Dynamics and stability of Hilfer-Hadamard type fractional differential equations with boundary conditions

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1 Journl Nonliner Anlyi nd Appliction 208 No Avilble online t Volume 208, Iue, Yer 208 Article ID jn-00386, 3 Pge doi:0.5899/208/jn Reerch Article Dynmic nd tbility of Hilfer-Hdmrd type frctionl differentil eqution with boundry condition D. Vivek, K. Kngrjn, S. Hrikrihnn Deprtment of Mthemtic, Sri Rmkrihn Miion Vidyly College of Art nd Science, Coimbtore , Tmilndu, Indi. Copyright 208 c D. Vivek, K. Kngrjn nd Hrikrihnn. Thi i n open cce rticle ditributed under the Cretive Common Attribution Licene, which permit unretricted ue, ditribution, nd reproduction in ny medium, provided the originl work i properly cited. Abtrct In thi pper, we invetigte the exitence, uniquene nd Ulm tbilitie of olution for Hilfer-Hdmrd type frctionl differentil eqution with boundry condition in weighted pce of continuou function. The exitence reult rely on Schefer fixed point theorem. The Bnch contrction principle i lo conidered to obtin uniquene nd tbility reult. An exmple i provided to illutrte the uefulne of the obtined reult. Keywor: Boundry vlue problem; Hilfer-Hdmrd frctionl derivtive; Exitence; Ulm tbility; Fixed point. Introduction In thi pper, we prove the exitence, uniquene nd Ulm tbilitie nlyi of olution of Hilfer-Hdmrd type frctionl differentil eqution with boundry condition of the form { + xt = f t,xt, t J := [,T ], I γ + x =, I γ. + xt = b, γ = α + β αβ, where H D α,β + i the Hilfer-Hdmrd frctionl derivtive, 0 < α <, 0 β nd let X be Bnch pce, f : J X X i given continuou function. It i een tht eqution. i equivlent to the integrl eqution xt = Γγ logtγ + b I β α + f T,xT + Γα log t Γ2β logt γ+2β 2 Γγ + 2β logt 2β α f,x,.2 Differentil eqution of frctionl order hve recently proved to be vluble tool in the modeling of mny phenomen in vriou fiel of cience nd engineering. There h been ignificnt development in frctionl differentil FDE nd prtil differentil eqution in recent yer; ee the monogrph of of Hilfer [2], Kilb [24] nd Podlubny [26]. There re ome work on FDE with Hdmrd frctionl derivtive, even if it h been tudied mny Correponding uthor. Emil ddre: peppyvivek@gmil.com. 4

