of Differential Equations, Vol. 200(200), No. 74, pp. 0. ISSN: URL: or ftp ejde.math.swt.edu (login: ftp) Sufficient conditions for functions to form Riesz bases in L 2 and applications to nonlinear boundary-value problems Peter E. Zhidkov Abstract We find sufficient conditions for systems of functions to be Riesz bases in L 2(0, ). Then we improve a theorem presented in [3] by showing that a standard system of solutions of a nonlinear boundary-value problem, normalized to, is a Riesz basis in L 2(0, ). The proofs in this article use Bari s theorem. Introduction Early results in the study of basis properties of eigenfunctions of nonlinear ordinary differential operators can be found in the monograph by Makhmudov [5]. Because of its difficulty and the small number of publications on this question, basis properties has been established only for very simple nonlinear ordinary differential equations. Among the results in this direction, we have the following. In [7, 8], Zhidkov presents an analysis of the equation u + f(u 2 )u = λu, u = u(x), x (0, ), u(0) = u() = 0, 0 u2 (x) dx =, where λ is a spectral parameter, f(s) is a smooth nondecreasing function for s 0, and all quantities are real. In these two publications, it is proved that the eigenfunctions {u n } (n = 0,, 2,...) of this problem have precisely n zeros in (0, ). Furthermore, each eigenfunction is unique up to the coefficient ±. The main result states that the sequence of eigenfunctions {u n } (n = 0,, 2,...) is a Bari basis in L 2 = L 2 (0, ), i.e., it is a basis and there exists an orthonormal basis {e n } (n = 0,, 2,...) in L 2 for which n=0 u n e n 2 L 2 <. Note that in [7] there are some errors which have been corrected in [9]. In [0, ], a modified version of the above nonlinear eigenvalue problem is studied and similar basis properties for their eigenfunctions are obtained. In Mathematics Subject Classifications: 4A58, 42C5, 34L0, 34L30. Key words: Riesz basis, infinite sequence of solutions, nonlinear boundary-value problem. c 200 Southwest Texas State University. Submitted September 24, 200. Published December 4, 200.

2 2 Riesz bases in L 2 EJDE 200/74 [2], an analog to the Fourier transform associated with an eigenvalue problem for a nonlinear ordinary differential operator on a half-line is considered. The aim in the present publication is to improve the result in [3], where the following nonlinear problem is considered: u = f(u 2 )u, u = u(x), x (0, ), (.) u(0) = u() = 0. (.2) Here there is no spectral parameter, and all variables are real. For the rest of this article, we will assume that (F) The function f(u 2 )u is a continuously differentiable for u R, f(0) 0, and f(+ ) =. It is well known now (and partially proved in [3]) that under assumption (F): For each integer n 0 problem (.) (.2) has a solution u n which possesses precisely n zeros in (0, ) and that generally speaking this solution is not unique. Definition A sequence {u n } (n = 0,, 2,...) of solutions to (.) (.2) is called standard if the solution u n has precisely n zeros in (0, ). The main result in [3] states that there exists s 0 < 0 such that for s < s 0 any standard sequence of solutions {u n } is a basis in H s (0, ). In addition, the sequence {u n / u n Hs (0,)} is a Riesz basis in H s (0, ). Here H s (0, ) is the usual Sobolev space with negative index s. In the present paper, we improve this result by showing the above properties of a standard system {u n } in L 2 (see Theorem.3 below), by first obtaining a general result on bases in L 2 (see Theorem. below). We believe that this result is of a separate interest. Notation By c, C, C, C 2, C, C,... we denote positive constants. By L 2 (a, b) we denote the standard Lebesgue space of square integrable functions on the interval (a, b). In this space we introduce the standard inner product, and norm: (g, h) L2(a,b) = b a g(x)h(x)dx, g L2(a,b) = (g, g) /2 L 2(a,b). For short notation we will use (, ) and respectively. Let l 2 be the space of square summable sequences of real numbers. For a Banach space X with a norm X, let L(X; X) be the linear space of linear bounded operators acting from X into X, equipped with the norm A L(X;X) = sup Ax X. x X: x X = We also set = L(L2;L 2) for short notation. Now, for convenience of readers, we define some well-known terms.

