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Existence and uniqueness for p -type fractional neutral differential equations
Nonlinear Analysis 71 (2009) 2724–2733
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Existence and uniqueness for p-type fractional neutral differential equationsI Yong Zhou ∗ , Feng Jiao, Jing Li Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, PR China
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Article history: Received 23 November 2008 Accepted 15 January 2009 Keywords: Fractional neutral differential equations p-function Existence Uniqueness
abstract In this paper, the Cauchy initial value problem is discussed for the p-type fractional neutral functional differential equations and various criteria on existence and uniqueness are obtained. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction 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 and partial differential equations involving fractional derivatives, see the monographs of Kilbas et al. [1], Miller and Ross [2], Podlubny [3] and the papers [4–16] and the references therein. In [10], Lakshmikantham initiates the basic theory for fractional functional differential equations. In [8], El-Sayed discusses a class of nonlinear functional differential equations of arbitrary orders. In [5], Benchohra et al. consider the IVP for a particular class of fractional neutral functional differential equations with infinite delay. In [16], Zhou studies the existence and uniqueness of the fractional functional differential equations with unbounded delay. Let J ⊂ R. Denote C (J , Rn ) as the Banach space of all continuous functions from J into Rn with the norm kxk = supt ∈J |x(t )|, where | · | denotes a suitable complete norm. Let C = C ([−1, 0], Rn ) denote the space of continuous functions on [−1, 0]. For any element φ ∈ C , define the norm kφk∗ = supθ∈[−1,0] |φ(θ )|. Consider the IVP of p-type fractional neutral functional differential equations of the form Dq g (t , xt ) = f (t , xt ),
(1)
xt0 = ϕ,
(2)
(t0 , ϕ) ∈ Ω ,
where Dq is Caputo’s fractional derivative of order 0 < q < 1, Ω is an open subset of [0, ∞) × C and g , f : Ω → Rn are given functionals satisfying some assumptions that will be specified later. xt ∈ C is defined by xt (θ ) = x(p(t , θ )), where −1 ≤ θ ≤ 0, p(t , θ ) is a p-function. In this paper, we obtain various criteria on existence and uniqueness for IVP (1)–(2).
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: [email protected] (Y. Zhou).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.105
Y. Zhou et al. / Nonlinear Analysis 71 (2009) 2724–2733
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2. Preliminaries In this section, we introduce definitions and preliminary facts which are used throughout this paper. Definition 2.1 ([3]). The fractional integral of order µ with the lower limit t0 for a function f is defined as I µ f (t ) =
Z
1
0 (µ)
f (s)
t
t0
(t − s)1−µ
ds,
t > t0 , µ > 0,
provided the right-hand side is pointwise defined on [t0 , ∞), where 0 is the gamma function. Definition 2.2 ([3]). Caputo’s derivative of order µ with the lower limit t0 for a function f : [0, ∞) → R can be written as µ
D f (t ) =
1
0 (n − µ)
Z
f (n) (s)
t t0
(t − s)µ+1−n
ds = I n−µ f (n) (t ),
t > t0 , 0 ≤ n − 1 < µ < n.
Obviously, Caputo’s derivative of a constant is equal to zero. Definition 2.3 ([17]). A function p ∈ C (J × [−1, 0], R) is called a p-function if it has the following properties (i) (ii) (iii) (iv)
p(t , 0) = t, p(t , −1) is a nondecreasing function of t, there exists a σ ≥ −∞ such that p(t , θ ) is an increasing function for θ for each t ∈ (σ , ∞), p(t , 0) − p(t , −1) > 0 for t ∈ (σ , ∞).
Throughout the following text we suppose t ∈ (σ , ∞). Definition 2.4 ([17]). Let t0 ≥ 0, A > 0 and x ∈ C ([p(t0 , −1), t0 + A], Rn ). For any t ∈ [t0 , t0 + A], we define xt by xt (θ ) = x(p(t , θ )),
−1 ≤ θ ≤ 0,
so that xt ∈ C = C ([−1, 0], Rn ). Note that the frequently used symbol ‘‘xt ’’ (in [10,17,18], xt (θ ) = x(t + θ ), where −r ≤ θ ≤ 0, r > 0, r = const) in the theory of functional differential equations with bounded delay is a partial case of the above definition. Indeed, in this case we can put p(t , θ ) = t + r θ , θ ∈ [−1, 0]. Definition 2.5. A function x is said to be a solution of IVP (1)–(2) on [p(t0 , −1), t0 + α], if there are t0 ≥ 0, α > 0, such that (i) x ∈ C ([p(t0 , −1), t0 + α], Rn ) and (t , xt ) ∈ Ω , for t ∈ [t0 , t0 + α], (ii) xt0 = ϕ , (iii) g (t , xt ) is differentiable and (1) holds almost everywhere on [t0 , t0 + α]. We need the following lemma relative to p-function before we proceed further, which is taken from [17]. Lemma 2.1 ([17]). Suppose that p(t , θ ) is a p-function. For A > 0, τ ∈ (σ , ∞) (τ may be σ if σ > −∞), let x ∈ C ([p(τ , −1), τ + A], Rn ) and ϕ ∈ C ([−1, 0], Rn ). Then we have (i) xt is continuous in t on [τ , τ + A] and p˜ (t , θ ) = p(τ + t , θ ) − τ is also a p-function, (ii) if p(τ + t , −1) < τ for t > 0, then there exists −1 < s(τ , t ) < 0 such that p(τ + t , s(τ , t )) = τ and p(τ + t , −1) ≤ p(τ + t , θ ) ≤ τ τ ≤ p(τ + t , θ ) ≤ τ + t
for −1 ≤ θ ≤ s(τ , t ), for s(τ , t ) ≤ θ ≤ 0.
Moreover, s → 0 uniformly in τ as t → 0, (iii) there exists a function η ∈ C ([p(τ , −1), τ ], Rn ) such that
η(p(τ , θ )) = ϕ(θ ) for −1 ≤ θ ≤ 0. It is well known that a neutral functional differential equation (NFDE for short) is one in which the derivatives of the past history or derivatives of functionals of the past history are involved as well as the present state of the system. In other words, 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 concept of atomic. For a detailed discussion on atomic concept we refer the reader to the books [17,18].
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Let gφ0 denote the Fréchet derivative of g with respect to the second argument, then we have the following lemma. Lemma 2.2 ([17,18]). Suppose that g (t , φ) is atomic at zero on Ω . Then there are a continuous n × n matrix function A(t , φ) with detA(t , φ) 6= 0 on Ω and a functional L(t , φ, ψ) which is linear in ψ such that gφ0 (t , φ)ψ = A(t , φ)ψ(0) + L(t , φ, ψ). Moreover, there exists a continuous function γ : Ω × [0, 1] → R+ with γ (t , φ, 0) = 0 such that for every s ∈ [0, 1] and ψ with (t , ψ) ∈ Ω , ψ(θ ) = 0 for −1 ≤ θ ≤ −s,
|L(t , φ, ψ)| ≤ γ (t , φ, s)kψk∗ . Lemma 2.3 (Krasnoselskii’s Fixed Point Theorem). Let X be a Banach space, let E be a bounded closed convex subset of X and let S , T be maps of E into X such that Sz + T w ∈ E for every pair z , w ∈ E. If S is a contraction and T is completely continuous, then the equation Sz + Tz = z has a solution on E. 3. Main results Assume that the functional f : Ω → Rn satisfies the following conditions.
(H1 ) f (t , φ) is Lebesgue measurable with respect to t for any (t , φ) ∈ Ω , (H2 ) f (t , φ) is continuous with respect to φ for any (t , φ) ∈ Ω , 1
(H3 ) there exist a constant q1 ∈ (0, q) and a L q1 -integrable function m such that |f (t , φ)| ≤ m(t ) for any (t , φ) ∈ Ω . For each (t0 , ϕ) ∈ Ω , let p˜ (t , θ ) = p(t0 + t , θ ) − t0 . Define the function η ∈ C ([˜p(0, −1), ∞), Rn ) by
η(˜p(0, θ )) = ϕ(θ ) η(t ) = ϕ(0)
for θ ∈ [−1, 0], for t ∈ [0, ∞).
Let x ∈ C ([p(t0 , −1), t0 + α], Rn ), α < A and let x(t0 + t ) = η(t ) + z (t )
for p˜ (0, −1) ≤ t ≤ α.
(3)
In order to prove our main results, we need the following lemmas. Lemma 3.1. x(t ) is a solution of IVP (1)–(2) on [p(t0 , −1), t0 + α] if and only if z (t ) satisfies the relation
g (t0 + t , η˜ t + z˜t ) − g (t0 , ϕ) = z0 = 0,
˜
1
Z
0 (q)
t
(t − s)q−1 f (t0 + s, η˜ s + z˜s )ds,
for t ∈ [0, α],
0
(4)
where η˜ t (θ ) = η(˜p(t , θ )), z˜t (θ ) = z (˜p(t , θ )), for −1 ≤ θ ≤ 0. Proof. Since xt is continuous in t, xt is a measurable function, therefore according to conditions (H1 ) and (H2 ), f (t , xt ) 1
is Lebesgue measurable on [t0 , t0 + α]. Direct calculation gives that (t − s)q−1 ∈ L 1−q1 [t0 , t ], for t ∈ [t0 , t0 + α] and q1 ∈ (0, q). In light of Hölder inequality, we obtain that (t − s)q−1 f (s, xs ) is Lebesgue integrable with respect to s ∈ [t0 , t ] for all t ∈ [t0 , t0 + α], and t
Z
|(t − s)q−1 f (s, xs )|ds ≤ k(t − s)q−1 k t0
1
L 1−q1 [t0 ,t ]
kmk
1
L q1 [t0 ,t0 +α]
,
where
Z kH kLp [J ] =
|H (t )|p dt
1p
,
J
for any Lp -integrable function H : J → R. Hence x(t ) is the solution of IVP (1)–(2) if and only if it satisfies the relation
g (t , xt ) − g (t0 , xt0 ) = xt0 = ϕ,
1
0 (q)
Z
t
(t − u)q−1 f (u, xu )du, t0
for t ∈ [t0 , t0 + α],
Y. Zhou et al. / Nonlinear Analysis 71 (2009) 2724–2733
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or setting u = t0 + s,
g (t0 + t , xt0 +t ) − g (t0 , xt0 ) = xt0 = ϕ.
