Asymptotic behavior of intermediate points in certain mean value theorems. III

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1 Stud. Univ. Bbeş-Bolyi Mth. 59(2014), No. 3, Asymptotic behvior of intermedite points in certin men vlue theorems. III Tiberiu Trif Abstrct. The pper is devoted to the study of the symptotic behvior of intermedite points in certin men vlue theorems of integrl nd differentil frctionl clculus. Mthemtics Subject Clssifiction (2010): 26A33, 26A24, 41A60, 41A80. Keywords: men vlue theorems of frctionl clculus, Cputo derivtive, Riemnn-Liouville integrl, symptotic pproximtion. 1. Introduction Let, b R such tht < b, let f : [, b] R be continuous function, nd let g L 1 [, b] such tht g does not chnge its sign in [, b]. Then, ccording to the first men vlue theorem of integrl clculus (see, for instnce, [11, Theorem 85.6], or [6]), for every x (, b] there exists ξ x (, x) such tht f(t)g(t)dt = f(ξ x ) g(t)dt. It ws proved in [15, Theorem 2.2] tht ξ x k + 1 x + x = n n + k + 1 if, in ddition, the functions f nd g stisfy the following conditions: (1.1) (i) there exists positive integer n such tht f is n times differentible t, with f (j) () = 0 for 1 j n 1 nd f (n) () 0; (ii) g C[, b] nd there exists nonnegtive integer k such tht g is k times differentible t with g (j) () = 0 for 0 j k 1 nd g (k) () 0. Regrding (ii) we notice tht the continuity of g is utomticlly ssured if k 2, t lest on smll intervl [, + h] [, b] (this clerly suffices when deling with the it (1.1)).

2 280 Tiberiu Trif Further, let n be positive integer, nd let f : [, b] R be function whose derivtive f (n) exists on [, b]. Then, ccording to the Lgrnge-Tylor men vlue theorem, for every x (, b] there exists some ξ x (, x) such tht f(x) T n 1 (f; )(x) = f (n) (ξ x ) (x ) n, n! where T m (h; ) denotes the mth Tylor polynomil ssocited with h nd T m (h; )(x) := h() + h ()(x ) + + h(m) () (x ) m, m! provided tht h is m times differentible t. It ws proved by A. G. Azpeiti [4] tht ( ) 1/p ξ x n + p x + x = (1.2) p if, in ddition, f stisfies the following conditions: (i) there exists positive integer p such tht f C n+p [, b]; (ii) f (n+j) () = 0 for 1 j < p ; (iii) f (n+p) () 0. This result ws generlized by U. Abel [1], who derived for ξ x complete symptotic expnsion of the form c k ξ x = + k! (x )k (x ). k=1 Azpeiti s result ws generlized lso by T. Trif [14], who obtined the symptotic behvior of the intermedite point in the Cuchy-Tylor men vlue theorem. For other results concerning the symptotic behvior of the intermedite points in certin men vlue theorems the reder is referred to [2, 7, 8, 9, 19]. The purpose of our pper is to estblish symptotic formuls, tht re similr to (1.1) nd (1.2), but in the frmework of frctionl clculus. 2. Frctionl men vlue theorems of integrl clculus K. Diethelm [6, Theorem 2.1] generlized the first men vlue theorem of integrl clculus to the frmework of frctionl clculus. Recll tht given α > 0, the Riemnn- Liouville frctionl primitive of order α of function f : [, b] R is defined by (see [12] or [5]) J α f(x) := 1 (x t) α 1 f(t)dt, Γ(α) provided the right side is pointwise defined on [, b]. Theorem 2.1. ([6, Theorem 2.1]) Let α > 0, let f : [, b] R be continuous function, nd let g L 1 [, b] be function which does not chnge its sign on [, b]. Then for lmost every x (, b] there exists some ξ x (, x) such tht J α (fg)(x) = f(ξ x )J α g(x). (2.1)