3 EJDE 200/74 Peter E. Zhidkov 3 Definition A system {e n } L 2 (a, b) is called a basis in L 2 (a, b) if for any g L 2 (a, b) there exists a unique sequence {a n } of real numbers such that g = n=0 a ne n in L 2 (a, b). There are several definitions of Riesz bases. In accordance with the classical paper by N. K. Bari [], where this concept was introduced for the first time, we use the following definition. Definition A basis {e n } in L 2 (a, b) is called a Riesz basis in this space when the series n=0 a ne n, with real coefficients a n, converges in L 2 (a, b) if and only if n=0 a2 n <. Remark It is proved in [] (see also [8]) that if {e n } is a Riesz basis in L 2 (a, b) in the sense of this definition, then there exist constants 0 < c < C such that c a 2 n n=0 2 a n e n n=0 L 2(a,b) C for all a = (a 0, a, a 2,...) l 2. These estimates have been often used to define Riesz bases. Definition A system of functions {g n } in L 2 (a, b) is called ω-linearly independent in L 2 (a, b) when n=0 a ng n = 0, with a n are real numbers, holds in L 2 (a, b) if and only if 0 = a 0 = a = a 2 =.... Definition Two systems of functions {h n } and {e n } in L 2 (a, b) are called quadratically close in L 2 (a, b), if n=0 h n e n 2 L <. 2(a,b) Results Theorem. Let {h n } be a system of real-valued, three-times continuously differentiable functions. Assume that for each integer n 0 the following holds: ( ) ( (a) h n x+ n+ = hn (x) and h n 2(n+) +x) ( = h n 2(n+) x) for all x R (b) h n(x) > 0, h n(x) 0, and h n (x) 0 for all x ( ) 0, 2(n+) ( (c) There exist 0 < c < C such that c < h n 2(n+)) < C for all n. Then, the system {h n } is a Riesz basis in L 2. Remark Clearly, it follows from Theorem. that if a system of functions {h n } satisfies all the conditions of this theorem, except maybe (c), then it is a basis in L 2. The next result follows from Theorem. by taking h n (x) = h((n + )x). Theorem.2 Let h(x) be a real-valued three-times continuously differentiable function satisfying: n=0 a 2 n

4 4 Riesz bases in L 2 EJDE 200/74 (a) h( + x) = h(x) and h(/2 + x) = h(/2 x) for all x R (b) h (x) > 0, h (x) 0 and h (x) 0 for all x (0, /2) Then, the sequence of functions h n (x) = h((n + )x), where n = 0,, 2,..., is a Riesz basis in L 2. The following statement also follows from Theorem., when applied to problem (.) (.2). Theorem.3 Let assumption (F) be valid and f(u 2 ) + 2u 2 f (u 2 ) 0 for all sufficiently large u. Let {u n } be an arbitrary standard sequence of solutions of (.) (.2). Then, the sequence { u n u n } is a Riesz basis in L2. To prove this theorem in Section 3, we exploit the following theorem. Theorem.4 (Bari s Theorem) Let {e n } be a Riesz basis in L 2 (a, b) and let a system {h n } L 2 (a, b) be ω-linearly independent and quadratically close to {e n } in L 2 (a, b). Then, the system {h n } is a Riesz basis in L 2 (a, b). This theorem, in a weaker form, was proved by N. K. Bari in []. In its current form it is proved, for example, in [4] and in [8]. We conclude the introduction by pointing out that the concept of a Riesz basis appeared for the first time in the middle of last century in the papers of N. K. Bari, as a result of developments in the general theory of orthogonal series and bases in infinite-dimensional spaces. Currently, this concept has important applications in areas such as wavelet analysis. Readers may consult [2, 6] for theoretical aspects of this field and [3] for applied aspects. 2 Proof of Theorem. Let e n (x) = 2 sin π(n + )x, n = 0,, 2,..., so that {e n } is an orthonormal basis in L 2. Lemma 2. Let g satisfy condition (a) of Theorem. with n 0 and let g be positive in (0, n+ ). Then in the expansion g( ) = c m e m ( ), m=0 understanding in the sense of L 2, one has c 0 =... = c n = 0 and c n > 0. Proof We follow the arguments in the proof of a similar statement in [3]. We have the above expansion in L 2 (0, n+ ) with c m = 0 if m (n + )(l + ) for all integers l 0 (this occurs because the functions {e (n+)(m+) } m form an orthogonal basis in L 2 (0, n+ )). Therefore, c 0 =... = c n = 0. We observe that each e (n+)(m+) becomes zero at the points n+, 2 n+,...,.

