EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR CAPUTO-HADAMARD SEQUENTIAL FRACTIONAL ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

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1 Electronic Journal of Differential Equation, Vol ), No. 36, pp.. ISSN: URL: or EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR CAPUTO-HADAMARD SEQUENTIAL FRACTIONAL ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS BASHIR AHMAD, SOTIRIS K. NTOUYAS Abtract. In thi article, we tudy the exitence and uniquene of olution for Hadamard-type equential fractional order neutral functional differential equation. The Banach fixed point theorem, a nonlinear alternative of Leray- Schauder type and Kranoelki fixed point theorem are ued to obtain the deired reult. Example illutrating the main reult are preented. An initial value integral condition cae i alo dicued.. Introduction Thi work i concerned with the exitence and uniquene of olution to the following initial value problem IVP) of Caputo-Hadamard equential fractional order neutral functional differential equation D α [D β yt) gt, y t )] = ft, y t ), t J := [, b],.) yt) = φt), t [ r, ],.2) D β y) = η R,.3) where D α, D β are the Caputo-Hadamard fractional derivative, 0 < α, β <, f, g : J C[ r, 0], R) R are given function and φ C[ r, ], R). For any function y defined on [ r, b] and any t J, we denote by y t the element of C r := C[ r, 0], R) defined by y t θ) = yt θ), θ [ r, 0]. Functional differential equation are found to be of central importance in many dicipline uch a control theory, neural network, epidemioy, etc. [7]. In analyzing the behavior of real population, delay differential equation are regarded a effective tool. Since the delay term can be finite a well a infinite in nature, one need to tudy thee two cae independently. Moreover, the delay term may appear in the derivative involved in the given equation. A it i difficult to formulate uch a problem, an alternative approach i followed by conidering neutral functional differential equation. On the other hand, fractional derivative are capable to decribe hereditary and memory effect in many procee and material. So the tudy of neutral functional differential equation in preence of fractional 200 Mathematic Subject Claification. 34A08, 34K05. Key word and phrae. Fractional differential equation; exitence; fixed point; functional fractional differential equation; Caputo-Hadamard fractional differential equation. c 207 Texa State Univerity. Submitted November, 206. Publihed February 2, 207.

2 2 B. AHMAD, S. K. NTOUYAS EJDE-207/36 derivative contitute an important area of reearch. For more detail, ee the text [27]. In recent year, there ha been a ignificant development in fractional calculu, and initial and boundary value problem of fractional differential equation, ee the monograph of Kilba et al. [9], Lakhmikantham et al. [2], Miller and Ro [22], Podlubny [23], Samko et al. [24], Diethelm [] and a erie of paper [, 2, 3, 4, 0, 2, 3, 8, 25, 26] and the reference therein. One can notice that much of the work on the topic involve Riemann-Liouville and Caputo type fractional derivative. Beide thee derivative, there i an other fractional derivative introduced by Hadamard in 892 [6], which i known a Hadamard derivative and differ from aforementioned derivative in the ene that the kernel of the integral in it definition contain arithmic function of arbitrary exponent. A detailed decription of Hadamard fractional derivative and integral can be found in [7, 8, 9] and reference cited therein. In [6], the author tudied an initial value problem IVP) for Riemman-Liouville type fractional functional and neutral functional differential equation with infinite delay. Recently, initial value problem for fractional order Hadamard-type functional and neutral functional differential equation and incluion were repectively invetigated in [3, 5], while an IVP for retarded functional Caputo type fractional impulive differential equation with variable moment wa dicued in [4]. In thi paper, we invetigate a new cla of Hadamard-type equential fractional neutral functional differential equation. Our tudy i baed on fixed point theorem due to Banach and Kranoelkii [20], and nonlinear alternative of Leray-Schauder type [5]. The ret of thi paper i organized a follow: in Section 2 we recall ome ueful preliminarie. In Section 3 we dicu the exitence and uniquene of olution for the problem.)-.3), while the exitence reult for the problem are preented in Section 4. Example are contructed in Section 5 for illutrating the obtained reult. Finally, a generalization involving initial value integral condition i decribed in Section Preliminarie In thi ection, we introduce notation, definition, and preliminary fact that we need in the equel. By CJ, R) we denote the Banach pace of all continuou function from J into R with the norm Alo C r i endowed with norm y := up{ yt) : t J}. φ C := up{ φθ) : r θ 0}. Definition 2. [9]). The Hadamard derivative of fractional order q for a function g : [, ) R i defined a D q gt) = t d Γn q) dt ) n t ) n q g) d, n < q < n, n = [q], where [q] denote the integer part of the real number q and ) = e ).

3 EJDE-207/36 EXISTENCE AND UNIQUENESS OF SOLUTIONS 3 Definition 2.2 [9]). The Hadamard fractional integral of order q for a function g i defined a I q gt) = t ) q g) d, q > 0, Γq) provided the integral exit. Lemma 2.3. The function y C 2 [ r, b], R) i a olution of the problem D α [D β yt) gt, y t )] = ft, y t ), yt) = φt), t [ r, ], D β y) = η R, t J := [, b], if and only if φt), if t [ r, ], t)β φ) η g, φ))) yt) = Γβ) t ) Γα) t α g,y ) d ) Γαβ) t αβ f,y ) d, if t [, b]. Proof. The olution of Hadamard differential equation in 2.) can be written a D β yt) gt, y t ) = Γα) 2.) 2.2) t ) α f, y ) d c, 2.3) where c R i arbitrary contant. Uing the condition D β y) = η we find that c = η g, φ)). Then we obtain t)β yt) = η g, φ)) Γβ ) Γβ) t t Γα β) t ) β g, y ) d ) αβ f, y ) d c 2. From the above equation we find c 2 = φ) and 2.2) i proved. The convere follow by direct computation. 3. Exitence and uniquene reult In thi ection, we etablih the exitence and uniquene of a olution for the IVP.).3). Definition 3.. A function y C 2 [ r, b], R), i aid to be a olution of.).3) if y atifie the equation D α [D β yt) gt, y t )] = ft, y t ) on J, the condition yt) = φt) on [ r, ] and D β y) = η. The next theorem give u a uniquene reult uing the aumption A) there exit l > 0 uch that ft, u) ft, v) l u v C, for t J and every u, v C r ; A2) there exit a nonnegative contant k uch that gt, u) gt, v) k u v C, for t J and every u, v C r.

4 4 B. AHMAD, S. K. NTOUYAS EJDE-207/36 Theorem 3.2. Aume that A), A2) hold. If k b) α Γα ) l b)αβ <, 3.) Γα β ) then there exit a unique olution for IVP.).3) on the interval [ r, b]. Proof. Conider the operator N : C[ r, b], R) C[ r, b], R) defined by φt), if t [ r, ], t)β φ) η g, φ)) Ny)t) = Γβ) t ) Γα) t α 3.2) g,y ) d ) Γαβ) t αβ f,y ) d, if t J. To how that the operator N i a contraction, let y, z C[ r, b], R). Then we have Ny)t) Nz)t) t ) α g, y ) g, z ) d Γα) t t ) αβ f, y ) f, z ) d Γα β) k t ) α y z C d Γα) l t t ) αβ y z C d Γα β) Conequently we obtain Ny) Nz) [ r,b] k t)α Γα ) y z l t)αβ [ r,b] Γα β ) y z [ r,b]. [ k b) α Γα ) l ] b)αβ y z [ r,b], Γα β ) which, in view of 3.), implie that N i a contraction. Hence N ha a unique fixed point by Banach contraction principle. Thi, in turn, how that problem.).3) ha a unique olution on [ r, b]. 4. Exitence reult In thi ection, we etablih our exitence reult for the IVP.).3). The firt reult i baed on Leray-Schauder nonlinear alternative. Lemma 4. Nonlinear alternative for ingle valued map [5]). Let E be a Banach pace, C a cloed, convex ubet of E, U an open ubet of C and 0 U. Suppoe that F : U C i a continuou, compact that i, F U) i a relatively compact ubet of C) map. Then either i) F ha a fixed point in U, or ii) there i a u U the boundary of U in C) and λ 0, ) with u = λf u). For the next theorem we need the following aumption: A3) f, g : J C r R are continuou function;

5 EJDE-207/36 EXISTENCE AND UNIQUENESS OF SOLUTIONS 5 A4) there exit a continuou nondecreaing function ψ : [0, ) 0, ) and a function p CJ, R ) uch that ft, u) pt)ψ u C )for each t, u) J C r ; A5) there exit contant d < Γα ) b) α and d 2 0 uch that gt, u) d u C d 2, t J, u C r. A6) there exit a contant M > 0 uch that ) M where M 0 d b)α Γα) >, d2 b)α Γα) ψm) p Γαβ) b)αβ M 0 = φ C [ η d φ C d 2 ] Γβ ). Theorem 4.2. Under aumption A3) A6) hold, IVP.).3) ha at leat one olution on [ r, b]. Proof. We hall how that the operator N : C[ r, b], R) C[ r, b], R) defined by 3.2) i continuou and completely continuou. Step : N i continuou. Let {y n } be a equence uch that y n y in C[ r, b], R). Then Ny n )t) Ny)t) Γα) Γα β) b t Γα) Γα β) t ) α g, y n ) g, y ) d t b g, y n.) g, y. ) Γα) ) αβ f, yn ) f, y ) d ) α up g, y n ) g, y ) d [,b] t ) αβ up b f, y n.) f, y. ) Γα β) b b)α g, y n. ) g, y. ) Γα ) [,b] t b)αβ f, y n. ) f, y. ). Γα β ) Since f, g are continuou function, we have Ny n ) Ny) f, y n ) f, y ) d ) α d t ) αβ d b)α g, y n. ) g, y. ) Γα ) a n. b)αβ f, y n. ) f, y. ) Γα β ) 0

6 6 B. AHMAD, S. K. NTOUYAS EJDE-207/36 Step 2: N map bounded et into bounded et in C[ r, b], R). Indeed, it i ufficient to how that for any θ > 0 there exit a poitive contant l uch that for each y B θ = {y C[ r, b], R) : y θ}, we have Ny) l. By A4) and A5), for each t J, we have Thu Ny)t) φ C [ η d φ C d 2 ] Γβ ) t ) α g, y ) d Γα) t t ) αβ f, y ) d Γα β) φ C [ η d φ C d 2 ] Γβ ) d y [ r,b] d t 2 t ) α d Γα) ψ y [ r,b]) p t ) αβ d Γα β) φ C [ η d φ C d 2 ] Γβ ) d y [ r,b] d 2 Γα ) b) α ψ y [ r,b]) p b) αβ. Γα β ) Ny) φ C [ η d φ C d 2 ] Γβ ) d θ d 2 Γα ) b)α ψθ) p Γα β ) b)αβ := l. Step 3: N map bounded et into equicontinuou et of C[ r, b], R). Let t, t 2 J, t < t 2, B θ be a bounded et of C[ r, b], R) a in Step 2, and let y B θ. Then Ny)t 2 ) Ny)t ) η d φ C d 2 [ t 2 ) β t ) β ] Γβ ) t [ t ) α 2 t ) α ] g, y ) d Γα) 2 t ) 2 α g, y ) d Γα) t t [ t ) αβ 2 t ) αβ ] f, y ) d Γα β) t2 t ) 2 αβ f, y ) d Γα β) t η d φ C d 2 [ t 2 ) β t ) β ] Γβ )

7 EJDE-207/36 EXISTENCE AND UNIQUENESS OF SOLUTIONS 7 d θ d 2 Γα) d θ d 2 Γα) ψθ) p Γα β) ψθ) p Γα β) 2 t 2 [ t ) α 2 t ) α 2 d [ t ) αβ 2 t ) αβ 2 d t η d φ C d 2 [ t 2 ) β t ) β ] Γβ ) t ) α ] d t ) αβ ] d d θ d 2 Γα ) [ t 2 ) α t ) α t 2 /t α ] ψθ) p Γα β ) [ t2 ) αβ t ) αβ t2 /t αβ ]. A t t 2 the right-hand ide of the above inequality tend to zero. The equicontinuity for the cae t < t 2 0 and t 0 t 2 i obviou. A a conequence of Step to 3, it follow by the Arzelá-Acoli theorem that N : C[ r, b], R) C[ r, b], R) i continuou and completely continuou. Step 4: We how that there exit an open et U C[ r, b], R) with y λny) for λ 0, ) and y U. Let y C[ r, b], R) and y = λny) for ome 0 < λ <. Then, for each t J, we have t)β yt) = λ φ) η g, φ))) Γβ ) Γα) Γα β) t ) αβ f, y ) d By our aumption, for each t J, we obtain ). yt) φ C [ η d φ C d 2 ] Γβ ) d y [ r,b] d t 2 t ) α d Γα) t t ) αβ p)ψ y C ) d Γα β) t ) α g, y ) d φ C [ η d φ C d 2 ] Γβ ) d y [ r,b] d 2 b) α Γα ) p ψ y [ r,b] ) b) αβ, Γα β ) which can be expreed a M 0 ) d b)α Γα) y [ r,b]. d2 b)α Γα) ψ y [ r,b] ) p Γαβ) b)αβ

8 8 B. AHMAD, S. K. NTOUYAS EJDE-207/36 In view of A6), there exit M uch that y [ r,b] M. Let u et U = {y C[ r, b], R) : y [ r,b] < M}. Note that the operator N : U C[ r, b], R) i continuou and completely continuou. From the choice of U, there i no y U uch that y = λny for ome λ 0, ). Conequently, by the nonlinear alternative of Leray-Schauder type Lemma 4.), we deduce that N ha a fixed point y U which i a olution of the problem.)-.3). Thi complete the proof. The econd exitence reult i baed on Kranoelkii fixed point theorem. Lemma 4.3 Kranoelkii fixed point theorem [20]). Let S be a cloed, bounded, convex and nonempty ubet of a Banach pace X. Let A, B be the operator uch that a) Ax Bx S whenever x, y S; b) A i compact and continuou; c) B i a contraction mapping. Then there exit z S uch that z = Az Bz. Theorem 4.4. Aume that A2) and A3) hold. In addition we aume that A7) ft, x) µt), gt, x) νt), for all t, x) J R, and µ, ν CJ, R ). Then problem.)-.3) ha at leat one olution on [ r, b], provided k b) α <. 4.) Γα ) Proof. We define the operator G and G 2 by 0, if t [ r, ], G yt) = t)β η g, φ)) Γβ) t ) Γα) t α g,y ) d, if t J. φt), if t [ r, ], G 2 yt) = φ) Γαβ) t ) αβ f,y ) d, if t J. Setting up t [,b] µt) = µ, up t [,b] νt) = ν and chooing ρ φ C [ η ν ] Γβ ) ν b)α Γα ) 4.2) 4.3) µ b) αβ Γα β ), 4.4) we conider B ρ = {y C[ r, b], R) : y ρ}. For any y, z B ρ, we have G yt) G 2 zt) t)β up {η g, φ)) t [,b] Γβ ) Γα) φ) Γα β) t φ C [ η ν ] Γβ ) ρ. ) αβ f, z ) d ν b)α Γα ) t ) α g, y ) d } µ b) αβ Γα β ) Thi how that G yg 2 z B ρ. Uing 4.) it i eay to ee that G i a contraction mapping.

9 EJDE-207/36 EXISTENCE AND UNIQUENESS OF SOLUTIONS 9 Continuity of f implie that the operator G 2 i continuou. Alo, G 2 i uniformly bounded on B ρ a b) αβ G 2 y φ C µ Γα β ). Now we prove the compactne of the operator G 2. We define f = up ft, y) <, t,y) [,b] B ρ and conequently, for t, t 2 [, b], t < t 2, we have G 2 yt 2 ) G 2 yt ) f t t ) αβ 2 t ) αβ d Γα β) f t2 t ) αβ 2 d Γα β) t f Γα β ) [ t 2 ) αβ t ) αβ t 2 /t αβ ], which i independent of y and tend to zero a t 2 t 0. Thu, G 2 i equicontinuou. So G 2 i relatively compact on B ρ. Hence, by the Arzelá-Acoli theorem, G 2 i compact on B ρ. Thu all the aumption of Lemma 4.3 are atified. So the concluion of Lemma 4.3 implie that the problem.)-.3) ha at leat one olution on [ r, b] 5. Example In thi ection we give an example to illutrate the uefulne of our main reult. Let u conider the fractional functional differential equation, D /2[ D 3/4 yt) e t y t ] C y t C 8 e t = y t C ) 2 y t C ) e t, 5.) t J := [, e], Let ft, x) = yt) = φt), t [ r, ], 5.2) D 3/4 y) = /2. 5.3) x e t x ), gt, x) = 2 x) 8 e t, t, x) [, e] [0, ). x For x, y [0, ) and t J, we have ft, x) ft, y) = x 2 x y x y = y 2 x) y) x y, 2 and gt, x) gt, y) = e t x 8 e t x y = e t x y y 8 e t x) y) e x y. ee 8)

10 0 B. AHMAD, S. K. NTOUYAS EJDE-207/36 Hence condition A) and A2) hold with l = /2 and k = e ee8) repectively. k b)α Γα) l b)αβ Since Γαβ) <, therefore, by Theorem 3.2, problem 5.)-5.3) ha a unique olution on [ r, e]. Alo ft, x) 2e t )/2 = µt), gt, x) e t )/8 e t ) = νt) and k b) α /Γα ) = 2e )/ πee 8) <. Clearly the aumption of Theorem 4.4 are atified. Conequently, by the concluion of Theorem 4.4, there exit a olution of the problem 5.)-5.3) on [ r, e]. 6. Initial value integral condition cae The reult of thi paper can be extended to the cae of an initial value integral condition of the form D β y) = b h, y )d, 6.) where h : J C[ r, 0], R) R i a given function. In thi cae η will be replaced with b h, y )d in 3.2) and the tatement of the exitence and uniquene reult for the problem.).2) 6.) can be formulated a follow. Theorem 6.. Aume that the condition A) and A2) hold. Further, we uppoe that A8) there exit a nonnegative contant m uch that ht, u) ht, v) m u v C, for t J and every u, v C r. Then the problem.)-.2)-6.) ha a unique olution on [ r, b] if mb ) b) β Γβ ) k b)α Γα ) l b)αβ Γα β ) <. We do not provide the proof of the above theorem a it i imilar to that of Theorem 3.2. The ana form of the exitence reult: Theorem 4.2 and 4.4 for the problem.)-.2)-6.) can be contructed in a imilar manner. Reference [] R. P. Agarwal, S. Hritova, D. O Regan; A urvey of Lyapunov function, tability and impulive Caputo fractional differential equation, Fract. Calc. Appl. Anal., 9 206), [2] B. Ahmad, S. K. Ntouya; Nonlocal fractional boundary value problem with lit-trip boundary condition, Fract. Calc. Appl. Anal., 8 205), [3] B. Ahmad, S. K. Ntouya; Initial value problem of fractional order Hadamard-type functional differential equation, Electron. J. Differential Equ., Vol ), No. 77, 9 pp. [4] B. Ahmad, S. K. Ntouya, J. Tariboon; Fractional differential equation with nonlocal integral and integerfractional-order Neumann type boundary condition, Mediterr. J. Math., 3 206), [5] B. Ahmad, S. K. Ntouya; Initial value problem for functional and neutral functional Hadamard type fractional differential incluion, Mikolc Math. Note, 7 206), [6] M. Benchohra, J. Henderon, S. K. Ntouya, A. Ouahab; Exitence reult for fractional order functional differential equation with infinite delay, J. Math. Anal. Appl ), [7] P. L. Butzer, A. A. Kilba, J. J. Trujillo; Compoition of Hadamard-type fractional integration operator and the emigroup property, J. Math. Anal. Appl ), [8] P. L. Butzer, A. A. Kilba, J. J. Trujillo; Fractional calculu in the Mellin etting and Hadamard-type fractional integral, J. Math. Anal. Appl., ), -27.

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