3 Asymptotic behvior of intermedite points 281 Moreover, if α 1 or g C[, b], then the existence of ξ x is ssured for every x (, b]. In the specil cse when g(x) 1, the bove men vlue theorem tkes the following form. Corollry 2.2. ([6, Corollry 2.2]) If α > 0 nd f : [, b] R is continuous function, then for every x (, b] there exists some ξ x (, x) such tht J α f(x) = f(ξ x ) (x )α Γ(α + 1). (2.2) We point out tht there is misprint in the sttement of [6, Corollry 2.2], where Γ(α) ppers insted of Γ(α + 1). In wht follows we intend to prove frctionl version of the second men vlue theorem of integrl clculus. For reder s convenience we recll first the second men vlue theorem for Lebesgue integrls, which is usully stted for Riemnn integrls (see, for instnce, [3, Theorem ] or [18, Theorem 1]). Theorem 2.3. Let f : [, b] [0, ) be nondecresing function, nd let g L 1 [, b]. Then for every x (, b] there exists some ξ x [, x] such tht f(t)g(t)dt = f(x 0) ξ x g(t)dt. The frctionl version of Theorem 2.3 cn be formulted s follows. Theorem 2.4. Let α > 0, let f : [, b] [0, ) be nondecresing function, nd let g L 1 [, b]. Then for lmost every x (, b] there exists some ξ x [, x] such tht J α (fg)(x) = f(x 0)J α ξ x g(x). (2.3) Moreover, if α 1 or g C[, b], then the existence of ξ x is ssured for every x (, b]. Proof. Let x (, b]. Under the ssumptions of the theorem we hve J α (fg)(x) = 1 Γ(α) (x t) α 1 f(t)g(t)dt = f(t)h(t)dt, where h : (, x) R is the function defined by h(t) := (x t) α 1 g(t)/γ(α). If α 1 or g C[, b], then h L 1 [, x]. By Theorem 2.3 it follows tht there exists ξ x [, x] such tht J α (fg)(x) = f(t)h(t)dt = f(x 0) ξ x h(t)dt = f(x 0)J α ξ x g(x). If 0 < α < 1 nd g is supposed only Lebesgue integrble, then the bove rgument still works, but the integrbility of h holds only for lmost ll x (, b] (see [17, Theorem 4.2 (d)]).

4 282 Tiberiu Trif 3. Asymptotic behvior of intermedite points in frctionl men vlue theorems of integrl clculus The purpose of this section is to investigte the symptotic behvior of the point ξ x in (2.1) nd (2.2) s the intervl [, x] shrinks to zero. More precisely, we prove tht ξ x under certin dditionl ssumptions on f nd g, the it exists nd x + x we find its vlue. In the proof of the min result of this section we need the following Lemm 3.1. Let α > 0, let p be nonnegtive integer, nd let ω : [, b] R be continuous function such tht ω(x) 0 s x +. Then (x t) α 1 (t ) p ω(t)dt = o ( (x ) p+α) (x +). Proof. Indeed, for every x (, b] we hve (x t) α 1 (t ) p ω(t)dt = Γ(α)J α (ωg)(x), where g : [, b] [0, ) is defined by g(t) := (t ) p. According to Theorem 2.1 there exists ξ x (, x) such tht whence J α (ωg)(x) = ω(ξ x )J α g(x) = ω(ξ x) (x t) α 1 (t ) p dt Γ(α) B(p + 1, α) = ω(ξ x ) (x ) p+α, Γ(α) (x t) α 1 (t ) p ω(t)dt = ω(ξ x )B(p + 1, α)(x ) p+α. Since ω(x) 0 s x +, we obtin the conclusion. Theorem 3.2. Let α be positive rel number nd let f, g : [, b] R be functions stisfying the following conditions: (i) f C[, b] nd there is positive integer n such tht f is n times differentible t with f (j) () = 0 for 1 j n 1 nd f (n) () 0; (ii) g C[, b], g does not chnge its sign in some intervl [, + h] [, b], nd there is nonnegtive integer k such tht g is k times differentible t with g (j) () = 0 for 0 j k 1 nd g (k) () 0. Then the point ξ x in (2.1) stisfies ξ x x + x = (k + 1)(k + 2) (k + n) n (α + k + 1)(α + k + 2) (α + k + n). Proof. Without loosing the generlity we my ssume tht f() = 0. Indeed, otherwise we replce f by the function t [, b] f(t) f(). Note tht if ξ x stisfies (2.1), then ξ x stisfies lso ( (f ) ) f() g (x) = ( f(ξ x ) f() ) J α g(x). J α

