MEAN VALUE PROBLEMS OF FLETT TYPE FOR A VOLTERRA OPERATOR

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1 Electronic Journl of Differentil Equtions, Vol. 213 (213, No. 53, pp ISSN: URL: or ftp ejde.mth.txstte.edu MEAN VALUE PROBLEMS OF FLETT TYPE FOR A VOLTERRA OPERATOR CEZAR LUPU Dedicted to the memory of Professor Ion Cucurezenu Abstrct. In this note we give generliztion of men vlue problem which cn be viewed s problem relted to Volterr opertors. This problem cn be seen s generliztion of result concerning the zeroes of Volterr opertor in the Bnch spce of continuous functions with null integrl on compct intervl. 1. Introduction Men vlue theorems hve lwys been n importnt tool in mthemticl nlysis. It is worth mentioning the pioneering contributions of Fermt, Rolle, Lgrnge, Cuchy, Drboux nd others. A vrition of Lgrnge s men vlue theorem with Rolle type condition ws given by Flett [5] in 1958 nd it ws lter extended in [17] nd generlized in [13, 14]. Theorem 1.1. Let f : [, b] R be continuous function on [, b], differentible on (, b nd f ( = f (b. Then there exists c (, b such tht f f(c f( (c =. c For instnce, Trhn [17] extended Theorem 1.1 by replcing the condition f ( = f (b with (f ( m(f (b m >, where m = f(b f( b. Moreover, Meyers [11] proved in the sme condition, f ( = f (b tht there exists c (, b such tht f f(b f(c (c =. b c Riedel nd Shoo [16] removed the boundry ssumption on the derivtives nd proved the following Theorem 1.2. Let f : [, b] R be differentible on [, b]. Then, there exists point c (, b such tht f(c f( = (c f (c 1 2 f (b f ( (c 2. b 2 Mthemtics Subject Clssifiction. 26A24, 26A36, 28A15. Key words nd phrses. Flett s men vlue theorem; Volterr opertor; integrl eqution; differentible function. c 213 Texs Stte University - Sn Mrcos. Submitted November 1, 212. Published Februry 18,

2 2 C. LUPU EJDE-213/53 Moreover, Theorems 1.1 nd 1.2 were used in [3] nd [7] in proving Hyers-Ulm stbility results for Flett s nd Shoo-Riedel s points. Another results of Flett-type ppers lso in [15], Theorem 1.3. If f : [, b] R is twice differentible function such tht f ( = f (b, then there exists c (, b such tht f(c f( = (c f (c (c 2 f (c. 2 Theorems 1.1 nd 1.3 hve been generlized by Pwlikowsk [14]. Moreover, Molnrov [13] gve new proof for the generlized Flett men vlue theorem of Pwlikowsk using only Theorem 1.1. Furthermore, Molnrov estblishes Trhn-type condition in the generl cse. For more detils, see [13]. We lso point out other contributions in this direction in [1, 2, 3, 7, 1]. In [6] it is proved tht Volterr opertor hs t lest one zero in the spce of functions hving the integrl equl to zero without ssuming differentibility of the function. The sme problem is lso discussed in [12] in the setting of C 1 positive functions with nonnegtive derivtive. More exctly, in [12] it is showed tht for f, g rel-vlued continuous functions on [, 1], the problem φ(xg(xdx = g(xdx φ(x, hs solution c (, 1 for clss of weight functions. As stted bove, the sme problem hs been studied in [6] for smller clss of weight functions. Curiously, both ides of proofs from [6] nd [12] cn be found in [1, pge 6.], In this pper, ssuming differentibility, we prove more generl result from the one provided in [6] nd [12] by employing n extension of Flett s theorem (Theorem 1.1. The min result nd some consequences re given in the section tht follows. 2. Min results The following lemm is n extension of Theorem 1.1 from the previous section. Lemm 2.1. Let u, v : [, b] R be two differentible functions on [, b] with v (x for ll x [, b] nd Then there exists c (, b such tht Proof. Define w : [, b] R by w(x = u ( v ( = u (b v (b. u(c u( v(c v( = u (c v (c. { u(x u( v(x v(, u ( v (, if x if x = Clerly, w is continuous on [, b], nd by Weierstrss theorem w ttins its bounds. If w does not ttin its bounds simultneously in nd b, then it follows tht there exists x (, b extremum point. By Fermt s theorem, we hve w (x = ; i.e., u(x u( v(x v( = u (x v (x.