2 Journl of Nonliner Anlyi nd Appliction 208 No yer go ee for exmple [2, 6, 7]. Recently, everl work reporting Hilfer type of eqution hve been publihed. See [2, 6, 7] for more exmple nd remrk concerning Hilfer frctionl derivtive. The tbility of functionl eqution w originlly ried by Ulm in 940 in tlk given t Wiconin Univerity. The problem poed by Ulm w the following: Under wht condition doe there exit n dditive mpping ner n pproximtely dditive mpping? The firt nwer to the quetion of Ulm w given by Hyer in 94 in the ce of Bnch pce. Therefter, thi type of tbility i clled the Ulm-Hyer tbility [5, 3, 4, 7]. In 978, Ri [7] provided remrkble generliztion of the Ulm-Hyer tbility of mpping by conidering vrible. The concept of tbility for functionl eqution rie when we replce the functionl eqution by n inequlity which ct perturbtion of the eqution. The tbility propertie of ll kin of eqution hve ttrcted the ttention of mny mthemticin. In prticulr, the Ulm-Hyer tbility nd Ulm-Hyer-Ri tbility hve been tken up by number of mthemticin nd the tudy of thi re h developed to be one of the centrl ubject in the mthemticl nlyi re. For more detil on the Ulm- Hyer tbility nd Ulm-Hyer-Ri tbility of differentil eqution, ee [3, 9, 28, 29]. Recently, Kim et. l. [6] invetigted well-poedne nd tbility for differentil eqution with Hilfer- Hdmrd frctionl derivtive. In thi pper we minly focu on developing the theory of dynmic nd tbility for FDE vi Hilfer-Hdmrd derivtive. The problem of the exitence of olution for FDE with boundry condition h been recently treted in the literture in [2, 4, 8, 9, 20, 8]. The ret of thi pper i orgnized follow. In Section 2, we give ome bic definition nd reult concerning the Hilfer-Hdmrd frctionl derivtive. In Section 3, we preent our min reult by uing Schefer fixed point theorem. In ection 4, we introduce four type of Ulm tbility definition for FDE: Ulm-Hyer tbility, generlized Ulm-Hyer tbility, Ulm-Hyer-Ri tbility nd generlized Ulm-Hyer-Ri tbility. We preent the four type of Ulm tbility reult for FDE.. 2 Prerequiite Thi ection i devoted to bic definition nd lemm from [, 0,, 5, 2, 22, 23] the theory of Hilfer frctionl derivtive which re ued in ubequent ection. Definition 2.. Let C[J,X] denote the Bnch pce of continuou function on [,T ] with the norm x C := up{xt : t J}. We denote L {R + }, the pce of Lebegue integrble function on J. By C γ,log [J,X] nd Cγ,log [J,X], we denote the weighted pce of continuou function defined by C γ,log [J,X] := { f t : J X logt γ f t C[J,X]}, with norm f Cγ,log = logt γ f t C, nd Moreover, C 0 γ,log [J,X] := C γ,log[j,x]. f C n γ,log = n k=0 f k C + f n Cγ,log, n N. Now, we give ome reult nd propertie of Hdmr frctionl clculu. Interntionl Scientific Publiction nd Conulting Service

3 Journl of Nonliner Anlyi nd Appliction 208 No Definition 2.2. [2, 7] The Hdmrd frctionl integrl of order α for function h i defined I α + ht = log t α h, α > 0, Γα provided the integrl exit. Notice tht for ll α,α,α 2 > 0 nd ech h C[J,X], we hve I α + h C[J,X], nd I α + I α 2 + ht = I α +α 2 + ht;for.e. t J. Definition 2.3. [2, 7] The Hdmrd derivtive of frctionl order α for function h : [, X i defined HD α + ht = t d n log t n α h, n < α < n, n = α +, Γn α dt where α denote the integer prt of rel number α nd log = log e. Let α 0,], γ [0, nd h C γ,log [J,X]. Then the following expreion le to the left invere opertor follow. H D α + I α + ht = ht; for ll t [,b]. Moreover, if I α + h C γ,log [J,X], then the following compoition I α + H D α + ht = ht I α + h + logt α ; Γα for ll t [,b]. In [2], R. Hilfer tudied ppliction of generlized frctionl opertor hving the Riemnn-Liouville nd Cputo derivtive pecific ce ee lo [22, 5]. Definition 2.4. Hilfer-Hdmrd derivtive. Let 0 < α <, 0 β, h L {R + }, I α β + Cγ,log [J,X]. The Hilfer-Hdmrd frctionl derivtive of order α nd type β of h i defined H D α,β + ht = I β α d + dt I α β + h t; for.e. t J. 2.3 Propertie: Let 0 < α <, 0 β, γ = α + β αβ, nd h L {R + }.. The opertor H D α,β + ht cn be written H D α,β + ht = I β α d + dt I γ + h Moreover, the prmeter γ tifie t = I β α + H D γ + h t; for.e. t J. 0 < γ, γ α, γ > β, γ < β α. 2. The generliztion 2.3 for β = 0, coincide with the Hdmrd Riemnn-Liouville derivtive nd for β = with the Hdmrd Cputo derivtive. 3. If H D β α + h exit nd in L {R + }, then H D α,β + I α + ht = HD α,0 + = H D α +, nd H D α, + = c HD α +. I β α + H D β α + h t; for.e. t J. Furthermore, if h C γ,log [J,X] nd I β α + h Cγ,log [J,X], then H D α,β + I α + ht = ht; for.e. t J. Interntionl Scientific Publiction nd Conulting Service

