EXISTENCE AND UNIQUENESS RESULT OF SOLUTIONS TO INITIAL VALUE PROBLEMS OF FRACTIONAL DIFFERENTIAL EQUATIONS OF VARIABLE-ORDER

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1 Journal of Fractional Calculus and Applications, Vol. 4(1) Jan. 213, pp ISSN: EXISENCE AND UNIQUENESS RESUL OF SOLUIONS O INIIAL VALUE PROBLEMS OF FRACIONAL DIFFERENIAL EQUAIONS OF VARIABLE-ORDER SHUQIN ZHANG Abstract. In this work, an initial value problem is discussed for a fractional differential equation of variable-order. By means of some analysis techniques and Arzela-Ascoli theorem, existence result of solution is obtained; Using the upper solutions and lower solutions and monotone iterative method, uniqueness existence results of solutions are obtained. 1. Introduction he fractional operators (fractional derivatives and integrals) refer to the differential and integral operators of arbitrary order, and fractional differential equations refer to those containing fractional derivatives. he former are the generalization of integer-order differential and integral operators and the latter, the generalization of differential equations of integer order. he fractional operators of variable order, which fall into a more complex category, are the derivatives and integrals whose order is the function of certain variables. In recent years, fractional operators and fractional differential equations of variable order have been applied in engineering more and more frequently, For the examples and details, see [1]-[14] and the references therein. heir extensive applications urgently call for systematic studies on the existence, uniqueness of solutions to initial value problems of these equations. Research in this area is at the trailblazing stage and has so far but produced a very limited number of published papers dealing with relatively simple problems with limited methods, such as [15], [16]. 2 Mathematics Subject Classification. 26A33, 34A12. Key words and phrases. fractional calculus of variable-order, fractional differential equation, Arzela-Ascoli theorem, existence of solution, lower solution, upper solution, monotone iterative, uniqueness of solution. Submitted July. 6, 212. Published Jan. 3, 213. Research supported by the NNSF of China ( ), the Ministry of Education for New Century Excellent alent Support Program(NCE-1-725) and the Fundamental Research Funds for the Central Universities(29QS6). 82

2 JFCA-213/4 EXISENCE AND UNIQUENESS RESUL OF SOLUIONS 83 he following are several definitions of fractional integral and fractional derivatives of variable-order, which can be founded in [14]: a+ f(t) a (t s) p(t) 1 f(s)ds, p(t) >, t > a, (1) Γ(p(t)) where Γ( ) denotes the Gamma function, < a < +, provided that the righthand side is pointwise defined. a+ f(t) a provided that the right-hand side is pointwise defined. a + f(t) a provided that the right-hand side is pointwise defined. a+ f(t) dn dt n In p(t) a+ f(t) dn dt n (t s) p(s) 1 f(s)ds, p(t) >, t > a, (2) Γ(p(s)) (t s) p(t s) 1 f(s)ds, p(t) >, t > a, (3) Γ(p(t s)) a (t s) n 1 p(t) f(s)ds, t > a, (4) Γ(n p(t)) where n 1 < p(t) < n, t > a, n N, provided that the right-hand side is pointwise defined. a+ f(t) dn dt n In p(t) a+ f(t) dn dt n a (t s) n 1 p(s) f(s)ds, t > a, (5) Γ(n p(s)) where n 1 < p(t) < n, t > a, n N, provided that the right-hand side is pointwise defined. a f(t) dn + dt n In p(t) a+ f(t) dn dt n a (t s) n 1 p(t s) f(s)ds, t > a, (6) Γ(n p(t s)) where n 1 < p(t) < n, t > a, n N, provided that the right-hand side is pointwise defined. In particular, when p(t) is a constant function, i.e. p(t) q(q is a finite positive constant), then a+, a+ are the usual Riemann-Liouville fractional calculus[?]. It is well known that fractional calculus D q a+, I q a+ have some very important properties, which play very important role in considering existence of solutions of fractional differential equation denoted by D q a+, by means of nonlinear analysis method. Such as, the following some properties, which can be founded in [17]: Proposition 1.1.([17]) he equality I γ a+i δ a+f(t) I γ+δ a+ f(t), γ >, δ > holds for f L(a, b). Proposition 1.2.([17]) he equality D γ a+i γ a+f(t) f(t), γ > holds for f L(a, b). Proposition 1.3.([17]) Let α >. hen the differential equation has unique solution D α a+u u(t) c 1 (t a) α 1 + c 2 (t a) α c n (t a) α n, c i R, i 1, 2,, n, here n 1 < α n. Proposition 1.4.([17]) Let α >, u L(a, b), Da+u α L(a, b). following equality holds I α a+d α a+u(t) u(t) + c 1 (t a) α 1 + c 2 (t a) α c n (t a) α n, hen the