1
t
Z
0 (q)
(t − s)q−1 f (t0 + s, xt0 +s )ds,
for t ∈ [0, α],
(5)
0
In view of (3), we have xt0 +t (θ ) = x(p(t0 + t , θ )) = x(˜p(t , θ ) + t0 ) = η(˜p(t , θ )) + z (˜p(t , θ ))
= η˜ t (θ ) + z˜t (θ ),
for t ∈ [0, α].
In particular xt0 (θ ) = η˜ 0 (θ ) + z˜0 (θ ). Hence xt0 = ϕ if and only if z˜0 = 0 according to η˜ = ϕ . It is clear that x(t ) satisfies (5) if and only if z (t ) satisfies (4). This completes the proof. For any δ, ξ > 0, let E (δ, ξ ) = {z ∈ C ([˜p(0, −1), δ], Rn ) : z˜0 = 0, k˜zt k∗ ≤ ξ for t ∈ [0, δ]}, which is a bounded closed convex subset of the Banach space C ([˜p(0, −1), δ], Rn ) endowed with supremum norm k · k. Lemma 3.2. Suppose Ω ⊆ R × C is open, W ⊂ Ω is compact. For any a neighborhood V 0 ⊂ Ω of W , there is a neighborhood V 00 ⊂ V 0 of W and there exist positive numbers δ and ξ such that (t0 + t , η˜ t + λ˜zt ) ∈ V 0 with 0 ≤ λ ≤ 1 for any (t0 , ϕ) ∈ V 00 , t ∈ [0, δ] and z ∈ E (δ, ξ ). The proof of Lemma 3.2 is similar to that of (iii) of Lemma 2.1.8 in [17], thus it is omitted. Suppose g is atomic at 0 on Ω . Define two operators S and T on E (α, β) as follows
(Sz )(t ) = 0, for t ∈ [˜p(0, −1), 0], A(t0 + t , η˜ t )(Sz )(t ) = g (t0 , ϕ) − g (t0 + t , η˜ t + z˜t ) + gφ0 (t0 + t , η˜ t )˜zt − L(t0 + t , η˜ t , z˜t ),
for t ∈ [0, α]
(6)
and
(Tz )(t ) = 0,
for t ∈ [˜p(0, −1), 0], Z t 1 (t − s)q−1 f (t0 + s, η˜ s + z˜s )ds, A(t0 + t , η˜ t )(Tz )(t ) = 0 (q) 0
for t ∈ [0, α],
(7)
where A(t0 + t , η˜ t ), L(t0 + t , η˜ t , z˜t ) are functions described in Lemma 2.2. It is clear that the operator equation z = Sz + Tz
(8)
has a solution z ∈ E (α, β) if and only if z is a solution of (4). Therefore the existence of a solution of IVP (1)–(2) is equivalent to determining α, β > 0 such that S + T has a fixed point on E (α, β). We are now in a position to prove the following existence results, and the proof is based on Krasnoselskii’s fixed point theorem. Theorem 3.1. Suppose g : Ω → Rn is continuous together with its first Fréchet derivative with respect to the second argument, and g is atomic at 0 on Ω . f : Ω → Rn satisfies conditions (H1 )–(H3 ). W ⊂ Ω is a compact set. Then there exist a neighborhood V ⊂ Ω of W and a constant α > 0 such that for any (t0 , ϕ) ∈ V , IVP (1)–(2) has a solution which exists on [p(t0 , −1), t0 + α]. Proof. As we have mentioned above, we only need to discuss operator equation (8). For any (t , φ) ∈ Ω , the property of the matrix function A(t , φ) which is nonsingular and continuous on Ω implies that its inverse matrix A−1 (t , φ) exists and is continuous on Ω . Let V0 ⊂ Ω be the neighborhood of W , suppose that there is an M > 0 such that
|A−1 (t 0 , φ)| ≤ M for every (t 0 , φ) ∈ V0 .
(9)
Note the complete continuity of the function (m(t ))
α0 > 0 satisfying Z t0 +α0 q1 1 (m(s)) q1 ds ≤ N.
1 q1
, hence, for a given positive number N, there must exist a number
(10)
t0
Due to the continuity of functions γ and gφ0 described in Lemma 2.2, there exist a neighborhood V1 ⊂ Ω of W and constants h1 > 0, h2 ∈ (0, 1] such that
|γ (t0 + t , η˜ t , −s)| = |γ (t0 + t , η˜ t , −s) − γ (t0 + t , η˜ t , 0)| < |gφ0 (t0 + t , η˜ t + ψ) − gφ0 (t0 + t , η˜ t )| <
1 8M
1 4M
,
,
whenever (t0 + t , η˜ t ), (t0 + t , η˜ t + ψ) ∈ V1 and kψk∗ < h1 , −s ∈ [0, h2 ].
(11) (12)
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Let V2 = V0 ∩ V1 . According to Lemma 3.2, we can find a neighborhood V ⊂ V2 of W and positive numbers α1 and β with α1 < α0 and β ≤ h1 such that (t0 + t , η˜ t + λ˜zt ) ∈ V2 with 0 ≤ λ ≤ 1 for any (t0 , ϕ) ∈ V , t ∈ [0, α1 ] and z ∈ E (α1 , β). Let h(t0 + t , η˜ t , z˜t ) = g (t0 + t , η˜ t + z˜t ) − g (t0 + t , η˜ t ) − gφ0 (t0 + t , η˜ t )˜zt . Then we have
Z 1 0 0 |h(t0 + t , η˜ t , z˜t )| = gφ (t0 + t , η˜ t + λ˜zt )dλ − gφ (t0 + t , η˜ t ) z˜t 0 Z 1 ≤ [gφ0 (t0 + t , η˜ t + λ˜zt ) − gφ0 (t0 + t , η˜ t )]dλ k˜zt k∗ .
(13)
0
According to (9), (12) and (13), for any (t0 , ϕ) ∈ V , we have
|A−1 (t0 + t , η˜ t ) h(t0 + t , η˜ t , z˜t )| ≤
β 8
.
(14)
On the other hand, for any z , w ∈ E (α1 , β) and t ∈ [0, α1 ]
kλ˜zt + (1 − λ)w ˜ t k∗ ≤ kλ˜zt k∗ + k(1 − λ)w ˜ t k∗ ≤ λβ + (1 − λ)β = β, thus, (t0 + t , η˜ t + λ˜zt + (1 − λ)w ˜ t ) ∈ V2 , and
|h(t0 + t , η˜ t , z˜t ) − h(t0 + t , η˜ t , w ˜ t )| = |g (t0 + t , η˜ t + z˜t ) − g (t0 + t , η˜ t + w ˜ t ) − gφ0 (t0 + t , η˜ t )(˜zt − w ˜ t )| Z 1 = gφ0 (t0 + t , η˜ t + w ˜ t + λ(˜zt − w ˜ t ))dλ − gφ0 (t0 + t , η˜ t ) (˜zt − w ˜ t ) 0 Z 1 0 0 ≤ [gφ (t0 + t , η˜ t + λ˜zt + (1 − λ)w ˜ t ) − gφ (t0 + t , η˜ t )]dλ k˜zt − w ˜ t k∗ . (15) 0
From (9), (12) and (15), we have
|A−1 (t0 + t , η˜ t )[h(t0 + t , η˜ t , z˜t ) − h(t0 + t , η˜ t , w ˜ t )]| ≤
1 8
k˜zt − w ˜ t k∗ .
