Eigenfunction Expansions for a Sturm Liouville Problem on Time Scales

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1 Interntionl Journl of Difference Equtions (IJDE). ISSN Volume 2 Number 1 (2007), pp Reserch Indi Publictions Eigenfunction Expnsions for Sturm Liouville Problem on Time Scles Gusein Sh. Guseinov Deprtment of Mthemtics, Atilim University, Incek, Ankr, Turkey E-mil: guseinov@tilim.edu.tr Abstrct In this pper we investigte Sturm Liouville eigenvlue problem on time scles. Existence of the eigenvlues nd eigenfunctions is proved. Men squre convergent nd uniformly convergent expnsions in the eigenfunctions re estblished. AMS subject clssifiction: 34L10. Keywords: Time scle, delt nd nbl derivtives nd integrls, Green s function, completely continuous opertor, eigenfunction expnsion. 1. Introduction Let T be time scle nd,b T be fixed points with <bsuch tht (, b) is not empty. Throughout, ll the intervls re time scle intervls. For stndrd notions nd nottions connected to time scles clculus we refer to [4, 5]. In this study we del with the simple Sturm Liouville eigenvlue problem y (t) = λy(t), t (, b), (1.1) y() = y(b) = 0. (1.2) Some spects of Sturm Liouville eigenvlue problems on time scles hve lredy been considered in the literture (see [1, 6]). In the present pper we re concerned with eigenfunction expnsions (generlized Fourier nlysis) for problem (1.1), (1.2). In our dicussion n importnt role is plyed by certin new type integrtion by prts formuls on time scles, estblished recently by the uthor [7, 9]. These formuls contin delt Received Februry 4, 2007; Accepted April 1, 2007

2 94 Gusein Sh. Guseinov nd nbl derivtives nd integrls t the sme time nd they re elborted in Section 2. Next in Section 3 it is shown, by using the Hilbert Schmidt theorem on symmetric completely continuous opertors, tht the eigenvlue problem (1.1), (1.2) hs system of eigenfunctions tht forms n orthonorml bsis for n pproprite Hilbert spce. This yields men squre convergent (tht is, convergent in n L 2 -metric) expnsions in eigenfunctions. Finlly, in Section 4 uniformly convergent expnsions in eigenfunctions re obtined when the expnded functions stisfy some smoothness conditions. 2. Integrtion by Prts Formuls The im of this section is to present two integrtion by prts formuls on time scles, given below in Theorem 2.4. These formuls will be employed in the subsequent sections. They were recently estblished by the uthor in [9] (see lso [7]). First we formulte theorem which gives reltionship between the delt nd nbl derivtives. For its proof see [3, Theorem 2.5 nd Theorem 2.6]. The derivtives t the end points of intervls re understood to be one-sided derivtives. Theorem 2.1. (i) If f :[,b] R is continuous on [,b] nd -differentible on [,b) with continuous f, then f is -differentible on (, b] nd f (t) = f (ρ(t)) for ll t (, b]. (ii) If f :[,b] R is continuous on [,b] nd -differentible on (, b] with continuous f, then f is -differentible on [,b) nd f (t) = f (σ (t)) for ll t [,b). The next theorem (see [9] nd [7]) gives reltionship between the delt nd nbl integrls. Theorem 2.2. Let f :[,b] R be continuous function. Then (i) f (t) t = f(ρ(t)) t, (ii) f(t) t = f (σ (t)) t. Proof. We only prove (i) s (ii) cn be proved similrly. Tke n rbitrry prtition P of [,b]: P ={t 0,t 1,...,t n } [,b], = t 0 <t 1 < <t n = b.