5 EJDE 200/74 Peter E. Zhidkov 5 Furthermore, due to condition (a) of Theorem. the function g is odd with respect to these points and each function e (n+)(m+) (x) is odd. Thus this expansion also holds in each space L 2 ( n+, 2 n+ ), L 2( 2 n+, 3 n+ ),..., L 2( n n+, ). Finally, c n > 0 because e n (x) and g(x) are of the same sign everywhere. Due to Lemma 2., we have the following sequence of expansions: h n ( ) = a n me m ( ) in L 2, (2.) m=0 with a n 0 =... = a n n = 0 and a n n > 0, for n = 0,, 2,.... Lemma 2.2 Under the assumptions of Theorem., the coefficients in (2.) satisfy (a n n) a n (n+)(m+) π (m + ) 2 2 for all n and m. In addition, a n (n+)(m+) = 0 if m = 2l + for l = 0,, 2,.... Proof The second claim of this lemma is obvious because e (n+)(2l+2) (x) is odd with respect to the middles of the intervals (0, n+ ), ( n+, 2 n+ ),..., ( n n+, ) and the function h n (x) is even so that a n (n+)(2l+2) = (e (n+)(2l+2), h n ) = 0. Let us prove the first claim. Due to the properties of the functions h n and e (n+)(m+), with m = 2l, we have (a n n) a n (n+)(m+) = 0 h n(x) sin π(n + )(m + )xdx 0 h n(x) sin π(n + )xdx /2(n+) h 0 n (x) sin π(n + )(m + )xdx = /2(n+) h 0 n (x) sin π(n + )xdx /2(n+) h = (m + ) 0 n(x) cos π(n + )(m + )xdx /2(n+) h 0 n(x) cos π(n + )xdx = (m + ) 0 h s π(m+)s n( 2(n+) ) cos 2 ds 0 h s πs n( 2(n+) ) cos 2 ds. Due to the conditions of Theorem., h n( concave function on (0, ). Therefore, 0 h s πs n( ) cos 2(n + ) 2 ds h n(0) 0 s 2(n+) ) is a positive non-increasing ( s) cos πs 2 ds = 4 π 2 h n(0). Using the same properties of h n, one can easily see on its graph that h s π(m + )s /(m+) n( ) cos ds h π(m + )s 0 2(n + ) 2 n(0) cos ds = π(m + ) h n(0).

6 6 Riesz bases in L 2 EJDE 200/74 Detailed arguments leading to a similar estimate are considered in [3]. easily obtain now We which completes the proof. (a n n) a n (n+)(m+) π 2 (m + ) 2, From the conditions on Theorem., we have 0 < c a n n C. Then to prove this theorem, it suffices to prove that the system {h n } with h n = (a n n) h n is a Riesz basis in L 2. Lemma 2.3 Let {g n } be a sequence of functions such that g n satisfies condition (a) on Theorem. and is positive in (0, n+ ). Then the system {g n} is ω- linearly independent in L 2. The proof of this lemma is rather simple. We refer the reader to the proof of similar statements in [8,, 3]. Let b n m = (a n n) a n m, and let Id be the unit operator in L 2. For positive integers m, let B m be the operator mapping e n into b n (n+)(m+) e (n+)(m+), B m L(L 2 ; L 2 ). Also let B = m= B m. Then for each m, Furthermore, by Lemma 2, B m sup b n (n+)(m+) = b m. n b m π 2 m= (2l + ) 2 π 2 l= /2 (2x + ) 2 dx = π/8 ; hence, B L(L 2 ; L 2 ) and B π/8 <. Therefore, the operator A = Id +B has a bounded inverse A = Id + n= ( )n B n. Note also that Ae n = h n. Hence, as proved in [4], {h n } is a Riesz basis in L 2 (0, ). For the convenience of the reader, we present a short proof of this statement. Take an arbitrary v L 2 and let u = A v = n=0 c ne n L 2 where c n are real coefficients. Then, n=0 c2 n < because {e n } is an orthonormal basis in L 2. Since the series n=0 c ne n converges in L 2, we have v = Au = n=0 c nae n = n=0 c nh n where all infinite sums also converge in L 2. Therefore, in view of Lemma 2.3, the system {h n } is a basis in L 2 and, if n=0 c2 n <, then the series n=0 c nh n converges in L 2. Conversely, let a series u = n=0 c nh n converge in L 2. Then A u = n=0 c ne n in L 2 ; hence n=0 c2 n <. Thus, {h n } is a Riesz basis in L 2, and the proof of Theorem. is complete. 3 Proof of Theorem.3 As was proved in [3], any solution u n of problem (.) (.2), that possesses precisely n zeros in (0, ), satisfies condition (a) of Theorem.. In addition,