(16)
By (ii) of Lemma 2.1, we can also choose α2 < α1 such that for t ∈ [0, α2 ], −s(0, t ) ∈ [0, h2 ]. From (9) and (11), we have
|A−1 (t0 + t , η˜ t )||L(t0 + t , η˜ t , z˜t )| ≤ |A−1 (t0 + t , η˜ t )|γ (t0 + t , η˜ t , −s(0, t ))k˜zt k∗ ≤
1 4
k˜zt k∗ ,
(17)
whenever t ∈ [0, α2 ] and z ∈ E (α2 , β). Now consider the expression g (t0 , ϕ) − g (t0 + t , η˜ t ). Since g is continuous in Ω and noting the facts that η˜ t is continuous in t and η˜ 0 = ϕ , there exists a constant α3 < α2 such that
|g (t0 , ϕ) − g (t0 + t , η˜ t )| <
β 8M
,
(18)
whenever t ∈ [0, α3 ]. Set
(
1
α = min α3 , (1 + b) 1+b
0 (q)β 2MN
1
(1−q1 )(1+b)
) ,
(19)
where b = 1−q ∈ (−1, 0). 1 Now we show that for any (t0 , ϕ) ∈ V , S + T has a fixed point on E (α, β), where S and T are defined as in (6) and (7) respectively. The proof is divided into three steps. q −1
Step I. Sz + T w ∈ E (α, β) whenever z , w ∈ E (α, β). Obviously, for every pair z , w ∈ E (α, β), (Sz )(t ) and (T w)(t ) are continuous in t ∈ [0, α]. From (14), (17) and (18), for t ∈ [0, α], we have
|(Sz )(t )| ≤ |A−1 (t0 + t , η˜ t )|{|g (t0 , ϕ) − g (t0 + t , η˜ t )| + |L(t0 + t , η˜ t , z˜t )| + |g (t0 + t , η˜ t ) − g (t0 + t , η˜ t + z˜t ) + gφ0 (t0 + t , η˜ t )˜zt |} β ≤ . 2
Y. Zhou et al. / Nonlinear Analysis 71 (2009) 2724–2733
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For t ∈ [0, α], by using (9), (10), (19) and Hölder inequality, we have
Z t q −1 |(T w)(t )| ≤ |A (t0 + t , η˜ t )| (t − s) f (t0 + s, η˜ s + w ˜ s )ds 0 (q) 0 Z t 1−q1 Z t0 +α q1 1 1 M (m(s)) q1 ds ≤ ((t − s)q−1 ) 1−q1 ds 0 (q) 0 t0 1−q1 1 MN α 1+b ≤ 0 (q) 1 + b 1
−1
≤
β 2
.
(20)
Thus |(Sz )(t ) + (T w)(t )| ≤ β i.e. Sz + T w ∈ E (α, β), whenever z , w ∈ E (α, β). Step II. S is a contraction mapping from E (α, β) into itself whose contraction constant is independent of (t0 , ϕ) ∈ V . For any z , w ∈ E (α, β), w ˜ 0 − z˜0 = 0. Hence (ii) of Lemmas 2.1 and 2.2 are applicable to w ˜ t − z˜t . For every pair z , w ∈ E (α, β), from (16) and (17) and noting the fact that sup k˜zt − w ˜ t k∗ = sup 0≤t ≤α
sup |z (˜p(t , θ )) − w(˜p(t , θ ))|
0≤t ≤α −1≤θ≤0
= sup
sup
0≤t ≤α p˜ (t ,−1)≤s≤t
=
sup p˜ (0,−1)≤s≤α
|z (s) − w(s)|
|z (s) − w(s)|
= kz − wk, we have
kSz − S wk =
sup p˜ (0,−1)≤t ≤α
|(Sz )(t ) − (S w)(t )|
= sup |(Sz )(t ) − (S w)(t )| 0≤t ≤α
≤ sup |A−1 (t0 + t , η˜ t )|[|L(t0 + t , η˜ t , w ˜ t − z˜t )| + |h(t0 + t , η˜ t , z˜t ) − h(t0 + t , η˜ t , w ˜ t )|] 0≤t ≤α 1 1 ≤ + sup k˜zt − w ˜ t k∗ 8
≤
3 8
4
0≤t ≤α
kz − wk.
Therefore S is a contraction mapping from E (α, β) into itself whose contraction constant is independent of (t0 , ϕ) ∈ V . Step III. Now we show that T is a completely continuous operator. For any z ∈ E (α, β) and 0 ≤ t1 < t2 ≤ α , we get
Z 1 |(Tz )(t2 ) − (Tz )(t1 )| = A−1 (t0 + t2 , η˜ t2 ) 0 (q) 1
− A (t0 + t1 , η˜ t1 ) 0 (q) Z −1 1 = A (t0 + t2 , η˜ t2 ) 0 (q) −1
+ A (t0 + t2 , η˜ t2 ) −1
− A−1 (t0 + t2 , η˜ t2 ) + A (t0 + t2 , η˜ t2 ) −1
1
0 (q) 1
0 (q) 1
0 (q)
t2
(t2 − s)q−1 f (t0 + s, η˜ s + z˜s )ds
0 t1
Z
(t1 − s)
q −1
0 t2
f (t0 + s, η˜ s + z˜s )ds
(t2 − s)q−1 f (t0 + s, η˜ s + z˜s )ds
t1
Z
t1
(t2 − s)q−1 f (t0 + s, η˜ s + z˜s )ds
0
Z
t1
(t1 − s)q−1 f (t0 + s, η˜ s + z˜s )ds
0
Z
t1
(t1 − s)q−1 f (t0 + s, η˜ s + z˜s )ds
0 t1
− A (t0 + t1 , η˜ t1 ) (t1 − s) f (t0 + s, η˜ s + z˜s )ds 0 (q) 0 Z |A−1 (t0 + t2 , η˜ t2 )| t2 q−1 (t2 − s) f (t0 + s, η˜ s + z˜s )ds ≤ 0 (q) t1 −1
1
Z
q −1
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Y. Zhou et al. / Nonlinear Analysis 71 (2009) 2724–2733
Z |A−1 (t0 + t2 , η˜ t2 )| t1 q −1 q −1 ˜ [( t − s ) − ( t − s ) ] f ( t + s , η ˜ + z ) ds 2 1 0 s s 0 (q) 0 Z |A−1 (t0 + t2 , η˜ t2 ) − A−1 (t0 + t1 , η˜ t1 )| t1 q −1 ˜ + ( t − s ) f ( t + s , η ˜ + z ) ds 1 0 s s 0 (q) 0 |A−1 (t0 + t2 , η˜ t2 )| |A−1 (t0 + t2 , η˜ t2 ) − A−1 (t0 + t1 , η˜ t1 )| = (I1 + I2 ) + I3 , 0 (q) 0 (q) +
where t2
Z
I1 =
t
(t2 − s)q−1 f (t0 + s, η˜ s + z˜s )ds ,
Z 1t 1 I2 = [(t2 − s)q−1 − (t1 − s)q−1 ]f (t0 + s, η˜ s + z˜s )ds , 0 Z t 1 I3 = (t1 − s)q−1 f (t0 + s, η˜ s + z˜s )ds . 0
By using the analogous argument performed in (20), we can conclude that I1 ≤ I3 ≤
N
(1 + b)1−q1 N
(1 + b)
1−q1
1−q1 (t2 − t1 )1+b , t1 1+b
1−q1
,
and t1
Z
|(t2 − s)
q −1
I2 ≤
− (t1 − s)
1−q1 Z
1
q−1 1−q1
|
t0 + t1
ds
0
1 q1
|f (s, xs )| ds
q1
t0 t1
Z ≤N
((t1 − s)b − (t2 − s)b )ds
1−q1
0
= ≤
N
(1 + b)
1−q1
N
(1 + b)
1−q1
t1 1+b − t2 1+b + (t2 − t1 )1+b
1−q1
1−q1 (t2 − t1 )1+b ,
where b = 1−q ∈ (−1, 0). Therefore 1 q −1
|(Tz )(t2 ) − (Tz )(t1 )| ≤
1−q1 |A−1 (t0 + t2 , η˜ t2 )| 2N (t2 − t1 )1+b 1 − q 0 (q) (1 + b) 1 1−q1 |A−1 (t0 + t2 , η˜ t2 ) − A−1 (t0 + t1 , η˜ t1 )| N + t1 1+b . 1 − q 0 (q) ( 1 + b) 1
Since A−1 (t0 + t , η˜ t ) is continuous in t ∈ [0, α], then {Tz ; z ∈ E (α, β)} is equicontinuous. In addition, T is continuous from the condition (H2 ) and {Tz ; z ∈ E (α, β)} is uniformly bounded from (20), thus T is a completely continuous operator by Ascoli–Arzela Theorem. Therefore, by Lemma 2.3, for every (t0 , ϕ) ∈ V , S + T has a fixed point on E (α, β). Hence, IVP (1)–(2) has a solution defined on [p(t0 , −1), t0 + α]. The proof is complete. Corollary 3.1. Suppose that (t0 , ϕ) ∈ Ω is given, g , f are defined as in Theorem 3.1. Then there exists a solution of IVP (1)–(2). Corollary 3.2. Suppose that Ω , f are defined as in Theorem 3.1. If (t0 , ϕ) ∈ Ω is given, then the IVP relative to fractional p-type retarded differential equations of the form
Dq x(t ) = f (t , xt ), xt0 = ϕ
has a solution. The following existence and uniqueness result for IVP (1)–(2) is based on Banach’s contraction principle.
Y. Zhou et al. / Nonlinear Analysis 71 (2009) 2724–2733
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Theorem 3.2. Suppose (t0 , ϕ) ∈ Ω is given, g is defined as in Theorem 3.1. f : Ω → Rn satisfies the condition (H3 ) and
(H4 ) f (t , xt ) is measurable for every (t , xt ) ∈ Ω , (H5 ) let A > 0, there exists a nonnegative function ` : [0, A] → [0, ∞) continuous at t = 0 and `(0) = 0 such that for any (t , xt ), (t , yt ) ∈ Ω , t ∈ [t0 , t0 + A], we have Z t (t − s)q−1 [f (s, xs ) − f (s, ys )]ds ≤ `(t − t0 ) sup kxs − ys k∗ . t ≤s≤t t0
0
Then IVP (1)–(2) has a unique solution. Proof. According to the argument of Theorem 3.1, it suffices to prove that S + T has a unique fixed point on E (α, β), where α, β > 0 sufficiently small. Now, choose α ∈ (0, A] such that (19) holds and c=
3 8
+ sup 0≤s≤α
|A−1 (t0 + s, η˜ s )||`(s)| < 1. 0 (q)
(21)
Obviously, S + T is a mapping from E (α, β) into itself. Using the same argument as that of Theorem 3.1, for any z , w ∈ E (α, β), t ∈ [0, α], we get
|(Sz )(t ) − (S w)(t )| ≤
3 8
kz − wk,
and
Z Z t |A−1 (t0 + t , η˜ t )| t q −1 q −1 ˜ ( t − s ) f ( t + s , η ˜ + z ) ds − ( t − s ) f ( t + s , η ˜ + w ˜ ) ds 0 s s 0 s s 0 (q) 0 0 |A−1 (t0 + t , η˜ t )| ≤ |`(t )| sup k˜zs − w ˜ s k∗ 0 (q) 0≤s≤t
|(Tz )(t ) − (T w)(t )| ≤
sup |A−1 (t0 + s, η˜ s )||`(s)|
≤
0≤s≤α
kz − wk.
0 (q)
Therefore
|(S + T )z (t ) − (S + T )w(t )| ≤
3 8
+ sup 0≤s≤α
|A−1 (t0 + s, η˜ s )||`(s)| kz − wk 0 (q)
= c kz − wk. Hence, we have
k(S + T )z − (S + T )wk ≤ c kz − wk, where c < 1. By applying Banach’s contraction principle, we know that S + T has a unique fixed point on E (α, β). The proof is complete. Corollary 3.3. Suppose the condition (H5 ) of Theorem 3.2 is replaced by the following condition 1
(H5 )0 let A > 0, there exist q2 ∈ (0, q) and a real-valued function `1 ∈ L q2 [t0 , t0 + A] such that for any (t , xt ), (t , yt ) ∈ Ω , t ∈ [t0 , t0 + A], we have |f (t , xt ) − f (t , yt )| ≤ `1 (t ) sup kxs − ys k∗ . t0 ≤s≤t
Then the result of Theorem 3.2 holds. 1
Proof. It suffices to prove that the condition (H5 ) of Theorem 3.2 holds. Note that `1 ∈ L q2 [t0 , t0 + A], let K = k`1 k Then for any (t , xt ), (t , yt ) ∈ Ω we have
Z t Z t (t − s)q−1 [f (s, xs ) − f (s, ys )]ds ≤ (t − s)q−1 |f (s, xs ) − f (s, ys )|ds t0 t0 Z t ≤ (t − s)q−1 `1 (s) ds sup kxs − ys k∗ t0 ≤s≤t
t0
≤
K
(1 + b1 )1−q2
(t − t0 )(1+b1 )(1−q2 ) sup kxs − ys k∗ , t0 ≤s≤t
1
L q2 [t0 ,t0 +A]
.
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Y. Zhou et al. / Nonlinear Analysis 71 (2009) 2724–2733
where b1 = 1−q ∈ (−1, 0). Let 2 q −1
`(t − t0 ) =
K
(1 + b1 )1−q2
(t − t0 )(1+b1 )(1−q2 ) .
Obviously, ` : [0, A] → [0, ∞) is continuous at t = 0 and `(0) = 0. Then the condition (H5 ) of Theorem 3.2 holds. This completes the proof. The next result is concerned with the uniqueness of solutions. Theorem 3.3. Suppose that g are defined as in Theorem 3.1 and the condition (H5 )0 of Corollary 3.3 holds. If x is a solution of IVP (1)–(2), then x is unique. Proof. Suppose (for contradiction) x and y are the solutions of IVP (1)–(2) on [p(t0 , −1), t0 + A] with x 6= y, let t1 = inf{t ∈ [t0 , t0 + A] : x(t ) 6= y(t )}. Then t0 ≤ t1 < t0 + A and x(t ) = y(t )
for p(t0 , −1) ≤ t < t1 ,
which implies that xt (θ ) = x(p(t , θ )) = y(p(t , θ )) = yt (θ ),
t0 ≤ t < t1 , −1 ≤ θ ≤ 0.
(22)
Choose α > 0 such that t1 + α < t0 + A. According to (i) of Definition 2.5, we have
{(t , xt ), t1 ≤ t ≤ t1 + α} ∪ {(t , yt ), t1 ≤ t ≤ t1 + α} ⊂ Ω . On the one hand, x and y satisfy (1)–(2) on [t0 , t0 + A], thus from (22) and the condition (H5 )0 , for t ∈ [t0 , t1 + α], we have
Z t q −1 (t − s) [f (s, xs ) − f (s, ys )]ds |g (t , xt ) − g (t , yt )| ≤ 0 (q) t0 Z 1 t q −1 (t − s) [f (s, xs ) − f (s, ys )]ds = 0 (q) t1 Z t 1 (t − s)q−1 `1 (s) ds sup kxs − ys k∗ ≤ 0 (q) t1 t0 ≤s≤t 1
≤
K
0 (q)(1 + b1 )1−q2
α (1+b1 )(1−q2 )
sup t1 ≤s≤t1 +α
kxs − ys k∗ ,
(23)
where b1 = 1−q ∈ (−1, 0), K = k`1 k 1 . 2 L q2 [t0 ,t0 +A] On the other hand, since g (t , φ) is continuously differentiable in φ , we have q −1
g (t , xt ) − g (t , yt ) = gφ0 (t , yt )(xt − yt ) + kkxt − yt k∗
(24)
with k → 0 as kxt − yt k∗ → 0. By the hypothesis that g (t , φ) is atomic at 0 on Ω , there exist a nonsingular continuous matrix function A(t , yt ) and a function L(t , yt , ψ) which is linear in ψ such that gφ0 (t , yt )ψ = A(t , yt )ψ(0) + L(t , yt , ψ).
(25)
Moreover, there is a positive real-valued continuous function γ (t , yt , −s) such that for every s ∈ [−1, 0],
|L(t , yt , ψ)| ≤ γ (t , yt , −s)kψk∗
(26)
if ψ(θ) = 0 for −1 ≤ θ ≤ s. Hence for every t ∈ [t1 , t1 + α], by (ii) of Lemma 2.1, there is s(t1 , t − t1 ) ∈ [−1, 0] with s(t1 , t − t1 ) → 0 as t → t1 such that
|L(t , yt , xt − yt )| ≤ γ (t , yt , −s(t1 , t − t1 ))kxt − yt k∗ . From (24)–(26), it follows that g (t , xt ) − g (t , yt ) = A(t , yt )(x(t ) − y(t )) + L(t , yt , xt − yt ) + kkxt − yt k∗ , therefore
|x(t ) − y(t )| ≤ |A−1 (t , yt )|{|g (t , xt ) − g (t , yt )| + γ (t , yt , −s(t1 , t − t1 ))kxt − yt k∗ + kkxt − yt k∗ }.
Y. Zhou et al. / Nonlinear Analysis 71 (2009) 2724–2733
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Let M1 = max{|A−1 (t , yt )| : t1 ≤ t ≤ t1 + α}. Then by relation (23), for t ∈ [t1 , t1 + α], we have
|x(t ) − y(t )| ≤ c1 where c1 = M1
h
sup t1 ≤s≤t1 +α
K 0 (q)(1+b1 )1−q2
kxs − ys k∗ ,
i α (1+b1 )(1−q2 ) + γ (t , yt , −s(t1 , t − t1 )) + k .
Noting that sup
kxs − ys k∗ =
t1 ≤s≤t1 +α
sup
sup |x(p(s, θ )) − y(p(s, θ ))|
t1 ≤s≤t1 +α −1≤θ≤0
=
sup
sup
t1 ≤s≤t1 +α p(s,−1)≤ρ≤s
=
sup
p(t1 ,−1)≤s≤t1 +α
|x(ρ) − y(ρ)|
|x(s) − y(s)|,
we have sup
p(t1 ,−1)≤s≤t1 +α
|x(s) − y(s)| ≤ c1
sup
p(t1 ,−1)≤s≤t1 +α
|x(s) − y(s)|.
Choose α so small that c1 < 1. Thus sup
p(t1 ,−1)≤s≤t1 +α
|x(s) − y(s)| = 0,
i.e. x(t ) ≡ y(t ),
for t1 ≤ t ≤ t1 + α,
contradicting the definition of t1 . Hence the proof is complete.
