Nonlinear Analysis 71 (2009) 3249–3256

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Nonlinear Analysis

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Existence and uniqueness for fractional neutral differential equationswith infinite delayI

Yong Zhou ∗, Feng Jiao, Jing LiDepartment of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, PR China

a r t i c l e i n f o

Article history:Received 8 December 2008Accepted 26 January 2009

Keywords:Fractional neutral differential equationsInfinite delayExistenceUniqueness

a b s t r a c t

In this paper, the Cauchy initial value problem is discussed for the fractional neutralfunctional differential equations with infinite delay and various criteria on existence anduniqueness are obtained.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper, we consider the Cauchy initial value problem (IVP for short) of fractional neutral functional differentialequations with infinite delay of the form

Dqg(t, xt) = f (t, xt), t ∈ [t0,∞), (1)xt0 = ϕ, (t0, ϕ) ∈ [0,∞)×Ω, (2)

where Dq is Caputo’s fractional derivative of order 0 < q < 1, Ω is an open subset of B and g, f : [t0,∞) × Ω → Rn aregiven functionals satisfying some assumptions that will be specified later. B is called a phase space that will be defined later(see Section 2).If x : (−∞, A)→ Rn, A ∈ (0,∞), then for any t ∈ [0, A) define xt by xt(θ) = x(t + θ), for θ ∈ (−∞, 0].Fractional differential equations have gained considerable importance due to their application in various sciences, such

as physics, mechanics, chemistry, engineering, etc. In recent years, there has been a significant development in ordinary andpartial differential equations involving fractional derivatives, see the monographs of Kilbas et al. [1], Miller and Ross [2],Podlubny [3] and the papers of Agarwal et al. [4,5], Ahmad and Nieto [6], Araya and Lizama [7], Belmekki and Nieto [8],Bonilla et al. [9], Chang and Nieto [10,11], Daftardar-Gejji and Jafari [12], Daftardar-Gejji and Bhalekar [13], Delbosco andRodino [14], Diethelm [15], El-Borai [16], El-Sayed [17], Gafiychuk et al. [18], Ibrahim and Momani [19], Jaradat et al. [20],Kosmatov [21], Lakshmikantham and Vatsala [22,23], Muslim [24], Salem [25], Vasundhara Devi and Lakshmikantham [26]and the references therein. However, there are few works on the initial value problems of fractional functional differentialequations. In [17], El-Sayed discusses a class of nonlinear functional differential equations of arbitrary orders. In [27],Lakshmikantham initiates the basic theory for fractional functional differential equations. In [28], Benchohra et al. considerthe IVP for a particular class of fractional neutral functional differential equations with infinite delay. In [29,30], Zhou et al.investigate the existence and uniqueness for p-type fractional functional differential equations.

I Research supported by National Natural Science Foundation of PR China and Research Fund of Hunan Provincial Education Department (08A071).∗ Corresponding author.E-mail address: yzhou@xtu.edu.cn (Y. Zhou).

0362-546X/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2009.01.202

3250 Y. Zhou et al. / Nonlinear Analysis 71 (2009) 3249–3256

In this paper, we discuss existence and uniqueness of solutions for IVP (1)–(2) on a class of comparatively comprehensivephase spaces. We establish various criteria on existence and uniqueness of solutions for IVP (1)–(2).

2. Preliminaries

In this section, we introduce definitions and preliminary facts which are used throughout this paper.Let J ⊂ R. Denote by C(J, Rn) the Banach space of all continuous functions from J into Rn with the norm

‖x‖ = supt∈J|x(t)|,

where | · | denotes a suitable complete norm.Denote by BC(J, Rn) the Banach space of all continuous and bounded functions from J into Rn with the norm ‖ · ‖.

Definition 2.1 ([3]). The fractional integral of order µwith the lower limit t0 for a function f is defined as

Iµf (t) =1

Γ (µ)

∫ t

t0

f (s)(t − s)1−µ

ds, t > t0, µ > 0,

provided the right side is point-wise defined on [t0,∞), where 0 is the gamma function.

