INITIAL VALUE PROBLEMS OF FRACTIONAL ORDER HADAMARD-TYPE FUNCTIONAL DIFFERENTIAL EQUATIONS

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1 Electronic Journal of Differential Equation, Vol ), No. 77, pp. 9. ISSN: URL: or ftp ejde.math.txtate.edu INITIAL VALUE PROBLEMS OF FRACTIONAL ORDER HADAMARD-TYPE FUNCTIONAL DIFFERENTIAL EQUATIONS BASHIR AHMAD, SOTIRIS K. NTOUYAS Abtract. The Banach fixed point theorem and a nonlinear alternative of Leray-Schauder type are ued to invetigate the exitence and uniquene of olution for fractional order Hadamard-type functional and neutral functional differential equation.. Introduction Thi article concern the exitence of olution for initial value problem IVP for hort) of fractional order functional and neutral functional differential equation. In the firt problem, we conider fractional order functional differential equation: D α yt) = ft, y t ), for each t J = [, b], 0 < α <,.) yt) = φt), t [ r, ],.2) where D α i the Hadamard fractional derivative, f : J C[ r, 0], R) R i a given function and φ C[ r, ], R) with φ) = 0. For any function y defined on [ r, b] and any t J, we denote by y t the element of C[ r, 0], R) and i defined by y t θ) = yt + θ), θ [ r, 0]. Here y t ) repreent the hitory of the tate from time t r up to the preent time t. The econd problem i devoted to the tudy of fractional neutral functional differential equation: D α [yt) gt, y t )] = ft, y t ), t J,.3) yt) = φt), t [ r, ],.4) where f and φ are a in problem.).2), and g : J C[ r, 0], R) R i a given function uch that g, φ) = 0. Functional and neutral functional differential equation arie in a variety of area of biological, phyical, and engineering application, ee, for example, the book [5, 7] and the reference therein. Differential equation of fractional order have recently proved to be valuable tool in the modeling of many phenomena 2000 Mathematic Subject Claification. 34A08, 34K05. Key word and phrae. Fractional differential equation; functional differential equation; Hadamard fractional differential equation; exitence; fixed point. c 205 Texa State Univerity - San Marco. Submitted January 0, 205. Publihed March 26, 205.

2 2 B. AHMAD, S. K. NTOUYAS EJDE-205/77 in variou field of cience and engineering. Indeed, we can find numerou application in variou field uch a vicoelaticity, electrochemitry, control, porou media, electromagnetic, etc. ee [6, 8, 20, 2]). Fractional differential equation involving Riemann-Liouville and Caputo type fractional derivative have extenively been tudied by everal reearcher [, 2, 3, 4, 5, 6, 2, 22, 23]. However, the literature on Hadamard type fractional differential equation i not enriched yet. The fractional derivative due to Hadamard, introduced in 892 [4], differ from the aforementioned derivative in the ene that the kernel of the integral in the definition of Hadamard derivative contain logarithmic function of arbitrary exponent. A detailed decription of Hadamard fractional derivative and integral can be found in [9, 0, ] and reference cited therein. The IVP.).2) and.3).4) in the cae of infinite delay and Riemann- Liouville fractional derivative wa tudied in [8]. IVP for hybrid Hadamard fractional differential equation wa tudied in [7]. Here we tudy the problem involving Hadamard-type fractional derivative. Our approach i baed on the Banach fixed point theorem and nonlinear alternative of Leray-Schauder type [3]. 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.).2), while the exitence reult for the problem.3).4) are preented in Section 4. Finally, an example i given in Section 5 for illutration of the reult. 2. 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 y := up{ yt) : t J}. Alo C[ r, 0], R) i endowed with the norm φ C := up{ φθ) : r θ 0}. Definition 2. [6]). The Hadamard derivative of fractional order q for a function g : [, ) R i defined a D q gt) = t d ) n log t ) n q g) d, n < q < n, n = [q] +, Γn q) dt where [q] denote the integer part of the real number q and log ) = log e ). Definition 2.2 [6]). The Hadamard fractional integral of order q for a function g i defined a I q gt) = log t ) q g) d, q > 0, Γq) provided the integral exit. 3. Functional differential equation Definition 3.. A function y C[ r, b], R), i aid to be a olution of.).2) if y atifie the equation D α yt) = ft, y t ) on J, and the condition yt) = φt) on [ r, ].