3 84 SHUQIN ZHANG JFCA-213/3 c i R, i 1, 2,, n, here n 1 < α n. But, in general, these properties don t hold for fractional calculus of variableorder a+, a+ defined by (1)-(6). For example, a+ I q(t) a+ f(t) +q(t) a+ f(t), p(t) >, q(t) >, f L(a, b), (7) where a+ denote one of fractional integrals defined by (1)-(3). 2, t 2 Example 1.1. Let p(t) t, t 6, q(t) 1, 2 < t 3, t 6. We calculate + + Iq(t) + f(t) (t s) p(t) 1 Γ(p(t)) we see that + Iq(t) + f(t) t f(t) and Ip(t)+q(t) + defined by (1). (t s) p(t) 1 Γ(p(t)) (t s) p(t) 1 Γ(p(t)) s s s t, 3 < t 6, (s τ) q(s) 1 f(τ)dτds Γ(q(s)) f(t) 1, (s τ) q(s) 1 (t s) p(t) 1 s dτds + Γ(q(s)) 2 Γ(p(t)) (s τ) 2 1 Γ(2) (t s) p(t) 1 s 2 (t s) p(t) 1 s ds + 2Γ(p(t)) 2 Γ(p(t)) +q(t) + f(t) 3 +q(t) + f(t) t3 we see easily that (3 s) 3 1 s Γ(3) (t s) p(t) 1 s dτds + 2 Γ(p(t)) (t s) p(t)+q(t) 1 Γ(p(t) + q(t)) f(s)ds, 3 (3 s) 3 1 s ds + 2 Γ(3) (3 s) 3 1 s ds 5 Γ(3) , (3 s) p(3)+q(3) 1 3 f(s)ds Γ(p(s) + q(3)) + Iq(t) + f(t) t3 +q(t) + f(t) t3. (s τ) q(s) 1 dτds, Γ(q(s)) (s τ) 1 1 dτds Γ(1) (3 s) ds 27 Γ(3 + 1) 8 According to (7), we can see that Proposition don t hold for a+ and a+ defined by (1)-(6). Example 1.2. Let < p(t) < 1, t >. By calculating, we have that Γ(p(t)) + Ip(t) + 1 d dt I1 p(t) d dt t p(t) (t s) p(t) 1 ds (p(t))γ(p(t)), t >, 1 Γ(1 p(t)) (t s) p(t) s p(s) ds 1, p(s)γ(p(s)) (s τ) q(s) 1 dτds Γ(q(s)) (s τ) q(s) 1 dτds Γ(q(s))

4 JFCA-213/4 EXISENCE AND UNIQUENESS RESUL OF SOLUIONS 85 and + Dp(t) + 1 d Ip(t) ( d dt (1 p(t))γ(1 p(t)) ) 1. dt I1 p(t) t 1 p(t) Remark 1.1. For fractional integral of variable-order defined by (5)-(6), we can t easily calculate out fractional integral a+ of some functions f(t), for example, we don t know that what a+ 1 (t s) p(s) 1 a Γ(p(s)) ds and a+ 1 a (t s) p(t s) 1 Γ(p(t s)) ds equal. herefore, without those properties like as Propositions 1.1, 12, 1.3 and 1.4, ones can not transform a fractional differential equation of variable-order into an equivalent integral equation, so that one can consider existence of solutions of a fractional differential equation of variable-order, by means of nonlinear functional analysis method. here also has more complex fractional calculus of variable-order, whose order function p(t) of (1)-(6) are replaced by p(t, f(t)), please see [1], [15], [16]. In [15], authors considered the solution existence of the following variable order fractional differential equations { p(t,x(t)) D c+ x(t) f(t, x(t)), c < t b (8) x(c) x, where D p(t,x(t)) c+ is a fractional derivative of variable-order defined by D p(t,x(t)) c+ x(t) d dt c (t s) p(s,x(s)) x(s)ds, t > c. (9) Γ(1 p(s, x(s))) In [15], authors transformed (8) into one equivalent integral equation x(t) I p(t,x(t)) c+ f(t, x(t)) c (t s) p(s,x(s)) 1 f(s, x(s))ds, c t b, (1) Γ(p(s, x(s))) and then, using approximated method, authors obtained existence result of solution for problem (8) with 1 2 q(t, x) 1, c t b, x R. In my opinion, the problem and analysis techniques are interesting and meaning, but, there had a critical wrong, that is, the transformation( from (8) to (1)) is unsuitable, because fractional calculus D p(t,x(t)) c+, I p(t,x(t)) c+ don t usually have properties like Propositions 1.3 and 1.4. Among of those analysis, the sequence (6) has little mistakes, since ones can t know whether such sequence exists. As well, the initial value condition x(c) x isn t suitable when x, because, when p(t, x(t)) is a constant function, i.e. p(t) q( < q < 1 is a finite positive constant), then D p(t,x(t)) c+ is the usual Riemann-Liouville fractional derivative D q c+. From [17], we know that Riemann-Liouville fractional derivative of order < q < 1 of constant x is not zero, but is x (t c) q Γ(1 q), t > c, as a result, fractional differential equation(involving Riemann-Liouville fractional derivative) can not have such x(c) x initial value condition, but is initial value condition (t c) 1 q x(t) tc or I 1 q c+ x(t) tc ( (t c) 1 q x(t) tc and I 1 q c+ x(t) tc can transform each other, see [17]). Hence, in some degree, problem (8) is not a suitable problem(expect x ). In this paper, based on characters of fractional derivative of variable-order, by means of some analysis techniques, we will consider existence and uniqueness of

5 86 SHUQIN ZHANG JFCA-213/3 solution to initial value problems for fractional differential equation of variableorder(ivp for short) { q(t,x(t)) D + x(t) f(t, x), < t, < < + (11) x(), where D q(t,x(t)) + denotes fractional derivative of variable-order defined by (9), < q(t, x(t)) q < 1, t, x R, and f : [, ] R R is a continuous function. he rest of this paper is organized as follows. In Section 2, we give some preparation results. In Section 3, the existence results of solutions for IVP (11) are presented. In section 4, uniqueness existence result od solution for a particular case of IVP(11). 2. Preliminaries We assume that (H 1 ) q : [, ] R (, q ], < q < 1, is a continuous function; (H 2 ) f : [, ] R R is a continuous function. It follows from the continuity of compose functions that Γ(q(t, x(t))) is continuous on [, ], when q satisfies assumption condition (H 1 ). he following results will play very important role in proving our existence result of solution to problems (11). Let δ is an arbitrary small positive constant. Lemma 2.1. Let (H 1 ) hold. And let x n, x C[, ], assume that x n (t) x(t), t [, ] as n, then (t s) q(s,x n(s)) Γ(1 q(s, x n (s))) x n(s)ds as n. Proof. We see that hus, for < < +, we let We let (t s) q(s,x(s)) x(s)ds, t [δ, ], (12) Γ(1 q(s, x(s))) if < 1, then q(s,x(s)) q, (13) if 1 < < +, then q(s,x(s)) < 1. (14) M max t x(t) +1, M 1 max t x n(t) +1, L max{ q, 1}. (15) max t, x n M 1 1 Γ(1 q(t, x n (t))) +1. By the convergence of x n, for (1 q )ε 3L (ε is arbitrary small positive number), there exists N N such that x n (t) x(t) < (1 q )ε 3L, t [, ], n N. (16) Since (t s) q(s,x(s)), δ t s, is continuous with respect to its exponent ε q(s, x(s)), for 3ML, when n n, it holds (t s) q(s,xn(s)) (t s) q(s,x(s)) ε <, δ t s, (17) 3ML

6 JFCA-213/4 EXISENCE AND UNIQUENESS RESUL OF SOLUIONS 87 also, by continuity of 1 Γ(1 q(s,x(s))), for (1 q )ε 3M, when n n, it holds 1 Γ(1 q(s, x n (s))) 1 Γ(1 q(s, x(s))) < (1 q )ε 3M, s. (18) Hence, from (13)-(18), we have that + + L(1 q )ε 3L (t s) q(s,x n(s)) t δ Γ(1 q(s, x n (s))) x n(s)ds (t s) q(s,x n(s)) Γ(1 q(s, x n (s))) x n(s) x(s) ds (t s) q(s,xn(s)) (t s) q(s,x(s)) x(s) ds Γ(1 q(s, x n (s))) (t s) q(s,x(s)) Γ(1 q(s, x(s))) x(s)ds (t s) q(s,x(s)) 1 Γ(1 q(s, x n (s))) 1 Γ(1 q(s, x(s))) x(s) ds (t s) q(s,x n(s)) ds + MLε 3ML ds < + M(1 q )ε 3M (1 q )ε 3 + (1 q )ε 3 (1 q )ε 3 (t s) q(s,x(s)) ds q(s,xn(s)) ( t s ) q(s,x n(s)) ds + ε 3 q(s,x(s)) ( t s ) q(s,x(s)) ds ( t s ds ) q + ε 3 (1 q )ε (t s) q ds + ε 3 1 q 3 ε (t 1 q δ 1 q ) + ε ε (t δ) q 3 ε 1 q + ε 1 q ε q q ε 3 + ε 3 + ε 3 ε, ds + (1 q )ε 3 ds + (1 q )ε 3 1 q ds (t 1 q δ 1 q ) 3 1 q ( t s ) q ds (t s) q ds which implies that (12) holds. By the similar arguments, we can know that Lemma 2.2. Let (H 2 ) hold. And let x n, x C[, ], assume that x n (t) x(t), t [, ] as n, then f(s, x n (s))ds f(s, x(s))ds, t [δ, ], (19) as n. Proof. By the convergence of x n, for ζ >, there exists N N such that x n (t) x(t) < ζ, t [, ], n N,