3 Eigenfunction Expnsions for Sturm Liouville Problem on Time Scles 95 Let us set for ech i {1,...,n} M i = sup{f(t): t [t i 1,t i )}, M i = sup{f(ρ(t)): t (t i 1,t i ]} nd form upper Drboux -sum U(f,P) nd upper Drboux -sum U (f ρ,p)by U(f,P) = M i (t i t i 1 ), i=1 U (f ρ,p)= M i (t i t i 1 ), i=1 respectively, where f ρ denotes the function f ρ (t) = f(ρ(t)). Then, since f is continuous nd f ρ is left-dense continuous, we get tht f is -integrble over [,b) nd f ρ is -integrble over (, b] nd tht (see [8]) f (t) t = inf P U(f,P), f(ρ(t)) t = inf P U (f ρ,p). (2.1) On the other hnd, it is not difficult to see tht from continuity of f on [,b] it follows tht M i = M i for ny i {1,...,n} nd hence U(f,P) = U (f ρ,p)for ll prtitions P of [,b]. Therefore from (2.1) we get the sttement (i) of the theorem. Remrk 2.3. Another proof of Theorem 2.2 cn be given by using Theorem 2.1. Indeed, let F :[,b] R be -ntiderivtive for f on [,b], tht is, F is continuous on [,b], -differentible on [,b), nd F (t) = f(t)for ll t [,b). Then we hve, using Theorem 2.1(i), F (t) = F (ρ(t)) = f(ρ(t)) for t (, b], so tht F is t the sme time -ntiderivtive for f ρ on [,b]. Therefore f(ρ(t)) t = F(b) F() = f (t) t. The sttement Theorem 2.2(ii) cn be proved in similr mnner by using Theorem 2.1(ii). Now let us formulte nd prove the min result of this section. Theorem 2.4. Let f nd g be continuous functions on [,b]. Suppose tht f is differentible on [,b) with continuous nd bounded f nd g is -differentible on (, b] with continuous nd bounded g. Then f (t)g(t) t = f(t)g(t) b f(t)g (t) t, (2.2) f (t)g(t) t = f(t)g(t) b f(t)g (t) t. (2.3)

4 96 Gusein Sh. Guseinov Proof. It is enough to prove (2.2) s (2.3) is modifiction of (2.2). To prove (2.2) note tht by the product rule for -derivtive we hve (fg) (t) = f (t)g(t) + f (σ (t))g (t). Further, -integrting both sides of the lst eqution we get f(t)g(t) b = f (t)g(t) t + On the other hnd, using Theorem 2.1(ii) nd Theorem 2.2(ii) we hve f (σ (t))g (t) t = f (σ (t))g (σ (t)) t = f (σ (t))g (t) t. (2.4) f(t)g (t) t. (2.5) Substituting (2.5) into the right-hnd side of (2.4) we rrive t (2.2). 3. Men Squre Convergent Expnsions Denote by H the Hilbert spce of ll rel -mesurble functions y : (, b] R such tht y(b) = 0 in the cse b is left-scttered, nd tht with the inner product (sclr product) nd the norm y,z = y 2 (t) t <, y = { y,y = y(t)z(t) t } y (t) t. Next denote by D the set of ll functions y H stisfying the following three conditions: (i) y is continuous on (, b], y(b) = 0, there exists y() := lim y(t) nd y() = 0. t + (ii) y is continuously -differentible on (, b), there exist (finite) limits y () := lim t y (t) nd y (b) := lim + t b y (t). (iii) y is -differentible on (, b] nd y H.

5 Eigenfunction Expnsions for Sturm Liouville Problem on Time Scles 97 Obviously D is liner subset dense in H. Now we define the opertor A : D H H s follows. The domin of definition of A is D nd we put (Ay)(t) = y (t), t (, b], for y D. Definition 3.1. A complex number λ is clled n eigenvlue of problem (1.1), (1.2) if there exists nonidenticlly zero function y D such tht y (t) = λy(t), t (, b). The function y is clled n eigenfunction of problem (1.1), (1.2), corresponding to the eigenvlue λ. We see tht the eigenvlue problem (1.1), (1.2) is equivlent to the eqution Theorem 3.2. We hve Ay = λy, y D, y = 0. (3.1) Ay, z = y, Az for ll y, z D, (3.2) [ Ay, y = y (t) ] 2 t for ll y D. (3.3) Proof. Using integrtion by prts formuls (2.2), (2.3) we hve for ll y,z D Ay, z = y (t)z(t) t = y (t)z(t) b + y (t)z (t) t = y (t)z(t) b +y(t)z (t) b y(t)z (t) t = y(t)z (t) t = y,az, where we hve used the boundry conditions u() = u(b) = 0 for functions u D. Simultneously we hve lso got Ay, y = y (t)y(t) b + [ y (t) ] 2 [ t = y (t) ] 2 t. The theorem is proved. Reltion (3.2) shows tht the opertor A is symmetric (self-djoint), while (3.3) shows tht it is positive: Ay, y > 0 for ll y D, y = 0.