3 EJDE-205/77 INITIAL VALUE PROBLEMS OF FRACTIONAL ORDER 3 Our firt exitence reult for.).2) i baed on the Banach contraction principle. Theorem 3.2. Let f : J C[ r, 0], R) R. Aume that H0) there exit l > 0 uch that If [ r, b]. ft, u) ft, v) l u v C, llog b)α Γα+) for t J and every u, v C[ r, 0], R). <, then there exit a unique olution for.).2) on the interval Proof. Tranform problem.).2) into a fixed point problem. operator N : C[ r, b], R) C[ r, b], R) defined by Conider the Ny)t) = { φt), if t [ r, ], ) log t α f,y ) d, if t [, b]. t Γα) Let y, z C[ r, b], R). Then, for t J, Conequently, Ny)t) Nz)t Γα) l Γα) log t ) α f, y ) f, z ) d log t ) α y d z C l Γα) y z [ r,b] llog t)α Γα + ) y z [ r,b]. log t ) α d 3.) Ny) Nz) [ r,b] llog b)α Γα + ) y z [ r,b], which implie that N i a contraction, and hence N ha a unique fixed point by Banach contraction principle. Our econd exitence reult for.).2) i baed on the nonlinear alternative of Leray-Schauder. Lemma 3.3 Nonlinear alternative for ingle valued map [3]). 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). Theorem 3.4. Aume that the following hypothee hold: H) f : J C[ r, 0], R) R i a continuou function; H2) there exit a continuou nondecreaing function ψ : [0, ) 0, ) and a function p C[, b], R + ) uch that ft, u) pt)ψ u C ) for each t, u) [, b] C[ r, 0], R);

4 4 B. AHMAD, S. K. NTOUYAS EJDE-205/77 H3) there exit a contant M > 0 uch that M ψm) p log b) α Γα+) >. Then.).2) ha at leat one olution on [ r, b]. Proof. We conider the operator N : C[ r, b], R) C[ r, b], R) defined by 3.). We hall how that the operator N 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). Let η > 0 uch that y n η. Then Ny n )t) Ny)t) log t ) α f, yn ) f, y ) d Γα) Γα) b log t ) α up f, y n.) f, y. ) Γα) Since f i a continuou function, we have [,b] b log b)α f, y n. ) f, y. ). αγα) f, y n ) f, y ) d log t ) α d Ny n ) Ny) log b)α f, y n. ) f, y. ) Γα + ) 0 a n. Step 2: N map bounded et into bounded et in C[ r, b], R). Indeed, it i ufficient to how that for any η > 0 there exit a poitive contant l uch that for each y B η = {y C[ r, b], R) : y η }, we have Ny) l. By H2), for each t [, b], we have Ny)t) log t ) α f, y ) d Γα) Thu ψ y [ r,b]) p Γα) log t ψ y [ r,b]) p log b) α. Γα + ) Ny) ψη ) p Γα + ) log b)α := l. ) α d Step 3:: N map bounded et into equicontinuou et of C[ r, b], R). Let t, t 2 [, b], t < t 2, B η be a bounded et of C[ r, b], R) a in Step 2, and let y B η. Then Ny)t 2 ) Ny)t ) [ log t ) α 2 log t ) α ] f, y ) d Γα) + 2 log t ) 2 α f, y ) d Γα) t

5 EJDE-205/77 INITIAL VALUE PROBLEMS OF FRACTIONAL ORDER 5 ψη ) p Γα) + ψη ) p Γα) [ log t ) α 2 log t ) α 2 d. 2 t log t ) α ] d 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. In 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 <. Thu, for each t [, b] yt) = λ Γα) log t ) α f, y ) d ). By aumption H2), for each t J, we obtain yt) log t ) α p)ψ y C ) d Γα) p ψ y [ r,b] ) log b) α, Γα + ) which can be expreed a y [ r,b] ψ y [ r,b] ) p log b) α Γα+). In view of H4), 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 3.3), we deduce that N ha a fixed point y U which i a olution of.)-.2). Thi complete the proof. 4. Neutral functional differential equation In thi ection, we etablih the exitence reult for.3).4). Definition 4.. A function y C[ r, b], R), i aid to be a olution of.3).4) if y atifie the equation D α [yt) gt, y t )] = ft, y t ) on J, and the condition yt) = φt) on [ r, ]. Theorem 4.2 Uniquene reult). Aume that H0) and the following condition hold: A) there exit a nonnegative contant c uch that gt, u) gt, v) c u v C, for every u, v C[ r, 0], R). If llog b)α c + <, 4.) Γα + ) then there exit a unique olution for.3).4) on the interval [ r, b].