7 88 SHUQIN ZHANG JFCA-213/3 by the continuity of f, for ε (where ε is arbitrary small number), when n N, it holds f(s, x n (s)) f(s, x(s)) < ε, s [, ]. hus, we have that < ε ds ε (t δ) ε ε, (f(s, x n (s)) f(s, x(s)))ds f(s, x n (s)) f(s, x(s)) ds which implies that (19) holds. Lemma 2.3. Assume that (H 1 ) hold. hen for arbitrary fixed x C[, ], the following expression holds lim δ (t s) q(s,x(s)) Γ(1 q(s, x(s))) x(s)ds Proof. For arbitrary fixed x C[, ], we let M max x(t) + 1, L max t t, x M (t s) q(s,x(s)) x(s)ds. (2) Γ(1 q(s, x(s))) 1 Γ(1 q(t, x(t))) + 1. hus, for arbitrary fixed function x C[, ], for ε >, take δ ( ε(1 q ) then, when < δ < δ, by (14, (15),(16), we have that < t δ t δ ML (t s) q(s,x(s)) Γ(1 q(s, x(s))) x(s)ds (t s) q(s,x(s)) Γ(1 q(s, x(s))) x(s)ds q(s,x(s)) Γ(1 q(s, x(s))) (t s t δ ML q 1 q δ 1 q ( t s ) q ds ML q 1 q δ 1 q ε, (t s) q(s,x(s)) Γ(1 q(s, x(s))) x(s)ds ) q(s,x(s)) x(s)ds ML ) 1 q 1 q, which implies that (2) holds. Remark 2.1. Assume that (H 1 ), (H 2 ) hold. Obviously, we can know that: for the arbitrary fixed function x C[, ], lim δ x(s)ds x(s)ds, lim δ f(s, x(s))ds f(s, x(s))ds. (21)

8 JFCA-213/4 EXISENCE AND UNIQUENESS RESUL OF SOLUIONS 89 Lemma 2.4.([17]) Let [a, b]( < a < b < + ) be a finite interval and let AC[a, b] be the space of functions which are absolutely continuous on [a, b]. It is known that AC[a, b] coincides with the space of primitives of Lebesgue summable functions: f(t) AC[a, b] f(t) c + φ(s)ds, φ L(a, b), c R, (22) and therefore an absolutely continuous function f(t) has a summable derivative f (t) φ(t) almost everywhere on [a, b]. 3. Existence result In this section, we will consider the existence of solution for IVP (11), by means of some analysis techniques and Arzela-Ascoli theorem. By the definition of fractional derivative defined by (9), we see that problem (11) is equivalent to x() and the following expression (t s) q(s,x(s)) Γ(1 q(s, x(s))) x(s)ds c + f(s, x(s))ds, t [, ], (23) where c R. heorem 3.1. Assume that (H 1 ), (H 2 ) hold. hen IVP (11) exists one solution x C[, ]. Proof. In order to obtain the existence result of solution IVP (11), we firstly verify the following sequence has convergent subsequence, x k (t), t δ, x k 1 (t) + (t s) q(s,x k 1 (s)) Γ(1 q(s,x k 1 (s))) x k 1(s)ds f(s, x k 1 (s))ds, δ < t, k 1, 2,, where x (t), t [δ, ], δ is arbitrary small number. In order to apply Arzela-Ascoli theorem to consider the existence of convergent subsequence of sequence x k defined by (24), firstly, we prove the uniformly bounded of sequence x k on [, ]. We find that x k is uniformly bounded on [, δ]. Now, we will verify sequence x k is uniformly bounded on [δ, ]. Let M max t f(s, ) + 1. Since x is uniformly bounded on [, ], then, for t [δ, ], we have that x 1 (t) x (t) + M f(s, )ds ds M. M 1, (t s) q(s,x (s)) Γ(2 q(s, x (s))) x (s)ds f(s, )ds which implies that x 1 is uniformly bounded on [δ, ], together with x 1 (t) for t [, δ], we obtain that x 1 is uniformly bounded on [, ]. (24)

9 9 SHUQIN ZHANG JFCA-213/3 1 Let M f max t, x1 M 1 f(t, x 1 ) +1, L max t, x1 M 1 Γ(1 q(t,x + 1(t))) 1. From (13), (14), (15), for t [δ, ], we have that x 2 (t) x 1 (t) + M 1 + M 1 L M 1 + M 1 L (t s) q(s,x 1(s)) Γ(1 q(s, x 1 (s))) x 1(s) ds + f(s, x 1 (s)) ds q(s,x1(s)) ( t s ) q(s,x1(s)) ds + M f ( δ) ( t s ) q ds + M f M 1 + M 1L q 1 q (t 1 q δ 1 q ) + M f M 1 + M 1L 1 q + M f. M 2, which implies that x 2 is uniformly bounded on [δ, ], together with x 2 (t) for t [, δ], we obtain that x 2 is uniformly bounded on [, ]. Continuous this process, we can obtain that sequence x k is uniformly bounded on [, ]. Now, we consider the equicontinuous of sequence x k on [, ]. Obviously, x is equicontinuous on [, ]. Firstly, we can know that function k(t) a t b t, t ( 1, ), < a < b < 1, is decreasing. Indeed, since ln a < ln b < and a t > b t >, we have that k (t) a t ln a b t ln b < b t ln a b t ln b b t (ln a ln b) <, which implies that k(t) is a decreasing. hus, for l(s) ( t 1 s (where < t1 s < t2 s ) q(s,x(s)) ( t 2 s ) q(s,x(s)) < 1), we may look l(s) as the same type as k(s), then l(s) is decreasing with respect to its exponent q(s, x(s)). As well, in the next analysis, we will use the Minkowsk s inequality: for a, b non negative, and any R, it holds (a + b) R c R (a R + b R ), where c R max{1, 2 R 1 }. (25) As a result, for a, b non negative, and any < r < 1, it follows from (25) that (a + b) r c r (a r + b r ) max{1, 2 r 1 }(a r + b r ) a r + b r. (26) We let M max t f(s, ) + 1. For ε >, t 1, t 2 [, ], t 1 < t 2, we consider result in two cases. Case I: t 1 δ < t 2. We take η 1,I ε M, when t 2 t 1 < η 1,I, we have that x 1 (t 2 ) x 1 (t 1 ) 2 δ M 2 δ M(t 2 δ) f(s, )ds ds M(t 2 t 1 ) < Mη 1,I ε.

