Ahmad and Nieto 9 studied some existence results for boundary value problem ... the existence and uniqueness theorem for the nonlinear fractional mixed ...
Hindawi Publishing Corporation International Journal of Differential Equations Volume 2011, Article ID 304570, 15 pages doi:10.1155/2011/304570
Research Article Existence and Uniqueness Theorem of Fractional Mixed Volterra-Fredholm Integrodifferential Equation with Integral Boundary Conditions Shayma Adil Murad,1 Hussein Jebrail Zekri,2 and Samir Hadid3 1
Department of Mathematics, Faculty of Science, Dohuk University, Kurdistan, Iraq Department of Mathematics, Faculty of Science, Zakho University, Kurdistan, Iraq 3 Department of Mathematics and Science, College of Education and Basic Sciences, Ajman University of Science and Technology, UAE 2
Correspondence should be addressed to Samir Hadid, samir hadid@yahoo.com Received 7 May 2011; Accepted 24 May 2011 Academic Editor: Shaher Momani Copyright q 2011 Shayma Adil Murad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the existence and uniqueness of the solutions of mixed Volterra-Fredholm type integral equations with integral boundary condition in Banach space. Our analysis is based on an application of the Krasnosel’skii fixed-point theorem.
1. Introduction In the last century, notable contributions have been made to both the theory and applications of the fractional differential equations. For the theory part, Momani and Hadid have investigated the local and global existence theorem of both fractional differential equation and fractional integrodifferential equations; see 1–6. Fractional-order differential equations have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Integrodifferential equations with integral boundary conditions are often encountered in various applications; it is worthwhile mentioning the applications of those conditions in the study of population dynamics and cellular systems. For a detailed description of the integral boundary conditions, we refer the reader to a recent paper 7. In 8, Tidke studied the problem of existence of global solutions to nonlinear mixed Volttera-Fredholm integrodifferential equations with nonlocal condition. Ahmad and Nieto 9 studied some existence results for boundary value problem involving a nonlinear integrodifferential equation of fractional order with integral equation.
2
International Journal of Differential Equations
Very recently N Gu´er´ekata 10 discussed the existence of solutions of fractional abstract differential equations with nonlocal initial condition. Anguraj et al. 11 studied the existence and uniqueness theorem for the nonlinear fractional mixed Volterra-Fredholm integrodifferential equation with nonlocal initial condition. Motivated by these works, we study in this paper the existence of solution of boundary value problem for fractional integrodifferential equations in the case 1 < α ≤ 2 in Banach spaces by using Banach and Krasnosel’skii fixed-point theorems.
2. Preliminaries First of all, we recall some basic definitions; see 12–15. Definition 2.1. For a function f given on the interval a, b, the Caputo fractional order derivative of f is defined by
t α aD
1 ft Γn − α
t
t − sn−α−1 f n sds,
2.1
a
where n α 1 and α denotes the integer part of α. Lemma 2.2. Let α > 0, then t −α t α aD aD
yt yt c0 c1 t c2 t2 · · · cn−1 tn−1 ,
2.2
for some ci ∈ R, i 0, 1, . . . , n − 1, n α 1. Definition 2.3. Let f be a function which is defined almost everywhere a.e on a, b, for α > 0, we define b −α aD
f
1 Γα
b
b − tα−1 ftdt,
2.3
a