4 Journl of Nonliner Anlyi nd Appliction 208 No If H D γ + h exit nd in L {R + }, then I α + H D α,β + h t = I γ + H D γ + h t = ht I γ + h + logt γ ; for.e. t J. Γγ In order to olve our problem, the following pce re preented { } C α,β γ,log [J,X] = f C γ,log [J,X], H D α,β + f C γ,log [J,X], nd It i obviou tht C γ γ,log [J,X] = { f C γ,log [J,X], H D γ + f C γ,log [J,X] }. C γ γ,log [J,X] Cα,β γ,log [J,X]. Lemm 2.. Let α > 0, 0 β, o the homogeneou differentil eqution with Hilfer-Hdmrd frctionl order h olution + ht = 0 ht = c 0 logt γ + c logt γ+2β 2 + c 2 logt γ+22β c n logt γ+n2β n+. Corollry 2.. [] Let h C γ,log [J,X]. Then the liner problem h unique olution x L {R + } given by + xt = ht, t J := [,b], I γ + xt t= = x 0, γ = α + β αβ, xt = x 0 Γγ logtγ + Γα log t α h. Lemm 2.2. Let f : J X X be function uch tht f,x C γ,log [J,X] for ny x C γ,log [J,X]. A function x C γ γ,log [J,X] i olution of the integrl eqution.2 if nd only if x i olution of the Hilfer-Hdmrd frctionl BVP Proof. Aume x tifie.2. Then Lemm 2. implie tht From 2.5, imple clcultion give + xt = f t,xt, t J := [,T ], 2.4 I γ + x =, I γ + xt = b, γ = α + β αβ. 2.5 xt = c 0 logt γ + c logt γ+2β 2 + Γα log t α f,x. c 0 = Γγ, c = b I β α Γ2β + f T,xT Γγ + 2β logt 2β. Hence, we get eqution.2. Converly, it i cler tht if x tifie eqution.2, then eqution hold. Interntionl Scientific Publiction nd Conulting Service

5 Journl of Nonliner Anlyi nd Appliction 208 No Exitence nd uniquene reult In thi ection, we obtin the exitence nd uniquene of olution the problem.. For thi, let u mke the following condition. C f : J X X i continuou function. C2 There exit contnt L > 0 uch tht for ny u,u X, nd t J. f t,u f t,u L u u, C3 The function f : J X X i completely continuou nd there exit function µt L {R + } uch tht f t,u µt, t J, u X. C4 There exit n increing function φ C γ,log [J,X] nd there exit λ φ > 0 uch tht for ny t J I α + φt λ φ φt. Theorem 3.. Exitence of olution Aume tht the condition C,C3 re tified. Then the problem. h t let one olution defined on J. Proof. Conider the opertor P : C γ,log [J,X] C γ,log [J,X] defined by Pxt = Γγ logtγ + b I β α + f T,xT + Γα log t It i obviou tht the opertor P i well defined. Step : P i continuou. Let x n be equence uch tht x n x in C γ,log [J,X]. Then for ech t J, Γ2β logt γ+2β 2 Γγ + 2β logt 2β α f,x. 3.6 Px n t Pxtlogt γ logt γ log t α f,xn f,x Γα Γ2β T + log T β α f,x n f,x logt 2β Γγ + 2β Γ β α logt 2β logt γ+α Γ2β logt β α + f,x n f,x Γα + Γγ + 2β Γ2 β α C γ,log. Since f i continuou, then we hve Px n Px C γ,log 0 n. Step 2: P mp bounded et into bounded et in C γ,log [J,X]. Indeed, it i enough to how tht for q > 0, there exit poitive contnt l uch tht x B q { x C γ,log [J,X] : x q }, Interntionl Scientific Publiction nd Conulting Service