5 Asymptotic behvior of intermedite points 283 We notice lso tht (2.1) is equivlent to (x t) α 1 f(t)g(t)dt = f(ξ x ) By the Tylor expnsions of f nd g we hve f(t) = f (n) () n! g(t) = g(k) () k! (t ) n + ω(t)(t ) n, (t ) k + ε(t)(t ) k, (x t) α 1 g(t)dt. (3.1) where ω nd ε re continuous functions on [, b] stisfying ω(t) 0 nd ε(t) 0 s t +. Therefore we hve (x t) α 1 f(t)g(t) = f (n) ()g (k) () (x t) α 1 (t ) n+k n! k! +(x t) α 1 (t ) n+k γ(t), where γ is continuous on [, b] nd γ(t) 0 s t +. By pplying Lemm 3.1 we deduce tht (x t) α 1 f(t)g(t)dt (3.2) = f (n) ()g (k) () n! k! s x +. On the other hnd, since B(α, n + k + 1) (x ) n+k+α + o ( (x ) n+k+α) (x t) α 1 g(t) = g(k) () (x t) α 1 (t ) k + (x t) α 1 (t ) k ε(t), k! by Lemm 3.1 we get (x t) α 1 g(t)dt = g(k) () k! s x +. Tking into ccount tht f(ξ x ) = f (n) () n! nd tht 0 < ξ x < x, we obtin f(ξ x ) B(α, k + 1)(x ) k+α + o ( (x ) k+α) (ξ x ) n + ω(ξ x )(ξ x ) n (x t) α 1 g(t)dt (3.3) = f (n) ()g (k) () B(α, k + 1)(ξ x ) n (x ) k+α n! k! +o ( (x ) n+k+α)

6 284 Tiberiu Trif s x +. By (3.1), (3.2) nd (3.3) we conclude tht f (n) ()g (k) () n! k! = f (n) ()g (k) () n! k! B(α, k + 1)(ξ x ) n (x ) k+α B(α, n + k + 1) (x ) n+k+α + o ( (x ) n+k+α) s x +. Multiplying both sides by n! k!(x ) n k α / ( f (n) ()g (k) () ) we get ( ξx x ) n = = s x +, whence the conclusion. B(α, n + k + 1) + o(1) B(α, k + 1) (k + 1)(k + 2) (k + n) (α + k + 1)(α + k + 2) (α + k + n) + o(1) Corollry 3.3. Let α > 0, nd let f : [, b] R be function stisfying the condition (i) in Theorem 3.2. Then the point ξ x in (2.2) stisfies ξ x x + x = n! n (α + 1)(α + 2) (α + n). In the specil cse when α = 1, then Theorem 3.2 nd Corollry 3.3 coincide with erlier results obtined by T. Trif [15, Theorem 2.2] nd B. Zhng [20, Theorem 4], respectively. Unfortuntely, we were not ble to prove result similr to those stted in Theorem 3.2 nd Corollry 3.3, but concerning the symptotic behvior of the point ξ x in formul (2.3). 4. Frctionl men vlue theorems of differentil clculus K. Diethelm [6] nd P. Guo, C. P. Li, nd G. R. Chen [10] extended recently lso the clssicl Lgrnge nd Lgrnge-Tylor men vlue theorems to the frmework of frctionl clculus. Let α > 0, nd let f : [, b] R be given function. The Cputo frctionl derivtive of order α of f is defined by ( ) D f α := D α f T α 1 (f; ), where denotes the ceiling function tht rounds up to the nerest integer, while D α is the Riemnn-Liouville differentil opertor, defined by D α f := D α J α α f. In the bove formul D m denotes the clssicl differentil opertor of order m, nd J 0 f := f.

7 Asymptotic behvior of intermedite points 285 Theorem 4.1. ([6, Theorem 2.3], [10, Theorem 3]) Let α > 0, nd let f C α 1 [, b] be function such tht D α f C[, b]. Then for every x (, b] there exists some ξ x (, x) such tht f(x) T α 1 (f; )(x) 1 (x ) α = Γ(α + 1) Dα f(ξ x ). (4.1) In the specil cse when 0 < α 1, then T α 1 (f; )(x) = f(), nd Theorem 4.1 tkes the following form. Corollry 4.2. ([6, Corollry 2.4]) Let 0 < α 1, nd let f C[, b] be function such tht D f α C[, b]. Then for every x (, b] there exists some ξ x (, x) such tht f(x) f() (x ) α = 1 Γ(α + 1) Dα f(ξ x ). (4.2) Theorem 4.3. ([10, Theorem 4]) Let α > 0, nd let f, g C α 1 [, b] be functions such tht D f, α D g α C[, b]. Then for every x (, b] there exists some ξ x (, x) such tht ( ) ( ) D f(ξ α x ) g(x) T α 1 (g; )(x) = D g(ξ α x ) f(x) T α 1 (f; )(x). (4.3) Corollry 4.4. ([10, Corollry 3.6]) Let 0 < α 1, nd let f, g C[, b] be functions such tht D α f, D α g C[, b]. Then for every x (, b] there exists some ξ x (, x) such tht D α f(ξ x ) ( g(x) g() ) = D α g(ξ x ) ( f(x) f() ). (4.4) Remrk 4.5. Let α > 0, nd let f C α 1 [, b] such tht D f α C[, b]. Further, let g : [, b] R be the function defined by g(t) := (t ) α. If n := α, then n 1 < α n, nd T α 1 (g; )(x) = T n 1 (g; )(x) = 0. On the other hnd, for every y (, b) one hs D g(y) α = dn dy n J n α = Γ(α + 1). ( g T α 1 (g; ) )(y) = dn dy n J n α Therefore, in this cse (4.3) reduces to (4.1), while (4.4) reduces to (4.2). In other words, Theorem 4.3 coincides with Theorem 4.1, while Corollry 4.4 coincides with Corollry 4.2 in the specil cse when g(t) := (t ) α. It should be mentioned tht similr frctionl men vlue theorems, but involving the Riemnn-Liouville frctionl derivtive insted of the Cputo frctionl derivtive hve been obtined by other uthors (see, for instnce, [10], [13], [16]). g(y) 5. Asymptotic behvior of intermedite points in frctionl men vlue theorems of differentil clculus Theorem 5.1. Let α > 0 be non-integer number, let n := α, nd let f : [, b] R be function stisfying the following conditions: (i) there exists nonnegtive integer p such tht f C n+p [, b];

8 286 Tiberiu Trif (ii) f (n+j) () = 0 for 0 j < p ; (iii) f (n+p) () 0. Then the point ξ x in (4.1) stisfies ξ x x + x = ( (n + p α)b(α + 1, n + p α) ) 1 n+p α. Proof. We note first tht since f C n+p [, b], the derivtive f (n 1) must be bsolutely continuous on [, b]. By [5, Theorem 3.1] it follows tht Due to (ii), by the Tylor expnsion of f we hve f(x) T n 1 (f; )(x) = f (n+p) () (n + p)! D α f = J n α D n f. (5.1) (x ) n+p + ω(x)(x ) n+p, (5.2) where ω is continuous on [, b] nd stisfies ω(x) 0 s x +. On the other hnd, by the Tylor expnsion of f (n) we hve f (n) (t) = f (n+p) () p! (t ) p + ε(t)(t ) p, (5.3) where ε is continuous on [, b] nd stisfies ε(t) 0 s t +. Tking into ccount (5.1), equlity (4.1) cn be rewritten s f(x) T n 1 (f; )(x) = By (5.3) nd Lemm 3.1 we find tht (x ) α Γ(α + 1)Γ(n α) ξx (ξ x t) n α 1 f (n) (t)dt. (5.4) whence ξx (ξ x t) n α 1 f (n) (t)dt = f (n+p) () p! = + ξx ξx (ξ x t) n α 1 (t ) p dt (ξ x t) n α 1 (t ) p ε(t)dt Γ(n α) Γ(n + p + 1 α) f (n+p) ()(ξ x ) n+p α + o ( (ξ x ) n+p α), (x ) α Γ(α + 1)Γ(n α) ξx = f (n+p) ()(x ) α (ξ x ) n+p α Γ(α + 1)Γ(n + p + 1 α) (ξ x t) n α 1 f (n) (t)dt (5.5) + o ( (x ) n+p)