provided that the integral Lebesgue exists. Theorem 2.4 Krasnosel’skii fixed point theorem. Let M be a closed-convex bounded nonempty subset of a Banach space X. Let A and B be two operators such that i Ax By M, whenever x, y ∈ M, ii A is compact and continuous; iii B is a contraction mapping, then there exists z ∈ M such that z Az Bz. Let X be a Banach space with the norm · . Let C 0, T , X be Banach space ofall
International Journal of Differential Equations
3
continuous functions ψ : 0, T → X, with supermum norm ψ sup{ψs : s ∈ 0, T }. Consider the fractional mixed Volttera-Fredholm integrodifferential equation with boundary conditions, which has the form D yt f α
t t, yt,
T
k t, s, ys ds, 0
y0 − y 0
T
h1 t, s, ys ds ,
2.4
0
g ys ds,
yT − y T
0
T
h ys ds,
2.5
0
where 1 < α ≤ 2, Dα is the Caputo fractional derivative and the nonlinear functions f : 0, T × X × X × X → X, k, h1 : 0, T × 0, T × X → X and g, h : X → X satisfy the following hypotheses: H1 there exists constants G1 , G2 such that hy ≤ G1 ,gy ≤ G2 for y ∈ X, H2 there exists constants b1 , b2 such that hx − hy ≤ b1 x − y and gx − g y ≤ b2 x − y ,
∀x, y ∈ X,
2.6
H3 there exists continuous functions p : 0, T → R 0, ∞ and p1 : 0, T → R such t t that 0 kt, s, x − kt, s, yds ≤ pt x − y and 0 kt, s, yds ≤ p1 t y, for every t, s ∈ 0, T and x, y ∈ X, H4 there exists continuous functions q : 0, T → R and q1 : 0, T → R T T such that 0 h1 t, s, x − h1 t, s, yds ≤ qt x − y and 0 h1 t, s, yds ≤ q1 ty for every t, s ∈ 0, T andx, y ∈ X H5 there exists continuous function L : 0, T → R , and N1 is positive constant such that ft, x1 , y1 , z1 − ft, x2 , y2 , z2 ≤ LtKx1 − x2 y1 − y2 z1 − z2 and N1 supt∈0,T ft, 0, 0, 0, for every t ∈ 0, T and x1 , y1 , z1 , x2 , y2 , z2 ∈ X, where K : R → 0, ∞ is continuous nondecreasing function satisfying Kγtx ≤ γtKx, where γ is a continuous function γ : 0, T → R . Lemma 2.5. Let 1 < α ≤ 2 and f : J × X → X, where J 0, T , be a continuous function, then the solution of fractional differential equation 2.4 with the boundary condition 2.5 is
T 1 t h ys ds 1 − g ys ds T 0 0 T s T α−1 1 t T − s − f s, ys, k s, τ, yτ dτ, h1 s, τ, yτ dτ T Γα 0 0 0 s T 1 t T T − sα−2 f s, ys, k s, τ, yτ dτ, h1 s, τ, yτ dτ ds T 0 Γα − 1 0 0 s T t t − sα−1 f s, ys, k s, τ, yτ dτ, h1 s, τ, yτ dτ ds. Γα 0 0 0
1 t yt T
T
2.7
4
International Journal of Differential Equations
Proof. By Lemma 2.2, we reduce the problem 2.4-2.5 to an equivalent integral equation α
yt t0 I f C1 C2 t, s t T t − sα−1 f s, ys, k s, τ, yτ dτ, yt h1 s, τ, yτ dτ ds C1 C2 t. Γα 0 0 0
2.8
In view of the relations c Dα I α yt yt and I α I β yt I α β yt, for α, β > 0, we obtain
y t
t 0
t − sα−2 f Γα − 1
s s, ys,
T
k s, τ, yτ dτ, 0
h1 s, τ, yτ dτ ds C2 .
2.9
0
Applying the boundary condition 2.5, we find that
y0 C1 ,
yT
T 0
T − sα−1 f Γα
s s, ys,
k s, τ, yτ dτ,
T
0
h1 s, τ, yτ dτ ds
0
C1 C2 T,
y 0 C2 ,
y T
T 0
T − sα−2 f Γα − 1
s s, ys, 0
k s, τ, yτ dτ,
T
h1 s, τ, yτ dτ ds
0
C2 , 2.10 that is, 1 T h ys ds − g ys ds T 0 0 T s T 1 T − sα−1 − f s, ys, k s, τ, yτ dτ, h1 s, τ, yτ dτ ds T 0 Γα 0 0 s T 1 T T − sα−2 f s, ys, k s, τ, yτ dτ, h1 s, τ, yτ dτ ds, T 0 Γα − 1 0 0
1 C2 T
T
T
T 1 h ys ds 1 − g ys ds T 0 0 T T s 1 T − sα−1 − f s, ys, k s, τ, yτ dτ, h1 s, τ, yτ dτ ds T 0 Γα 0 0 s T 1 T T − sα−2 f s, ys, k s, τ, yτ dτ, h1 s, τ, yτ dτ ds. T 0 Γα − 1 0 0
1 C1 T
2.11
International Journal of Differential Equations
5
Therefore the solution of 2.4-2.5 is
yt
T 1 t h ys ds 1 − g ys ds T 0 0 T T s α−1 1 t T − s − f s, ys, k s, τ, yτ dτ, h1 s, τ, yτ dτ ds T Γα 0 0 0 s T 1 t T T − sα−2 f s, ys, k s, τ, yτ dτ, h1 s, τ, yτ dτ ds T 0 Γα − 1 0 0 s T t t − sα−1 f s, ys, k s, τ, yτ dτ, h1 s, τ, yτ dτ ds, Γα 0 0 0 1 t T
T
2.12
which completes the proof.