7 EJDE 200/74 Peter E. Zhidkov 7 u n is strictly monotone (u n(x) 0) in the interval (0, 2(n+) ). Let u > 0 be an arbitrary number such that f(u 2 ) < 0 and f(u 2 ) + 2u 2 f (u 2 ) 0 for all u u. Let {u n } be an arbitrary standard system of solutions of problem (.) (.2). We assume that u n (x) > 0 for x (0, n+ ) for each n which is possible without loss of generality due to the invariance of (.) when u(x) is replaced by u(x). Due to the standard comparison theorem max u [0,un(/2(n+))] f(u 2 ) + as n ; hence u n ( 2(n+) ) + as n. For n sufficiently large, we denote by x n (0, 2(n+) ) the point for which u n(x n ) = u. Then u n ( 2(n + ) ) u = /2(n+) u n(x)dx = u n( x n )( x n 2(n + ) x n) for some x n (x n, 2(n+) ). Since u n(x n ) u n( x n ) (because f(u 2 ) < 0 for u > u and, therefore, u n(x) < 0 for x (x n, 2(n+) )), we derive u n(x n ) 3 2 u n( )(n + ) 2(n + ) for all sufficiently large n. Since in view of (.), sup n max x [0,xn] u n(x) C, we have min x [0,xn] u n(x) u n ( 2(n+) )(n + ) for all sufficiently large n. Therefore, 0 < x n (n + ) [ u n ( 2(n + ) )] (3.) for all sufficiently large n. Using u n and n large, we now want to construct a function h n that satisfies the conditions of Theorem.. Introduce the linear function l n (x) = u x n x which is equal to 0 at x = 0 and to u = u n (x n ) at x = x n. Multiply (.), with u = u n, by 2u n(x) and integrate the result from 0 to x. Then {[u n(x)] 2 + F (u 2 n(x))} = 0, x R, (3.2) where F (s) = s 0 f(t)dt. Due to condition (F), F (u2 ) + as u, therefore, without loss of generality, we can assume that u > 0 and is large enough so that uf(u 2 ) > uf(u 2 ) and F (u 2 ) > F (u 2 ) for all u [0, u). Then from (3.2), it follows that for all sufficiently large n. By (3.3), we have u n(x n ) < u n(x), x [0, x n ), (3.3) therefore, u = 0 u n(x)dx > x n u n(x n ); u n(x n ) < u x n = l n(x) (3.4)

8 8 Riesz bases in L 2 EJDE 200/74 for all sufficiently large n. Take a sufficiently small (0, xn 2 ) and define a continuous function ω (x) equal to u n (x) for x [x n, 2(n+) ] such that u n (x) ω (x) 0 for x [x n, x n ] and ω (x) = 0 for x [0, x n ). We define g (x) to be equal to u n (x) for x [x n, 2(n+) ], and for x [0, x n) to be given by the rules: g (x) = u n(x n ) g (x) = u n(x n ) g (x) = u n (x n ) x x x ω (t)dt, g (t)dt, (3.5) g (t)dt. Then g (x) is three times continuously differentiable in [0, 2(n+) ] and satisfies condition (b) of Theorem.. It is easy to see that if > 0 is sufficiently small, then g (x n ) and g (x n ) are arbitrary close to u n (x n ) and u n(x n ), respectively, and g (x) is arbitrary close to u n(x n ) for all x [0, x n ]. Now, due to our choice of u > 0, for > 0 and sufficiently small, g (0) is arbitrary close to u n (x n ) x n u n(x n ) + x2 n 2 u n(x n ). (3.6) This expression is negative because 0 = u n (0) = u n (x n ) x n u n(x n ) + dx u n(t)dt 0 x where the last term in the right-hand side of this equality is larger than the last term in (3.6), due to our choice of u and (.). We have defined a function g (x) satisfying g (0) < 0. Take now a sufficiently small (0, xn 2 ) and a continuous function ω 2(x) 0 which is equal to u n (x) for x [x n, 2(n+) ] and to 0 for x [0, x n ), such that x n ω 2 (x)dx = u n(x n ). Then, defining the function g 2 (x) just as g (x) in (3.5) with the substitution of ω 2 in place of ω and of g 2 in place of g, we get that if > 0 is sufficiently small, then g 2 (x n ) and g 2(x n ) are arbitrary close, respectively, to u n (x n ) and u n(x n ), and g 2 (x) = 0 for 0 x x n. Therefore, due to (3.4), g 2 (0) > 0 if > 0 is sufficiently small, for all sufficiently large n. We have defined a function g 2 (x) satisfying g 2 (0) > 0. Now, consider the family of functions g λ (x) = λg (x) + ( λ)g 2 (x) where λ [0, ]. Clearly, there exists a unique λ 0 (0, ) such that g λ0 (0) = 0. Extend g λ0 (x) continuously on the entire real line by the rules: g λ0 ( n + + x) = g λ 0 (x), g λ0 ( 2(n + ) + x) = g λ 0 ( 2(n + ) x)