3 EJDE-213/53 MEAN VALUE PROBLEMS 3 If w ttins its bounds in nd b, then we hve the following situtions: or w( w(x w(b (2.1 w(b w(x w( (2.2 for ll x [, b]. We cn ssume without loss of generlity tht (2.1 holds. Moreover, possibly replcing u by u nd v by v, we cn lso dmit tht v (x >, x [, b]. This enbles us to estblish the following inequlity: u(x u( + w(b(v(x v(, for ll x [, b]. Now, for ll x [, b] we obtin u(b u(x u(b u( w(b(v(x v( u(b u( = v(b v(x v(b v(x v(b v( = w(b. Pssing to the limit s x to b, we obtin w( = u ( v ( = u (b v (b = lim u(b u(x x b,x<b v(b v(x w(b. This implies in view of (2.1 tht w is constnt, nd therefore w =. Remrk. Since v (x for ll x, v is diffeomorphism. Let u(v 1 (x = U(x, v( = A, v(b = B. Then we hve U (x = u (v 1 (x/v (v 1 (x so the constrint reds U (A = U (B. Thus, by Flett s men vlue theorem (Theorem 1.1, U(C U(A = U (C, C A for some C (A, B. But if C = v 1 (c, U(C U(A C A = u(c u( v(c v( = U (C = u (c v (c. This remrk yields n lterntive proof of Lemm 2.1. The Volterr opertor is defined for function f(t L 2 ([, 1] (the spce of Lebesgue squre-integrble function on [, 1], nd vlue t [, 1], s V (f(x = x f(tdt. It is well-known tht V is bounded liner opertor between Hilbert spces, with Hermitin djoint V (f(x = x f(tdt. This opertor hs been studied intensively in the lst decdes becuse it is the simplest opertor tht exhibits rnge of phenomen which cn rise when one leves the norml or finite-dimensionl cses. Moreover, the Volterr opertor is well-known s qusinilpotent, but not nilpotent opertor with no eigenvlues. For continuous rel-vlued function Ψ nd φ : [, 1] R differentible with φ (x for ll x (, 1. Let V Ψ be the function mpping given by V Ψ(t = Ψ(xdx nd similrly define Let V φ Ψ(t = φ(xψ(xdx. C 1 ([, b] := {φ : [, b] R : φ C 1 ([, b]; φ (x, x [, b], φ( = }.

4 4 C. LUPU EJDE-213/53 Denote C null ([, b] the spce of continuous functions hving null integrl on the intervl [, b]. Now, we re redy to prove the min results of this note. Theorem 2.2. Let f C null ([, b] nd g C 1 ([, b], with g (x for ll x [, b]. Then there exists c (, b such tht V g f(c = g( V f(c. Proof. The conclusion sks to prove the existence of c (, b such tht f(xg(xdx = g( Let us consider the functions u, v : [, b] R given by u(t = f(xg(xdx g(t v(t = g(t,. for ll t [, b]., Now, it is esy to see tht u (t = g (t. By the Lemm 2.1, there exists c (, b such tht u(c u( v(c v( = u (c v (c, which is equivlent to f(xg(xdx g(c = g (c g(c g( g (c which finlly reduces to f(xg(xdx g(c nd the conclusion follows. = g(c + g(, Remrk. In the sme setting s Theorem 2.2 if we pply Meyers [11] men vlue theorem we obtin the existence of c (, b such tht which is equivlent to (b cg (c (b cu (c = u(b u(c = (g(c (b cg (c nd this cn be rewritten s f(xg(xdx g(c = (g(c g (c(b cv f(c = V g f(c. f(xg(xdx,, Corollry 2.3. If f C null ([, b] nd g C 1 ([, b], then there is c (, b such tht f(xg(xdx =. The bove corollry is evident from Theorem 2.2.

5 EJDE-213/53 MEAN VALUE PROBLEMS 5 Theorem 2.4. If f, g re continuous rel-vlued functions on [, 1], then there exists x (, 1 such tht V φ f(x = φ( g(xdx V φ g(x ( V f(x g(xdx V g(x Proof. Let us consider the functions u, v : [, 1] R, u(t = (φ(tv f(t V φ f(t g(xdx (φ(tv g(t V φ g(t v(t = φ(t., Clerly, these two functions stisfy the conditions from the Lemm 2.1, nd thus, there exists x (, 1 such tht which is equivlent to x which is rewritten s φ(x u(x u( v(x v( = u (x v (x, ( x = φ( x g(xdx g(xdx φ(xg(xdx x g(x, V φ f(x = φ( g(xdx V φ g(x ( V f(x g(xdx V g(x. Corollry 2.5 ([6]. If φ( =, then there exists x (, 1 such tht V φ g(x = g(xdx V φ f(x, The bove corollry follows immeditely from Theorem 2.4. Corollry 2.6 ([9]. If f, g : [, 1] R re two continuous functions, then there exists x 1 (, 1 such tht x1 xg(xdx = x1 g(xdx x. The proof of the bove corollry follows by pplying Corollry 2.5 wiht φ(x = x.