3. The Main Result Theorem 3.1. If the hypotheses (H1)–(H5) are satisfied, then the fractional integrodifferential equation 2.4-2.5 has a unique solution on J. Proof. Define F : C → C by
T 1 t h ys ds 1 − g ys ds T 0 0 T s T α−1 1 t T − s − f s, ys, k s, τ, yτ dτ, h1 s, τ, yτ dτ ds T Γα 0 0 0 s T 1 t T T − sα−2 f s, ys, k s, τ, yτ dτ, h1 s, τ, yτ dτ ds T 0 Γα − 1 0 0 s T t t − sα−1 f s, ys, k s, τ, yτ dτ, h1 s, τ, yτ dτ ds. Γα 0 0 0
1 t Fyt T
T
3.1
We show that F has a fixed point on Br. This fixed point is then a solution of 2.4-2.5. Firstly, we show that FBr ⊂ Br, where Br {y ∈ C : y ≤ r}. For y ∈ Br, we have T
T h ys ds 1 − 1 t g ys ds T 0 0 s T T α−2 1 t T − s h1 s, τ, yτ dτ ds f s, ys, k s, τ, yτ dτ, T 0 Γα − 1 0 0
Fyt ≤ 1 t T
6
International Journal of Differential Equations s T 1 t T T − sα−1 h1 s, τ, yτ dτ ds f s, ys, k s, τ, yτ dτ, T Γα 0 0 0 s t T t − sα−1 h1 s, τ, yτ dτ ds, f s, ys, k s, τ, yτ dτ, Γα 0 0 0
T h ys ds 1 − 1 t g ys ds T 0 0 T T S 1 t T − sα−2 k s, τ, yτ dτ, h1 s, τ, yτ dτ f s, ys, T 0 Γα − 1 0 0 −fs, 0, 0, 0 fs, 0, 0, 0ds S T 1 t T T − sα−1 k s, τ, yτ dτ, h1 s, τ, yτ dτ f s, ys, T Γα 0 0 0 −fs, 0, 0, 0 fs, 0, 0, 0ds S t T t − sα−1 k s, τ, yτ dτ, h1 s, τ, yτ dτ f s, ys, Γα 0 0 0 −fs, 0, 0, 0 fs, 0, 0, 0ds,
Fyt ≤ 1 t G1 T 1 − 1 t G2 T T T S T T T − sα−2 1 t k s, τ, yτ dτ, h1 s, τ, yτ dτ f s, ys, T 0 Γα − 1 0 0 1 t T T − sα−2 fs, 0, 0, 0ds −fs, 0, 0, 0ds T 0 Γα − 1 t S T t − sα−1 k s, τ, yτ dτ, h1 s, τ, yτ dτ f s, ys, Γα 0 0 0 t t − sα−1 fs, 0, 0, 0ds −fs, 0, 0, 0ds Γα 0 S T
T 1 t T − sα−1 k s, τ, yτ dτ, h1 s, τ, yτ dτ f s, ys, T Γα 0 0 0
T 1 t T − sα−1 fs, 0, 0, 0ds −fs, 0, 0, 0ds T Γα 0 T
1 t T − sα−2 1 t 1 t ≤ G1 T 1 − G2 T Ls T T T 0 Γα − 1
Fyt ≤ 1 t T
T
International Journal of Differential Equations S T K ys k s, τ, yτ dτ h1 s, τ, yτ dτ ds 0 0 1 tN1 T
T 0
1 t T − sα−2 ds Γα − 1 T
T 0
7
T − sα−1 Ls Γα
T s K ys h1 s, τ, yτ dτ ds k s, τ, yτ dτ 0 0 1 tN1 T
T 0
T − sα−1 ds Γα
t 0
t − sα−1 Ls Γα
T s K ys h1 s, τ, yτ dτ ds k s, τ, yτ dτ 0 0
N1
t 0
t − sα−1 ds Γα
1 t 1 tN1 ≤ 1 tG1 1 − G2 T T T T 0
T − sα−2 ds Γα − 1
1 t T
1 t T t 0
T 0
T 0
T 0
t T − sα−1 t − sα−1 ds N1 ds Γα Γα 0
T − sα−2 LsK y p1 sy q1 sy ds Γα − 1 T − sα−1 LsK y p1 sy q1 sy ds Γα
t − sα−1 LsK y p1 sy q1 sy ds, Γα
Fyt ≤ 1 tG1 1 − 1 t G2 T T Tα Tα 1 tN1 T α−1 N1 T Γα Γα 1 Γα 1 1 t T
T 0
T − sα−2 Ls 1 p1 s q1 s K y ds Γα − 1
T
1 t T − sα−1 Ls 1 p1 s q1 s K y ds T Γα 0 t t − sα−1 Ls 1 p1 s q1 s K y ds. Γα 0
3.2
8
International Journal of Differential Equations
Since we have M1 sup{Lt 1 p1 t q1 t; t ∈ 0, T }, and 1 − 1 t/T < 1 − 1/T , we get
Tα Tα 1 1 tN1 T α−1 N1 ≤ 1 tG1 1 − G2 T T T Γα Γα 1 Γα 1 T 1 tM1 T T − sα−1 1 tM1 T − sα−2 K y ds K y ds T Γα − 1 T Γα 0 0 t α−1 t − s M1 K y ds Γα 0
Tα Tα 1 1 tN1 T α−1 ≤ 1 tG1 1 − G2 T N1 T T Γα Γα 1 Γα 1 T 1 tM1 Kr T T − sα−1 1 tM1 Kr T − sα−2 ds ds T T Γα 0 Γα − 1 0 t t − sα−1 M1 Kr ds Γα 0
Tα Tα 1 1 tN1 T α−1 N1 ≤ 1 tG1 1 − G2 T T T Γα Γα 1 Γα 1 T α−1 Tα M1 KrT α 1 tM1 Kr T Γα Γα 1 Γα 1 T α−1 Tα 1 t ≤ 1 tG1 T − 1G2 N1 M1 Kr T Γα Γα 1 N1 M1 Kr
Tα , Γα 1
Fyt ≤ 1 T G1 T − 1G2 1 T N1 M1 Kr T N1 M1 Kr
T α−1 Tα Γα Γα 1
Tα Γα 1
≤ G1 1 T G2 T − 1
C0 N1 M1 Kr , Γα 1T 2−α
where C0 2T 2 T αT 1. Now, take x, y ∈ C and for each t ∈ 0, T , we obtain T
T hx − h y ds 1 − 1 t gx − g y ds T 0 0 T 1 t T − sα−1 T Γα 0
Fxt − Fyt ≤ 1 t T
3.3
International Journal of Differential Equations s T ks, τ, xτdτ, h1 s, τ, xτdτ f s, xs, 0 0 −f
s
T
k s, τ, yτ dτ,
s, ys, 0
0
9
h1 s, τ, yτ dτ ds
1 t T T − sα−2 T 0 Γα − 1 s T ks, τ, xτdτ , h1 s, τ, xτdτ f s, xs, 0 0
−f
s
T
k s, τ, yτ dτ,
s, ys, 0
0
h1 s, τ, yτ dτ ds
t
t − sα−1 Γα 0 s T h1 s, τ, xτdτ f s, xs, ks, τ, xτdτ , 0 0
−f
s
T
k s, τ, yτ dτ,
s, ys, 0
0
h1 s, τ, yτ dτ ds,
3.4
by using H1–H5, we get Fxt − Fyt ≤ b1 1 t T
T
T x − yds b2 1 − 1 t x − y ds T 0 0
1 t T T − sα−2 Ls T 0 Γα − 1 s K xs − ys ks, τ, xτ − k s, τ, yτ dτ
0
T h s, τ, xτ − h1 s, τ, yτ dτ ds 0 1 1 t T T − sα−1 Ls T Γα 0 s K xs − ys ks, τ, xτ − k s, τ, yτ dτ 0
t T t − sα−1 h1 s, τ, xτ − h1 s, τ, yτ dτ ds Ls 0 Γα 0
10
International Journal of Differential Equations s K xs − ys ks, τ, xτ − k s, τ, yτ dτ 0
T h1 s, τ, xτ − h1 s, τ, yτ dτ 0 b1 1 t T
≤
1 t T 1 t T t 0
1 t T t 0
T x − yds b2 1 − 1 t x − yds T 0 0
0
T 0
T − sα−2 LsK x − y psx − y qsx − y ds Γα − 1 T − sα−1 LsK x − y psx − y qsx − y ds Γα
