Volume 2016, Article ID ama0299, 25 pages ISSN 2307-7743
http://scienceasia.asia
ON BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH
SPACES
SABRI THABET MOTHANA, MACHINDRA B. DHAKNE
Abstract. This paper is devoted to study the existence and uniqueness of solutions of
fractional integro-differential equations involving Caputo derivative with boundary value conditions of higher order in Banach spaces. A new generalized singular type Gronwall’s inequality is given by us to obtain priori bounds. New sufficient conditions for the exis-tence and uniqueness of solution are established by means of fractional calculus, fixed point theorems and H¨older inequality. Examples are provided to illustrate the main results.
1. Introduction
Fractional differential equations is a generalization of ordinary differential equations and integration to arbitrary non-integer orders. In the last few decades, fractional order model-s are found to be more adequate than integer order modelmodel-s for model-some real world problemmodel-s. There has been a significant progress in the investigation of fractional differential and partial differential equations in the recent years; see the monographs of Kilbas et al. [10], Miller and Ross [15], Podlubny [16], Samko et al. [17] and the references given therein. Recently, some basic theory for the initial value problems of fractional differential equations involving a Riemann-Liouville differential operator of order α ∈ (0,1) has been developed by Lak-shmikantham and Vatsala in [11, 12, 13, 14]. Fractional differential equations have been proved to be valuable tools in the modelling of many natural phenomena in various fields of engineering, physics, chemistry, aerodynamics, electrodynamics of complex medium, see for example [8, 10]. In [23], authors studied a hyperplastic and fractional derivative viscoelastic model to describe infant brain tissue under conditions consistent with the development of hydrocephalus. Agarwal et al. [2] established sufficient conditions for the existence and uniqueness of solutions for various classes of initial and boundary value problem for fraction-al differentifraction-al equations and inclusions involving the Caputo fractionfraction-al derivative in finite dimensional spaces. Recently, some fractional differential equations and optimal controls in Banach spaces were studied by Balachandran and Park [3], El-Borai [5], Henderson and Ouahab [6], Hernandez et al. [7], Wang et al. [20] and Wang et al. [21, 22].
Key words and phrases. Fractional integro-differential equations, boundary value problems, existence and uniqueness, generalized singular type Gronwall’s inequality, fixed point theorems.
c
2016 Science Asia
Karthikeyan and Trujillo [9], Wang et al. [19] and Yang et al. [24] have extended the work in [1] from real lineRto the abstract Banach spaceX by using more general assumptions on the nonlinear function f. Very recently, authors [4] have studied boundary value problems for fractional integro-differential equations
cDαy(t) = f t, y(t),(Sy)(t)
, t∈J = [0, T], α∈(n−1, n), y(0) =y0, y0(0) =y01, y00(0) =y02, . . . , y(n−2)(0) =y
n−2 0 ,
y(n−1)(T) =yT,
where cDα is the Caputo fractional derivative of order α, f : J ×X ×X → X, and S is a
linear integral operator given by (Sy)(t) =R0tk(t, s)y(s)ds, where k ∈C(J ×J,R+).
We are motivated by the works in [9, 19, 24] and influenced by Chalishajar and Karthikeyan [4]. Our aim in this paper is to consider the following more general boundary value problem for fractional integro-differential equation
(1)
cDαx(t) =f t, x(t),(Sx)(t)
, t∈J = [0, T], α∈(n−1, n), x(0) =x0, x0(0) =x10, x00(0) =x20, . . . , x(n−2)(0) =xn
−2 0 ,
x(n−1)(T) =x
T,
where cDα is the Caputo fractional derivative of order α, f : J ×X×X → X is a given function satisfying some assumptions that will be specified later and x0, xi0(i= 1,2, . . . , n− 2, n≥3, n is an integer ),xT are elements of X, and S is a nonlinear integral operator given
by (Sx)(t) =R0tk t, s, x(s)ds, where k ∈C(J ×J×X, X).
We investigate existence and uniqueness results for the fractional boundary value problem (BVP for short), (1) by using fractional calculus, fixed point theorems and H¨older inequality. Compared the present paper with the work [4], there are at least three differences: (i) the operator (Sx)(t) is not linear but nonlinear operator; (ii) another singular Gronwall’s inequality is given by us (Lemma 3.2) to obtain the priori bounds; (iii) some new hypotheses applied on the function f. Our attempt is to generalize the results proved in [1, 4, 24].
This paper is organized as follows. In Section 2, we set forth some preliminaries. Section 3 introduces a new generalized singular type Gronwall inequality to establish the estimate for priori bounds. In Section 4, we prove our main results by applying Banach contraction principle and Schaefer’s fixed point theorem. Finally, in Section 5, applications of the main results are exhibited.
2. Preliminaries
LetX be a Banach space with the normk·k. We denote byC(J, X) the space ofX-valued continuous functions on J with the supremum normkxk∞ := sup{kx(t)k:t ∈J}. For
mea-surable functions m : J → R, define the norm kmkLp(J,R) = R
J|m(t)| pdt1p
,1 ≤ p < ∞,
whereLp(J,
R) the Banach space of all Lebesgue measurable functionsm withkmkLp(J,
R)<
∞.
Definition 2.1. The Riemann-Liouville fractional integral of orderα >0of a suitable func-tion h is defined by
Iaα+h(t) = 1 Γ(α)
Z t
a
(t−s)α−1h(s)ds,
where a∈R and Γ is the Gamma function.
Definition 2.2. For a suitable function hgiven on the interval [a, b], the Riemann-Liouville fractional derivative of order α >0 of h, is defined by
(Dαa+h)(t) = 1 Γ(n−α)
d dt
nZ t
a
(t−s)n−α−1h(s)ds,
where n= [α] + 1,[α] denotes the integer part of α.
Definition 2.3. For a suitable function h given on the interval [a, b], the Caputo fractional order derivative of order α >0 of h, is defined by
(cDαa+h)(t) = 1 Γ(n−α)
Z t
a
(t−s)n−α−1h(n)(s)ds,
where n= [α] + 1,[α] denotes the integer part of α.
Remark 2.1. (i) The Caputo derivative of a constant is equal to zero.
(ii) If his an abstract function with values inX, then integrals which appeared in Definitions 2.1, 2.2 and 2.3 are taken in Bochner’s sense.
Lemma 2.1. ([25]) Let α >0; then the differential equation cDαh(t) = 0, has the following general solution h(t) = c0+c1t+c2t2 +· · ·+cn−1tn−1, where ci ∈ R, i= 0,1,2, . . . , n−1,
where n= [α] + 1.
Lemma 2.2. ([25]) Let α >0; then
Iα(cDαh)(t) =h(t) +c0+c1t+c2t2+· · ·+cn−1tn−1,
For more details, see [10].
Definition 2.4. A function x ∈ C(J, X) with it’s α derivative existing on J is said to be a solution of the fractional BVP (1) if x satisfies the equation cDαx(t) =f t, x(t),(Sx)(t)
a.e. on J, and the conditions x(0) =x0, x0(0) =x10, x00(0) =x20, . . . ,
x(n−2)(0) =xn0−2, x(n−1)(T) = xT.
For the existence of solutions for the fractional BVP (1), we need the following auxiliary lemma.
Lemma 2.3. Let f : J → X be continuous. A function x ∈ C(J, X) is solution of the fractional integral equation
(2)
x(t) = 1 Γ(α)
Z t
0
(t−s)α−1f(s)ds
− t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nf(s)ds
+x0+x10t+
x2 0 2!t
2 +· · ·+ x
n−2 0 (n−2)!t
n−2+ xT
(n−1)!t
n−1,
if and only if x is a solution of the following fractional BVP
(3) cDαx(t) =f(t), t∈J = [0, T], α∈(n−1, n),
(4) x(0) =x0, x0(0) =x10, x
00
(0) =x20, . . . , x(n−2)(0) =x0n−2, x(n−1)(T) = xT.
Proof Assume that x satisfies fractional BVP (3)-(4); then by using Lemma 2.2 and Def. 2.1, we get
x(t) +c0+c1t+c2t2+· · ·+cn−1tn−1 = 1 Γ(α)
Z t
0
(t−s)α−1f(s)ds,
where ci ∈R, i= 0,1,2, . . . , n−1. That is:
(5) x(t) = 1 Γ(α)
Z t
0
(t−s)α−1f(s)ds−c0−c1t−c2t2− · · · −cn−2tn−2−cn−1tn−1. By applying first condition of (4) i.ex(0) =x0,we have
x0 =−c0 ⇒c0 =−x0. Now,
x0(t) = 1 Γ(α−1)
Z t
0
(t−s)α−2f(s)ds−c1−2c2t−. . .
−(n−2)cn−2tn−3−(n−1)cn−1tn−2, by using x0(0) =x1
0,we get
And,
x00(t) = 1 Γ(α−2)
Z t
0
(t−s)α−3f(s)ds−2c2−. . .
−(n−2)(n−3)cn−2tn−4−(n−1)(n−2)cn−1tn−3, by using x00(0) =x2
0, we have
x20 =−2c2 ⇒c2 =−
x2 0 2!. By continuing this process, we have
x(n−2)(t) = 1 Γ(α−n+ 2)
Z t
0
(t−s)α−n+1f(s)ds −(n−2)(n−3)(n−4). . .(2)(1)cn−2
−(n−1)(n−2)(n−3). . .(3)(2)cn−1t, by using x(n−2)(0) =xn−2
0 ,we have
xn0−2 =−(n−2)!cn−2 ⇒cn−2 =−
xn0−2
(n−2)!. Finally,
x(n−1)(t) = 1 Γ(α−n+ 1)
Z t
0
(t−s)α−nf(s)ds
−(n−1)(n−2)(n−3). . .(3)(2)(1)cn−1
= 1
Γ(α−n+ 1)
Z t
0
(t−s)α−nf(s)ds−(n−1)!cn−1, by using x(n−1)(T) = x
T,we obtain
xT =
1 Γ(α−n+ 1)
Z T
0
(T −s)α−nf(s)ds−(n−1)!cn−1. Hence,
cn−1 =−
xT
(n−1)! +
1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nf(s)ds.
Now, by substituting the values of ci(i= 0,1,2, . . . , n−1) in (5), we obtain
x(t) = 1 Γ(α)
Z t
0
(t−s)α−1f(s)ds
− t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nf(s)ds
+x0+x10t+
x20
2!t
2+· · ·+ x
n−2 0 (n−2)!t
n−2+ xT (n−1)!t
Conversely, assume that if x satisfies fractional integral equation (2), if t ∈ [0, T] then
x(0) =x0, x0(0) =x10, x00(0) =x20, . . . , x(n−2)(0) =x
n−2
0 , x(n−1)(T) = xT and applying Remark
2.1 (i)-(iii), we get (3) is also satisfied.
As a consequence of lemma 2.3,we have the following result which is useful in what follows. Lemma 2.4. Letf :J×X×X →Xbe continuous function. Then,x∈C(J, X)is a solution of the fractional integral equation
x(t) = 1 Γ(α)
Z t
0
(t−s)α−1f s, x(s),(Sx)(s)ds
− t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nf s, x(s),(Sx)(s)ds
+x0+x10t+
x20
2!t
2+· · ·+ x
n−2 0 (n−2)!t
n−2+ xT
(n−1)!t
n−1,
if and only if x is solution of the fractional BVP (1).
Lemma 2.5. (Bochner theorem) A measurable function f :J →X is Bochner integrable if kfk is Lebesgue integrable.
Lemma 2.6. (Mazur theorem, [18]) Let X be a Banach space. If U ⊂ X is relatively compact, then conv(U) is relatively compact and conv(U) is compact.
Lemma 2.7. (Ascoli-Arzela theorem) Let S = {s(t)} is a function family of continuous mappings s : [a, b] → X. If S is uniformly bounded and equicontinuous, and for any t∗ ∈
[a, b], the set{s(t∗)}is relatively compact, then, there exists a uniformly convergent function sequence {sn(t)}(n= 1,2, . . . , t∈[a, b]) in S.
Lemma 2.8. (Schaefer’s fixed point theorem) Let F : X → X be a completely continuous operator. If the set E(F) ={x∈ X :x=ηF x for some η ∈[0,1]} is bounded, then, F has fixed points.
3. A generalized singular type Gronwall’s inequality
In order to apply the Schaefer’s fixed point theorem to establish the existence of solutions, we need to introduce a new generalized singular Gronwall type inequality with mixed type singular integral operator. It plays a fundamental role in the study of BVP for nonlinear differential equations of fractional order.
We, first, state a generalized Gronwall inequality from [21].
Lemma 3.1. (Lemma 3.2, [21]) Let x∈C(J, X) satisfies the following inequality: kx(t)k ≤a+b
Z t
0
kx(θ)kλ1dθ+c
Z T
0
kx(θ)kλ2dθ+d
Z t
0
+e
Z T
0
kxθkλB4dθ, t∈J,
where λ1, λ3 ∈[0,1], λ2, λ4 ∈[0,1), a, b, c, d, e≥0 are constants and
kxθkB = sup0≤s≤θkx(s)k. Then there exists a constant L >0 such that
kx(t)k ≤L.
Using the above generalized Gronwall inequality, we can obtain the following new gener-alized singular type Gronwall inequality.
Lemma 3.2. Let x∈C(J, X) satisfies the following inequality: kx(t)k ≤ a+b
Z t
0
(t−s)α−1kx(s)kλds+c
Z T
0
(T −s)α−nkx(s)kλds
+d
Z t
0
(t−s)α−1kxskλBds+e
Z T
0
(T −s)α−nkxskλBds,
(6)
where α ∈ (n −1, n), λ ∈ [0,1− 1
p) for some 1 < p <
1
n−α,kxskB = sup0≤τ≤skx(τ)k and
a, b, c, d, e≥0 are constants.Then, there exists a constant L >0, such that kx(t)k ≤L.
Proof Let
y(t) =
(
1, kx(t)k ≤1, x(t), kx(t)k>1.
Using (6) and H¨older inequality, we get
kx(t)k ≤ ky(t)k ≤(a+ 1) +b
Z t
0
(t−s)α−1ky(s)kλds+c
Z T
0
(T −s)α−nky(s)kλds
+d
Z t
0
(t−s)α−1kyskλBds+e
Z T
0
(T −s)α−nkyskλBds
≤(a+ 1) +b
Z t
0
(t−s)p(α−1)ds
1pZ t
0
ky(s)kpλp−1ds
p−1
p
+c
Z T
0
(T −s)p(α−n)ds
p1 Z T
0
ky(s)kpλp−1ds
p−1
p
+dkyskλB
Z t
0
(t−s)α−1ds+ekyskλB
Z T
0
(T −s)α−nds
≤(a+ 1) +b
tp(α−1)+1
p(α−1) + 1
1pZ t
0
+c
Tp(α−n)+1
p(α−n) + 1
1pZ T
0
ky(s)kpλp−1ds
+dkyskλB
tα
α +ekysk
λ B
Tα−n+1 (α−n+ 1)
≤(a+ 1) +dkyskλB
Tα
α +ekysk
λ B
Tα−n+1 (α−n+ 1)
+b
Tp(α−1)+1
p(α−1) + 1
1pZ t
0
ky(s)kpλp−1ds
+c
Tp(α−n)+1
p(α−n) + 1
1pZ T
0
ky(s)kpλp−1ds,
where 0< pλp−1 <1.
Hence, by lemma 3.1 there exists a constant L >0, such that kx(t)k ≤L.
4. Main Results
For convenience, we list hypotheses that will be used in our further discussion.
• (H1) The function f : J × X ×X → X is measurable with respect to t on J and is continuous with respect tox on X.
• (H2) There exists a constant α1 ∈(0, α−n+ 1) and real-valued functions m1(t), m2(t)∈
Lα11(J,
R),such that
kf t, x(t),(Sx)(t)−f t, y(t),(Sy)(t)k ≤m1(t) kx(t)−y(t)k+kSx(t)−Sy(t)k
,
kk t, s, x(s)−k t, s, y(s)k ≤m2(t)kx(s)−y(s)k, for each s ∈[0, t], t∈J and all x, y ∈X.
•(H3) There exists a constantα2 ∈(0, α−n+ 1) and real-valued functionh(t)∈L 1
α2(J,R), such that kf t, x(t),(Sx)(t)k ≤h(t), for each t ∈J, and all x∈X.
For brevity, let M =km1+m1m2Tk
L
1
α1(J,
R)
and H =khk
L
1
α2(J,
R)
.
• (H4) There exist constantsλ ∈[0,1− 1
p) for some 1< p <
1
n−α and Nf, Nk>0,such that
kf t, x(t),(Sx)(t)k ≤Nf 1 +kx(t)kλ +k(Sx)(t)k
,
kk t, s, x(s)k ≤Nk 1 +kx(s)kλ
,
for each s ∈[0, t], t∈J and all x∈X. • (H5) For every t∈J, the sets
K1 =
(t−s)α−1f s, x(s),(Sx)(s)
K2 =
(t−s)α−nf s, x(s),(Sx)(s):x∈C(J, X), s∈[0, t] are relatively compact.
Now, we are in position to deal with our main results.
Theorem 4.1. Assume that (H1)-(H3) hold. If
(7) Ωα,T ,n =
M
Γ(α)
Tα−α1 (α−α1
1−α1)
1−α1 +
M
(n−1)!Γ(α−n+ 1)
Tα−α1 (α−α1−n+1
1−α1 )
1−α1 <1.
Then, the fractional BVP (1) has a unique solution on J.
Proof By making use of hypothesis (H3) and H¨older inequality, for eacht∈J, we have
Z t
0
k(t−s)α−1f s, x(s),(Sx)(s)
kds≤
Z t
0
(t−s)α−1kf s, x(s),(Sx)(s)
kds
≤
Z t
0
(t−s)α−1h(s)ds
≤
Z t
0
(t−s)1−α−1α2ds
1−α2Z t o
(h(s))α12ds
α2
≤H
"
−(t−s)α −1 1−α2+1 α−1
1−α2 + 1
#t
0
1−α2
≤H T
α−α2 (α−α2
1−α2) 1−α2. Thus, k(t−s)α−1f s, x(s),(Sx)(s)
k is Lebesgue integrable with respect to s ∈[0, t] for all
t∈J andx∈C(J, X).Then, (t−s)α−1f s, x(s),(Sx)(s)
is Bochner integrable with respect tos ∈[0, t] for allt ∈J due to lemma 2.5, and
Z T
0
k(T −s)α−nf s, x(s),(Sx)(s)kds≤
Z T
0
(T −s)α−nkf s, x(s),(Sx)(s)kds
≤
Z T
0
(T −s)α−nh(s)ds
≤
Z T
0
(T −s)1−α−αn2ds
1−α2Z T
o
(h(s))α12ds
α2
≤H T
α−α2−n+1 1−α2 α−α2−n+1
1−α2
!1−α2
≤H T
α−α2−n+1 (α−α2−n+1
1−α2 ) 1−α2. Thus, k(T −s)α−nf s, x(s),(Sx)(s)k is Lebesgue integrable with respect to s ∈[0, T] and
x ∈ C(J, X). Then, (T −s)α−nf s, x(s),(Sx)(s) is Bochner integrable with respect to
s∈[0, T] due to lemma 2.5.
Hence, the fractional BVP (1) is equivalent to the following fractional integral equation
x(t) = 1 Γ(α)
Z t
0
− t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nf s, x(s),(Sx)(s)
ds
+x0+x10t+
x2 0 2!t
2+· · ·+ x
n−2 0 (n−2)!t
n−2+ xT
(n−1)!t
n−1.
Now, let Br={x∈C(J, X) :kxk∞ ≤r}, where
(8)
r≥ HT
α−α2 Γ(α)(α−α2
1−α2)
1−α2 +
HTα−α2
(n−1)!Γ(α−n+ 1)
α−α2−n+ 1 1−α2
−(1−α2)
+kx0k+kx10kT +
kx2 0k 2! T
2+· · ·+ kx
n−2 0 k (n−2)!T
n−2+ kxTk
(n−1)!T
n−1.
Define the operator F onBr as follows:
(9)
(F(x))(t) = 1 Γ(α)
Z t
0
(t−s)α−1f s, x(s),(Sx)(s)ds
− t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nf s, x(s),(Sx)(s)
ds
+x0+x10t+
x2 0 2!t
2+· · ·+ x
n−2 0 (n−2)!t
n−2+ xT
(n−1)!t
n−1, t∈J.
Clearly, the solution of the fractional BVP (1) is the fixed point of the operator F on Br.
We shall use the Banach contraction principle to prove thatF has a fixed point. The proof is divided into two steps.
Step 1. F(x)∈Br for every x∈Br.
For every x∈Br and δ >0, by (H3) and H¨older inequality, we have
k(F(x))(t+δ)−(F(x))(t)k ≤
1 Γ(α)
Z t+δ
0
(t+δ−s)α−1f s, x(s),(Sx)(s)ds
− 1
Γ(α)
Z t
0
(t−s)α−1f s, x(s),(Sx)(s)ds
+
tn−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nf s, x(s),(Sx)(s)ds
− (t+δ)
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nf s, x(s),(Sx)(s)ds
+
x10(t+δ−t) + x 2 0
2![(t+δ)
2−t2] +. . .
+ x
n−2 0
(n−2)![(t+δ)
n−2−tn−2] + xT
(n−1)![(t+δ)
n−1−tn−1]
≤ 1
Γ(α)
Z t
0
(t+δ−s)α−1−(t−s)α−1kf s, x(s),(Sx)(s)kds
+ 1 Γ(α)
Z t+δ
t
(t+δ−s)α−1kf s, x(s),(Sx)(s)kds
+
(t+δ)n−1−tn−1 (n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nkf s, x(s),(Sx)(s)kds
+kx10k(t+δ−t) + kx 2 0k
2! [(t+δ) 2−
t2] +. . .
+ kx
n−2 0 k
(n−2)![(t+δ)
n−2−tn−2] + kxTk
(n−1)![(t+δ)
n−1−tn−1]
≤ 1
Γ(α)
Z t
0
[(t+δ−s)α−1−(t−s)α−1]h(s)ds
+ 1 Γ(α)
Z t+δ
t
(t+δ−s)α−1h(s)ds
+
(t+δ)n−1−tn−1 (n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nh(s)ds
+kx10k(t+δ−t) + kx 2 0k
2! [(t+δ) 2−
t2] +. . .
+ kx
n−2 0 k
(n−2)![(t+δ)
n−2−tn−2] + kxTk
(n−1)![(t+δ)
n−1−tn−1]
≤ 1
Γ(α)
Z t
0
(t+δ−s)α−1−(t−s)α−1 1 1−α2 ds
1−α2Z t
0
(h(s))α12ds
α2
+ 1 Γ(α)
Z t+δ
t
(t+δ−s)1−α−1α2ds
1−α2Z t+δ
t
(h(s))α12ds
α2
+
(t+δ)n−1−tn−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α −n
1−α2ds
1−α2Z T
0
(h(s))α12ds
α2
+kx1
0k(t+δ−t) +
kx2 0k
2! [(t+δ)
2−t2] +. . .
+ kx
n−2 0 k
(n−2)![(t+δ)
n−2−tn−2] + kxTk
(n−1)![(t+δ)
n−1−tn−1]
≤ 1
Γ(α)
Z t
0
h
(t+δ−s)α −1
1−α2 −(t−s)
α−1 1−α2
i
ds
1−α2Z t
0
(h(s))α12ds
α2
+ 1 Γ(α)
Z t+δ
t
(t+δ−s)α −1 1−α2ds
1−α2Z t+δ
t
(h(s))α12ds
α2
+
(t+δ)n−1−tn−1 (n−1)!Γ(α−n+ 1)
Z T
0
(T −s)1−α−αn2ds
1−α2Z T 0
(h(s))α12ds
+kx1
0k(t+δ−t) +
kx2 0k
2! [(t+δ)
2−t2] +. . .
+ kx
n−2 0 k
(n−2)![(t+δ)
n−2−tn−2] + kxTk
(n−1)![(t+δ)
n−1−tn−1]
≤ H
Γ(α)
−δα
−α2 1−α2 α−α2 1−α2
+(t+δ)
α−α2 1−α2 α−α2 1−α2
−t
α−α2 1−α2 α−α2 1−α2
!1−α2 + H
Γ(α)
δ
α−α2 1−α2 α−α2 1−α2
!1−α2
+
(t+δ)n−1−tn−1 (n−1)!Γ(α−n+ 1)
H T
α−α2−n+1 1−α2
α−α2−n+1 1−α2
!1−α2
+kx1
0k(t+δ−t) +
kx2 0k
2! [(t+δ)
2−t2] +. . .
+ kx
n−2 0 k
(n−2)![(t+δ)
n−2−tn−2] + kxTk
(n−1)![(t+δ)
n−1−tn−1].
It is obvious that the right-hand side of the above inequality tends to zero as δ→0. There-fore, F is continuous on J, that is, F(x)∈C(J, X). Moreover, for x∈Br and all t ∈J, by
using (8), we have
k(F(x))(t)k ≤ 1
Γ(α)
Z t
0
(t−s)α−1kf s, x(s),(Sx)(s)
kds
+ t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nkf s, x(s),(Sx)(s)kds
+kx0k+kx10kt+
kx2 0k 2! t
2
+· · ·+ kx
n−2 0 k (n−2)!t
n−2
+ kxTk (n−1)!t
n−1
≤ 1
Γ(α)
Z t
0
(t−s)α−1h(s)ds
+ t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nh(s)ds
+kx0k+kx10kt+
kx2 0k 2! t
2
+· · ·+ kx
n−2 0 k (n−2)!t
n−2
+ kxTk (n−1)!t
n−1
≤ 1
Γ(α)
Z t
0
(t−s)α −1 1−α2ds
1−α2Z t
0
(h(s))α12ds
α2
+ t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α −n
1−α2ds
1−α2Z T 0
(h(s))α12ds
α2
+kx0k+kx10kt+
kx2 0k 2! t
2
+· · ·+ kx
n−2 0 k (n−2)!t
n−2
+ kxTk (n−1)!t
≤ HT
α−α2 Γ(α)(α−α2
1−α2) 1−α2 +
HTα−α2 (n−1)!Γ(α−n+ 1)
α−α2−n+ 1 1−α2
−(1−α2)
+kx0k+kx10kT +
kx2 0k 2! T
2+· · ·+ kx
n−2 0 k (n−2)!T
n−2 + kxTk
(n−1)!T
n−1
≤r.