Definition 2.2 ([3]). Caputo’s derivative of order µwith the lower limit t0 for a function f : [0,∞)→ R can be written as

Dµf (t) =1

0(n− µ)

∫ t

t0

f (n)(s)(t − s)µ+1−n

ds = In−µf (n)(t), t > t0, n− 1 < µ < n.

Obviously, Caputo’s derivative of a constant is equal to zero.To describe fractional neutral functional differential equations with infinite delay, we need to discuss a phase space B

in a convenient way. We shall provide a general description of phase spaces of neutral differential equations with infinitedelay which is taken from [31].Let B be a real vector space either(i) of continuous functions that map (−∞, 0] to Rn with ϕ = ψ if ϕ(s) = ψ(s) on (−∞, 0] or(ii) of measurable functions that map (−∞, 0] to Rn with ϕ = ψ (or ϕ is equivalent to ψ) in B if ϕ(s) = ψ(s) almost

everywhere on (−∞, 0], and ϕ(0) = ψ(0).Let B be endowed with a norm ‖ · ‖B such that B is completed with respect to ‖ · ‖B. Thus B equipped with norm ‖ · ‖B is

a Banach space. We denote this space by (B, ‖ · ‖B) or simply by B, whenever no confusion arises.Let 0 ≤ a < A. If x : (−∞, A) → Rn is given such that xa ∈ B and x ∈ [a, A) → Rn is continuous, then xt ∈ B for all

t ∈ [a, A).This is a very weak condition that the common admissible phase spaces and BC satisfy. For more details of the phase

spaces, we refer the reader to [31,32].

Definition 2.3. A function x : (−∞, t0 + δ)→ Rn(t0 ∈ [0,∞), δ > 0) is said to be a solution of IVP (1)–(2) through (t0, ϕ)on [t0, t0 + δ), if

(i) xt0 = ϕ,(ii) x is continuous on [t0, t0 + δ),(iii) g(t, xt) is absolutely continuous on [t0, t0 + δ),(iv) (1) holds almost everywhere on [t0, t0 + δ).

Lemma 2.1 (Krasnoselskii’s Fixed Point Theorem). Let X be a Banach space, let E be a bounded closed convex subset of X andlet S,U be maps of E into X such that Sz + Uw ∈ E for every pair z, w ∈ E. If S is a contraction mapping and U is completelycontinuous, then the equation

Sz + Uz = z

has a solution on E.

3. Main results

LetΩ ⊆ B be an open set such that for any (t0, ϕ) ∈ [0,∞)×Ω , there exist constants δ1, γ1 > 0 so that xt ∈ Ω providedthat x ∈ A(t0, ϕ, δ1, γ1) and t ∈ [t0, t0 + δ1], where A(t0, ϕ, δ1, γ1) is defined as

A(t0, ϕ, δ1, γ1) =x : (−∞, t0 + δ1] → Rn, xt0 = ϕ, sup

t0≤t≤t0+δ1|x(t)− ϕ(0)| ≤ γ1

.

Y. Zhou et al. / Nonlinear Analysis 71 (2009) 3249–3256 3251

It is well known that a neutral functional differential equation (NFDE for short) is one in which the derivatives of thepast history or derivatives of functionals of the past history are involved as well as the present state of the system. In otherwords, in order to guarantee that Eq. (1) is NFDE, the coefficient of x(t) that is contained in g(t, xt) cannot be equal to zero.Then we need to introduce the generalized atomic concept.