9 EJDE 200/74 Peter E. Zhidkov 9 and denote the obtained function by h n (x). This function satisfies conditions (a) and (b) of Theorem.. In addition, by Theorem.(b), h n(x n ) h n(x) 0 for all x [0, x n ). So far, we have constructed h n for n sufficient large. For small values of n, we use arbitrary functions h n satisfying the conditions of Theorem.. Therefore the sequence {h n } (n = 0,, 2,...) satisfies the conditions (a) and (b) of Theorem.. Let α n = [h n ( 2(n+) )]. Then, by Theorem., the system {α n h n } is a Riesz basis in L 2. Furthermore, by Lemma 2.3, the system {α n u n } is ω-linearly independent in L 2. Also, due to (.) and by construction, there exists C > 0 such that u n(x) = max u [0,u] uf(u2 ) C and max x [0,x n] h n(x) = h n(x n ) = u n(x n ) = uf(u 2 ) C for all n sufficiently large. Hence, u n(x) h n(x) C 2 x n for all n sufficiently large and all x [0, x n ]. Hence, due to (3.), α n u n α n h n 2 C 3 x 4 n C 4 (n + ) 4 for all n sufficiently large. Therefore, the systems {α n u n } and {α n h n } are quadratically close in L 2. In view of Bari s Theorem, the proof of Theorem.3 is complete. Acknowledgment The author is thankful to Mrs. G. G. Sandukovskaya for editing the original manuscript. References [] Bari, N.K., Biorthogonal systems and bases in Hilbert space. Moskov. Gos. Univ. Učenye Zapiski, 95, 48, Matematika 4, (in Russian); Math. Rev., 953, 4, 289. [2] Cristensen, O. Frames, Riesz bases, and discrete Gabor/wavelet expansions. Bull. Amer. Math. Soc., 200, 38, No 3, [3] Different Perspectives on wavelets (edited by I. Daubechies), Proc. Symposia in Appl. Math. 47, Amer. Math. Soc., Providence, 993. [4] Gohberg, I.C. and Krein, M.G. Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Nauka, Moscow, 965 (in Russian). [5] Makhmudov, A.P., Foundations of nonlinear spectral analysis. Azerbaidjan State Univ. Publ., Baku, 984 (in Russian).

10 0 Riesz bases in L 2 EJDE 200/74 [6] Novikov, I.Ya. and Stechkin, S.B. Fundamentals of wavelet theory. Uspekhi Mat. Nauk, 998, 53, No 6, (in Russian). [7] Zhidkov, P.E., Completeness of systems of eigenfunctions for the Sturm- Liouville operator with potential depending on the spectral parameter and for one nonlinear problem. Sbornik: Mathematics, 997, 88, No 7, [8] Zhidkov, P.E. Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Springer-Verlag, Lecture Notes in Mathematics 756, Heidelberg, 200. [9] Zhidkov, P.E., Eigenfunction expansions associated with a nonlinear Schrödinger equation on a half-line. Preprint of the JINR, E , Dubna, 999. [0] Zhidkov, P.E., On the property of being a Riesz basis for the system of eigenfunctions of a nonlinear Sturm-Liouville-type problem. Sbornik: Mathematics, 2000, 9, No 3, [] Zhidkov, P.E., Basis properties of eigenfunctions of nonlinear Sturm- Liouville problems, Electron. J. Diff. Eqns., 2000, 2000, No 28, -3. [2] Zhidkov, P.E. An analog of the Fourier transform associated with a nonlinear one-dimensional Schrödinger equation. Preprint of the JINR, E , Dubna, 200; to appear in Nonlinear Anal.: Theory, Methods and Applications. [3] Zhidkov, P.E. On the property of being a basis for a denumerable set of solutions of a nonlinear Schrödinger-type boundary-value problem. Nonlinear Anal.: Theory, Methods and Applications, 200, 43, No 4, Peter E. Zhidkov Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 4980 Dubna (Moscow region), Russia zhidkov@thsun.jinr.ru

SELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY

SELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS