Mittag-Leffler-Hyers-Ulam stability of Hadamard type fractional integral equations

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1 INTERNATIONA BAKAN JOURNA OF MATHEMATICS IBJM (2018), VO. 1, NO. 1, Mittg-effler-Hyers-Ulm stbility of Hdmrd tye frctionl integrl equtions Nsrin Eghbli 1,, Vid Klvndi 2 Dertment of Mthemtics, Fculty of Mthemticl Sciences, University of Mohghegh Ardbili, , Ardbil, Irn 1 Dertment of Mthemtics, Fculty of Mthemticl Sciences, University of Mohghegh Ardbili, , Ardbil, Irn 2 Abstrct. In this er, we roose n roch to the Mittg-effler-Hyers-Ulm stbility of Hdmrd eqution. We rove tht the Hdmrd eqution is Mittg-effler-Hyers-Ulm stble on comct intervl Mthemtics Subject Clssifictions: 26A33, 34D10, 45N05 Key Words nd Phrses: Mittg-effler-Hyers-Ulm stbility; Hdmrd eqution; Fixed oint method 1. Introduction The frctionl differentil equtions re useful tools in the modelling of mny hysicl henomen nd rocesses in economics, chemistry, erodynmics, etc. (for more detils see [8]-[14] nd [17]). It lso serves s n excellent tool for the descrition of hereditry roerties of vrious mterils nd rocesses. For more detils on the frctionl clculus theory, one cn see the monogrhs of Kilbs et l. [8], Miller nd Ross [10], Podlubny [14]. The stbility of functionl equtions ws originlly rised by Ulm in 1940 in tlk given t Wisconsin University. The roblem osed by Ulm ws the following: Under wht conditions does there exist n dditive ming ner n roximtely dditive ming? (for more detils see [18]).The first nswer to the question of Ulm ws given by Hyers in 1941 in the cse of Bnch sces in [4]: et X 1, X 2 be two Bnch sces nd ε > 0. Then for every ming f : X 1 X 2 stisfying f(x + y) f(x) f(y) ε for ll x, y there exists unique dditive ming g : X 1 X 2 with the roerty f(x) g(x) ε, forll x X 1. Corresonding uthor. Emil ddress: nsrineghbli@gmil.com;eghbli@um.c.ir (Corresonding uthor) htt:// 34 c 2018 IBJM All rights reserved.

2 Nsrin Eghbli, Vid Klvndi 35 Therefter, this tye of stbility is clled the Hyers-Ulm stbility. In 1978, Rssis [15] rovided remrkble generliztion of the Hyers-Ulm stbility of ming by considering vribles. The concet of stbility for functionl eqution rises when we relce the functionl eqution by n inequlity which cts s erturbtion of the initil eqution. Alsin nd Ger were the first uthors who investigted the Hyers-Ulm stbility of differentil eqution [1]. Recently some uthors ([5], [6], [7], [16], [19], [22], [23], [24], [25], [26] nd [27]) extended the Ulm stbility roblem from n integer-order differentil eqution to frctionl-order differentil eqution. In 2013, J. Wng et l. [20] roved the Ulm stbility for Hdmrd differentil eqution. In this er, with lying the method used in [2] nd [3], we roose Mittg-effler-Hyers-Ulm stbility for the Hdmrd differentil eqution y(x) = Γ(α j + 1) (ln x )α j + 1 (ln x t ) f(t, y(t)) dt t, (1.1) where α (n 1, n], n = 1, 2,..., Γ(.) is the Gmm function,, b nd re fixed rel numbers such tht 0 < x b < + nd f : [, b] R R is continuous function. 2. Preliminries In this section, we introduce nottions, definitions, nd reliminry fcts which re used throughout this er. Definition 2.1. Given n intervl [, b] of R, then the frctionl order integrl of function h 1 ([, b], R) of order γ R + is defined by I γ h(t) = 1 t (t s) γ 1 h(s)ds, + Γ(γ) where Γ(.) is the Gmm function. In the sequel, we will use Bnch s fixed oint theorem in frmework of generlized comlete metric sce. For nonemty set X, we introduce the definition of the generlized metric on X. Definition 2.2. For function h given on the intervl [, b], the αth Riemnn-iouville frctionl order derivtive of h, is defined by (D α 1 h)(t) = + Γ(n α) ( d dt )n t (t s)(n ) h(s)ds, where n = [α] + 1 nd [α] denotes the integer rt of α. Definition 2.3. For function h given on the intervl [, b], the Cuto frctionl order derivtive of h, is defined by ( c D α 1 t h(t) = + Γ(n α) (t s)n h (n) (s)ds, where n = [α] + 1. Definition 2.4. A function d : X X [0, + ] is clled generlized metric on X if nd only if it stisfies the following three roerties: (1) d(x, y) = 0 if nd only if x = y; (2) d(x, y) = d(y, x) for ll x, y X; (3) d(x, z) d(x, y) + d(y, z) for ll x, y, z X.