Definition 3.1 ([31]). The functional g : [0,∞)×Ω → Rn is said to be generalized atomic onΩ , if

g(t, ϕ)− g(t, ψ) = K(t, ϕ, ψ)[ϕ(0)− ψ(0)] + L(t, ϕ, ψ)

where (t, ϕ, ψ) ∈ [0,∞)×Ω ×Ω , K : [0,∞)×Ω ×Ω → Rn×n and L : [0,∞)×Ω ×Ω → Rn satisfy(a) det K(t, ϕ, ϕ) 6= 0 for all (t, ϕ) ∈ [0,∞)×Ω,(b) for any (t0, ϕ) ∈ [0,∞) × Ω , there exist constants δ1, γ1 > 0, and k1, k2 > 0, with 2k2 + k1 < 1 such that for allx, y ∈ A(t0, ϕ, δ1, γ1), g(t, xt), K(t, xt , yt) and L(t, xt , yt) are continuous in t ∈ [t0, t0 + δ1], and

|K−1(t0, ϕ, ϕ)L(t, xt , yt)| ≤ k1 supt0≤s≤t

|x(s)− y(s)|,

|K−1(t0, ϕ, ϕ)K(t, xt , yt)− I| ≤ k2,

where I is the n× n unit matrix.

For a detailed discussion on the atomic concept we refer the reader to the books [31,33].The following existence result for IVP (1)–(2) is based on Krasnoselskii’s fixed point theorem.

Theorem 3.1 (Existence). Assume that g is generalized atomic onΩ , and that for any (t0, ϕ) ∈ [0,∞)×Ω , there exist constantsδ1, γ1 ∈ (0,∞), q1 ∈ (0, q) and a real-valued function m(t) ∈ L

1q1 [t0, t0 + δ1] such that

(H1) for any x ∈ A(t0, ϕ, δ1, γ1), f (t, xt) is measurable,(H2) for any x ∈ A(t0, ϕ, δ1, γ1), |f (t, xt)| ≤ m(t), for t ∈ [t0, t0 + δ1],(H3) f (t, φ) is continuous with respect to φ onΩ .Then IVP (1)–(2) has a solution.

Proof. We know that f (t, xt) is Lebesguemeasurable in [t0, t0+δ1] according to the condition (H1). Direct calculation givesthat (t − s)q−1 ∈ L

11−q1 [t0, t], for t ∈ [t0, t0 + δ1]. In light of the Hölder inequality and the condition (H2), we obtain that

(t − s)q−1f (s, xs) is Lebesgue integrable with respect to s ∈ [t0, t] for all t ∈ [t0, t0 + δ1], and∫ t

t0|(t − s)q−1f (s, xs)|ds ≤ ‖(t − s)q−1‖

L1

1−q1 [t0,t]‖m‖

L1q1 [t0,t0+δ1]

, (3)

where

‖F‖Lp[J] =(∫J|F(t)|pdt

) 1p

,

for any Lp-integrable function F : J → R.According to Definition 2.3, IVP (1)–(2) is equivalent to the following equation

g(t, xt) = g(t0, ϕ)+1

0(q)

∫ t

t0(t − s)q−1f (s, xs)ds for t ∈ [t0, t0 + δ1]. (4)

Let ϕ ∈ A(t0, ϕ, δ1, γ1) be defined as ϕt0 = ϕ, ϕ(t0 + t) = ϕ(0) for all t ∈ [0, δ1]. If x is a solution of IVP (1)–(2), letx(t0 + t) = ϕ(t0 + t)+ z(t), t ∈ (−∞, δ1], then we have xt0+t = ϕt0+t + zt , t ∈ [0, δ1]. Thus (4) implies that z satisfies theequation

g(t0 + t, ϕt0+t + zt) = g(t0, ϕ)+1

0(q)

∫ t

0(t − s)q−1f (t0 + s, ϕt0+s + zs)ds (5)

for 0 ≤ t ≤ δ1.Since g is generalized atomic onΩ , there exist a positive constant α > 1 and a positive function δ2(γ ) defined in (0, γ1],

such that for any γ ∈ (0, γ1], when 0 ≤ t ≤ δ2(γ ), we have

α(2k2 + k1) < 1, (6)

|K−1(t0, ϕ, ϕ)K(t0 + t, xt0+t , yt0+t)− I| ≤ k2, (7)

|I − K−1(t0 + t, ϕt0+t , ϕt0+t)K(t0, ϕ, ϕ)| ≤ minαk2, α − 1, (8)

|K−1(t0 + t, ϕt0+t , ϕt0+t)||g(t0 + t, ϕt0+t)− g(t0, ϕ)| ≤1− α(2k2 + k1)

2γ . (9)

3252 Y. Zhou et al. / Nonlinear Analysis 71 (2009) 3249–3256

Note the completely continuity of the function (m(t))1/q1 . Hence, for a given positive numberM , there must exist a numberh > 0, satisfying∫ t0+h

t0(m(s))

1q1 ds ≤ M.

For a given γ ∈ (0, γ1], choose

δ = min

δ1, δ2(γ ), h, (1+ β)

11+β

[[1− α(2k2 + k1)]0(q)γ2α|K−1(t0, ϕ, ϕ)|Mq1

] 1(1−q1)(1+β)

, (10)

where β = q−11−q1∈ (−1, 0).

For any (t0, ϕ) ∈ [0,∞)×Ω , define E(δ, γ ) as follows

E(δ, γ ) = z : (−∞, δ] → Rnis continuous; z(s) = 0 for s ∈ (−∞, 0] and ‖z‖ ≤ γ

where ‖z‖ = sup0≤s≤δ |z(t)|. Then E(δ, γ ) is a closed bounded and convex subset of Banach space BC((−∞, δ1], Rn).Now, on E(δ, γ ) define two operators S and U as follows

(Sz)(t) =

0, t ∈ (−∞, 0],K−1(t0 + t, ϕt0+t , ϕt0+t)[−g(t0 + t, ϕt0+t + zt)+ g(t0, ϕ)+K(t0 + t, ϕt0+t , ϕt0+t)z(t)], t ∈ [0, δ]

and

(Uz)(t) =

0, t ∈ (−∞, 0],

K−1(t0 + t, ϕt0+t , ϕt0+t)1

0(q)

∫ t

0(t − s)q−1f (t0 + s, ϕt0+s + zs)ds, t ∈ [0, δ]

where z ∈ E(δ, γ ).It is easy to see that the operator equation

z = Sz + Uz (11)

has a solution z ∈ E(δ, γ ) if and only if z is a solution of Eq. (5). Thus xt+t0 = ϕt0+t + zt is a solution of Eq. (1) on [0, δ].Therefore, the existence of a solution of IVP (1)–(2) is equivalent to determining δ, γ > 0 such that (11) has a fixed pointon E(δ, γ ).Now we show that S + U has a fixed point on E(δ, γ ). The proof is divided into three steps.

Step I. Sz + Uw ∈ E(δ, γ ) for every pair z, w ∈ E(δ, γ ).Obviously, for every pair z, w ∈ E(δ, γ ), (Sz)(t) and (Uw)(t) are continuous in t ∈ [0, δ], and for t ∈ [0, δ], by using the

Hölder inequality and (8), we have

|(Uw)(t)| ≤ |K−1(t0 + t, ϕt0+t , ϕt0+t)K(t0, ϕ, ϕ)K−1(t0, ϕ, ϕ)|

10(q)

∣∣∣∣∫ t

0(t − s)q−1f (t0 + s, ϕt0+s + ws)ds

∣∣∣∣≤ α|K−1(t0, ϕ, ϕ)|

10(q)

[∫ t

0[(t − s)q−1]

11−q1 ds

]1−q1[∫ t0+δ

t0(m(s))

1q1 ds

]q1≤ α|K−1(t0, ϕ, ϕ)|

Mq1

0(q)

[1

1+ βδ1+β

]1−q1≤1− α(2k2 + k1)

2γ , where β =

q− 11− q1

∈ (−1, 0), (12)

and

|(Sz)(t)| = |K−1(t0 + t, ϕt0+t , ϕt0+t)[−g(t0 + t, ϕt0+t + zt)+ g(t0 + t, ϕt0+t)− g(t0 + t, ϕt0+t)+ g(t0, ϕ)+ K(t0 + t, ϕt0+t , ϕt0+t)z(t)]|

