Nonlocal initial value problems for implicit differential equations with Hilfer Hadamard fractional derivative

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1 Nonlinear Analyi: Modelling and Control, Vol. 23, No. 3, ISSN Nonlocal initial value problem for implicit differential equation with Hilfer Hadamard fractional derivative Devaraj Vivek a, Kuppuamy Kanagarajan a, Elayed M. Elayed b,c, a Department of Mathematic, Sri Ramakrihna Miion Vidyalaya College of Art and Science, Coimbatore-64020, India peppyvivek@gmail.com; kanagarajank@gmail.com b Department of Mathematic, Faculty of Science, King Abdulaziz Univerity, Jeddah 2589, Saudi Arabia c Department of Mathematic, Faculty of Science, Manoura Univerity, Manoura 3556, Egypt emmelayed@yahoo.com Received: Augut 8, 207 / Revied: December 27, 207 / Publihed online: April 20, 208 Abtract. In thi paper, the Schaefer fixed-point theorem i ued to invetigate the exitence of olution to nonlocal initial value problem for implicit differential equation with Hilfer Hadamard fractional derivative. Then the Ulam tability reult i obtained by uing Banach contraction principle. An example i given to illutrate the application of the main reult. Keyword: Hilfer Hadamard fractional derivative, implicit differential equation, fixed point, generalized Ulam Hyer tability. Introduction Fractional differential equation FDE have been applied in many field uch a phyic, mechanic, chemitry, engineering etc. There ha been a ignificant development in ordinary differential equation involving fractional-order derivative, one can ee the monograph of Hilfer [9], Kilba [6] and Podlubny [8] and the reference therein. Moreover, Hilfer [9] tudied application of a generalized fractional operator having the Riemann Liouville and the Caputo derivative a pecific cae. Hilfer fractional derivative ha been receiving more and more attention in recent time; ee, for example, [8, 4, 2, 24]. Benchohra et al. [4, 5] tudied implicit differential equationide of fractional order in variou apect. Recently, ome mathematician have conidered FDE depending on the Hadamard fractional derivative [2, 6, 7]. Correponding author. c Vilniu Univerity, 208

2 342 D. Vivek et al. In thi paper, we conider the Hilfer Hadamard-type IDE with nonlocal condition of the form xt = f t, xt, H D α,β xt, 0 < α <, 0 β, t J := [, b], HI γ x = xτ i, i= α γ = α β αβ <, τ i [, b], where H D α,β i the Hilfer Hadamard fractional derivative of order α and type β. Let X be a Banach pace, f : J X X X i a given continuou function and H I γ i the left-ided mixed Hadamard integral of order γ. In paing, we remark that the application of nonlocal condition H I γ x = m i= xτ i in phyical problem yield better effect than the initial condition HI γ x = x 0. For ake of brevity, let u take xt := K x t = f t, xt, K x t. A new and important equivalent mixed-type integral equation for our ytem can be etablihed. We adopt ome idea in [24] to etablih an equivalent mixed-type integral equation where xt = Zlog tγ Γα Γα i= α K x d log t α K x d, 2 Z := Γγ m i= f Γγ c i γ i γ. In the theory of functional equation, there i ome pecial kind of data dependence [3, 3, 7, 20]. For the advanced contribution on Ulam tability for FDE, we refer the reader to [2, 23, 25, 26]. In thi paper, we tudy different type of Ulam tability: Ulam Hyer tability, generalized Ulam Hyer tability, Ulam Hyer Raia tability and generalized Ulam Hyer Raia tability for the IDE with Hilfer Hadamard fractional derivative. Moreover, the Ulam Hyer tability for FDE with Hilfer fractional derivative wa invetigated in [, 22]. The paper i organized a follow. A brief review of the fractional calculu theory i given in Section 2. In Section 3, we will prove the exitence and uniquene of olution for problem. In Section 4, we dicu the Ulam Hyer tability reult. Finally, an example i given in Section 5 to illutrate the uefulne of our main reult. i=

3 Nonlocal initial value problem for implicit differential equation Fundamental concept In thi ection, we introduce ome definition and preliminary fact, which are ued in thi paper. Let C[J, X] be the Banach pace of all continuou function from J into X with the norm x C = max { xt : t [, b] }. For 0 γ <, we denote the pace C γ,log [J, X] a C γ,log [J, X] := { ft : [, b] X log t γ ft C[J, X] }, where C γ,log [J, X] i the weighted pace of the continuou function f on the finite interval [, b]. Obviouly, C γ,log [J, X] i the Banach pace with the norm f Cγ,log = log t γ ft C. Meanwhile, Cγ,log n [J, X] := {f Cn [J, X]: f n pace with the norm C γ,log [J, X]} i the Banach n f C n γ,log = f k C f n Cγ,log, n N. i=0 Moreover, C 0 γ,log [J, X] := C γ,log[j, X]. Definition. See [26]. The Hadamard fractional integral of order α for a continuou function f i defined a HI α ft = Γα provided the integral exit. log t α f d, α > 0, Definition 2. See [26]. The Hadamard derivative of fractional order α for a continuou function f : [, X i defined a HD α ft = t d Γn α dt t n log t n α f d, n < α < n, where n = α, α denote the integer part of real number α, and log = log e. We review ome baic propertie of Hilfer Hadamard fractional derivative, which are ued for thi work. For detail, ee [8, 9,, 9, 24] and reference therein. Nonlinear Anal. Model. Control, 233:34 360

4 344 D. Vivek et al. Definition 3. See [5]. The Hilfer Hadamard fractional derivative of order 0 < α < and 0 β of function ft i defined by where D := d/dt. Remark. Clearly, ft = HI β α D HI β α f t, i The operator H D α,β alo can be rewritten a = H I β α D H I β α = H I β α H D γ, γ = α β αβ. ii Let β = 0, the Hadamard-type Riemann Liouville fractional derivative can be preented a H D α := H D α,0. iii Let β =, the Hadamard-type Caputo fractional derivative can be preented a c H D α := H I α D. Lemma. See [5]. If α, β > 0 and < b <, then and [HI α log β ] t = Γβ log tβα Γβ α [HD α log α ] t = Γβ Γβ α log tβ α. In particular, if β = and α 0, then the Hadamard fractional derivative of a contant i not equal to zero: HD α t = Γ α log t α, 0 < α <. Lemma 2. If α, β > 0 and f L J for t [, T ] there exit the following propertie: HI α H I β f t = HI αβ f t and HD α H I α f t = ft. In particular, if f C γ,log [J, X] or f C[J, X], then thee equalitie hold at t, b] or t [, b], repectively. Lemma 3. Let 0 < α <, 0 γ <. If f C γ,log [J, X] and H I α f C γ,log [J, X], then HI α H D α ft = ft HI α f log t α t, b]. Γα Lemma 4. If 0 γ < and f C γ,log [J, X], then HI α f := lim t H I α ft = 0, 0 γ < α.

5 Nonlocal initial value problem for implicit differential equation 345 Lemma 5. Let α, β > 0 and γ = α β αβ. If f C γ γ,log [J, X], then HI γ H D γ h = H I α H D α,β f, HD γ H I α f = HD β α ft. Lemma 6. Let f L J and H D β α f L J exit, then 3 Exitence reult H I α f = HI β α H D β α f. In thi ection, we introduce pace that help u to olve and reduce ytem to an equivalent integral equation 2: C α,β γ,log = { f C γ,log [J, X], H D α,β f C γ,log [J, X] } and C γ γ,log = { f C γ,log [J, X], H D γ f C γ,log [J, X] }. It i obviou that C γ γ,log [J, X] Cα,β γ,log [J, X]. Lemma 7. Let f : J X X X be a function uch that f, x, H D α,β x C γ,log [J, X] for any x C γ,log [J, X]. A function x C γ γ,log [J, X] i a olution of Hilfer-type fractional IDE: xt = f t, xt, H D α,β xt, 0 < α <, 0 β, t J, HI γ x = x 0, γ = α β αβ, if and only if x atifie the following Volterra integral equation: xt = x 0log t γ Γγ Γα log t α f, x, H D α,β x d. Further detail can be found in [5]. From Lemma 7 we have the following reult. Lemma 8. Let f : J X X X be a function uch that f, x, H D α,β x C γ,log [J, X] for any x C γ,log [J, X]. A function x C γ γ,log [J, X] i a olution of ytem if and only if x atifie the mixed-type integral 2. Proof. According to Lemma 7, a olution of ytem can be expreed by xt = H I γ x log t γ Γγ Γα log t α K x d. 3 Nonlinear Anal. Model. Control, 233:34 360

6 346 D. Vivek et al. Next, we ubtitute t = τ i into the above equation: xτ i = H I γ x γ Γγ Γα multiplying both ide of 4 by, we can write xτ i = H I γ x γ Γγ Γα Thu, we have HI γ x = which implie xτ i i= = H I γ x Γγ i= HI γ x = Γγ m Γα Z γ Γα i= log τ α i K x d, 4 i= α K x d. α K x d, α K x d. 5 Submitting 5 to 3, we derive that 2. It i probative that x i alo a olution of the integral equation 2 when x i a olution of. The neceity ha been already proved. Next, we are ready to prove it ufficiency. Applying H I γ to both ide of 2, we have HI γ xt = H I γ log t γ Z Γα uing Lemma and 2, HI γ xt = Γγ m Γα Z H I γ H I α K xt, i= i= α K x d α K x d I β α K x t. Since γ < β α, Lemma 4 can be ued when taking the limit a t : HI γ x = Γγ m Γα Z i= α K x d. 6

7 Nonlocal initial value problem for implicit differential equation 347 Subtituting t = τ i into 2, we have i= xτ i = Then we derive xτ i = Z Γα that i, Z Γα γ Γα i= Γα = Γα i= i= = Γγ m Γα Z i= i= xτ i = Γγ m Γα Z i= It follow 6 and 7 that i= α K x d. α K x d α K x d α K x d γ i= α K x d Z α K x d, HI γ x = xτ i. i= γ i= α K x d. 7 Now by applying H D γ to both ide of 2, it follow from Lemma and 5 that HD γ xt = H D β α K x t = H D β α ft, xt, H D α,β xt. 8 Since x C γ γ,log [J, X] and by the definition of Cγ γ,log [J, X], we have that HD γ x C γ,log [J, X], then H D β α f = D H I β α f C γ,log [J, X]. For f C γ,log [J, X], it i obviou that HI β α f C γ,log [J, X], then HI β α f C γ,log [J, X]. Thu, f and HI β α f atify the condition of Lemma 3. Next, by applying H I β α to both ide of 8 and uing Lemma 3, we can obtain xt = K x t HI β α K x log t β α, Γβ α where I β α K x = 0 i implied by Lemma 4. Nonlinear Anal. Model. Control, 233:34 360

8 348 D. Vivek et al. Hence, it reduce to H D α,β xt = K x t = ft, xt, H D α,β xt. The reult are proved completely. Firt, we lit the following hypothee to tudy the exitence and uniquene reult: H The function f : J X X X i continuou. H2 There exit l, p, q C γ,log [J, X] with l = up t J lt < uch that ft, u, v lt pt u qt v, t J, u, v X. H3 There exit poitive contant K, L > 0 uch that ft, u, v ft, u, v K u u L v v, u, v, u, v X, t J. The exitence reult for problem will be proved by uing the Schaefer fixed-point theorem. Theorem. Aume that H and H2 are atified. Then ytem ha at leat one olution in C γ γ,log [J, X] Cα,β γ,log [J, X]. Proof. For ake of clarity, we plit the proof into a equence of tep. Conider the operator N : C γ,log [J, X] C γ,log [J, X]. Nxt = Zlog tγ Γα Γα i= log t α K x d. α K x d It i obviou that the operator N i well defined. Step. N i continuou. Let x n be a equence uch that x n x in C γ,log [J, X]. Then for each t J, log t γ Nx n t Nxt Z Γα i= log t γ Γα α K xn K x d log t α Kxn K x d Z Bγ, α m i= αγ Kxn K xn Γα C γ,log log bα Bγ, α K xn K x Γα C γ,log Bγ, α Z αγ log b α Kxn K x Γα C γ,log, i=

9 Nonlocal initial value problem for implicit differential equation 349 where we ue the formula a log t α x d = t a log t α log a γ d x C γ log t a αγ Bγ, α x C γ,log. Since K x i continuou i.e., f i continuou, then we have Nx n Nx C γ,log 0 a n. Step 2. N map bounded et into bounded et in C γ,log [J, X]. Indeed, it i enough to how that η > 0, there exit a poitive contant l uch that x B η = {x C γ,log [J, X]: x η}, we have Nx C γ,log l. Nxtlog t γ Z Γα For computational work, we et A = A 2 = Z Γα i= log t γ Γα α Kx d d log t α K x d d := A A 2. 9 i= log t γ Γα α Kx d d, log t α K x d d, and by H2, Kx t = f t, xt, Kx t lt pt xt qt Kx t l p xt q K x t l p xt q. 0 We etimate A, A 2 term eparately. By 0 we have A Z l α q Γα αγ p Bγ, α x C γ,log, Γα i= A 2 l log b α γ log bα q p Γα Γα Bγ, α x C γ,log. 2 Nonlinear Anal. Model. Control, 233:34 360

10 350 D. Vivek et al. Bringing inequalitie and 2 into 9, we have Nxtlog t γ l q Z α log b αγ Γα i= p q Z αγ log b Bγ, α α x C γ Γα := l. i= Step 3. N map bounded et into equicontinuou et of C γ,log [J, X]. Let t, t 2 J, t 2 t and x B η. Uing the fact f i bounded on the compact et J B η thu up t,x J Bη K x t := C 0 <, we will get Nxt Nxt 2 C 0 Z Bγ, α m i= αγ log t γ log t 2 γ Γα C 0Bγ, α log t αγ log t 2 αγ Γα A t t 2, the right-hand ide of the above inequality tend to zero. A a conequence of Step 3 together with Arzela Acoli theorem, we can conclude that N : C γ,log [J, X] C γ,log [J, X] i continuou and completely continuou. Step 4. A priori bound. Now it remain to how that the et ω = { x C γ,log [J, X]: x = δnx, 0 < δ < } i bounded et. Let x ω, x = δnx for ome 0 < δ <. Thu, for each t J, we have [ Z xt = δ log tγ Γα Γα log t i= Thi implie by H2 that for each t J, we have α K x d ]. xtlog t γ Nxtlog t γ K x d

11 Nonlocal initial value problem for implicit differential equation 35 l q Z α log b αγ Γα i= p q Z αγ log b Bγ, α α x C γ Γα := R. i= Thi how that the et ω i bounded. A a conequence of Schaefer fixed-point theorem, we deduce that N ha a fixed point, which i a olution of ytem. The proof i completed. Our econd theorem i baed on the Banach contraction principle. Theorem 2. Aume that H and H3 are atified. If K Bγ, α Z αγ log b α <, 3 LΓα then ytem ha a unique olution. i= Proof. Let the operator N : C γ,log [J, X] C γ,log [J, X]. Nxt = Z log tγ Γα Γα i= log t α K x d. α K x d By Lemma 8 it i clear that the fixed point of N are olution of ytem. Let x, x 2 C γ,log [J, X] and t J, then we have Nx t Nx 2 t log t γ and Z Γα i= log t γ Γα α Kx K x2 d log t α K x K d x2 Kx t K x2 t = f t, x t, K x t f t, x 2 t, K x2 t 4 K x t x 2 t LKx t K x2 t K x t x 2 t. 5 L Nonlinear Anal. Model. Control, 233:34 360