3 Nsrin Eghbli, Vid Klvndi 36 The bove concet differs from the usul concet of comlete metric sce by the fct tht not every two oints in X hve necessrily finite distnce. One might cll such sce generlized comlete metric sce. We now introduce one of the fundmentl results of the Bnch s fixed oint theorem in generlized comlete metric sce. Theorem 2.5. et (X, d) be generlized comlete metric sce. Assume tht Λ : X X is strictly contrctive oertor with the ischitz constnt < 1. If there exists non negtive integer k such tht d(λ k+1 x, Λ k x) < for some x X, then the following roerties re true: () The sequence Λ n x convergences to fixed oint x of Λ; (b) x is the unique fixed oint of Λ in X = {y X d(λ k x, y) < }; (c) If y X, then d(y, x ) 1 d(λy, y) Mittg-effler-Hyers-Ulm stbility for the Hdmrd tye of frctionl integrl eqution In this section, we will study Mittg-effler-Hyers-Ulm-Rssis nd Mittg-effler-Hyers-Ulm stbility of the eqution (1.1) on comct intervl [, b]. et 0 < < b, 0 < < 1, n 1 < n, < nd M = 1 ( 1 α )1 (ln b )α. We introduce the following ssumtions: [H 1 ] : f : [, b] R R is continuous function nd for ny t [, b] nd y, z R, f(t, y) f(t, z) t y z. (3.1) [H 2 ] : There exists continuous function y : [, b] R stisfies y(x) Γ(α j + 1) (ln x )α j 1 for ll x [, b], nd ϕ : [, b] (0, + ) stisfies ( (ln x t ) f(t, y(t)) dt t ϕ(x)e q(x q ), (3.2) (ϕ(t)) 1 dt) Kϕ(x), (3.3) [H 3 ] : 0 < KM < 1. Definition 3.1. If for ech function y stisfying y(x) n Γ(α j+1) (ln x )α j 1 (ln x t ) f(t, y(t)) dt t ϕ(x)e q(x q ), where ϕ is nonnegtive function, there is solution y 0 of eqution (1.1) nd constnt c > 0 indeendent of y nd y 0 such tht y(x) y 0 (x) cϕ(x)e q (x q ), x [, b],

4 Nsrin Eghbli, Vid Klvndi 37 then the eqution (1.1) is clled Mittg-effler-Hyers-Ulm-Rssis stble. In the cse where ϕ tkes the form of constnt function, the eqution (1.1) is clled Mittg- effler-hyers-ulm stble. Theorem 3.2. Assume tht [H 1 ], [H 2 ] nd [H 3 ] re stisfied. Then there exists unique continuous function y 0 : [, b] R such tht y 0 (x) = Γ(α j + 1) (ln x )α j + 1 (ln x t ) f(t, y 0 (t)) dt t (3.4) nd for ll x [, b]. y(x) y 0 (x) cϕ(x)e q (x q ) (3.5) Proof. First, we consider of continuous functions nd introduce generlized metric on X s follows: X = {g : [, b] R g is continuous} (3.6) d(f, g) = inf{c [0, ] f(x) g(x) Cϕ(x) for ll x [, b]}. (3.7) It is obvious tht (X, d) is generlized comlete metric sce. et us define n oertor T : X X by (T y)(x) = Γ(α j + 1) (ln x )α j + 1 (ln x t ) f(t, y(t)) dt t, (3.8) for ny y X nd x [, b]. Clerly, T is well-defined oertor. we now ssert tht T is strictly contrctive on X. For f, g X, let C fg [0, ] be n rbitrry constnt with d(f, g) C fg, nd let us ssume tht f(x) g(x) C fg ϕ(x) (3.9) for ll x [, b]. From the definition of T in (3.8), we obtin (T g)(x) (T h)(x) = 1 1 t [f(t, g(t)) f(t, h(t))]dt 1 t t g(t) h(t) dt x C gh t 1 ϕ(t)dt x C gh (t 1 ) ϕ(t)dt C gh x ( [t 1 ] ϕ(t)dt 1 C gh [t ] 1 dt) 1 ( (ϕ(t)) 1 dt) KC gh ϕ(x) 1 t 1 dt) 1 ( KC gh ϕ(x) ( 1 d) 1 KC gh ϕ(x) 1 ] 1 [ 1 α KC gh ϕ(x) ( 1 α )1 (ln b )α This yields tht (T g)(x) (T h)(x) KMC gh ϕ(x),