t − sα−1 LsK x − y psx − y qsx − y ds Γα
1 t T
T
T
b1 1 t T
≤
ds
T
T x − yds b2 1 − 1 x − yds T 0 0
T 0
T 0
T − sα−2 Ls 1 ps qs K x − y ds Γα − 1 T − sα−1 Ls 1 ps qs K x − y ds Γα
t − sα−1 Ls 1 ps qs K x − y ds . Γα 3.5
Since we have Mt Lt1 pt qt, M∗ sup{Mt : t ∈ 0, T }, and, Let Kx − y ≤ wx − y, w > 0, then
T x − yds b2 1 − 1 x − yds T 0 0 α−2 T wM∗ 1 t T − s x − yds T 0 Γα − 1 w M∗ 1 t T T − sα−1 x − yds T Γα 0 t t − sα−1 x − yds wM∗ Γα 0 wM∗ C1 1 T x − y , ≤ b1 1 T b2 T − 1 Γα 1 T 2−α
Fxt − Fyt ≤ b1 1 t T
T
3.6
International Journal of Differential Equations
11
where C1 2T 2 T α1 T . As b1 1 T b2 T −1 wM∗ C1 1 T /Γα 1T 2−α < 1, therefore f is a contraction. Thus, the conclusion of the theorem is followed by the contraction mapping principle. Theorem 3.2. Assume that (H1)–(H5) hold with t T f t, yt, k t, τ, yτ dτ, h1 t, τ, yτ dτ ≤ ψt, 0 0
where ψt ∈ L1 J.
3.7
Then the boundary value problem 2.4-2.5 has at least one element on 0, T . Proof. Consider Br {y ∈ C : y ≤ r}. We define the operators A and B as
1 Axt Γα
t
f
t − s
f
s t, xs,
0
T ks, τ, xτdτ,
0
h1 s, τ, xτdτ ds,
0
T 1 t 1 t T T − sα−2 h ys ds 1 − g ys ds T T 0 0 0 Γα − 1 s T s, xs, ks, τ, xτdτ, h1 s, τ, xτdτ ds
1 t T
Bxt
α−1
T
0
1 t T
T 0
T − sα−1 f Γα
0
s s, xs,
T ks, τ, xτdτ,
0
h1 s, τ, xτdτ ds.
0
3.8
Let us observe that if x, y ∈ Br, then Ax By ∈ Br, s T α−1 t − s t Ax By f s, xs, ks, τ, xτdτ, h1 s, τ, xτdτ ds 0 Γα 0 0
T 1 t 1 t T T − sα−2 1 t T h ys ds 1 − g ys ds T T T 0 0 0 Γα − 1 s T 1 t f s, ys, k s, τ, yτ dτ , h1 s, τ, yτ dτ ds T 0 0 T s T T − sα−1 f s, ys, k s, τ, yτ dτ, h1 s, τ, yτ dτ ds Γα 0 0 0 t s T t − sα−1 ≤ ks, τ, xτdτ, h1 s, τ, xτdτ ds f s, xs, Γα 0 0 0
12
International Journal of Differential Equations T
T h ys ds 1 − 1 t g ys ds T 0 0 T α−2 1 t T − s T 0 Γα − 1 s T h1 s, τ, yτ dτ ds f s, ys, k s, τ, yτ dτ, 0 0
1 t T
1 t T T − sα−1 T Γα 0 T s k s, τ, yτ dτ, h1 s, τ, yτ dτ ds f s, ys, 0 0
≤ ψ L1
t
t − sα−1 ds 1 tG1 Γα 0
1 tψ L1 T T − sα−2 1 t ds 1− G2 T T T 0 Γα − 1 1 tψ L1 T T − sα−1 ds T Γα 0 α
ψ T 1 tT α−1 ψ L1 1 tT α ψ L1 1 L1 ≤ 1 tG1 1 − G2 T Γα 1 T T Γα T Γα 1 α ψ T 1 tT α−1 ψ L1 1 tT α ψ L1 L1 ≤ 1 T G1 T − 1G2 Γα 1 T Γα T Γα 1
≤ G1 1 T G2 T − 1