12 352 D. Vivek et al. By replacing 5 in inequality 4 we get Nx t Nx 2 t log t γ Z K Γα L Bγ, αlog ταγ x x 2 C γ,log i= log b γ Γα K Bγ, α LΓα Hence, Nx t Nx 2 t C γ,log K Bγ, α LΓα log b αγ K L Bγ, α x x 2 C γ,log Z αγ log b x α x 2 C γ,log. Z i= αγ log b x α x 2 C γ,log. i= From 3 it follow that N ha a unique fixed point, which i olution of ytem. The proof of Theorem 2 i completed. 4 Ulam Hyer Raia tability In thi ection, we conider the Ulam tability of Hilfer Hadamard-type IDE. The following obervation are taken from [4, 20]. Definition 4. Equation i Ulam Hyer table if there exit a real number C f > 0 uch that for each ɛ > 0 and for each olution z C γ γ,log [J, X] of the inequality H D α,β zt f t, zt, H D α,β zt ɛ, t J, 6 there exit a olution x C γ γ,log [J, X] of equation with zt xt C f ɛ, t J. Definition 5. Equation i generalized Ulam Hyer table if there exit ψ f C γ [,, [,, ψ f = 0, uch that for each olution z C γ γ,log [J, X] of the inequality H D α,β zt f t, zt, H D α,β zt ɛ, t J, there exit a olution x C γ γ,log [J, X] of equation with zt xt ψ f ɛ, t J.

13 Nonlocal initial value problem for implicit differential equation 353 Definition 6. Equation i Ulam Hyer Raia table with repect to ϕ C γ,log [J, X] if there exit a real number C f > 0 uch that for each ɛ > 0 and for each olution z C γ γ,log [J, X] of the inequality H D α,β zt f t, zt, H D α,β zt ɛϕt, t J, 7 there exit a olution x C γ γ,log [J, X] of equation with zt xt Cf ɛϕt, t J. Definition 7. Equation i generalized Ulam Hyer Raia table with repect to ϕ C γ,log [J, X] if there exit a real number C f,ϕ > 0 uch that for each olution z C γ γ,log [J, X] of the inequality H D α,β zt f t, zt, H D α,β zt ϕt, t J, 8 there exit a olution x C γ γ,log [J, X] of equation with zt xt Cf,ϕ ϕt, t J. Remark 2. A function z C γ γ,log [J, X] i a olution of inequality 6 if and only if there exit a function g C γ γ,log [J, X] which depend on olution z uch that i gt ɛ for all t J; ii H D α,β zt = ft, zt, H D α,β zt gt, t J. Remark 3. It i clear that i Definition 4 = Definition 5. ii Definition 6 = Definition 7. iii Definition 6 for ϕt = = Definition 4. One can have imilar remark for inequalitie 7 and 8. Lemma 9. Let 0 < α <, 0 β. If a function z C γ γ,log [J, X] i a olution of inequality 6, then x i a olution of the following integral inequality: zt A z Γα log t α K z d Zmclog b αγ log bα ɛ, Γα Γα where A z = Z log tγ Γα i= α K z d. Nonlinear Anal. Model. Control, 233:34 360

14 354 D. Vivek et al. Proof. Indeed, by Remark 2 we have that Then zt = zt = f t, zt, H D α,β zt gt = K z t gt. Zlog tγ Γα Γα From thi it follow that zt A z Γα Zlog t γ = Γα Zlog tγ Γα τi i= α K z d log t α K z d Γα i= i= log t Zmclog b αγ Γα α K z d α g d Γα α g d log bα Γα ɛ. log t α g d. Γα log t α g d α g d log t α g d We have imilar remark for the olution of inequalitie 7 and 8. The following generalized Gronwall inequalitie will be ued to deal with our ytem in the equence. Lemma 0. See [26]. Let v, w : [, b] [, be continuou function. If w i nondecreaing and there are contant k 0 and 0 < α < uch that then vt wt k vt wt [ n= log t α v d, t J = [, b], kγα n ] log t nα d w Γnα, t J. Remark 4. Under the aumption of Lemma 0, let wt be a nondecreaing function on J. Then we have vt wte α, kγαlog t α,