= |K−1(t0 + t, ϕt0+t , ϕt0+t)[−K(t0 + t, ϕt0+t + zt , ϕt0+t)z(t)− L(t0 + t, ϕt0+t + zt , ϕt0+t)− g(t0 + t, ϕt0+t)+ g(t0, ϕ)+ K(t0 + t, ϕt0+t , ϕt0+t)z(t)]|

= |K−1(t0 + t, ϕt0+t , ϕt0+t)[K(t0 + t, ϕt0+t , ϕt0+t)− K(t0 + t, ϕt0+t + zt , ϕt0+t)]z(t)

+ K−1(t0 + t, ϕt0+t , ϕt0+t)[−L(t0 + t, ϕt0+t + zt , ϕt0+t)− g(t0 + t, ϕt0+t)+ g(t0, ϕ)]|

= |K−1(t0 + t, ϕt0+t , ϕt0+t)K(t0, ϕ, ϕ)[K−1(t0, ϕ, ϕ)K(t0 + t, ϕt0+t , ϕt0+t)− I]z(t)

Y. Zhou et al. / Nonlinear Analysis 71 (2009) 3249–3256 3253

− K−1(t0 + t, ϕt0+t , ϕt0+t)K(t0, ϕ, ϕ)[K−1(t0, ϕ, ϕ)K(t0 + t, ϕt0+t + zt , ϕt0+t)− I]z(t)

+ K−1(t0 + t, ϕt0+t , ϕt0+t)[−L(t0 + t, ϕt0+t + zt , ϕt0+t)− g(t0 + t, ϕt0+t)+ g(t0, ϕ)]|

≤ |K−1(t0 + t, ϕt0+t , ϕt0+t)K(t0, ϕ, ϕ)|[(|K−1(t0, ϕ, ϕ)K(t0 + t, ϕt0+t , ϕt0+t)− I|

+ |K−1(t0, ϕ, ϕ)K(t0 + t, ϕt0+t + zt , ϕt0+t)− I|)|z(t)|

+ |K−1(t0, ϕ, ϕ)L(t0 + t, ϕt0+t + zt , ϕt0+t)|]

+ |K−1(t0 + t, ϕt0+t , ϕt0+t)||g(t0 + t, ϕt0+t)− g(t0, ϕ)|.

According to (6)–(9), we have

|(Sz)(t)| ≤ α(2k2 + k1)γ +1− α(2k2 + k1)

2γ =

1+ α(2k2 + k1)2

γ .

Therefore, |(Sz)(t)+ (Uw)(t)| ≤ γ for t ∈ [0, δ]. This means that Sz + Uw ∈ E(δ, γ )whenever z, w ∈ E(δ, γ ).Step II. S is a contraction mapping on E(δ, γ ).For any z, w ∈ E(δ, γ ), we obtain

|(Sz)(t)− (Sw)(t)| ≤ |K−1(t0 + t, ϕt0+t , ϕt0+t)||K(t0 + t, ϕt0+t , ϕt0+t)− K(t0 + t, ϕt0+t + zt , ϕt0+t + wt)||z(t)− w(t)|

+ |K−1(t0 + t, ϕt0+t , ϕt0+t)L(t0 + t, ϕt0+t + zt , ϕt0+t + wt)|

≤ |[I − K−1(t0 + t, ϕt0+t , ϕt0+t)K(t0, ϕ, ϕ)] − [K−1(t0 + t, ϕt0+t , ϕt0+t)K(t0, ϕ, ϕ)]

× [K−1(t0, ϕ, ϕ)K(t0 + t, ϕt0+t + zt , ϕt0+t + wt)− I] | |z(t)− w(t)|

+ |K−1(t0 + t, ϕt0+t , ϕt0+t)K(t0, ϕ, ϕ)K−1(t0, ϕ, ϕ)L(t0 + t, ϕt0+t + zt , ϕt0+t + wt)|