10 JFCA-213/4 EXISENCE AND UNIQUENESS RESUL OF SOLUIONS 91 Case II: δ t 1 < t 2. We take η 1,II ε M, when t 2 t 1 < η 1,II, we have that x 1 (t 2 ) x 1 (t 1 ) 1 δ 2 δ t 1 δ 2 δ M t 1 δ f(s, )ds f(s, ) ds ds M(t 2 t 1 ) < Mη 1,II ε. 2 δ f(s, )ds hese imply that x 1 (t) is equicontinuous on [, ], the same result can be obtained when t 2 < t 1. 1 We let M f max t, x1 M 1 f(s, x 1 ) +1, L max t, x1 M 1 Γ(1 q(s,x + 1(s))) 1. For 3ε 2 > (ε is arbitrary small number), t 1, t 2 [, ], t 1 < t 2, we consider result in two cases. Case I: t 1 δ < t 2. We take η 2,I min{η 1,I, ( when t 2 t 1 < η 2,I, by (13), (14), (15), (26), we have that x 1 (t 2 ) x 1 (t 1 ) x 1 (t 2 ) + 2 δ x 1 (t 2 ) + M 1 L x 1 (t 2 ) + M 1 L x 1 (t 2 ) + M 1 L (t 2 s) q(s,x 1(s)) Γ(1 q(s, x 1 (s))) x 1(s)ds 2 δ 2 δ 2 δ 2 δ (1 q )ε 4M 1L ) 1 q 1 q, f(s, x 1 )ds 2 δ (t 2 s) q(s,x1(s)) ds + M f ds q(s,x1(s)) ( t 2 s ) q(s,x1(s)) ds + M f (t 2 δ) ( t 2 s ) q ds + M f (t 2 δ) x 1 (t 2 ) + M 1L q 1 q (t 1 q 2 δ 1 q ) + M f (t 2 δ) x 1 (t 2 ) x 1 (t 1 ) + M 1L q 1 q ((t 2 δ + δ) 1 q δ 1 q ) + M f (t 2 δ) ε 4M f }, x 1 (t 2 ) x 1 (t 1 ) + M 1L q 1 q ((t 2 δ) 1 q + δ 1 q δ 1 q ) + M f (t 2 δ) x 1 (t 2 ) x 1 (t 1 ) + M 1L q 1 q (t 2 t 1 ) 1 q + M f (t 2 t 1 ) < x 1 (t 2 ) x 1 (t 1 ) + M 1L q 1 q η 1 q 2,I + M f η 2,I < ε + ε 4 + ε 4 3ε 2.

11 92 SHUQIN ZHANG JFCA-213/3 Case II: δ t 1 < t 2. We take η 2,II min{η 1,II, ( when t 2 t 1 < η 2,I, by (13), (14), (15), (26) we have that (1 q )ε 8M 1 L ) 1 q 1 q, ε 4M f }, x 1 (t 2 ) x 1 (t 1 ) 2 δ (t 2 s) q(s,x1(s)) x 1 (t 2 ) x 1 (t 1 ) + Γ(1 q(s, x 1 (s))) x 1(s)ds 1 δ (t 1 s) q(s,x1(s)) 2 δ Γ(1 q(s, x 1 (s))) x 1(s)ds f(s, x 1 )ds + 2 δ x 1 (t 2 ) x 1 (t 1 ) + (t 2 s) q(s,x1(s)) Γ(1 q(s, x 1 (s))) x 1(s) ds δ 2 δ t 1 δ t 1 δ 1 δ f(s, x 1 )ds 1 Γ(1 q(s, x 1 (s))) (t 2 s) q(s,x1(s)) (t 1 s) q(s,x1(s)) x 1 (s) ds f(s, x 1 (s)) ds x 1 (t 2 ) x 1 (t 1 ) + M 1 L M 1 L 2 δ t 1 δ x 1 (t 2 ) x 1 (t 1 ) + M 1 L +M 1 L 2 δ t 1 δ 1 δ 2 δ (t 2 s) q(s,x1(s)) ds + M f ds x 1 (t 2 ) x 1 (t 1 ) + M 1 L +M 1 L 2 δ t 1 δ 1 δ ((t 1 s) q(s,x1(s)) (t 2 s) q(s,x1(s)) )ds + t 1 δ q(s,x1(s)) (( t 1 s q(s,x1(s)) ( t 2 s ) q(s,x 1(s)) ds + M f (t 2 t 1 ) 1 δ (( t 1 s ) q ( t 2 s ) q ds + M f (t 2 t 1 ) ) q(s,x1(s)) ( t 2 s ) q(s,x1(s)) )ds ( t 2 s ) q )ds x 1 (t 2 ) x 1 (t 1 ) + M 1L q 1 q (t 1 q 1 δ 1 q + 2(t 2 t 1 + δ) 1 q t 1 q 2 δ 1 q ) +M f (t 2 t 1 ) x 1 (t 2 ) x 1 (t 1 ) + M 1L q 1 q (t 1 q 2 2δ 1 q + 2(t 2 t 1 ) 1 q + 2δ 1 q t 1 q +M f (t 2 t 1 ) x 1 (t 2 ) x 1 (t 1 ) + 2M 1L q 1 q (t 2 t 1 ) 1 q + M f (t 2 t 1 ) < x 1 (t 2 ) x 1 (t 1 ) + 2M 1L q 1 q η 1 q 2,II + M f η 2,II < ε + ε 4 + ε 4 3ε 2. 2 )