15 Nonlocal initial value problem for implicit differential equation 355 where E α, i the Mittag Leffler function defined by E α, z = k=0 z k Γkα, z C. Now we give the Ulam Hyer and Ulam Hyer Raia reult in thi equel. Theorem 3. Aume that H3 and 3 are atified. Then ytem i Ulam Hyer table. Proof. Let ɛ > 0, and let z C γ γ,log [J, X] be a function, which atifie inequality 6, and let x C γ γ,log [J, X] i the unique olution of the following Hilfer Hadamard-type IDE: xt = f t, xt, H D α,β xt, t J := [, b], HI γ x = H I γ z = where 0 < α <, 0 β. Uing Lemma 8, we obtain where A x = xτ i, i= xt = A x Γα Z log tγ Γα i= τ i [, b], γ = α β αβ, log t α K x d, α K x d. On the other hand, if m i= xτ i = m i= zτ i and H I γ z = H I γ x, then A x = A z. Indeed, A x A z Z log tγ log τ α i K x K z d Γα Z log tγ Γα K Z log tγ L i= i= α K L H I α xτi zτ i = 0. i= x z d Thu, A x = A z. Nonlinear Anal. Model. Control, 233:34 360

16 356 D. Vivek et al. Then we have xt = A z Γα log t α K x d. By integration of inequality 6 and applying Lemma 9, we obtain zt A z log t α K z d Γα Zmclog b αγ log bα ɛ. 9 Γα Γα For ake of brevity, we take U = Zmclog b αγ /Γα log b α /Γα. We have for any t J, zt xt zt A z Γα By uing 9 we have Γα zt A z Γα K LΓα log t α K z d log t α Kz K x d zt xt K Uɛ LΓα log t α K z d log t α z x d. log t α z x d, and applying Lemma 0 and Remark 4, we obtain zt xt K UE α, log bα ɛ L where Thu, ytem i Ulam Hyer table. := C f ɛ, K C f = UE α, log bα. L

17 Nonlocal initial value problem for implicit differential equation 357 Theorem 4. Aume that H3 and 3 are atified. Suppoe that there exit an increaing function ϕ C γ,log [J, X] and there exit λ ϕ > 0 uch that for any t J, HI α ϕt λ ϕϕt. Then ytem i generalized Ulam Hyer Raia table. Proof. Let ɛ > 0, and let z C γ γ,log [J, X] be a function, which atifie inequality 7, and let x C γ γ,log [J, X] be the unique olution of the following Hilfer Hadamard IDE: xt = f t, xt, H D α,β xt, t J := [, b], HI γ x = H I γ z = where 0 < α <, 0 β. Uing Lemma 8, we obtain where A z = xτ i, i= xt = A z Γα Z log tγ Γα By integration of 7 we obtain zt A z Γα ɛ Γα i= τ i [, b], γ = α β αβ, log t α K x d, log t α K z d α K z d. log t α ϕ d Zlog b γ mc ɛλ ϕ ϕt. 20 For ake of brevity, we take U = Zlog b γ mc. On the other hand, we have zt xt zt A z Γα K LΓα log t α K z d log t α z x d. Nonlinear Anal. Model. Control, 233:34 360

18 358 D. Vivek et al. By uing 20 we have zt xt K Uɛλϕ ϕt LΓα log t α z x d, and applying Lemma 0 and Remark 4, we obtain zt xt K Uɛλϕ ϕte α, log bα, t [, b]. L Thu, ytem i generalized Ulam Hyer Raia table. The proof i completed. 5 An example A an application of our reult, we conider the following problem of Hilfer Hadamard IDE: xt = HI γ = 2x [ t e log xt 9 e log t xt ] xt H D α,β, t J := [, e], 2 xt 3, γ = α β αβ Notice that thi problem i a particular cae of, where α = 2/3, β = /2, and chooe γ = 5/6. Set [ t e log u ft, u, v = 9 e log t u v ] for any u, v X. v Clearly, the function f atifie the condition of Theorem. For any u, v, u, v X and t J, ft, u, v ft, u, v u u v v. 0 0 Hence, condition H3 i atified with K = L = /0. Thu, from 3 we have K Bγ, α Z αγ log b α <, LΓα i= where Z = It follow from Theorem 2 that problem 2 22 ha a unique olution. From the above example the exitence and Ulam Hyer tability of Hilfer Hadamardtype IDE with nonlocal condition are verified by Theorem and 3.

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