6 98 Gusein Sh. Guseinov Therefore ll eigenvlues of the opertor A re rel nd positive nd ny two eigenfunctions corresponding to distinct eigenvlues re orthogonl. Besides, it cn esily be seen tht eigenvlues of problem (1.1), (1.2) re simple, tht is, to ech eigenvlue there corresponds single eigenfunction up to constnt fctor (eqution (1.1) cn not hve two linerly independent solutions stisfying y() = 0). Now we re going to prove the existence of eigenvlues for problem (1.1), (1.2). Note tht ker A ={y D : Ay = 0} consists only of the zero element. Indeed, if y D nd Ay = 0, then from (3.3) we hve y (t) = 0 for t [,b) nd hence y(t) = constnt on [,b]. Then using the condition y() = 0 (or y(b) = 0) we get tht y(t) 0. It follows tht the inverse opertor A 1 exists. To present its explicit form we introduce the Green function (see [2, 3, 9] nd [4, Sec.8.4]) G(t, s) = 1 b { (t )(b s) if t s, (s )(b t) if t s. (3.4) Then (A 1 u)(t) = G(t, s)u(s) s for ny u H. (3.5) The equtions (3.4) nd (3.5) imply tht A 1 is completely continuous (or compct) symmetric liner opertor in the Hilbert spce H. The eigenvlue problem (3.1) is equivlent (note tht λ = 0 is not n eigenvlue of A) to the eigenvlue problem Bu = µu, u H, u = 0, where B = A 1 nd µ = 1 λ. In other words, if λ is n eigenvlue nd y D is corresponding eigenfunction for A, then µ = λ 1 is n eigenvlue for B with the sme corresponding eigenfunction y; conversely, if µ = 0 is n eigenvlue nd u H is corresponding eigenfunction for B, then u D nd λ = µ 1 is n eigenvlue for A with the sme eigenfunction u. Note tht µ = 0 cnnot be n eigenvlue for B. In fct, if Bu = 0, then pplying to both sides A we get tht u = 0. Next we use the following well-known Hilbert Schmidt theorem (see, for exmple, [10, Sec.24.3]): For every completely continuous symmetric liner opertor B in Hilbert spce H there exists n orthonorml system {ϕ k } of eigenvectors corresponding to eigenvlues {µ k } (µ k = 0) such tht ech element f H cn be written uniquely in the form f = c k ϕ k + h, k

7 Eigenfunction Expnsions for Sturm Liouville Problem on Time Scles 99 where h ker B, tht is, Bh = 0. Moreover, Bf = k µ k c k ϕ k, nd if the system {ϕ k } is infinite, then lim µ k = 0 (k ). As corollry of the Hilbert Schmidt theorem we hve: If B is completely continuous symmetric liner opertor in Hilbert spce H nd if ker B ={0}, then the eigenvectors of B form n orthogonl bsis of H. Applying the corollry of the Hilbert Schmidt theorem to the opertor B = A 1 nd using the bove described connection between the eigenvlues nd eigenfuncions of A nd the eigenvlues nd eigenfunctions of B we obtin the following result. Theorem 3.3. For the eigenvlue problem (1.1), (1.2) there exists n orthonorml system {ϕ k } of eigenfunctions corresponding to eigenvlues {λ k }. Ech eigenvlue λ k is positive nd simple. The system {ϕ k } forms n orthonorml bsis for the Hilbert spce H. Therefore the number of the eigenvlues is equl to N = dim H. Any function f H cn be expnded in eigenfunctions ϕ k in the form f(t)= N c k ϕ k (t), (3.6) where c k re the Fourier coefficients of f defined by c k = f(t)ϕ k (t) t. (3.7) In the cse N = the sum in (3.6) becomes n infinite series nd it converges to the function f in metric of the spce H, tht is, in men squre metric: Note tht since [ b lim f(t) n [ f(t) 2 c k ϕ k (t)] t = 0. (3.8) 2 c k ϕ k (t)] t = we get from (3.8) the Prsevl equlity f 2 (t) t = f 2 (t) t ck 2, N ck 2. (3.9)