12 JFCA-213/4 EXISENCE AND UNIQUENESS RESUL OF SOLUIONS 93 hese imply that x 2 (t) is equicontinuous on [, ], the same result can be obtained when t 2 < t 1. Continue these process, we can obtain that x k, k 1, 2, is equicontinuous on [, ]. As well, by the arguments of equicontinuity of x k, we can know that x k C[, ], k 1, 2,. hen, from Arzela-Ascoli theorem, sequence x k exists a convergent subsequence x mk. From (24), x mk should satisfy, t δ, x mk (t) x mk 1 (t) + (t s) q(s,x m k 1 (s)) Γ(1 q(s,x mk 1 (s))) x m k 1 (s)ds f(s, x mk 1 (s))ds, δ < t. Now, we will prove that the continuous limit of x mk, denoted by x is one solution of IVP (11). Let k + in (27), by Lemmas 2.1, 2.2, we have that x (t) hus, we find that, x (t), t δ;, t δ, x (t) + (t s) q(s,x (s)) Γ(1 q(s,x (s))) x (s)ds f(s, x (s))ds, δ < t. (t s) q(s,x (s)) Γ(1 q(s, x (s))) x (s)ds (27) (28) f(s, x (s))ds, (29) δ < t. In order to verify x is one solution of IVP (11), we let δ in (29), by (2), (21), we obtain that x () ; (t s) q(s,x (s)) Γ(1 q(s, x (s))) x (s)ds f(s, x (s))ds, < t. (3) It follows from the continuity of f and Lemma 2.4 that f(s, x (s))ds AC[, ], consequently, from (3), we get f(s, x (s))ds (t s) q(s,x (s)) Γ(1 q(s, x (s))) x (s)ds AC[, ]. As a result, differential on both sides of the second expression in (3), we get D q(t,x (t)) + x (t) f(t, x ), < t, (31) together with x (), we see that x is one solution of IVP (11). hus, we complete this proof. 4. Unique results In this section, using the monotone iterative method, we will consider the existence and unique result of solution to the following particular case of IVP (11), { p(t) D + x(t) f(t, x), < t, < < + (32) x(), where + denotes fractional derivative of variable-order defined by (5), where p : [, ] (, p ], < p < 1, is a continuous function. We assume that (H 3 ) p : [, ] (, p ] is continuous, here < p < 1.

13 94 SHUQIN ZHANG JFCA-213/3 he following result will play very important role in our next analysis. Lemma 4.1. Let (H 3 ) hold. If x C[, ] and satisfies the relations + x + d x, t (, ] x() where + denotes fractional derivative of variable-order defined by (5), d > is a constant. hen x for t [, ]. Proof. We assume that x(t), t [, ] is false. hen from x(), there exists points t [, ], t (, ] such that, x(t ), x(t ) < ; and that x(t) for t [, t ], x(t) < for t (t, t ], and assume that t 1 is the first minimal point of x(t) on [t, t ]. It follows from the inequality of (33) that hence, we have + x(t), t [t, t ], D p(s) + x(s)ds, t [t, t ], t from the definition of fractional derivative of variable order defined by (5), we can obtain that d ds (I1 p(s) + x(s))ds I 1 p(t) + x(t) I 1 p(t) x(t ), t [t, t ]. (34) t On the other hand, for t [t, t ], we have I 1 p(t) + x(t) I 1 p(t) x(t ) (t s) p(s) (t s) p(s) x(s)ds + Γ(1 p(s)) < +, (t s) p(s) Γ(1 p(s)) x(s)ds t (t s) p(s) Γ(1 p(s)) x(s)ds (33) (t s) p(s) Γ(1 p(s)) x(s)ds which contradicts (34). herefore, we obtain that x(t), t [, ]. hus we complete this proof. For problem (32), we have the following definitions of upper and lower solutions. Definition 4.1. A function α C[, ] is called a upper solution of problem (32), if it satisfies { p(t) D + α(t) f(t, α), t (, ] (35) α() Analogously, function β C[, ] is called a lower solution of problem (32), if it satisfies { p(t) D + β(t) f(t, β), t (, ] (36) β() In what follows we assume that and define that sector α(t) β(t), t [, ], (37) α, β u C[, ]; α(t) u(t) β(t), t [, ].