6 6 B. AHMAD, S. K. NTOUYAS EJDE-205/77 Proof. Conider the operator N : C[ r, b], R) C[ r, b], R) defined by: φt), if t [ r, ], N y)t) = α f, 4.2) gt, y t ) + log ) t y )d, if t [, b]. Γα) To how that the operator N i a contraction, let y, z C[ r, b], R). Then we have N y)t) N z)t) gt, y t ) gt, z t ) + Γα) c y t z t C + l Γα) c y z [ r,b] + l Γα) y z [ r,b] c y z [ r,b] + Conequently we obtain f, y ) f, z ) log ) t α d log t ) α y z C d llog t)α Γα + ) y z [ r,b]. N y) N z) [ r,b] [ c + log t ) α d llog b)α ] y z [ r,b], Γα + ) which, in view of 4.), implie that N i a contraction. Hence N ha a unique fixed point by Banach contraction principle. Thi, in turn, how that the problem.3).4) ha a unique olution on [ r, b]. Theorem 4.3. Aume that H) H2) hold. Further we uppoe that H4) the function g i continuou and completely continuou, and for any bounded et B in C[ r, b], R), the et {t gt, y t ) : y B} i equicontinuou in C[, b], R), and there exit contant 0 d <, d 2 0 uch that gt, u) d u C + d 2, H5) there exit a contant M > 0 uch that t [, b], u C[ r, 0], R). d )M d 2 + p ψm) Γα+) log b) >. α Then.3).4) ha at leat one olution on [ r, b]. Proof. We conider the operator N : C[ r, b], R) C[ r, b], R) defined by 4.2) and how that the operator N i continuou and completely continuou. Uing H3), it uffice to how that the operator N 2 : C[ r, b], R) C[ r, b], R) defined by φt), t [ r, ], N 2 y)t) = α f, log ) t y ) d, t [, b], Γα) i continuou and completely continuou. The proof i imilar to that of Theorem 3.4. So we omit the detail.

7 EJDE-205/77 INITIAL VALUE PROBLEMS OF FRACTIONAL ORDER 7 We now how that there exit an open et U C[ r, b], R) with y λn y) for λ 0, ) and y U. Let y C[ r, b], R) and y = λn y) for ome 0 < λ <. Thu, for each t [, b], we have yt) = λ gt, y t ) + Γα) For each t J, it follow by H2) and H3) that which yield yt) d y t C + d 2 + Γα) In conequence, we obtain log ) t α f, ) y ) d. log t ) α p)ψ y C ) d d y t C + d 2 + p ψ y [ r,b] ) log b) α, Γα + ) d ) y [ r,b] d 2 + p ψ y [ r,b] ) log b) α. Γα + ) d ) y [ r,b] d 2 + p ψ y [ r,b]) Γα+) log b). α In view of H4), 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 u U uch that y = λn y for ome λ 0, ). Thu, by the nonlinear alternative of Leray-Schauder type Lemma 3.3), we deduce that N ha a fixed point y U which i a olution of problem.3)-.4). Thi complete the proof. 5. An example In thi ection we give an example to illutrate the uefulne of our main reult. Let u conider the fractional functional differential equation, Let D /2 y t C yt) =, t J := [, e], 5.) 2 + y t C ) yt) = φt), t [ r, ]. 5.2) ft, x) = x, t, x) [, e] [0, ). 2 + 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 llog b)α Hence the condition H0) hold with l = /2. Since Γα+) = π <, by Theorem 3.2, problem 5.)-5.2) ha a unique olution on [ r, e].

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9 EJDE-205/77 INITIAL VALUE PROBLEMS OF FRACTIONAL ORDER 9 Sotiri K. Ntouya Department of Mathematic, Univerity of Ioannina, 45 0 Ioannina, Greece. Nonlinear Analyi and Applied Mathematic NAAM)-Reearch Group, Department of Mathematic, Faculty of Science, King Abdulaziz Univerity, P.O. Box 80203, Jeddah 2589, Saudi Arabia addre: ntouya@uoi.gr