6 Journl of Nonliner Anlyi nd Appliction 208 No we hve Px C γ,log l. Pxtlogt γ Γγ + + logt γ Γα Γ2β b Γγ + 2β log t Γγ + Γ2β Γγ + 2β := l. α f,x b + Γ2β T log T Γ β α Γγ + 2β logt γ+α Γα + β α f,x logt 2β logt 2β logt β α µ Γ2 β α C γ,log Step 3: P mp bounded et into equicontinuou et of C γ,log [J,X]. Let t,t 2 J,t < t 2, B q be bounded et of C γ,log [J,X] in Step 2, nd x B q. Then, Pxt 2 Pxt 2 logt2 γ logt γ Γγ [ ] + b I β α Γ2β logt 2 γ+2β 2 logt γ+2β 2 + f T,xT Γγ + 2β logt 2β + t2 log t α 2 f,x Γα log t α f,x Γα logt2 γ logt γ Γγ [ ] logt β α + b Γ2 β α µ Γ2β logt 2 γ+2β 2 logt γ+2β 2 C γ,log Γγ + 2β logt 2β + log t α 2 µ Γα + t + C γ,log Γα log t α 2 log t α µ. A t t 2, the right hnd ide of the bove inequlity ten to zero. A conequence of tep to 3, together with Arzel-Acoli theorem, we cn conclude tht P : C γ,log [J,X] C γ,log [J,X] i continuou nd completely continuou. Step 4: A priori boun. Now it remin to how tht the et ω = { x C γ,log [J,X] : x = δpx, 0 < δ < } i bounded et. Let x ω, x = δpx for ome 0 < δ <. Thu for ech t J. We hve, [ xt = δ Γγ logtγ + b I β α Γ2β logt γ+2β 2 + f T,xT Γγ + 2β logt 2β + log t α ] f,x. Γα Interntionl Scientific Publiction nd Conulting Service

7 Journl of Nonliner Anlyi nd Appliction 208 No Thi implie by C3tht for ech t J, we hve xtlogt γ Pxtlogt γ Γγ + Γ2β b Γγ + 2β + logt γ log t Γα Γγ + Γ2β Γγ + 2β := R. α f,x b + Γ2β T log T Γ β α Γγ + 2β logt γ+α Γα + β α f,x logt 2β logt 2β logt β α µ Γ2 β α C γ,log tht µ C γ,log R. Thi how tht the et ω i bounded. A conequence of Schefer fixed point theorem, we deduce tht P h fixed point which i olution of problem.. Lemm 3.. Uniquene of olution Aume tht the condition C,C2 re tified. If Γ2β logt β α logt γ+α L + <, 3.7 Γγ + 2β Γ2 β α Γα + then the problem. h unique olution. Proof. Conider the opertor P : C γ,log [J,X] C γ,log [J,X]. Pxt = Γγ logtγ + b I β α + f T,xT + Γα log t α f,x. Γ2β logt γ+2β 2 Γγ + 2β logt 2β It i cler tht the fixed point of P re olution of.. Let x,y C γ,log [J,X] nd t J, then we hve Pxt Pytlogt γ Γ2β Γγ + 2β Γ β α + logt γ log t Γα Γ2β logt β α L Γγ + 2β Γ2 β α T log T α f,x f,y + logt γ+α Γα + β α f,x f,y logt 2β logt 2β x y C γ,log. Hence, Γ2β logt β α Px Py C γ,log L Γγ + 2β Γ2 β α logt γ+α + x y Γα + C γ,log. From 3.7, it follow tht P h unique fixed point which i olution of problem.. Interntionl Scientific Publiction nd Conulting Service