≤ (αk2 + αk2)|z(t)− w(t)| + αk1 sup0≤s≤t|z(s)− w(s)|

≤ α(2k2 + k1) sup0≤s≤t|z(s)− w(s)|,

where α(2k2 + k1) < 1, and therefore S is a contraction mapping on E(δ, γ ).Step III. Now we show that U is a completely continuous operator.For any z ∈ E(δ, γ ), 0 ≤ τ < t ≤ δ, we get

|(Uz)(t)− (Uz)(τ )| =∣∣∣∣K−1(t0 + t, ϕt0+t , ϕt0+t) 10(q)

∫ t

0(t − s)q−1f (t0 + s, ϕt0+s + zs)ds

− K−1(t0 + τ , ϕt0+τ , ϕt0+τ )1

0(q)

∫ τ

0(τ − s)q−1f (t0 + s, ϕt0+s + zs)ds

∣∣∣∣=

∣∣∣∣K−1(t0 + t, ϕt0+t , ϕt0+t) 10(q)∫ t

τ

(t − s)q−1f (t0 + s, ϕt0+s + zs)ds

+ K−1(t0 + t, ϕt0+t , ϕt0+t)1

0(q)

∫ τ

0(t − s)q−1f (t0 + s, ϕt0+s + zs)ds

− K−1(t0 + t, ϕt0+t , ϕt0+t)1

0(q)

∫ τ

0(τ − s)q−1f (t0 + s, ϕt0+s + zs)ds

+ K−1(t0 + t, ϕt0+t , ϕt0+t)1

0(q)

∫ τ

0(τ − s)q−1f (t0 + s, ϕt0+s + zs)ds

− K−1(t0 + τ , ϕt0+τ , ϕt0+τ )1

0(q)

∫ τ

0(τ − s)q−1f (t0 + s, ϕt0+s + zs)ds

∣∣∣∣≤|K−1(t0 + t, ϕt0+t , ϕt0+t)|

0(q)

∣∣∣∣∫ t

τ

(t − s)q−1f (t0 + s, ϕt0+s + zs)ds∣∣∣∣

+|K−1(t0 + t, ϕt0+t , ϕt0+t)|

0(q)

∣∣∣∣∫ τ

0[(t − s)q−1 − (τ − s)q−1]f (t0 + s, ϕt0+s + zs)ds

∣∣∣∣+|K−1(t0 + t, ϕt0+t , ϕt0+t)− K

−1(t0 + τ , ϕt0+τ , ϕt0+τ )|0(q)

×

∣∣∣∣∫ τ

0(τ − s)q−1f (t0 + s, ϕt0+s + zs)ds

∣∣∣∣

3254 Y. Zhou et al. / Nonlinear Analysis 71 (2009) 3249–3256

=|K−1(t0 + t, ϕt0+t , ϕt0+t)|

0(q)(I1 + I2)

+|K−1(t0 + t, ϕt0+t , ϕt0+t)− K

−1(t0 + τ , ϕt0+τ , ϕt0+τ )|0(q)

I3,

where

I1 =∣∣∣∣∫ t

τ

(t − s)q−1f (t0 + s, ϕt0+s + zs)ds∣∣∣∣,

I2 =∣∣∣∣∫ τ

0[(t − s)q−1 − (τ − s)q−1]f (t0 + s, ϕt0+s + zs)ds

∣∣∣∣,I3 =

∣∣∣∣∫ τ

0(τ − s)q−1f (t0 + s, ϕt0+s + zs)ds

∣∣∣∣.By using an analogous argument presented in (12), we can conclude that

I1 ≤Mq1

(1+ β)1−q1

[(t − τ)1+β

]1−q1,

I3 ≤Mq1

(1+ β)1−q1

(τ 1+β

)1−q1,

and

I2 ≤[∫ τ

0|(t − s)q−1 − (τ − s)q−1|

11−q1 ds

]1−q1[∫ t0+τ

t0|f (s, xs)|

1q1 ds

]q1≤ Mq1

[∫ τ

0((τ − s)β − (t − s)β)ds

]1−q1≤

Mq1

(1+ β)1−q1

[τ 1+β − t1+β + (t − τ)1+β

]1−q1≤

Mq1

(1+ β)1−q1

[(t − τ)1+β

]1−q1,

where β = q−11−q1∈ (−1, 0). Therefore

|(Uz)(t)− (Uz)(τ )| ≤|K−1(t0 + t, ϕt0+t , ϕt0+t)|

0(q)2Mq1

(1+ β)1−q1

[(t − τ)1+β

]1−q1+|K−1(t0 + t, ϕt0+t , ϕt0+t)− K

−1(t0 + τ , ϕt0+τ , ϕt0+τ )|0(q)

Mq1

(1+ β)1−q1

(τ 1+β

)1−q1.

Since the property of thematrix functionK(t0+t, ϕt0+t , ϕt0+t)which is nonsingular and continuous in t ∈ [0, δ] implies thatits inverse matrix K−1(t0 + t, ϕt0+t , ϕt0+t) exists and is continuous in t ∈ [0, δ], then Uz; z ∈ E(δ, γ ) is equicontinuous.On the other hand, U is continuous from the condition (H3) and Uz; z ∈ E(δ, γ ) is uniformly bounded from (12), thus Uis a completely continuous operator by the Ascoli–Arzela Theorem.Therefore, Krasnoselskii’s fixed point theorem shows that S + U has a fixed point on E(δ, γ ), and hence IVP (1)–(2) has

a solution x(t) = ϕ(0)+ z(t − t0) for all t ∈ [t0, t0 + δ]. This completes the proof.

Remark 3.1. If we replace the condition (H1) by(H1)′ f (t, φ) is measurable with respect to t on [t0, t0 + δ1], then we can also conclude that the result of Theorem 3.1

holds. In fact, for any x ∈ A(t0, ϕ, δ1, γ1), suppose xt0+t = ϕt0+t + zt , t ∈ [0, δ1], then, according to the definition of ϕt0+tand zt , we know that xt0+t is a measurable function. It follows that from (H1)

′ and (H3), f (t, xt) is measurable in t , wherex ∈ A(t0, ϕ, δ1, γ1) and satisfies xt0+t = ϕt0+t + zt , t ∈ [0, δ1].

Remark 3.2. If we replace the condition (H3) by a weaker condition(H3)′ for any x, y ∈ A(t0, ϕ, δ, γ )with supt0≤s≤t0+δ |x(s)− y(s)| → 0,∣∣∣∣∫ t

t0(t − s)q−1[f (s, xs)− f (s, ys)]ds

∣∣∣∣→ 0, t ∈ [t0, t0 + δ]

where δ satisfy (10), then we can also conclude that the result of Theorem 3.1 holds.

Y. Zhou et al. / Nonlinear Analysis 71 (2009) 3249–3256 3255

The following existence and uniqueness result for IVP (1)–(2) is based on Banach contraction principle.

Theorem 3.2 (Uniqueness). Assume that g is generalized atomic on Ω , and that for any (t0, ϕ) ∈ [0,∞) × Ω , there existconstants δ1, γ1 ∈ (0,∞), q1 ∈ (0, q) and a real-valued function m(t) ∈ L

1q1 [t0, t0 + δ1] such that conditions (H1)–(H2) of

Theorem 3.1 hold. Further assume that(H4) there exists a nonnegative function ` : [0, δ1] → [0,∞) continuous at t = 0 and `(0) = 0 such that for any

x, y ∈ A(t0, ϕ, δ1, γ1) we have∣∣∣∣∫ t

t0(t − s)q−1[f (s, xs)− f (s, ys)]ds

∣∣∣∣ ≤ `(t − t0) supt0≤s≤t

|x(s)− y(s)|, t ∈ [t0, t0 + δ1],

then IVP (1)–(2) has a unique solution.