9 Asymptotic behvior of intermedite points 287 s x +, becuse 0 < ξ x < x. Tking now into ccount (5.2), (5.4), nd (5.5), we find tht f (n+p) ()(x ) n+p Γ(n + p + 1) s x +. Multiplying both sides by = f (n+p) ()(x ) α (ξ x ) n+p α Γ(α + 1)Γ(n + p + 1 α) +o ( (x ) n+p) Γ(α + 1)Γ(n + p + 1 α)(x ) n p /f (n+p) () we get (ξx ) n+p α Γ(α + 1)Γ(n + p + 1 α) = + o(1) x Γ(n + p + 1) = (n + p α)b(α + 1, n + p α) + o(1) (x +), whence the conclusion. Corollry 5.2. Let 0 < α < 1, nd let f : [, b] R be function stisfying the following conditions: (i) there exists positive integer p such tht f C p [, b]; (ii) f (j) () = 0 for 1 j < p ; (iii) f (p) () 0. Then the point ξ x in (4.2) stisfies References ξ x x + x = ( (p α)b(α + 1, p α) ) 1 p α. [1] Abel, U., On the Lgrnge reminder of the Tylor formul, Amer. Mth. Monthly, 110(2003), [2] Abel, U., Ivn, M., The differentil men vlue of divided differences, J. Mth. Anl. Appl., 325(2007), [3] Asplund, E., Bungrt, L., A First Course in Integrtion, Holt, Rinehrt nd Winston, [4] Azpeiti, A.G., On the Lgrnge reminder of the Tylor formul, Amer. Mth. Monthly, 89(1982), [5] Diethelm, K., The Anlysis of Frctionl Differentil Equtions, Springer, [6] Diethelm, K., The men vlue theorems nd Ngumo-type uniqueness theorem for Cputo s frctionl clculus, Frct. Clc. Appl. Anl., 15(2012), [7] Duc, D.I., Properties of the intermedite point from the Tylor s theorem, Mth. Inequl. Appl., 12(2009), [8] Duc, D.I., Pop, O., On the intermedite point in Cuchy s men-vlue theorem, Mth. Inequl. Appl., 9(2006), [9] Duc, D.I., Pop, O. T., Concerning the intermedite point in the men vlue theorem, Mth. Inequl. Appl., 12(2009),

10 288 Tiberiu Trif [10] Guo, P., Li, C.P., Chen, G.R., On the frctionl men-vlue theorem, Internt. J. Bifurction Chos, 22(2012), no. 5, (6 pp.). [11] Heuser, H., Lehrbuch der Anlysis, Teil 1, 10th ed., Teubner, [12] Miller, K.S., Ross, B., An Introduction to the Frctionl Clculus nd Frctionl Differentil Equtions, John Wiley, [13] Pečrić, J. E., Perić, I., Srivstv, H.M., A fmily of the Cuchy type men-vlue theorems, J. Mth. Anl. Appl., 306(2005), [14] Trif, T., Asymptotic behvior of intermedite points in certin men vlue theorems, J. Mth. Inequl., 2(2008), [15] Trif, T., Asymptotic behvior of intermedite points in certin men vlue theorems. II, Studi Univ. Bbeş-Bolyi, Ser. Mth., 55(2010), no. 3, [16] Trujillo, J.J., Rivero, M., Bonill, B., On Riemnn-Liouville generlized Tylor s formul, J. Mth. Anl. Appl., 231(1999), [17] Wilson, J.H., Lebesgue Integrtion, Holt, Rinehrt nd Winston, [18] Witul, R., Hetmniok, E., S lot, D., A stronger version of the second men vlue theorem for integrls, Comput. Mth. Appl., 64(2012), [19] Xu, A., Cui, F., Hu, Z., Asymptotic behvior of intermedite points in the differentil men vlue theorem of divided differences with repetitions, J. Mth. Anl. Appl., 365(2010), [20] Zhng, B., A note on the men vlue theorem for integrls, Amer. Mth. Monthly, 104(1997), Tiberiu Trif Bbeş-Bolyi University Fculty of Mthemtics nd Computer Sciences 1, Kogălnicenu Street, Cluj-Npoc, Romni e-mil: ttrif@mth.ubbcluj.ro