8 Journl of Nonliner Anlyi nd Appliction 208 No Stbility nlyi In thi ection, we conider the Ulm tbility of Hilfer-Hdmrd type FDE with boundry condition.. We dopt the definition in [9, 27]. We conider the following inequlitie H D α,β + zt f t,zt ε, t J; 4.8 H D α,β + zt f t,zt εφt, t J; 4.9 H D α,β + zt f t,zt φt, t J. 4.0 Definition 4.. The eqution. i Ulm-Hyer tble if there exit rel number C f > 0 uch tht for ech ε > 0 nd for ech olution z C γ γ,log [J,X] of the inequlity 4.8 there exit olution x Cγ γ,log [J,X] of eqution. with zt xt C f ε, t J. Definition 4.2. The eqution. i generlized Ulm-Hyer tble if there exit ψ f C[0,,[0,,ψ f 0 = 0 uch tht for ech olution z C γ γ,log [J,X] of the inequlity 4.8 there exit olution x Cγ γ,log [J,X] of eqution. with zt xt ψ f ε, t J. Definition 4.3. The eqution. i Ulm-Hyer-Ri tble with repect to φ C γ,log [J,X] if there exit rel number C f > 0 uch tht for ech ε > 0 nd for ech olution z C γ γ,log [J,X] of the inequlity 4.9 there exit olution x C γ γ,log [J,X] of eqution. with zt xt C f εφt, t J. Definition 4.4. The eqution. i generlized Ulm-Hyer-Ri tble with repect to φ C γ,log [J,X] if there exit rel number C f,φ > 0 uch tht for ech olution z C γ γ,log [J,X] of the inequlity 4.0 there exit olution x C γ γ,log [J,X] of eqution. with zt xt C f,φ φt, t J. Remrk 4.. A function z C γ γ,log [J,X] i olution of the inequlity H D α,β + zt f t,zt ε, t J, if nd only if there exit function g C γ γ,log [J,X] which depend on olution x uch tht. gt ε, t J; 2. + zt = f t,zt + gt, t J. Remrk 4.2. Clerly,. Definition 4. Definition Definition 4.3 Definition 4.4. Lemm 4.. [30] Suppoe > α > 0, > 0 nd b > 0 nd uppoe ut i nonnegtive nd loclly integrl on [,+ with ut + b log t α u, t [,+. Then ] bγα n ut + log t nα Γnα, t [,+. [ n= Interntionl Scientific Publiction nd Conulting Service

9 Journl of Nonliner Anlyi nd Appliction 208 No Remrk 4.3. Under the umption of Lemm 4., let ut be nondecreing function on [,. Then we hve where E α, i the Mittg-leffler function defined by ut E α, bγαlogt α, E α, z = k=0 We redy to prove our tbility reult for problem.. z k Γkα +, z C. Theorem 4.. Aume tht the condition C,C2 nd 3.7 re tified, then the problem. i Ulm-Hyer tble. Proof. Let ε > 0 nd let z C γ γ,log [J,X] be function which tifie the inequlity: H D α,β + zt f t,zt ε, for ny t J, 4. nd let x C γ γ,log [J,X] be the unique olution of the following Hilfer-Hdmrd type BVP where 0 < α <, 0 β. Uing Lemm 2.2, we obtin where A x = + xt = f t,xt, t J := [,T ], I γ + x = I γ + z =, I γ + xt = I γ + zt = b γ = α + β αβ, xt = A x + T log t α f,x Γα Γγ logtγ + b I β α Γ2β logt γ+2β 2 + f T,xT Γγ + 2β logt 2β. On the other hnd, if I γ + xt = I γ + zt nd I γ + x = I γ + z, then A x = A z. Indeed, Thu, A x = A z. Then, we hve A x A z Γ2β Γγ + 2β = 0. xt = A z + Γα By integrtion of the inequlity 4., we obtin zt A z Γα We hve logt γ+2β 2 LI β α logt 2β + xt zt log t zt xt zt A z Γα + t Γα log t α f,x. α f,z log t εlogt α Γα + + L Γα εlogt α Γα +. log t α f,z α [ f,z f,x] log t α z x, Interntionl Scientific Publiction nd Conulting Service