5 Nsrin Eghbli, Vid Klvndi 38 for ech x [, b], tht is, d(t g, T h) KMC gh. Thus, it follows tht d(t g, T h) KMd(g, h) for ll g, h X. Note tht 0 < KM < 1. So the strictly continuous roerty is verified. et us tke g 0 X. From the continuous roerty of g 0 X nd T g 0 X there exists constnt 0 < C 1 < with (T g 0 )(x) g 0 (x) = n Γ(α j+1) (ln x )α j + 1 (ln x t ) f(t, g 0 (t)) dt t g 0(x) C 1 ϕ(x), for ll x [, b] since f nd g 0 re bounded on [, b], thus (3.7) imlies tht d(t g 0, g 0 ) <. Therefore, Theorem 2.1 () imlies tht there exists continuous function y 0 : [, b] R such tht T n g 0 y 0 in (X, d) s n nd such tht y 0 = T y 0, tht is y 0 stisfies eqution (3.4) for ny x [, b]. If g X, then g 0 nd g re continuous functions defined on comct intervl [, b]. Hence, there exists constnt C g > 0 with g 0 (x) g(x) C g ϕ(x) for ll x [, b]. This imlies tht d(g 0, g) < for every g X or equivlently {g X d(g 0, g) < } = X. Therefore, ccording to Theorem 2.1 (b), y 0 is unique continuous function with the roerty (3.4). Furthermore, it follows from (3.2) tht d((t y)(x), y(x)) ϕ(x)e q (x q ) for ll x [, b]. At lst, 1 d(y, y 0 ) 1 KM d(t y, y) 1 1 KM E q(x q )ϕ(x). which mens tht the inequlity (3.5) holds true for ll x [, b]. Now, we resent Mittg-effler-Hyers-Ulm stbility of the eqution (1.1). et 0 < < b, n 1 < α n. We need the following ssumtions: [H 1] : f : [, b] R R is continuous function nd for ny t [, b] nd y, z R, f(t, y) f(t, z) t y z. (3.10) [H 2] : There exists continuous function y : [, b] R stisfies y(x) for ll x [, b]. Γ(α j + 1) (ln x )α j + 1 [H 3] : M = ( 1 α )1 (ln b )α ( k+ ) tht 0 < M < 1 (ln x t ) f(t, y(t)) dt t εe q(x q ) (3.11) Theorem 3.3. Assume tht [H 1], [H 2] nd [H 3] re stisfied. Then, there exists unique continuous function y 0 : [, b] R such tht y 0 (x) = Γ(α j + 1) (ln x )α j + 1 (ln x t ) f(t, y 0 (t)) dt t (3.12) nd for ll x [, b]. 1 y(x) y 0 (x) ε 1 M E q(x q ). (3.13)