6 6 C. LUPU EJDE-213/53 3. Discussion nd some exmples In the proof of Theorem 2.2 we considered the uxiliry function ϕ : [, 1] R, ϕ(t = φ(x φ(t. If we pply Theorem 1.1 for f C null ([, 1], then there exists c (, 1 such tht ϕ(c ϕ( = cϕ (c which is equivlent to φ(x = (φ(c cφ (c. One cn remrk tht this result is completely different from Theorem 2.2. On the other hnd, if we consider f C([, 1] such tht φ(x = φ(1, then by Rolle s theorem there exists c (, 1 such tht ϕ ( c =, c =. Now, we present some exmples tht follow s consequences from Theorem 2.2 on intervl [, 1]. Exmple 3.1. If we replce functions f, g by their squres in Theorem 2.4, we hve the equlity x x φ(xf 2 (xdx g 2 (xdx φ(xg 2 (xdx g 2 (xdx ( x = φ( f 2 (xdx g 2 (xdx x g 2 (xdx f 2 (xdx. This equlity ctully sys tht given ny two continuous functions f, g we hve f L 2 φ (,x g L2 (,1 g L 2 φ (,x f L2 (,1 = φ(( f L2 (,x g L2 (,1 g L2 (,x f L2 (,1, where the quntities re the norms in their respective spces of (weighted squre integrble functions. Moreover, if φ( = we recover [12, Exmple 3]. Exmple 3.2. For i j consider f(x = P i (xp j (x, where P i, P j re orthogonl functions on [, 1]; i.e., P i (xp j (xdx =. Now, by Theorem 2.2, there is point c ij (, 1 such tht ij P i (xp j (xφ(xdx = φ( ij P i (xp j (xdx. If φ( = we obtin ij P i (xp j (xφ(xdx = which is precisely [12, Exmple 4]. Acknoledgments. The uthor is indebted to the deprtment of mthemtics of the Politehnic University of Buchrest for hospitlity during My-June, 212, when this work ws done.

7 EJDE-213/53 MEAN VALUE PROBLEMS 7 References [1] U. Abel, M. Ivn, T. Riedel; The men vlue of Flett nd divided differences, J. Mth. Anl. Appl. 295(24, 1 9. [2] D. Ckmk, A. Tiryki; Men vlue theorems for holomorphic functions, Electronic J. Diff. Eqn. 212(212, No. 34, 1 6. [3] M. Ds, T. Riedel, P. K. Shoo; Hyers-Ulm stbility of Flett s points, Appl. Mth. Lett. 16(23, [4] J. B. Diz, R. Vyborny; On some men vlue theorems for differentil clculus, Bull. Aus. Mth.Soc. 5(1971, [5] T. M. Flett; A men vlue problem, Mth. Gz. 42(1958, [6] R. Gologn, C. Lupu; An olympid problem: Zeroes of functions in the imge of Volterr opertor, Gz. Mt. (A-series, 27(29, [7] W. Lee, S. Xu, F. Ye; Hyers-Ulm stbility of Shoo-Riedel s point, Appl. Mth. Lett. 22(29, [8] I. Jedrzejewsk, B. Szkopinsk; On generliztions of Flett s Theorem, Rel Anl. Exchnge, 3(24, [9] C. Lupu, T. Lupu, Problem 1129, Amer. Mth. Mon., 114(27, 359. [1] C. Lupu, T. Lupu; Men vlue theorems for some liner integrl opertors, Electron. J. Diff. Eqn., 29(29, no. 117, [11] R. E. Meyers; Some elementry results relted to the men vlue theorem, The two-yer college Mthemtics Journl 8(1977, [12] A. B. Mingrelli, J. M. Pcheco, A. Plz; A men vlue property for pir of integrls, Act Mth. Univ. Comenine, 87(29, [13] J. Molnrov; On generlized Flett s men vlue theorem, preprint t cche/rxiv/pdf/117/ v1.pdf [14] I. Pwlikowsk; An extension of theorem of Flett, Demonstrtio Mth., 32 (1999, [15] T.L. Rădulescu, V.D. Rădulescu, T. Andreescu; Problems in Rel Anlysis: dvnced clculus on the rel xis, Springer Verlg, 29. [16] P.K. Shoo, T. Riedel; Men Vlue Theorems nd Functionl Equtions, World Scientific, River Edge, NJ, [17] D. H. Trhn; A new type of men vlue theorem, Mth. Mg., 39(1966, [18] R. Witul, E. Hetmniok, D. Slot; A stronger version of the second men vlue theorem for integrls, Computer Mth. Appl., in press t Cezr Lupu University of Pittsburgh, Deprtment of Mthemtics, Pittsburgh, PA 1526, USA. University of Criov, Deprtment of Mthemtics, Str. Alexndru Ion Cuz 13, RO 2585, Criov, Romni E-mil ddress: cel47@pitt.edu, lupucezr@gmil.com