10 Journl of Nonliner Anlyi nd Appliction 208 No nd to pply Lemm 4. nd Remrk 4.3, we obtin Thu, the eqution. i Ulm-Hyer tble. zt xt logt α E α, LlogT α Γα + := C f ε. Theorem 4.2. Aume tht the condition C, C2, C4 nd 3.7 re tified. Then, the problem. i generlized Ulm-Hyer-Ri tble. Proof. Let z C γ γ,log [J,X] be olution of the inequlity H D α,β + zt f t,zt εφt, t J, ε > 0, 4.2 nd let x C γ γ,log [J,X] be the unique olution of the following Hilfer-Hdmrd type BVP where 0 < α <, 0 β. Uing Lemm 2.2, we obtin where A z = + xt = f t,xt, t J := [,T ], I γ + z =, I γ + zt = b γ = α + β αβ, xt = A z + T log t α f,x Γα, Γγ logtγ + b I β α Γ2β logt γ+2β 2 + f T,zT Γγ + 2β logt 2β. By integrtion of the inequlity 4.2, we get zt A z Γα On the other hnd, we hve zt xt zt A z Γα ε log t α f,z ελ φφt L Γα ελ φ φt + By pplying Lemm 4. nd Remrk 4.3, we get log t L Γα log t α f,z α z x log t α z x. zt xt ελ φ φte α, LlogT α, t [,T ]. Thu, the eqution. i generlized Ulm-Hyer-Ri tble. 5 Exmple In thi ection, we preent n exmple to illutrte the theory reult. Interntionl Scientific Publiction nd Conulting Service

11 Journl of Nonliner Anlyi nd Appliction 208 No Exmple 5.. Let u conider conider the following Hilfer-Hdmrd type frctionl BVP + xt = xt, t J := [,e], xt I γ + x =, I γ + xe = 2, γ = α + β αβ. 5.5 Notice tht thi problem i prticulr ce of., where α = 2 3, β = 2 nd chooe γ = 5 6. Set f t,u = 0 u +u, for ny u X, nd t J. Clerly, the function f tifie condition of Theorem 3.. For ech u,v X nd t J. f t,u f t,v u v. 0 Hence, the condition C2 i tified with L = 0. Here T = e. Thu, condition from 3.7 Γ2β logt β α logt γ+α L <, Γγ + 2β Γ2 β α Γα + It follow from Lemm 3. tht the problem h unique olution. Moreover, Theorem 4. implie tht the problem i Ulm-Hyer tble. Acknowledgement The uthor re grteful to the referee for their creful reding of the mnucript nd vluble comment. The uthor thnk the help from editor too. Reference [] S. Abb, M. Benchohr, S. Sivundrm, Dynmic nd Ulm tbility for Hilfer type frctionl differentil eqution, Nonliner Studie, [2] B. Ahmd, S. K. Ntouy, Initil vlue problem for hybrid Hdmrd frctionl differentil eqution, Electronic Journl of Differentil Eqution, [3] B. Ahmd, J. J. Nieto, Exitence of olution for nonlocl boundry vlue problem of higher-order nonliner frctionl differentil eqution, Abtrct nd Applied Anlyi, [4] B. Ahmd, J. J. Nieto, Riemnn-Liouville frctionl differentil eqution with frctionl boundry condition, Fixed point Theory, [5] S. Andrá, J. J. Kolumbán, On the Ulm-Hyer tbility of firt order differentil ytem with nonlocl initil condition, Nonliner Anlyi, [6] P. L. Butzer, A. A. Kilb, J. J. Trujillo, Compoition of Hdmrd-type frctionl integrtion opertor nd the emigroup property, Journl of Mthemticl Anlyi nd Appliction, [7] P. L. Butzer, A. A. Kilb, J. J. Trujillo, Mellin trnform nlyi nd integrtion by prt for Hdmrd-type frctionl integrl, Journl of Mthemticl Anlyi nd Appliction, Interntionl Scientific Publiction nd Conulting Service

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