6 Nsrin Eghbli, Vid Klvndi 39 Proof. We consider the sce of continuous functions resented in (3.6) gin nd endowed with the generlized metric defined by d(f, g) = inf{c [0, ] f(x) g(x) CE q (x) for ll x [, b]}. (3.14) Define the sme oertor T in (3.8). We shll verify tht T is strictly contrctive on X. With the definition of (X, d), for ny g, h X, it is ossible to find C gh [0, + ] such tht f(x) g(x) C gh E q (x) (3.15) for ny x [, b]. We obtin (T g)(x) (T h)(x) = 1 1 t [f(t, g(t)) f(t, h(t))]dt 1 t t g(t) h(t) dt x C gh t 1 E q (t)dt x C gh t 1 t k k=0 Γ(kq+1) dt C gh ( t 1 ) 1 1 dt) 1 ( t k dt) C gh ( (t 1 ) ) 1 1 dt) 1 ( t k dt) C gh ( 1 t ) 1 dt) 1 ( t k dt) C gh ( 1 t 1 dt) 1 ( t k dt) C gh ( 1 d) 1 ( t k dt) C gh ( 1 α ( 1 ( 1 α (ln x ln ) 1 ) 1 ( x k+ C gh α )1 (ln x ln ) α ( C gh α )1 (ln b )α ( k+ ) x k C gh M x k C gh M E q (x). k+ k+ k+ ) x k Hence, we cn conclude tht d(t g, T h) M d(f, g) for ny g, h X, nd since 0 < M < 1, the strictly continuous roerty verified. Similrly s in the roof of Theorem 3.1, one cn derive the results. In the following Theorem we used the Bielecki s norm:. B ( x B = mx t J x(t) e θt, θ > 0, J R + ) to derive the similr Theorem 3.1 for the fundmentl eqution (1.1) vi the Bielecki s norm. We need the following ssumtions: [H 1 ] : 0 < M < 1 tht M = 1 ( 1 α )1 (ln b )α ( θ ). Theorem 3.4. Assume tht [H 1], [H 2] nd [H 1 ] re stisfied. Then eqution (1.1) is Mittg- effler-hyers-ulm stble vi bielecki s norm. Proof. Just like the discussion in Theorem 3.1, we only rove tht T defined in (3.8) is contrction ming on X with resect to the Bielecki s norm: (T g)(x) (T h)(x) = 1 1 t [f(t, g(t)) f(t, h(t))]dt 1 t t g(t) h(t) dt )

7 REFERENCES 40 x t 1 g(t) h(t) dt x eθt t 1 g(t) h(t) e θt dt x eθt t 1 g(t) h(t) B dt g(t) h(t) B eθt t 1 dt g(t) h(t) B( t 1 ) 1 1 dt) 1 ( e θt dt) g(t) h(t) B( (t 1 ) ) 1 1 dt) 1 ( e θt dt) g(t) h(t) B( 1 t ) 1 dt) 1 ( e θt dt) g(t) h(t) B( 1 t 1 dt) 1 ( e θt dt) g(t) h(t) B( 1 d 1 ( e θt dt) g(t) h(t) B( 1 α α (ln x ln ) 1 ) 1 ( θ (e θx θ e )) g(t) h(t) B( 1 α )1 (ln x ln ) α ( θ ) e θx g(t) h(t) B( 1 α )1 (ln b )α ( θ ) e θx g(t) h(t) B M e θx. Then (T g)(x) (T h)(x) e θx g(t) h(t) B M e θx e θx for ech x [, b], tht is, d(t g, T h) g(t) h(t) B M. Hence we cn conclude tht d(t g, T h) M d(f, g) for ny g, h X. By letting 0 < M < 1, the strictly continuous roerty is verified. Now by similr rocess with Theorem 3.1, we hve 1 1 d(y, y 0 ) 1 M d(t y, y) 1 M εe q (x q ), which mens tht eqution (1.1) is Mittg-effler-Hyers-Ulm stble vi the Bielecki s norm. References [1] Alsin, C. nd Ger, R. (1988). On some inequlities nd stbility results relted to the exonentil function, J. Inequl. Al. 2: [2] Brzdek, J. nd Eghbli, N. (2016). On roximte solutions of some delyed frctionl differentil equtions, Al. Mth. ett. 54: [3] Eghbli, N., Klvndi, V. nd Rssis, J. M. (2016). A fixed oint roch to the Mittg- effler-hyers-ulm stbility of frctionl integrl eqution, Oen Mth. 14: [4] Hyers, D. H. (1941). On the stbility of the liner functionl eqution, Proc. Nt. Acd. Sci. 27: [5] Ibrhim R. W. (2012). Generlized Ulm-Hyers stbility for frctionl differentil equtions, Int. J. Mth. 23(5):9. [6] Ibrhim R. W. (2012). Ulm stbility for frctionl differentil eqution in comlex domin, Abstr. Al. Anl. 2012:1-8. [7] Ibrhim R. W. (2012). Ulm-Hyers stbility for Cuchy frctionl differentil eqution in the unit disk, Abstr. Al. Anl. 2012:1-10. [8] Killbs, A. A., Srivstv, H. M. nd Trujilo, J. J. (2006). Theory nd Alictions of Frctionl Differentil Equtions, North-Hollnd Mthemtics Studies, Elsevier, Amsterdm, 204.

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