14 JFCA-213/4 EXISENCE AND UNIQUENESS RESUL OF SOLUIONS 95 We also assume that f satisfies the following condition f(t, x 1 ) f(t, x 2 ) d (x 1 x 2 ), α x 2 x 1 β, (38) where d is a constant and α, β C[, ] are lower and upper solutions of problem (32). Clearly this condition is satisfied with d, when f is monotone nondecreasing in u. In view of (38), the function F (t, x) d x + f(t, x) (39) is monotone nondecreasing in x for x α, β. We also suppose that there exists a constant d 1, such that f(t, x 1 ) f(t, x 2 ) d 1 (x 1 x 2 ), α x 2 x 1 β, (4) where α, β C[, ] are lower and upper solutions of problem (32). he following is existence and uniqueness theorem of solution for (32). heorem 4.1. Let (H 2 ), (H 3 ) hold. Assume that α, β C[, ] are lower and upper solutions of problem (32), such that (37) holds, f also satisfies (38). hen problem (32) exists one solution in the sector α, β, as well, if condition (4) holds, then (32) exists one unique solution in the sector α, β. proof. We see that (32) is equivalent to the following problem { p(t) D + v + d v d v + f(t, v), t (, ], (41) v(), where d is the constant in (38). his proof consists of six steps. Step 1. Constructing sequences {v (k) }, k 1, 2, as following { D p(t) + v(k) (t) + d v (k) d v (k 1) + f(t, v (k 1) ), t (, ], v (k) (). heorem 3.1 assures that the sequence {v (k) } is well defined, since for each k, we can obtain result from theorem 3.1 with f(t, v (k) ). d v (k 1) + f(t, v (k 1) ) d v (k) and q(t, v (k) ). p(t). Of particular interest is the sequence obtained from (42) with a upper solution or lower solution of problem (32) as the initial iteration. Denote the sequence with the initial iteration v () β by {v (k) } and the sequence with v () α by {v (k) }. Step 2. Monotone property of the two sequences. he sequences {v (k) }, {v (k) } constructed by (41) process the monotone property (42) α v (k) v (k+1) v (k+1) v (k) β, t (, ], (43) for every k 1, 2,. In fact, let r v () v (1). By (42), (36), (37), (38), and v () β, there has + r + d r ( + β + d β (d β + f(t, β)) + β f(t, β), t (, ], r(). In view of Lemma 4.1, r for t [, ], which leads to v (1) v () β, t [, ]. A similar argument using the property of a lower solution of (32) gives

15 96 SHUQIN ZHANG JFCA-213/3 v (1) v () α, t [, ]. Let r (1) v (1) v (1). By (42), (36), (37), (38), there has + r(1) + d r (1) d v () + f(t, v () ) (d v () + f(t, v () )) d (β α) + f(t, β) f(t, α), t (, ], r (1) () v (1) () v (1) (), Again, in view of Lemma 4.1, r (1) for t [, ], the above conclusion shows that Assume, by induction α v () v (1) v (1) v () β, t [, ] α v (k 1) v (k) v (k) v (k 1) β, t (, ]. (44) hen by (42), (44), (38), the function r (k) v (k) v (k+1) satisfies the relations + r(k) (t) + d r (k) d v (k 1) + f(t, v (k 1) ) (d v (k) + f(t, v (k) )), t (, ], r (k) (), In view of Lemma 4.1, r (k), that is v (k+1) v (k) for t [, ]. Similar reasoning gives v (k) v (k+1) and v (k+1) v (k+1) for t [, ]. Hence, the monotone property (43) follows from the principle of induction. Step 3. he two sequences constructed by (42) have pointwise limits and satisfy some relations, that is lim k v(k) (t) v(t), exists and satisfy the relation lim k v(k) (t) w(t), t [, ] (45) α v (k) v (k+1) w v v (k+1) v (k) β, t [, ] (46) for every k 1, 2,. In fact, By (43), we see that the upper sequence {v (k) } is monotone nonincreasing and is bounded from below and that the lower sequence {v (k) } is monotone nondecreasing and is bounded from above. herefore the pointwise limits exist and these limits are denoted by v and w as in (45). Moreover, by (43), the limits v, w satisfy (46). Step 4. o prove that v and w are solutions of initial value problem (32). Let v (k) be either v (k) or v (k) From the definition of fractional derivative + defined by (5), we see that the equation of (42) may be expressed as and I 1 p(t) + v (k) (t) + d I+v 1 (k) (t)) c + d I+v 1 (k 1) (t) + I+f(t, 1 v (k 1) (t)), where c R, that is (t s) p(s) Γ(1 p(s)) v(k) (s)ds + d Now, we consider the expression (t s) p(s) Γ(1 p(s)) v(k) (s)ds + d v (k) (s)ds c + d v (k 1) (s)ds + v (k) (s)ds c + d v (k 1) (s)ds + f(s, v (k 1) (s))ds. (47) f(s, v (k 1) (s))ds. (48)