Proof. According to the argument of Theorem 3.1, it suffices to prove that S+U has a unique fixed point on E(δ, γ ), whereδ, γ > 0 are sufficiently small. Now, choose δ ∈ (0, δ1), γ ∈ (0, γ1], such that (10) holds and that

c = α(2k2 + k1)+ sup0≤s≤δ

|K−1(t0 + s, ϕt0+s, ϕt0+s)||`(s)|0(q)

< 1. (13)

Obviously, S+U is amapping from E(δ, γ ) into itself. Using the same argument as that of Theorem3.1, for any z, w ∈ E(δ, γ ),we get

|(Sz)(t)− (Sw)(t)| ≤ α(2k2 + k1) sup0≤s≤δ|z(s)− w(s)|,

and

|(Uz)(t)− (Uw)(t)| ≤|K−1(t0 + t, ϕt0+t , ϕt0+t)|

0(q)

∣∣∣∣∫ t

0(t − s)q−1f (t0 + s, ϕt0+s + zs)ds

−

∫ t

0(t − s)q−1f (t0 + s, ϕt0+s + ws)ds

∣∣∣∣≤|K−1(t0 + t, ϕt0+t , ϕt0+t)|

0(q)|`(t)| sup

0≤s≤t|z(s)− w(s)|

≤

sup0≤s≤δ|K−1(t0 + s, ϕt0+s, ϕt0+s)||`(s)|

0(q)sup0≤s≤δ|z(s)− w(s)|.

Therefore

|(S + U)z(t)− (S + U)w(t)| ≤[α(2k2 + k1)+ sup

0≤s≤δ

|K−1(t0 + s, ϕt0+s, ϕt0+s)||`(s)|0(q)

]sup0≤s≤δ|z(s)− w(s)|

= c sup0≤s≤δ|z(s)− w(s)|.

Hence, we have

‖(S + U)z − (S + U)w‖ ≤ c‖z − w‖,

where c < 1. By applying Banach contraction principle, we know that S + U has a unique fixed point on E(δ, γ ). The proofis complete.

Corollary 3.1. If the condition (H4) of Theorem 3.2 is replaced by the following condition (H4)′ there exist q2 ∈ (0, q) and afunction `1 ∈ L

1q2 [t0, t0 + δ1], such that for any x, y ∈ A(t0, ϕ, δ1, γ1) we have

|f (t, xt)− f (t, yt)| ≤ `1(t) supt0≤s≤t

|x(s)− y(s)|, t ∈ [t0, t0 + δ1],

then the result of Theorem 3.2 holds.

Proof. It suffices to prove that the condition (H4) of Theorem 3.2 holds. Let N = ‖`1‖L1q2 [t0,t0+δ1]

. Then for any x, y ∈

A(t0, ϕ, δ1, γ1)we have∣∣∣∣∫ t

t0(t − s)q−1[f (s, xs)− f (s, ys)]ds

∣∣∣∣ ≤ ∫ t

t0(t − s)q−1|f (s, xs)− f (s, ys)|ds

3256 Y. Zhou et al. / Nonlinear Analysis 71 (2009) 3249–3256

≤

∫ t

t0(t − s)q−1`1(s)ds sup

t0≤s≤t|x(s)− y(s)|

≤N

(1+ β ′)1−q2(t − t0)(1+β

′)(1−q2) supt0≤s≤t

|x(s)− y(s)|,

where β ′ = q−11−q2∈ (−1, 0). Let

`(t − t0) =N

(1+ β ′)1−q2(t − t0)(1+β

′)(1−q2), t ∈ [t0, t0 + δ1].

Obviously, ` : [0, δ1] → [0,∞) continuous at t = 0 and `(0) = 0. Then the condition (H4) of Theorem 3.2 holds. Thiscompletes the proof.

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