Thus,kF(x)k∞≤r and we conclude that for all x∈Br, F(x)∈Br, that is,F :Br →Br.
Step 2. F is contraction mapping on Br.
Forx, y ∈Br and any t∈J, by using (7), (H2) and H¨older inequality, we have
k(F(x))(t)−(F(y))(t)k ≤ 1
Γ(α)
Z t
0
(t−s)α−1f s, x(s),(Sx)(s)
−f s, y(s),(Sy)(s)ds
+ t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nf s, x(s),(Sx)(s)
−f s, y(s),(Sy)(s)ds
≤ 1
Γ(α)
Z t
0
(t−s)α−1m1(s) kx(s)−y(s)k+k(Sx)(s)−(Sy)(s)k
ds
+ t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nm1(s) kx(s)−y(s)k+k(Sx)(s)−(Sy)(s)k
ds
≤ 1
Γ(α)
Z t
0
(t−s)α−1m1(s)
×
kx(s)−y(s)k+
Z s
0
k s, τ, x(τ)
−k(s, τ, y(τ))dτ
ds
+ t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nm1(s)
×
kx(s)−y(s)k+
Z s
0
k s, τ, x(τ)
−k(s, τ, y(τ))dτ
ds
≤ 1
Γ(α)
Z t
0
(t−s)α−1m1(s)
×
kx(s)−y(s)k+
Z s
0
m2(s)kx(τ)−y(τ)kdτ
ds
+ t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nm1(s)
×
kx(s)−y(s)k+
Z s
0
m2(s)kx(τ)−y(τ)kdτ
ds
≤ 1
Γ(α)
Z t
0
(t−s)α−1m1(s) kx−yk∞+m2(s)Tkx−yk∞
+ t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nm1(s) kx−yk∞+m2(s)Tkx−yk∞
ds
≤ kx−yk∞
Γ(α)
Z t
0
(t−s)α −1 1−α1ds
1−α1Z t
0
m1(s) +m1(s)m2(s)T
α1
1ds
α1
+ t
n−1kx−yk
∞
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)
α−n
1−α1ds
1−α1
×
Z T
0
m1(s) +m1(s)m2(s)T
1
α1ds
α1
≤ Mkx−yk∞
Γ(α)
t
α−α1
1−α1
α−α1 1−α1
!1−α1
+ Mkx−yk∞t
n−1
(n−1)!Γ(α−n+ 1)
T
α−α1−n+1 1−α1
α−α1−n+1 1−α1
!1−α1
≤ M
Γ(α)
Tα−α1 (α−α1
1−α1)
1−α1 +
M
(n−1)!Γ(α−n+ 1)
Tα−α1 (α−α1−n+1
1−α1 ) 1−α1
!
kx−yk∞.
= Ωα,T ,nkx−yk∞.
Thus, we have
kF(x)−F(y)k∞ ≤Ωα,T ,nkx−yk∞.
Since Ωα,T ,n <1, F is contraction. By Banach contraction principle, we can deduce that F
has a unique fixed point which is the unique solution of the fractional BVP (1). Our second main result is based on the well known Schaefer’s fixed point theorem. Theorem 4.2. Assume that(H1), (H4) and (H5) hold. Then the fractional BVP (1) has at least one solution on J .
Proof Transform the fractional BVP (1) into a fixed point problem. Consider the oper-ator F :C(J, X)→C(J, X) defined as (9). It is obvious that F is well defined due to (H1), H¨older inequality and the lemma 2.5.
For the sake of convenience, we subdivide the proof into several steps. Step 1. F is continuous operator.
Let{xn} be a sequence such that xn→x in C(J,X). Then for each t∈J, we have
k(F(xn))(t)−(F(x))(t)k
≤ 1
Γ(α)
Z t
0
(t−s)α−1f s, xn(s),(Sxn)(s)
−f s, x(s),(Sx)(s) ds
+ t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nf s, xn(s),(Sxn)(s)
≤ kf(., xn(.),(Sxn)(.))−f(., x(.),(Sx)(.))k∞
1 Γ(α)
Z t
0
(t−s)α−1ds
+
f ., xn(.),(Sxn)(.)
−f ., x(.),(Sx)(.) ∞tn−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nds
≤f ., xn(.),(Sxn)(.)
−f ., x(.),(Sx)(.)∞
tα αΓ(α) +
f ., xn(.),(Sxn)(.)
−f ., x(.),(Sx)(.)∞tn−1
(n−1)!Γ(α−n+ 1)
Tα−n+1 (α−n+ 1)
≤
Tα
Γ(α+ 1) +
Tα
(n−1)!Γ(α−n+ 2)
f ., xn(.),(Sxn)(.)
−f ., x(.),(Sx)(.) ∞.
Taking supremum, we get
kF xn−F xk∞
≤
Tα
Γ(α+ 1) +
Tα
(n−1)!Γ(α−n+ 2)
f ., xn(.),(Sxn)(.)
−f ., x(.),(Sx)(.)∞,
since f is continuous, we have
kF xn−F xk∞→0 as n → ∞.
Therefore,F is continuous operator.
Step 2. F maps bounded sets into bounded sets in C(J, X).
Indeed, it is enough to show that for any η∗ > 0, there exists a l > 0 such that for each
x∈Bη∗ ={x∈C(J, X) :kxk∞ ≤η∗}, we have kF xk∞≤l. For each t∈J, by (H4), we get
k(F(x))(t)k ≤ 1
Γ(α)
Z t
0
(t−s)α−1Nf 1 +kx(s)kλ+k(Sx)(s)k
ds
+ t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nNf 1 +kx(s)kλ+k(Sx)(s)k
ds
+kx0k+kx10kt+
kx2 0k 2! t
2
+· · ·+ kx
n−2 0 k (n−2)!t
n−2
+ kxTk (n−1)!t
n−1
≤ 1
Γ(α)
Z t
0
(t−s)α−1Nf
1 +kx(s)kλ+
Z s
0
kk s, τ, x(τ)kdτ
ds
+ t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
×Nf
1 +kx(s)kλ +
Z s
0
kk s, τ, x(τ)kdτ
ds
+kx0k+kx10kt+
kx2 0k 2! t
2
+· · ·+ kx
n−2 0 k (n−2)!t
n−2
+ kxTk (n−1)!t
n−1
≤ 1
Γ(α)
Z t
0
(t−s)α−1Nf
1 +kx(s)kλ+
Z s
0
Nk(1 +kx(τ)kλ)dτ
ds
+ t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−n
×Nf
1 +kx(s)kλ +
Z s
0
Nk(1 +kx(τ)kλ)dτ
ds
+kx0k+kx10kt+
kx2 0k 2! t
2+· · ·+ kx
n−2 0 k (n−2)!t
n−2 + kxTk
(n−1)!t
n−1
≤ 1
Γ(α)
Z t
0
(t−s)α−1Nf 1 +kxkλ∞+Nk(1 +kxkλ∞)T
ds
+ t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−n
×Nf 1 +kxkλ∞+Nk(1 +kxkλ∞)T
ds
+kx0k+kx10kt+
kx2 0k 2! t
2+· · ·+ kx
n−2 0 k (n−2)!t
n−2 + kxTk
(n−1)!t
n−1
≤ Nf(1 + (η
∗)λ)(1 +N kT)
Γ(α)
Z t
0
(t−s)α−1ds
+t
n−1N
f(1 + (η∗)λ)(1 +NkT)
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nds
+kx0k+kx10kt+
kx2 0k 2! t
2+· · ·+ kx
n−2 0 k (n−2)!t
n−2 + kxTk
(n−1)!t
n−1
≤ Nf(1 + (η
∗)λ)(1 +N kT)
Γ(α)
tα
α
+t
n−1N
f(1 + (η∗)λ)(1 +NkT)
(n−1)!Γ(α−n+ 1)
Tα−n+1
(α−n+ 1) +kx0k+kx10kt+
kx2 0k 2! t
2
+· · ·+ kx
n−2 0 k (n−2)!t
n−2
+ kxTk (n−1)!t
n−1
≤
1 Γ(α+ 1) +
1
(n−1)!Γ(α−n+ 2)
Nf(1 + (η∗)λ)(1 +NkT)Tα
+kx0k+kx10kT +
kx2 0k 2! T
2+· · ·+ kx
n−2 0 k (n−2)!T
n−2+ kxTk
(n−1)!T
n−1
where
l :=
1 Γ(α+ 1) +
1
(n−1)!Γ(α−n+ 2)
Nf(1 + (η∗)λ)(1 +NkT)Tα
+kx0k+kx10kT +
kx2 0k 2! T
2+· · ·+ kx
n−2 0 k (n−2)!T
n−2+ kxTk
(n−1)!T
n−1.
Thus, we have
k(F(x))(t)k ≤l and hence kF xk∞≤l.
Step 3. F maps bounded sets into equicontinuous sets of C(J, X).
Let 0≤t1 ≤t2 ≤T, x∈Bη∗.Using (H4), again we have
k(F(x))(t2)−(F(x))(t1)k
≤ 1
Γ(α)
Z t1 0
(t2−s)α−1−(t1−s)α−1
Nf 1 +kxkλ∞+Nk(1 +kxkλ∞)T
ds
+ 1 Γ(α)
Z t2
t1
(t2−s)α−1Nf 1 +kxkλ∞+Nk(1 +kxkλ∞)T
ds
+ t
n−1 2 −t
n−1 1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nNf 1 +kxkλ∞+Nk(1 +kxkλ∞)T
ds
+kx1
0k(t2 −t1) +
kx2 0k 2! (t
2 2−t
2
1) +· · ·+
kxn−2 0 k (n−2)!(t
n−2 2 −t
n−2 1 ) +
kxTk
(n−1)!(t
n−1 2 −t
n−1 1 )
≤ Nf(1 + (η
∗)λ)(1 +N kT)
Γ(α)
Z t1 0
(t2−s)α−1−(t1−s)α−1
ds
+Nf(1 + (η
∗)λ)(1 +N kT)
Γ(α)
Z t2
t1
(t2−s)α−1ds
+(t
n−1 2 −tn
−1
1 )Nf(1 + (η∗)λ)(1 +NkT)
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nds
+kx1
0k(t2 −t1) +
kx2 0k 2! (t
2
2−t21) +· · ·+
kxn0−2k
(n−2)!(t
n−2 2 −tn
−2 1 ) +
kxTk
(n−1)!(t
n−1 2 −tn
−1 1 )
≤ Nf(1 + (η
∗)λ)(1 +N kT)
Γ(α+ 1) (t
α
2 −t
α
1) +
(tn2−1−tn1−1)Nf(1 + (η∗)λ)(1 +NkT)
(n−1)!Γ(α−n+ 2) T
α−n+1
+kx1
0k(t2 −t1) +
kx2 0k 2! (t
2 2−t
2
1) +· · ·+
kxn−2 0 k (n−2)!(t
n−2 2 −t
n−2 1 ) +
kxTk
(n−1)!(t
n−1 2 −t
As t2 →t1, the right-hand side of the above inequality tends to zero and since x is an arbi-trary in Bη∗, F is equicontinuous.
Now, let {xn}, n= 1,2, . . . be a sequence on Bη∗, and
(F xn)(t) = (F1xn)(t) + (F2xn)(t) + (F3x)(t), t∈J, where
(F1xn)(t) =
1 Γ(α)
Z t
0
(t−s)α−1f s, xn(s),(Sxn)(s)
ds, t∈J,
(F2xn)(t) = −
tn−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nf s, xn(s),(Sxn)(s)
ds, t∈J,
(F3x)(t) =x0+x10t+
x20
2!t
2+· · ·+ x
n−2 0 (n−2)!t
n−2 + xT
(n−1)!t
(n−1), t ∈J.
In view of hypothesis (H5) and lemma 2.6, the set convK1 is compact. For any t∗ ∈J, (F1xn)(t∗) =
1 Γ(α)
Z t∗
0
(t∗−s)α−1f s, xn(s),(Sxn)(s)
ds
= 1 Γ(α)klim→∞
k
X
i=1
t∗ k
t∗− it
∗
k
α−1
f
it∗ k , xn(
it∗
k ),(Sxn)( it∗
k )
= t
∗
Γ(α)ζn1,
where
ζn1 = lim
k→∞
k
X
i=1 1
k
t∗− it
∗
k
α−1
f
it∗ k , xn(
it∗
k ),(Sxn)( it∗
k )
.
Now, we have {(F1xn)(t)}is a function family of continuous mappingsF1xn:J →X, which
is uniformly bounded and equicontinuous. As convK1 is convex and compact, we know
ζn1 ∈ convK1. Hence, for any t∗ ∈ J = [0, T], the set {(F1xn)(t∗)}, is relatively compact.
Therefore by lemma 2.7, every {(F1xn)(t)} contains a uniformly convergent subsequence
{(F1xnk)(t)}, k = 1,2, . . . , on J. Thus,{F1x:x∈Bη∗}is relatively compact.
Set
(F2xn)(t) =−
tn−1
(n−1)!Γ(α−n+ 1)
Z t
0
(t−s)α−nf s, xn(s),(Sxn)(s)
For any t∗ ∈J,
(F2xn)(t∗) =−
(t∗)n−1
(n−1)!Γ(α−n+ 1)
Z t∗
0
(t∗−s)α−nf s, xn(s),(Sxn)(s)
ds
=− (t
∗)n−1
(n−1)!Γ(α−n+ 1)
× lim
k→∞
k
X
i=1
t∗ k
t∗− it
∗
k
α−n
f
it∗ k , xn(
it∗
k ),(Sxn)( it∗
k )
=− (t
∗)n
(n−1)!Γ(α−n+ 1)ζn2,
where
ζn2 = lim
k→∞
k
X
i=1 1
k
t∗−it
∗
k
α−n
f
it∗ k , xn(
it∗
k ),(Sxn)( it∗
k )
.
Now, we have{(F2xn)(t)}is a function family of continuous mappingsF2xn :J →X,which
is uniformly bounded and equicontinuous. As convK2 is convex and compact, we know
ζn2 ∈ convK2. Hence, for any t∗ ∈ J = [0, T], the set {(F2xn)(t∗)}, is relatively compact.
Therefore by lemma 2.7, every {(F2xn)(t)} contains a uniformly convergent subsequence
{(F2xnk)(t)}, k = 1,2, . . . , on J. Particularly, {(F2xn)(t)} contains a uniformly convergent
subsequence {(F2xnk)(t)}, k= 1,2, . . . , on J. Thus,{F2x:x∈Bη∗} is relatively compact.
Obviously, the set {F3x : x ∈ Bη∗} is relatively compact. As a result, the set {F x : x ∈
Bη∗} is relatively compact.
As a consequence of steps 1-3, we can conclude that F is continuous and completely contin-uous.
Step 4. A priori bounds.
Now it remains to show that the set
E(F) = {x∈C(J, X) :x=ηF x for someη ∈[0,1]},
is bounded
x(t) = η
1 Γ(α)
Z t
0
(t−s)α−1f s, x(s),(Sx)(s)ds
− t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nf s, x(s),(Sx)(s)ds
+ x0+x10t+
x2 0 2!t
2
+· · ·+ x
n−2 0 (n−2)!t
n−2
+ xT (n−1)!t
(n−1)
.
Using (H4), for each t∈J, we have
kx(t)k ≤ k(F(x))(t)k ≤ 1
Γ(α)
Z t
0
(t−s)α−1Nf 1 +kx(s)kλ+Nk(1 +kxskλB)T
ds
+ t
n−1
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−n
×Nf 1 +kx(s)kλ+Nk(1 +kxskλB)T
ds
+kx0k+kx10kt+
kx2 0k 2! t
2+· · ·+ kx
n−2 0 k (n−2)!t
n−2+ kxTk
(n−1)!t
n−1
≤ Nf
Γ(α)
Z t
0
(t−s)α−1ds+ Nf Γ(α)
Z t
0
(t−s)α−1kx(s)kλds
+ NfNkT Γ(α)
Z t
0
(t−s)α−1ds+ NfNkT Γ(α)
Z t
0
(t−s)α−1kxskλBds
+ t
n−1N
f
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nds
+ t
n−1N
f
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nkx(s)kλds
+ t
n−1N
fNkT
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nds
+ t
n−1N
fNkT
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nkxskλBds
+kx0k+kx10kt+
kx2 0k 2! t
2+· · ·+ kx
n−2 0 k (n−2)!t
n−2+ kxTk
(n−1)!t
n−1
≤ NfT
α
Γ(α+ 1) +
NfNkTα+1
Γ(α+ 1) +
NfTα
(n−1)!Γ(α−n+ 2) +
NfNkTα+1
(n−1)!Γ(α−n+ 2) +kx0k+kx10kT +
kx2 0k 2! T
2+· · ·+ kx
n−2 0 k (n−2)!T
n−2+ kxTk
(n−1)!T
n−1
+ Nf Γ(α)
Z t
0
(t−s)α−1kx(s)kλds+NfNkT Γ(α)
Z t
0
+ T
n−1N
f
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nkx(s)kλds
+ T
nN fNk
(n−1)!Γ(α−n+ 1)
Z T
0
(T −s)α−nkxskλBds.
By lemma 3.2, there exists a N >0 such that kx(t)k ≤N, t∈J.
Thus for every t∈J, we have kxk∞ ≤N.This show that the set E(F) is bounded.
As a consequence of Schaefer’s fixed point theorem, we deduce that F has a fixed point that
is solution of fractional BVP (1).
5. Examples
In this section, we give tow examples to illustrate the usefulness of our main results.
Example 5.1.
(10)
cDαx(t) = e−µt
1+et
cos(t)|x(t)|
1+|x(t)| + Rt
0
e−µt(s+|x(s)|)
2(1+s+|x(s)|)ds
, t ∈J1, α∈(3,4),
x(0) = 0, x0(0) = 0, x00(0) = 0, x000(1) = 0, where µ >0 is constant.
TakeX1 = [0,∞), J1 = [0,1] and so T = 1.
Set
f1 t, x(t),(Sx)(t)
= e
−µt
1 +et
cos(t)|x(t)|
1 +|x(t)| + (Sx)(t)
, k1(t, s, x(s)) =
e−µt(s+|x(s)|) 2(1 +s+|x(s)|). Letx1, x2 ∈C(J1, X1) and t ∈[0,1],we have
|k1(t, s, x1(s))−k1(t, s, x2(s))| ≤
e−µt
2
|x1(s)| − |x2(s)|
(1 +s+|x1(s)|)(1 +s+|x2(s)|)
≤ e
−µt
2 |x1(s)−x2(s)|,
and
f1 t, x1(t),(Sx1)(t)
−f1 t, x2(t),(Sx2)(t)
≤ e
−µt
1 +et
|cos(t)|
|x1(t)| 1 +|x1(t)|
− |x2(t)|
1 +|x2(t)|
+
(Sx1)(t)−(Sx2)(t)
≤ e −µt 2
|x1(t)| − |x2(t)| 1 +|x1(t)|
1 +|x2(t)|
+
(Sx1)(t)−(Sx2)(t) ! ≤ e −µt 2
x1(t)−x2(t) +
(Sx1)(t)−(Sx2)(t)
.
Also, for all x∈C(J1, X1) and each t∈J1, we have
|f1 t, x(t),(Sx)(t)
| ≤ e
−µt
1 +et
|x(t)|
1 +|x(t)|
+ e −µt 2 Z t 0
s+|x(s)|
(1 +s+|x(s)|)
ds ≤ e −µt 2
1 + e
−µt
2
.
Fort ∈J1, β∈(0, α−3), we have
m1(t) = e −µt
2 ∈L 1
β(J
1,R), m2(t) = e −µt
2 ∈L 1
β(J
1,R), h(t) = e −µt
2
1 + e−2µt∈Lβ1(J
1,R) and
M =ke−µt
2 +
e−µt
2
e−µt
2 kLβ1(J
1,R)
.
Choosing someµ >0 large enough and suitableβ ∈(0, α−3),one can arrive at the following inequality
Ωα,β =
M
Γ(α) 1 (α1−−ββ)1−β +
M
3!Γ(α−3) 1
(α−1−β−β3)1−β <1.
All the assumptions in Theorem 4.1 are satisfied, and therefore, the fractional BVP 10 has a unique solution on J1.
Example 5.2. (11)
cDαx(t) = tν
1+et
cos(t)|x(t)|λ
1+|x(t)| + Rt
0
tν|x(s)|λ
2(1+|x(s)|)ds
, t∈J1, α∈(3,4),
x(0) = 0, x0(0) = 0, x00(0) = 0, x000(1) = 0, ν >−α, λ ∈[0,1− 1
p),1< p <
1 4−α.
TakeX1 = [0,∞), J1 = [0,1] and so T = 1
Set
f2 t, x(t),(Sx)(t)
= t
ν
1 +et
cos(t)|x(t)|λ
1 +|x(t)| + (Sx)(t)
, k2(t, s, x(s)) =
tν|x(s)|λ
2(1 +|x(s)|),
For all x∈C(J1, X1) and each t∈J1 = [0,1], we have
|k2(t, s, x(s))| ≤ 1 2
1 +|x(s)|λ,
and
f2 t, x(t), Sx(t)
≤
tν
1 +et
|x(t)|λ
1 +|x(t)|
+
(Sx)(t)
≤ t
ν
2 1 +|x(t)|
λ
+|(Sx)(t)|
≤ 1
2 1 +|x(t)|
λ+|(Sx)(t)|
.
Now, since ν >−α, we have
Z t
0
(t−s)α−1f2 s, x(s), Sx(s)
ds ≤
Z t
0
(t−s)α−1sν
|x(s)|λ +s ν
2
Z s
0
|x(τ)|λdτ
ds
≤M1λ
Z t
0
(t−s)α−1
sν +s 2ν
2
ds
≤M1λΓ(α)Iα
tν +t 2ν
2
≤ M
λ
1Γ(α)Γ(1 +ν) Γ(1 +ν+α) +
Mλ
1Γ(α)Γ(1 + 2ν) 2Γ(1 + 2ν+α) ,
and
Z t
0
(t−s)α−4f2 s, x(s), Sx(s)
ds≤
Z t
0
(t−s)α−4sν
|x(s)|λ+s
ν
2
Z s
0
|x(τ)|λdτ
ds
≤M1λ
Z t
0
(t−s)α−4
sν +s 2ν
2
ds
≤M1λΓ(α−3)Iα−3
tν+ t 2ν
2
≤ M
λ
1Γ(α−3)Γ(1 +ν) Γ(ν+α−2) +
Mλ
As a result, the sets
K11 =
n
(t−s)α−1f2 s, x(s), Sx(s)
:x∈C(J1, X1), s∈[0, t]
o
,
K12 =
n
(t−s)α−4f2 s, x(s), Sx(s)
:x∈C(J1, X1), s∈[0, t]
o
,
are bounded which implies thatK11, K12 are relatively compact. Thus, all the assumptions in Theorem 4.2 satisfied, and, hence, the fractional BVP 11 has at least one solution on J1.
References
[1] R. P. Agarwal, M. Benchohra and S. Hamani, Boundary value problems for fractional Differential equations, Georgian Math. J., 16(2009), 401-411.
[2] R. P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109(2010), 973-1033. [3] K. Balachandran and J. Y. Park, Nonlocal Cauchy problem for abstract fractional semilinear evolution
equations, Nonlinear Anal., 71(2009), 4471-4475.
[4] N. D. Chalishajar and K. Karthikeyan, Existence and uniqueness results for boundary value problems of higher order fractional integro-differential equations involving Gronwall’s inequality in Banach spaces, Acta Math. Sci., 33B(2013), 1-16.
[5] M. M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fractls, 14(2002), 433-440.
[6] J. Henderson and A. Ouahab, Fractional functional differential inclusions with finite delay, Nonlinear Anal., 70(2009), 2091-2105.
[7] E. Hernandez, D. O’Regan and K. Balachandran, On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Anal., 73(2010), 3462-3471.
[8] R. Hilfer, Applications of Fractional Calculus in Physics, Singapore, World Scientific, (2000).
[9] K. Karthikeyan and J. J. Trujillo, Existenes and uniqueness results for fractional integrodifferential equations with boundary value conditions, Commun. Nonlinear Sci. Numer. Simulat., 17(2012), 4037-4043.
[10] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam, Elsevier, (2006).
[11] V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal., 69(2008), 3337-3343.
[12] V. Lakshmikantham and A. S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal., 11(2007), 395-402.
[13] V. Lakshmikantham and A. S. Vatsala, General uniqueness and monotone iterative technique for frac-tional differential equations, Appl. Math. Lett., 21(2008), 828-834.
[14] V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal., 69(2008), 2677-2682.
[15] K. S. Miller and B. Ross, An introduction to the fractional calculus and differential equations, New York, Wiley, (1993).
[16] I. Podlubny, Fractional differential equations, San Diego, Academic Press, (1999).
[18] I. Stakgold and M. Holst, Green’s functions and boundary value problems, 3rd ed. Canada, Wiley, (2011).
[19] J. R. Wang, L. Lv and Y. Zhou, Boundary value problems for fractional differential equations involoving Caputa derivative in Banach spaces, J. Appl. Math. Comput., 38(2012), 209-224.
[20] J. R. Wang, W. Wei and Y. L. Yang, Fractional nonlocal integrodifferential equations of mixed type with time-varying generating operators and optimal control, Opusc. Math., 30(2010), 217-234.
[21] J. R. Wang, X. Xiang, W. Wei and Q. Chen, The generalized Grownwall inequality and its applications to periodic solutions of integrodifferential impulsive periodic system on Banach space, J. Inequal. Appl., 2008(2008), Art. ID430521, 22 pp.
[22] J. R. Wang, Y. Yang and W. Wei, Nonlocal impulsive problems for fractional differential equations with time-varying generating operators in Banach spaces, Opusc. Math., 30(2010), 361-381.
[23] K. P. Wilkie, C. S. Drapaca and S. Sivaloganathan, A nonlinear viscoelastic fractional derivative model of infant hydrocephalus, Appl. Math. Compu., 217(2011), 8693-8704.
[24] Y. L. Yang, L. Lv and J. R. Wang, Existence results for boundary value problems of high order differ-ential equations involving Caputo derivative, J. Appl. Math. Comput., 38(2012), 565-583.
[25] S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. Differ. Equ., 2006(2006), 1-12.