16 JFCA-213/4 EXISENCE AND UNIQUENESS RESUL OF SOLUIONS 97 where δ > is arbitrary small number. By Lemma 2.1, we know that (t s) p(s) Γ(1 p(s)) v(k) (s)ds C[δ, ] with v () α or v () β. Let k in (48) and apply the Lemmas 2.2, 2.3 and the dominated convergence theorem, v satisfies the integral equation that is (t s) p(s) Γ(1 p(s)) v(s)ds+d (t s) p(s) Γ(1 p(s)) v(s)ds c + Now let δ in (49), by (2), (21), we get (t s) p(s) Γ(1 p(s)) v(s)ds c + v(s)ds c+d v(s)ds + f(s, v(s))ds, f(s, v(s))ds. (49) f(s, v(s))ds. (5) It follows from the continuity of f and Lemma 2.4 that c + f(s, v(s))ds AC[, ], consequently, from (5), we get c + f(s, v(s))ds (t s) p(s) v(s)ds AC[, ]. Γ(1 p(s)) As a result, differential on two sides of (5), we have that + v(t) f(t, v(t)), t (, ]. (51) We also Let k in the second expression of (42), it holds v(), this together with (51), we know that v(t) is a solution of (32). his proves that the upper sequence {v (k) } converges to a solution v of problem (32), the lower sequence {v (k) } converges to a solution w of problem (32), and satisfies relation v(t) w(t), t [, ]. Step 5 If condition (4) holds, then v w is unique solution of problem (32). It is sufficient to prove v(t) w(t), t [, ], by v(t) w(t), t [, ] obtained in Step 4. In fact, by (32) and (4), the function r w v satisfies the relations { D p(t) + r (f(t, v) f(t, w)) d 1r, t (, ], r(), then, Lemma 4.1 implies that r(t), t [, ], this proves w v, therefore, we obtain that v w is unique solution of problem (32). hus, we complete this proof. References [1] H.G. Sun, W. Chen, Y.Q. Chen, Variable-order fractional differential operators in anomalous diffusion modeling, Physica A, 388(29) [2] C.F.M. Coimbra, Mechanics with variable-order differential operators, Annalen der Physik, 12(23) [3] H. Sheng, H.G. Sun, Y.Q. Chen,. Qiu, Synthesis ofmultifractional Gaussian noises based on variable-order fractional operators, Signal Processing, 91(211) [4] C.C. seng, Design of variable and adaptive fractional order FIR differentiators, Signal Processing, 86(26) [5] H. Sheng, H. G. Sun, C. Coopmans, Y.Q. Chen, G.W. Bohannan, Physical experimental study of variable-order fractional integrator and differentiator, in: Proceedings of the 4th IFAC Workshop on Fractional Differentiation and its Applications(FDA 1), 21.

17 98 SHUQIN ZHANG JFCA-213/3 [6] H.G. Sun, W. Chen, H. Wei, Y.Q. Chen, A comparative study of constant-order and variableorder fractional models in characterizing memory property of systems, Eur. Phys. J. Special opics, 193(211) [7].. Hartley, C.F. Lorenzo, Fractional system identification: An approach using continuous order distributions, NASA/M, 4(1999) [8] C.H. Chan, J.J. Shyu, R.H.H. Yang, A new structure for the design of wideband variable fractional-order FIR differentiator, Signal Processing, 9(21) [9] C.C.seng, Series expansion design for variable fractional order integrator and Differentiator using logarithm, Signal Processing, 88(28) [1] C.F. Lorenzo,.. Hartley, Variabe order and distributed order fractional operators, Nonlinear Dynamics, 29(22) [11] C.H. Chan, J.J.Shyu, R.H.H. Yang, Iterative design of variable fractional-order IIR differintegrators, Signal Processing, 9(21) [12] C.H. Chan, J.J. Shyu, R. H.H. Yang, An iterative method for the design of variable fractionalorder FIR differintegrators, Signal Processing, 89(29) [13] B. ROSS, Fractional integration operator of variable-order in the Hölder space H λ(x), Internat. J. Math. and Math. Sci., 18(1995) [14] D. Valério, J.S. Costa, Variable-order fractional derivatives and their numerical approximations, Signal Processing, 91(211) [15] A. Razminia, A.F. Dizaji, V.J. Majd, Solution existence for non-autonomous variable-order fractional differential equations, Mathematical and Computer Modelling, 55(212) [16] R. Lin, F. Liu, V. Anh, I. urner, Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation, Applied Mathematics and Computation, 2(29) [17] A. A. Kilbas, H. M. Srivastava, J. J. rujillo, heory and Applications of Fractional Differential Equations, Elsevier B. V., Amsterdam, 26. Shuin Zhang Department of Mathematics, China University of Mining and echnology, Beijing, 183, China address: zsqjk@163.com