References [1] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam, 2006. [2] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. [3] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1993. [4] R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations, Acta Appl. Math. (2009), doi:10.1007/s10440-008-9356-6. [5] M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl. 338 (2008) 1340–1350. [6] Varsha Daftardar-Gejji, Sachin Bhalekar, Boundary value problems for multi-term fractional differential equations, J. Math. Anal. Appl. 345 (2008) 754–765. [7] Mahmoud M. El-Borai, Semigroups and some nonlinear fractional differential equations, Appl. Math. Comput. 149 (2004) 823–831. [8] A.M.A. El-Sayed, Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal. 33 (1998) 181–186. [9] Nickolai Kosmatov, Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear Anal. (2009), doi:10.1016/j.na.2008.03.037. [10] V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal. 69 (2008) 3337–3343. [11] V. Lakshmikantham, A.S. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Lett. 21 (2008) 828–834. [12] V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. 69 (2008) 2677–2682. [13] M. Muslim, Existence and approximation of solutions to fractional differential equations, Math. Comput. Modelling (2009), doi:10.1016/j.mcm.2008.07.013. [14] H.A.H. Salem, On the existence of continuous solutions for a singular system of non-linear fractional differential equations, Appl. Math. Comput. 198 (2008) 445–452. [15] J. Vasundhara Devi, V. Lakshmikantham, Nonsmooth analysis and fractional differential equations, Nonlinear Anal. (2009), doi:10.1016/j.na.2008.09.003. [16] Yong Zhou, Existence and uniqueness of fractional functional differential equations with unbounded delay, Int. J. Dyn. Syst. Differ. Equs. 1 (4) (2008) 239–244. [17] V. Lakshmikantham, L. Wen, B. Zhang, Theory of Differential Equations with Unbounded Delay, Kluwer Academic Publishers, Dordrecht, 1994. [18] J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Existence and uniqueness for p-type fractional neutral differential equationsI Yong Zhou ∗ , Feng Jiao, Jing Li Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, PR China
article
info
Article history: Received 23 November 2008 Accepted 15 January 2009 Keywords: Fractional neutral differential equations p-function Existence Uniqueness
abstract In this paper, the Cauchy initial value problem is discussed for the p-type fractional neutral functional differential equations and various criteria on existence and uniqueness are obtained. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction 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 and partial differential equations involving fractional derivatives, see the monographs of Kilbas et al. [1], Miller and Ross [2], Podlubny [3] and the papers [4–16] and the references therein. In [10], Lakshmikantham initiates the basic theory for fractional functional differential equations. In [8], El-Sayed discusses a class of nonlinear functional differential equations of arbitrary orders. In [5], Benchohra et al. consider the IVP for a particular class of fractional neutral functional differential equations with infinite delay. In [16], Zhou studies the existence and uniqueness of the fractional functional differential equations with unbounded delay. Let J ⊂ R. Denote C (J , Rn ) as the Banach space of all continuous functions from J into Rn with the norm kxk = supt ∈J |x(t )|, where | · | denotes a suitable complete norm. Let C = C ([−1, 0], Rn ) denote the space of continuous functions on [−1, 0]. For any element φ ∈ C , define the norm kφk∗ = supθ∈[−1,0] |φ(θ )|. Consider the IVP of p-type fractional neutral functional differential equations of the form Dq g (t , xt ) = f (t , xt ),
(1)
xt0 = ϕ,
(2)
(t0 , ϕ) ∈ Ω ,
where Dq is Caputo’s fractional derivative of order 0 < q < 1, Ω is an open subset of [0, ∞) × C and g , f : Ω → Rn are given functionals satisfying some assumptions that will be specified later. xt ∈ C is defined by xt (θ ) = x(p(t , θ )), where −1 ≤ θ ≤ 0, p(t , θ ) is a p-function. In this paper, we obtain various criteria on existence and uniqueness for IVP (1)–(2).
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: [email protected] (Y. Zhou).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.105
Y. Zhou et al. / Nonlinear Analysis 71 (2009) 2724–2733
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2. Preliminaries In this section, we introduce definitions and preliminary facts which are used throughout this paper. Definition 2.1 ([3]). The fractional integral of order µ with the lower limit t0 for a function f is defined as I µ f (t ) =
Z
1
0 (µ)
f (s)
t
t0
(t − s)1−µ
ds,
t > t0 , µ > 0,
provided the right-hand side is pointwise defined on [t0 , ∞), where 0 is the gamma function. Definition 2.2 ([3]). Caputo’s derivative of order µ with the lower limit t0 for a function f : [0, ∞) → R can be written as µ
D f (t ) =
1
0 (n − µ)
Z
f (n) (s)
t t0
(t − s)µ+1−n
ds = I n−µ f (n) (t ),
t > t0 , 0 ≤ n − 1 < µ < n.
Obviously, Caputo’s derivative of a constant is equal to zero. Definition 2.3 ([17]). A function p ∈ C (J × [−1, 0], R) is called a p-function if it has the following properties (i) (ii) (iii) (iv)
p(t , 0) = t, p(t , −1) is a nondecreasing function of t, there exists a σ ≥ −∞ such that p(t , θ ) is an increasing function for θ for each t ∈ (σ , ∞), p(t , 0) − p(t , −1) > 0 for t ∈ (σ , ∞).
Throughout the following text we suppose t ∈ (σ , ∞). Definition 2.4 ([17]). Let t0 ≥ 0, A > 0 and x ∈ C ([p(t0 , −1), t0 + A], Rn ). For any t ∈ [t0 , t0 + A], we define xt by xt (θ ) = x(p(t , θ )),
−1 ≤ θ ≤ 0,
so that xt ∈ C = C ([−1, 0], Rn ). Note that the frequently used symbol ‘‘xt ’’ (in [10,17,18], xt (θ ) = x(t + θ ), where −r ≤ θ ≤ 0, r > 0, r = const) in the theory of functional differential equations with bounded delay is a partial case of the above definition. Indeed, in this case we can put p(t , θ ) = t + r θ , θ ∈ [−1, 0]. Definition 2.5. A function x is said to be a solution of IVP (1)–(2) on [p(t0 , −1), t0 + α], if there are t0 ≥ 0, α > 0, such that (i) x ∈ C ([p(t0 , −1), t0 + α], Rn ) and (t , xt ) ∈ Ω , for t ∈ [t0 , t0 + α], (ii) xt0 = ϕ , (iii) g (t , xt ) is differentiable and (1) holds almost everywhere on [t0 , t0 + α]. We need the following lemma relative to p-function before we proceed further, which is taken from [17]. Lemma 2.1 ([17]). Suppose that p(t , θ ) is a p-function. For A > 0, τ ∈ (σ , ∞) (τ may be σ if σ > −∞), let x ∈ C ([p(τ , −1), τ + A], Rn ) and ϕ ∈ C ([−1, 0], Rn ). Then we have (i) xt is continuous in t on [τ , τ + A] and p˜ (t , θ ) = p(τ + t , θ ) − τ is also a p-function, (ii) if p(τ + t , −1) < τ for t > 0, then there exists −1 < s(τ , t ) < 0 such that p(τ + t , s(τ , t )) = τ and p(τ + t , −1) ≤ p(τ + t , θ ) ≤ τ τ ≤ p(τ + t , θ ) ≤ τ + t
for −1 ≤ θ ≤ s(τ , t ), for s(τ , t ) ≤ θ ≤ 0.
Moreover, s → 0 uniformly in τ as t → 0, (iii) there exists a function η ∈ C ([p(τ , −1), τ ], Rn ) such that
η(p(τ , θ )) = ϕ(θ ) for −1 ≤ θ ≤ 0. It is well known that a neutral functional differential equation (NFDE for short) is one in which the derivatives of the past history or derivatives of functionals of the past history are involved as well as the present state of the system. In other words, 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 concept of atomic. For a detailed discussion on atomic concept we refer the reader to the books [17,18].
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Y. Zhou et al. / Nonlinear Analysis 71 (2009) 2724–2733
Let gφ0 denote the Fréchet derivative of g with respect to the second argument, then we have the following lemma. Lemma 2.2 ([17,18]). Suppose that g (t , φ) is atomic at zero on Ω . Then there are a continuous n × n matrix function A(t , φ) with detA(t , φ) 6= 0 on Ω and a functional L(t , φ, ψ) which is linear in ψ such that gφ0 (t , φ)ψ = A(t , φ)ψ(0) + L(t , φ, ψ). Moreover, there exists a continuous function γ : Ω × [0, 1] → R+ with γ (t , φ, 0) = 0 such that for every s ∈ [0, 1] and ψ with (t , ψ) ∈ Ω , ψ(θ ) = 0 for −1 ≤ θ ≤ −s,
|L(t , φ, ψ)| ≤ γ (t , φ, s)kψk∗ . Lemma 2.3 (Krasnoselskii’s Fixed Point Theorem). Let X be a Banach space, let E be a bounded closed convex subset of X and let S , T be maps of E into X such that Sz + T w ∈ E for every pair z , w ∈ E. If S is a contraction and T is completely continuous, then the equation Sz + Tz = z has a solution on E. 3. Main results Assume that the functional f : Ω → Rn satisfies the following conditions.
(H1 ) f (t , φ) is Lebesgue measurable with respect to t for any (t , φ) ∈ Ω , (H2 ) f (t , φ) is continuous with respect to φ for any (t , φ) ∈ Ω , 1
(H3 ) there exist a constant q1 ∈ (0, q) and a L q1 -integrable function m such that |f (t , φ)| ≤ m(t ) for any (t , φ) ∈ Ω . For each (t0 , ϕ) ∈ Ω , let p˜ (t , θ ) = p(t0 + t , θ ) − t0 . Define the function η ∈ C ([˜p(0, −1), ∞), Rn ) by
η(˜p(0, θ )) = ϕ(θ ) η(t ) = ϕ(0)
for θ ∈ [−1, 0], for t ∈ [0, ∞).
Let x ∈ C ([p(t0 , −1), t0 + α], Rn ), α < A and let x(t0 + t ) = η(t ) + z (t )
for p˜ (0, −1) ≤ t ≤ α.
(3)
In order to prove our main results, we need the following lemmas. Lemma 3.1. x(t ) is a solution of IVP (1)–(2) on [p(t0 , −1), t0 + α] if and only if z (t ) satisfies the relation
g (t0 + t , η˜ t + z˜t ) − g (t0 , ϕ) = z0 = 0,
˜
1
Z
0 (q)
t
(t − s)q−1 f (t0 + s, η˜ s + z˜s )ds,
for t ∈ [0, α],
0
(4)
where η˜ t (θ ) = η(˜p(t , θ )), z˜t (θ ) = z (˜p(t , θ )), for −1 ≤ θ ≤ 0. Proof. Since xt is continuous in t, xt is a measurable function, therefore according to conditions (H1 ) and (H2 ), f (t , xt ) 1
is Lebesgue measurable on [t0 , t0 + α]. Direct calculation gives that (t − s)q−1 ∈ L 1−q1 [t0 , t ], for t ∈ [t0 , t0 + α] and q1 ∈ (0, q). In light of Hölder inequality, we obtain that (t − s)q−1 f (s, xs ) is Lebesgue integrable with respect to s ∈ [t0 , t ] for all t ∈ [t0 , t0 + α], and t
Z
|(t − s)q−1 f (s, xs )|ds ≤ k(t − s)q−1 k t0
1
L 1−q1 [t0 ,t ]
kmk
1
L q1 [t0 ,t0 +α]
,
where
Z kH kLp [J ] =
|H (t )|p dt
1p
,
J
for any Lp -integrable function H : J → R. Hence x(t ) is the solution of IVP (1)–(2) if and only if it satisfies the relation
g (t , xt ) − g (t0 , xt0 ) = xt0 = ϕ,
1
0 (q)
Z
t
(t − u)q−1 f (u, xu )du, t0
for t ∈ [t0 , t0 + α],
Y. Zhou et al. / Nonlinear Analysis 71 (2009) 2724–2733
2727
or setting u = t0 + s,
g (t0 + t , xt0 +t ) − g (t0 , xt0 ) = xt0 = ϕ.
1
t
Z
0 (q)
(t − s)q−1 f (t0 + s, xt0 +s )ds,
for t ∈ [0, α],
(5)
0
In view of (3), we have xt0 +t (θ ) = x(p(t0 + t , θ )) = x(˜p(t , θ ) + t0 ) = η(˜p(t , θ )) + z (˜p(t , θ ))
= η˜ t (θ ) + z˜t (θ ),
for t ∈ [0, α].
In particular xt0 (θ ) = η˜ 0 (θ ) + z˜0 (θ ). Hence xt0 = ϕ if and only if z˜0 = 0 according to η˜ = ϕ . It is clear that x(t ) satisfies (5) if and only if z (t ) satisfies (4). This completes the proof. For any δ, ξ > 0, let E (δ, ξ ) = {z ∈ C ([˜p(0, −1), δ], Rn ) : z˜0 = 0, k˜zt k∗ ≤ ξ for t ∈ [0, δ]}, which is a bounded closed convex subset of the Banach space C ([˜p(0, −1), δ], Rn ) endowed with supremum norm k · k. Lemma 3.2. Suppose Ω ⊆ R × C is open, W ⊂ Ω is compact. For any a neighborhood V 0 ⊂ Ω of W , there is a neighborhood V 00 ⊂ V 0 of W and there exist positive numbers δ and ξ such that (t0 + t , η˜ t + λ˜zt ) ∈ V 0 with 0 ≤ λ ≤ 1 for any (t0 , ϕ) ∈ V 00 , t ∈ [0, δ] and z ∈ E (δ, ξ ). The proof of Lemma 3.2 is similar to that of (iii) of Lemma 2.1.8 in [17], thus it is omitted. Suppose g is atomic at 0 on Ω . Define two operators S and T on E (α, β) as follows
(Sz )(t ) = 0, for t ∈ [˜p(0, −1), 0], A(t0 + t , η˜ t )(Sz )(t ) = g (t0 , ϕ) − g (t0 + t , η˜ t + z˜t ) + gφ0 (t0 + t , η˜ t )˜zt − L(t0 + t , η˜ t , z˜t ),
for t ∈ [0, α]
(6)
and
(Tz )(t ) = 0,
for t ∈ [˜p(0, −1), 0], Z t 1 (t − s)q−1 f (t0 + s, η˜ s + z˜s )ds, A(t0 + t , η˜ t )(Tz )(t ) = 0 (q) 0
for t ∈ [0, α],
(7)
where A(t0 + t , η˜ t ), L(t0 + t , η˜ t , z˜t ) are functions described in Lemma 2.2. It is clear that the operator equation z = Sz + Tz
(8)
has a solution z ∈ E (α, β) if and only if z is a solution of (4). Therefore the existence of a solution of IVP (1)–(2) is equivalent to determining α, β > 0 such that S + T has a fixed point on E (α, β). We are now in a position to prove the following existence results, and the proof is based on Krasnoselskii’s fixed point theorem. Theorem 3.1. Suppose g : Ω → Rn is continuous together with its first Fréchet derivative with respect to the second argument, and g is atomic at 0 on Ω . f : Ω → Rn satisfies conditions (H1 )–(H3 ). W ⊂ Ω is a compact set. Then there exist a neighborhood V ⊂ Ω of W and a constant α > 0 such that for any (t0 , ϕ) ∈ V , IVP (1)–(2) has a solution which exists on [p(t0 , −1), t0 + α]. Proof. As we have mentioned above, we only need to discuss operator equation (8). For any (t , φ) ∈ Ω , the property of the matrix function A(t , φ) which is nonsingular and continuous on Ω implies that its inverse matrix A−1 (t , φ) exists and is continuous on Ω . Let V0 ⊂ Ω be the neighborhood of W , suppose that there is an M > 0 such that
|A−1 (t 0 , φ)| ≤ M for every (t 0 , φ) ∈ V0 .
(9)
Note the complete continuity of the function (m(t ))
α0 > 0 satisfying Z t0 +α0 q1 1 (m(s)) q1 ds ≤ N.
1 q1
, hence, for a given positive number N, there must exist a number
(10)
t0
Due to the continuity of functions γ and gφ0 described in Lemma 2.2, there exist a neighborhood V1 ⊂ Ω of W and constants h1 > 0, h2 ∈ (0, 1] such that
|γ (t0 + t , η˜ t , −s)| = |γ (t0 + t , η˜ t , −s) − γ (t0 + t , η˜ t , 0)| < |gφ0 (t0 + t , η˜ t + ψ) − gφ0 (t0 + t , η˜ t )| <
1 8M
1 4M
,
,
whenever (t0 + t , η˜ t ), (t0 + t , η˜ t + ψ) ∈ V1 and kψk∗ < h1 , −s ∈ [0, h2 ].
(11) (12)
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Y. Zhou et al. / Nonlinear Analysis 71 (2009) 2724–2733
Let V2 = V0 ∩ V1 . According to Lemma 3.2, we can find a neighborhood V ⊂ V2 of W and positive numbers α1 and β with α1 < α0 and β ≤ h1 such that (t0 + t , η˜ t + λ˜zt ) ∈ V2 with 0 ≤ λ ≤ 1 for any (t0 , ϕ) ∈ V , t ∈ [0, α1 ] and z ∈ E (α1 , β). Let h(t0 + t , η˜ t , z˜t ) = g (t0 + t , η˜ t + z˜t ) − g (t0 + t , η˜ t ) − gφ0 (t0 + t , η˜ t )˜zt . Then we have
Z 1 0 0 |h(t0 + t , η˜ t , z˜t )| = gφ (t0 + t , η˜ t + λ˜zt )dλ − gφ (t0 + t , η˜ t ) z˜t 0 Z 1 ≤ [gφ0 (t0 + t , η˜ t + λ˜zt ) − gφ0 (t0 + t , η˜ t )]dλ k˜zt k∗ .
(13)
0
According to (9), (12) and (13), for any (t0 , ϕ) ∈ V , we have
|A−1 (t0 + t , η˜ t ) h(t0 + t , η˜ t , z˜t )| ≤
β 8
.
(14)
On the other hand, for any z , w ∈ E (α1 , β) and t ∈ [0, α1 ]
kλ˜zt + (1 − λ)w ˜ t k∗ ≤ kλ˜zt k∗ + k(1 − λ)w ˜ t k∗ ≤ λβ + (1 − λ)β = β, thus, (t0 + t , η˜ t + λ˜zt + (1 − λ)w ˜ t ) ∈ V2 , and
|h(t0 + t , η˜ t , z˜t ) − h(t0 + t , η˜ t , w ˜ t )| = |g (t0 + t , η˜ t + z˜t ) − g (t0 + t , η˜ t + w ˜ t ) − gφ0 (t0 + t , η˜ t )(˜zt − w ˜ t )| Z 1 = gφ0 (t0 + t , η˜ t + w ˜ t + λ(˜zt − w ˜ t ))dλ − gφ0 (t0 + t , η˜ t ) (˜zt − w ˜ t ) 0 Z 1 0 0 ≤ [gφ (t0 + t , η˜ t + λ˜zt + (1 − λ)w ˜ t ) − gφ (t0 + t , η˜ t )]dλ k˜zt − w ˜ t k∗ . (15) 0
From (9), (12) and (15), we have
|A−1 (t0 + t , η˜ t )[h(t0 + t , η˜ t , z˜t ) − h(t0 + t , η˜ t , w ˜ t )]| ≤
1 8
k˜zt − w ˜ t k∗ .
(16)
By (ii) of Lemma 2.1, we can also choose α2 < α1 such that for t ∈ [0, α2 ], −s(0, t ) ∈ [0, h2 ]. From (9) and (11), we have
|A−1 (t0 + t , η˜ t )||L(t0 + t , η˜ t , z˜t )| ≤ |A−1 (t0 + t , η˜ t )|γ (t0 + t , η˜ t , −s(0, t ))k˜zt k∗ ≤
1 4
k˜zt k∗ ,
(17)
whenever t ∈ [0, α2 ] and z ∈ E (α2 , β). Now consider the expression g (t0 , ϕ) − g (t0 + t , η˜ t ). Since g is continuous in Ω and noting the facts that η˜ t is continuous in t and η˜ 0 = ϕ , there exists a constant α3 < α2 such that
|g (t0 , ϕ) − g (t0 + t , η˜ t )| <
β 8M
,
(18)
whenever t ∈ [0, α3 ]. Set
(
1
α = min α3 , (1 + b) 1+b
0 (q)β 2MN
1
(1−q1 )(1+b)
) ,
(19)
where b = 1−q ∈ (−1, 0). 1 Now we show that for any (t0 , ϕ) ∈ V , S + T has a fixed point on E (α, β), where S and T are defined as in (6) and (7) respectively. The proof is divided into three steps. q −1
Step I. Sz + T w ∈ E (α, β) whenever z , w ∈ E (α, β). Obviously, for every pair z , w ∈ E (α, β), (Sz )(t ) and (T w)(t ) are continuous in t ∈ [0, α]. From (14), (17) and (18), for t ∈ [0, α], we have
|(Sz )(t )| ≤ |A−1 (t0 + t , η˜ t )|{|g (t0 , ϕ) − g (t0 + t , η˜ t )| + |L(t0 + t , η˜ t , z˜t )| + |g (t0 + t , η˜ t ) − g (t0 + t , η˜ t + z˜t ) + gφ0 (t0 + t , η˜ t )˜zt |} β ≤ . 2
Y. Zhou et al. / Nonlinear Analysis 71 (2009) 2724–2733
2729
For t ∈ [0, α], by using (9), (10), (19) and Hölder inequality, we have
Z t q −1 |(T w)(t )| ≤ |A (t0 + t , η˜ t )| (t − s) f (t0 + s, η˜ s + w ˜ s )ds 0 (q) 0 Z t 1−q1 Z t0 +α q1 1 1 M (m(s)) q1 ds ≤ ((t − s)q−1 ) 1−q1 ds 0 (q) 0 t0 1−q1 1 MN α 1+b ≤ 0 (q) 1 + b 1
−1
≤
β 2
.
(20)
Thus |(Sz )(t ) + (T w)(t )| ≤ β i.e. Sz + T w ∈ E (α, β), whenever z , w ∈ E (α, β). Step II. S is a contraction mapping from E (α, β) into itself whose contraction constant is independent of (t0 , ϕ) ∈ V . For any z , w ∈ E (α, β), w ˜ 0 − z˜0 = 0. Hence (ii) of Lemmas 2.1 and 2.2 are applicable to w ˜ t − z˜t . For every pair z , w ∈ E (α, β), from (16) and (17) and noting the fact that sup k˜zt − w ˜ t k∗ = sup 0≤t ≤α
sup |z (˜p(t , θ )) − w(˜p(t , θ ))|
0≤t ≤α −1≤θ≤0
= sup
sup
0≤t ≤α p˜ (t ,−1)≤s≤t
=
sup p˜ (0,−1)≤s≤α
|z (s) − w(s)|
|z (s) − w(s)|
= kz − wk, we have
kSz − S wk =
sup p˜ (0,−1)≤t ≤α
|(Sz )(t ) − (S w)(t )|
= sup |(Sz )(t ) − (S w)(t )| 0≤t ≤α
≤ sup |A−1 (t0 + t , η˜ t )|[|L(t0 + t , η˜ t , w ˜ t − z˜t )| + |h(t0 + t , η˜ t , z˜t ) − h(t0 + t , η˜ t , w ˜ t )|] 0≤t ≤α 1 1 ≤ + sup k˜zt − w ˜ t k∗ 8
≤
3 8
4
0≤t ≤α
kz − wk.
Therefore S is a contraction mapping from E (α, β) into itself whose contraction constant is independent of (t0 , ϕ) ∈ V . Step III. Now we show that T is a completely continuous operator. For any z ∈ E (α, β) and 0 ≤ t1 < t2 ≤ α , we get
Z 1 |(Tz )(t2 ) − (Tz )(t1 )| = A−1 (t0 + t2 , η˜ t2 ) 0 (q) 1
− A (t0 + t1 , η˜ t1 ) 0 (q) Z −1 1 = A (t0 + t2 , η˜ t2 ) 0 (q) −1
+ A (t0 + t2 , η˜ t2 ) −1
− A−1 (t0 + t2 , η˜ t2 ) + A (t0 + t2 , η˜ t2 ) −1
1
0 (q) 1
0 (q) 1
0 (q)
t2
(t2 − s)q−1 f (t0 + s, η˜ s + z˜s )ds
0 t1
Z
(t1 − s)
q −1
0 t2
f (t0 + s, η˜ s + z˜s )ds
(t2 − s)q−1 f (t0 + s, η˜ s + z˜s )ds
t1
Z
t1
(t2 − s)q−1 f (t0 + s, η˜ s + z˜s )ds
0
Z
t1
(t1 − s)q−1 f (t0 + s, η˜ s + z˜s )ds
0
Z
t1
(t1 − s)q−1 f (t0 + s, η˜ s + z˜s )ds
0 t1
− A (t0 + t1 , η˜ t1 ) (t1 − s) f (t0 + s, η˜ s + z˜s )ds 0 (q) 0 Z |A−1 (t0 + t2 , η˜ t2 )| t2 q−1 (t2 − s) f (t0 + s, η˜ s + z˜s )ds ≤ 0 (q) t1 −1
1
Z
q −1
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Y. Zhou et al. / Nonlinear Analysis 71 (2009) 2724–2733
Z |A−1 (t0 + t2 , η˜ t2 )| t1 q −1 q −1 ˜ [( t − s ) − ( t − s ) ] f ( t + s , η ˜ + z ) ds 2 1 0 s s 0 (q) 0 Z |A−1 (t0 + t2 , η˜ t2 ) − A−1 (t0 + t1 , η˜ t1 )| t1 q −1 ˜ + ( t − s ) f ( t + s , η ˜ + z ) ds 1 0 s s 0 (q) 0 |A−1 (t0 + t2 , η˜ t2 )| |A−1 (t0 + t2 , η˜ t2 ) − A−1 (t0 + t1 , η˜ t1 )| = (I1 + I2 ) + I3 , 0 (q) 0 (q) +
where t2
Z
I1 =
t
(t2 − s)q−1 f (t0 + s, η˜ s + z˜s )ds ,
Z 1t 1 I2 = [(t2 − s)q−1 − (t1 − s)q−1 ]f (t0 + s, η˜ s + z˜s )ds , 0 Z t 1 I3 = (t1 − s)q−1 f (t0 + s, η˜ s + z˜s )ds . 0
By using the analogous argument performed in (20), we can conclude that I1 ≤ I3 ≤
N
(1 + b)1−q1 N
(1 + b)
1−q1
1−q1 (t2 − t1 )1+b , t1 1+b
1−q1
,
and t1
Z
|(t2 − s)
q −1
I2 ≤
− (t1 − s)
1−q1 Z
1
q−1 1−q1
|
t0 + t1
ds
0
1 q1
|f (s, xs )| ds
q1
t0 t1
Z ≤N
((t1 − s)b − (t2 − s)b )ds
1−q1
0
= ≤
N
(1 + b)
1−q1
N
(1 + b)
1−q1
t1 1+b − t2 1+b + (t2 − t1 )1+b
1−q1
1−q1 (t2 − t1 )1+b ,
where b = 1−q ∈ (−1, 0). Therefore 1 q −1
|(Tz )(t2 ) − (Tz )(t1 )| ≤
1−q1 |A−1 (t0 + t2 , η˜ t2 )| 2N (t2 − t1 )1+b 1 − q 0 (q) (1 + b) 1 1−q1 |A−1 (t0 + t2 , η˜ t2 ) − A−1 (t0 + t1 , η˜ t1 )| N + t1 1+b . 1 − q 0 (q) ( 1 + b) 1
Since A−1 (t0 + t , η˜ t ) is continuous in t ∈ [0, α], then {Tz ; z ∈ E (α, β)} is equicontinuous. In addition, T is continuous from the condition (H2 ) and {Tz ; z ∈ E (α, β)} is uniformly bounded from (20), thus T is a completely continuous operator by Ascoli–Arzela Theorem. Therefore, by Lemma 2.3, for every (t0 , ϕ) ∈ V , S + T has a fixed point on E (α, β). Hence, IVP (1)–(2) has a solution defined on [p(t0 , −1), t0 + α]. The proof is complete. Corollary 3.1. Suppose that (t0 , ϕ) ∈ Ω is given, g , f are defined as in Theorem 3.1. Then there exists a solution of IVP (1)–(2). Corollary 3.2. Suppose that Ω , f are defined as in Theorem 3.1. If (t0 , ϕ) ∈ Ω is given, then the IVP relative to fractional p-type retarded differential equations of the form
Dq x(t ) = f (t , xt ), xt0 = ϕ
has a solution. The following existence and uniqueness result for IVP (1)–(2) is based on Banach’s contraction principle.
Y. Zhou et al. / Nonlinear Analysis 71 (2009) 2724–2733
2731
Theorem 3.2. Suppose (t0 , ϕ) ∈ Ω is given, g is defined as in Theorem 3.1. f : Ω → Rn satisfies the condition (H3 ) and
(H4 ) f (t , xt ) is measurable for every (t , xt ) ∈ Ω , (H5 ) let A > 0, there exists a nonnegative function ` : [0, A] → [0, ∞) continuous at t = 0 and `(0) = 0 such that for any (t , xt ), (t , yt ) ∈ Ω , t ∈ [t0 , t0 + A], we have Z t (t − s)q−1 [f (s, xs ) − f (s, ys )]ds ≤ `(t − t0 ) sup kxs − ys k∗ . t ≤s≤t t0
0
Then IVP (1)–(2) has a unique solution. Proof. According to the argument of Theorem 3.1, it suffices to prove that S + T has a unique fixed point on E (α, β), where α, β > 0 sufficiently small. Now, choose α ∈ (0, A] such that (19) holds and c=
3 8
+ sup 0≤s≤α
|A−1 (t0 + s, η˜ s )||`(s)| < 1. 0 (q)
(21)
Obviously, S + T is a mapping from E (α, β) into itself. Using the same argument as that of Theorem 3.1, for any z , w ∈ E (α, β), t ∈ [0, α], we get
|(Sz )(t ) − (S w)(t )| ≤
3 8
kz − wk,
and
Z Z t |A−1 (t0 + t , η˜ t )| t q −1 q −1 ˜ ( t − s ) f ( t + s , η ˜ + z ) ds − ( t − s ) f ( t + s , η ˜ + w ˜ ) ds 0 s s 0 s s 0 (q) 0 0 |A−1 (t0 + t , η˜ t )| ≤ |`(t )| sup k˜zs − w ˜ s k∗ 0 (q) 0≤s≤t
|(Tz )(t ) − (T w)(t )| ≤
sup |A−1 (t0 + s, η˜ s )||`(s)|
≤
0≤s≤α
kz − wk.
0 (q)
Therefore
|(S + T )z (t ) − (S + T )w(t )| ≤
3 8
+ sup 0≤s≤α
|A−1 (t0 + s, η˜ s )||`(s)| kz − wk 0 (q)
= c kz − wk. Hence, we have
k(S + T )z − (S + T )wk ≤ c kz − wk, where c < 1. By applying Banach’s contraction principle, we know that S + T has a unique fixed point on E (α, β). The proof is complete. Corollary 3.3. Suppose the condition (H5 ) of Theorem 3.2 is replaced by the following condition 1
(H5 )0 let A > 0, there exist q2 ∈ (0, q) and a real-valued function `1 ∈ L q2 [t0 , t0 + A] such that for any (t , xt ), (t , yt ) ∈ Ω , t ∈ [t0 , t0 + A], we have |f (t , xt ) − f (t , yt )| ≤ `1 (t ) sup kxs − ys k∗ . t0 ≤s≤t
Then the result of Theorem 3.2 holds. 1
Proof. It suffices to prove that the condition (H5 ) of Theorem 3.2 holds. Note that `1 ∈ L q2 [t0 , t0 + A], let K = k`1 k Then for any (t , xt ), (t , yt ) ∈ Ω we have
Z t Z t (t − s)q−1 [f (s, xs ) − f (s, ys )]ds ≤ (t − s)q−1 |f (s, xs ) − f (s, ys )|ds t0 t0 Z t ≤ (t − s)q−1 `1 (s) ds sup kxs − ys k∗ t0 ≤s≤t
t0
≤
K
(1 + b1 )1−q2
(t − t0 )(1+b1 )(1−q2 ) sup kxs − ys k∗ , t0 ≤s≤t
1
L q2 [t0 ,t0 +A]
.
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Y. Zhou et al. / Nonlinear Analysis 71 (2009) 2724–2733
where b1 = 1−q ∈ (−1, 0). Let 2 q −1
`(t − t0 ) =
K
(1 + b1 )1−q2
(t − t0 )(1+b1 )(1−q2 ) .
Obviously, ` : [0, A] → [0, ∞) is continuous at t = 0 and `(0) = 0. Then the condition (H5 ) of Theorem 3.2 holds. This completes the proof. The next result is concerned with the uniqueness of solutions. Theorem 3.3. Suppose that g are defined as in Theorem 3.1 and the condition (H5 )0 of Corollary 3.3 holds. If x is a solution of IVP (1)–(2), then x is unique. Proof. Suppose (for contradiction) x and y are the solutions of IVP (1)–(2) on [p(t0 , −1), t0 + A] with x 6= y, let t1 = inf{t ∈ [t0 , t0 + A] : x(t ) 6= y(t )}. Then t0 ≤ t1 < t0 + A and x(t ) = y(t )
for p(t0 , −1) ≤ t < t1 ,
which implies that xt (θ ) = x(p(t , θ )) = y(p(t , θ )) = yt (θ ),
t0 ≤ t < t1 , −1 ≤ θ ≤ 0.
(22)
Choose α > 0 such that t1 + α < t0 + A. According to (i) of Definition 2.5, we have
{(t , xt ), t1 ≤ t ≤ t1 + α} ∪ {(t , yt ), t1 ≤ t ≤ t1 + α} ⊂ Ω . On the one hand, x and y satisfy (1)–(2) on [t0 , t0 + A], thus from (22) and the condition (H5 )0 , for t ∈ [t0 , t1 + α], we have
Z t q −1 (t − s) [f (s, xs ) − f (s, ys )]ds |g (t , xt ) − g (t , yt )| ≤ 0 (q) t0 Z 1 t q −1 (t − s) [f (s, xs ) − f (s, ys )]ds = 0 (q) t1 Z t 1 (t − s)q−1 `1 (s) ds sup kxs − ys k∗ ≤ 0 (q) t1 t0 ≤s≤t 1
≤
K
0 (q)(1 + b1 )1−q2
α (1+b1 )(1−q2 )
sup t1 ≤s≤t1 +α
kxs − ys k∗ ,
(23)
where b1 = 1−q ∈ (−1, 0), K = k`1 k 1 . 2 L q2 [t0 ,t0 +A] On the other hand, since g (t , φ) is continuously differentiable in φ , we have q −1
g (t , xt ) − g (t , yt ) = gφ0 (t , yt )(xt − yt ) + kkxt − yt k∗
(24)
with k → 0 as kxt − yt k∗ → 0. By the hypothesis that g (t , φ) is atomic at 0 on Ω , there exist a nonsingular continuous matrix function A(t , yt ) and a function L(t , yt , ψ) which is linear in ψ such that gφ0 (t , yt )ψ = A(t , yt )ψ(0) + L(t , yt , ψ).
(25)
Moreover, there is a positive real-valued continuous function γ (t , yt , −s) such that for every s ∈ [−1, 0],
|L(t , yt , ψ)| ≤ γ (t , yt , −s)kψk∗
(26)
if ψ(θ) = 0 for −1 ≤ θ ≤ s. Hence for every t ∈ [t1 , t1 + α], by (ii) of Lemma 2.1, there is s(t1 , t − t1 ) ∈ [−1, 0] with s(t1 , t − t1 ) → 0 as t → t1 such that
|L(t , yt , xt − yt )| ≤ γ (t , yt , −s(t1 , t − t1 ))kxt − yt k∗ . From (24)–(26), it follows that g (t , xt ) − g (t , yt ) = A(t , yt )(x(t ) − y(t )) + L(t , yt , xt − yt ) + kkxt − yt k∗ , therefore
|x(t ) − y(t )| ≤ |A−1 (t , yt )|{|g (t , xt ) − g (t , yt )| + γ (t , yt , −s(t1 , t − t1 ))kxt − yt k∗ + kkxt − yt k∗ }.
Y. Zhou et al. / Nonlinear Analysis 71 (2009) 2724–2733
2733
Let M1 = max{|A−1 (t , yt )| : t1 ≤ t ≤ t1 + α}. Then by relation (23), for t ∈ [t1 , t1 + α], we have
|x(t ) − y(t )| ≤ c1 where c1 = M1
h
sup t1 ≤s≤t1 +α
K 0 (q)(1+b1 )1−q2
kxs − ys k∗ ,
i α (1+b1 )(1−q2 ) + γ (t , yt , −s(t1 , t − t1 )) + k .
Noting that sup
kxs − ys k∗ =
t1 ≤s≤t1 +α
sup
sup |x(p(s, θ )) − y(p(s, θ ))|
t1 ≤s≤t1 +α −1≤θ≤0
=
sup
sup
t1 ≤s≤t1 +α p(s,−1)≤ρ≤s
=
sup
p(t1 ,−1)≤s≤t1 +α
|x(ρ) − y(ρ)|
|x(s) − y(s)|,
we have sup
p(t1 ,−1)≤s≤t1 +α
|x(s) − y(s)| ≤ c1
sup
p(t1 ,−1)≤s≤t1 +α
|x(s) − y(s)|.
Choose α so small that c1 < 1. Thus sup
p(t1 ,−1)≤s≤t1 +α
|x(s) − y(s)| = 0,
i.e. x(t ) ≡ y(t ),
for t1 ≤ t ≤ t1 + α,
contradicting the definition of t1 . Hence the proof is complete.
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