8 100 Gusein Sh. Guseinov Remrk 3.4. Above in the definition of the Hilbert spce H we required the condition y(b) = 0 for functions y : (, b] R in H in the cse b is left-scttered. This is needed to ensure tht D is dense in H. It is lso needed for vlidity of the men squre convergent expnsion (3.6) for ny function f in H, since in the cse b is left-scttered (3.6) must be held t t = b s pointwise equlity (ccording to (3.8)) nd then from ϕ k (b) = 0 we necessrily get f(b)= 0. Note lso tht the condition y(b) = 0 for H is necessry to gurntee the equlity H=D in the discrete cse T = Z. Remrk 3.5. It is esy to see tht the dimension of the spce H is finite if nd only if the time scle intervl (, b) consists of finite number of points, nd in this cse dim H is equl to the number of points in the intervl (, b). Remrk 3.6. If we denote by ϕ(t,λ) the solution of eqution (1.1) stisfying the initil conditions ϕ(,λ) = 0, ϕ (, λ) = 1, then the eigenvlues of problem (1.1), (1.2) will coincide with the zeros of the function ϕ(b,λ) (chrcteristic function of problem (1.1), (1.2)). So we hve proved existence of zeros of ϕ(b,λ) by proving existence of eigenvlues of problem (1.1), (1.2). It is possible (see [1]) to prove existence of zeros of ϕ(b,λ) directly nd to get in this wy existence of the eigenvlues. 4. Uniformly Convergent Expnsions In this section we prove the following result (we ssume tht dim H =, since in the cse dim H < the series becomes finite sum). Theorem 4.1. Let f :[,b] R be continuous function stisfying the boundry conditions f() = f(b) = 0 nd such tht it hs -derivtive f (t) everywhere on [,b), except t finite number of points t 1,t 2,...,t m, the -derivtive being continuous everywhere except t these points, t which f hs finite limits from the left nd right. Besides ssume tht f is bounded on [,b) \{t 1,t 2,...,t m }. Then the series where c k = c k ϕ k (t), (4.1) converges uniformly on [,b] to the function f. f(t)ϕ k (t) t, (4.2) Proof. We employ method pplied in the cse of the usul (T = R) Sturm Liouville problem by Steklov [11]. First for simplicity we ssume tht the function f is differentible everywhere on [,b) nd tht f is continuous nd bounded on [,b).

9 Eigenfunction Expnsions for Sturm Liouville Problem on Time Scles 101 Consider the functionl J(y) = [ y (t) ] 2 t so tht we hve J(y) 0. Substituting in the functionl J(y) y = f(t) c k ϕ k (t), where c k re defined by (4.2), we obtin ( ) { } 2 b J f c k ϕ k = f (t) c k ϕk (t) t [ = f (t) ] 2 2f (t) c k ϕk (t) + c k c l ϕk (t)ϕ l (t) t = + k,l=1 [ f (t) ] 2 t 2 c k f (t)ϕk (t) t c k c l ϕk (t)ϕ l (t) t. (4.3) k,l=1 Next, pplying integrtion by prts formul (2.2), we get f (t)ϕk (t) t = f(t)ϕ k (t) b f(t)ϕk (t) t = λ k f(t)ϕ k (t) t = λ k c k, ϕ k (t)ϕ l (t) t = ϕ k (t)ϕ l (t) b = λ l ϕ k (t)ϕ l (t) t = λ l δ kl, ϕ k (t)ϕl (t) t where δ kl is the Kronecker symbol nd where we hve used the boundry conditions f()= f(b)= 0, ϕ k () = ϕ k (b) = 0, nd the eqution ϕk (t) = λ k ϕ k (t). Therefore we hve from (4.3) ( ) [ J f c k ϕ k = f (t) ] 2 t λ k ck 2.

10 102 Gusein Sh. Guseinov Since the left-hnd side is nonnegtive we get the inequlity λ k ck 2 [ f (t) ] 2 t (4.4) nlogous to Bessel s inequlity, nd the convergence of the series on the left follows. All the terms of this series re nonnegtive, since λ k > 0. Note tht the proof of (4.4) is entirely unchnged if we ssume tht the function f stisfies only the conditions stted in the theorem. Indeed, when integrting by prts, it is sufficient to integrte over the intervls on which f is continuous nd then dd ll these integrls (the integrted terms vnish by f()= f(b)= 0 nd the fct tht f, ϕ k, nd ϕk re continuous on [,b]). We now show tht the series c k ϕ k (t) (4.5) is uniformly convergent on the intervl [,b]. Obviously from this the uniformly convergence of series (4.1) will follow. Using the integrl eqution ϕ k (t) = λ k G(t, s)ϕ k (s) s which follows from ϕ k = λ k A 1 ϕ k by (3.5), we cn rewrite (4.5) s where λ k c k g k (t), (4.6) g k (t) = G(t, s)ϕ k (s) s cn be regrded s the Fourier coefficient of G(t, s) s function of s. By using inequlity (4.4), we cn write λ k gk 2 (t) [ G s (t, s) ] 2 s, (4.7) where G s (t, s) is the delt derivtive of G(t, s) with respect to s. The function ppering under the integrl sign is bounded (see (3.4)), nd it follows from (4.7) tht λ k gk 2 (t) M,

11 Eigenfunction Expnsions for Sturm Liouville Problem on Time Scles 103 where M is constnt. Now replcing λ k by λ k λk, we pply the Cuchy Schwrz inequlity to the segment of series (4.6): m+p λ k c k g k (t) m+p λ k ck 2 m+p λ k gk 2(t) k=m k=m k=m m+p λ k ck 2 M, nd this inequlity, together with the convergence of the series with terms λ k c 2 k (see (4.4)), t once implies tht series (4.6), nd hence series (4.5) is uniformly convergent on the intervl [,b]. Denote the sum of series (4.1) by f 1 (t): f 1 (t) = k=m c k ϕ k (t). (4.8) Since the series in (4.8) is uniformly convergent on [,b], we cn multiply both sides of (4.8) by ϕ l (t) nd then -integrte it term-by-term to get f 1 (t)ϕ l (t) t = c l. Therefore the Fourier coefficients of f 1 nd f re the sme. Then the Fourier coefficients of the difference f 1 f re zero nd pplying the Prsevl equlity (3.9) to the function f 1 f we get tht f 1 f = 0, so tht the sum of series (4.1) is equl to f(t). Remrk 4.2. The proofs of Theorem 3.3 nd Theorem 4.1 cn esily be generlized to the cse of eqution [ p(t)y (t) ] + q(t)y(t) = λy(t), where p is continuously -differentible, p(t) > 0, nd q is continuous with q(t) 0. References [1] Rvi P. Agrwl, Mrtin Bohner, nd Ptrici J.Y. Wong, Sturm Liouville eigenvlue problems on time scles, Appl. Mth. Comput., 99(2-3): , [2] Dougls R. Anderson, Gusein Sh. Guseinov, nd Jon Hoffcker, Higher-order self-djoint boundry-vlue problems on time scles, J. Comput. Appl. Mth., 194(2): , [3] F. Merdivenci Atici nd Gusein Sh. Guseinov, On Green s functions nd positive solutions for boundry vlue problems on time scles, J. Comput. Appl. Mth., 141(1-2):75 99, Dynmic equtions on time scles.

12 104 Gusein Sh. Guseinov [4] Mrtin Bohner nd Alln Peterson, Dynmic equtions on time scles, Birkhäuser Boston Inc., Boston, MA, An introduction with pplictions. [5] Mrtin Bohner nd Alln Peterson, Advnces in dynmic equtions on time scles, Birkhäuser Boston Inc., Boston, MA, [6] Chun Jen Chyn, John M. Dvis, Johnny Henderson, nd Willim K.C.Yin, Eigenvlue comprisons for differentil equtions on mesure chin, Electron. J. Differentil Equtions, pges No. 35, 7 pp. (electronic), [7] Metin Gürses, Gusein Sh. Guseinov, nd Burcu Silindir, Integrble equtions on time scles, J. Mth. Phys., 46(11):113510, 22, [8] Gusein Sh. Guseinov, Integrtion on time scles, J. Mth. Anl. Appl., 285(1): , [9] Gusein Sh. Guseinov, Self-djoint boundry vlue problems on time scles nd symmetric Green s functions, Turkish J. Mth., 29(4): , [10] A.N. Kolmogorov nd S.V. Fomin, Introductory rel nlysis, Revised English edition. Trnslted from the Russin nd edited by Richrd A. Silvermn. Prentice- Hll Inc., Englewood Cliffs, N.Y., [11] V.A. Steklov, Osnovnye zdchi mtemticheskoi fiziki, Nuk, Moscow, second edition, 1983, Edited nd with prefce by V. S. Vldimirov.

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