C2 T α−2 ψ , L1 Γα 1 3.9
where C2 2T 2 T α 1 T . Now we prove that Bx is contraction mapping, s T h1 s, τ, x1 τdτ f s, x1 s, ks, τ, x1 τdτ , 0 0 0 s T h1 s, τ, x2 τdτ ds −f s, x2 s ks, τ, x2 τdτ , 0 0 s T 1 t T T − sα−2 h1 s, τ, x1 τdτ f s, x1 s, ks, τ, x1 τdτ, T 0 Γα − 1 0 0 s T h1 s, τ, x2 τdτ ds −f s, x2 s, ks, τ, x2 τdτ , 0 0
1 t Bx1 − Bx2 ≤ T
T
T − sα−1 Γα
International Journal of Differential Equations
1 t T T − sα−2 Ls 1 ps qs Kx1 − x2 ds ≤ T Γα − 1 0 T T − sα−1 Ls 1 ps qs Kx1 − x2 ds . Γα 0
13
3.10 Let Kx1 − x2 ≤ wx1 − x2 , we obtain
T T T − sα−2 T − sα−1 1 twM∗ ds ds , Bx1 − Bx2 ≤ x1 − x2 T Γα 0 Γα − 1 0
T α−1 Tα 1 T wM∗ Bx1 − Bx2 ≤ x1 − x2 T Γα Γα 1 ≤
wM∗ 1 T α T x1 − x2 . Γα 1T 2−α
3.11
It is clear that B is contraction mapping, since xt is continuous, then Ax is continuous s T 1 t α−1 h1 s, τ, xτdτ ds t − s f s, xs, ks, τ, xτdτ, Axt Γα 0 0 0 t t − sα−1 3.12 ≤ ψ L1 ds Γα 0 T α ψ L1 . Axt ≤ Γα 1 Hence, A is uniformly bounded on Br. Now, let us prove that Axt is equicontinuous, let t1 , t2 ∈ 0, T and x ∈ Br. Using the fact that f is bounded on the compact set J × Br, thus s T supt,s∈J×Br fs, xs, 0 ks, τ, xτdτ, 0 h1 s, τ, xτdτ c0 < ∞,we get s 1 t1 t1 − sα−1 f s, xs, ks, τ, xτdτ, Axt1 − Axt2 Γα 0 0 T
h1 s, τ, xτdτ ds −
0
f
s s, xs,
t2
1 Γα
0
T ks, τ, xτdτ,
0
t2 − sα−1
0
t 1 1 ≤ t1 − sα−1 − t2 − sα−1 Γα 0
h1 s, τ, xτdτ ds
14
International Journal of Differential Equations s T f s, xs, ks, τ, xτdτ, h1 s, τ, xτdτ ds 0 0 s T t2 α−1 t2 − s f s, xs, ks, τ, xτdτ , h1 s, τ, xτdτ ds t1 0 0 c0 2t2 − t1 α t1 α − t2 α . Γα 1
≤
3.13 So A is relatively compact. By Arzela-Ascoli theorem, A is compact. Now we conclude the result of the theorem of Krasnosel’skii theorem. Example 3.3. Consider the following fractional mixed Volterra-Fredholm integrodifferential equation:
y
1.5
1 1 t 10 10 yt
t 0
yt
1
10 e|yt| t
dt 0
yte−t 2 dt, 10 yt
3.14
with integral boundary conditions
y0 − y 0
1 0
1 dt, 10 yt
y1 − y 1
1 0
1 dt. 10 e−|yt|
3.15
Here, 1 g yt ≤ 10 yt 1 h yt 10 e−|yt| ≤
1 , 10
gx − g y ≤ 1 x − y, 100
1 , 10
hx − h y ≤ 1 x − y, 100 t 1 yt, k t, s, y ds ≤ 10 t 0 t 1 yt, h1 t, s, y ds ≤ 10 t 0
t 1 x − y , kt, s, x − k t, s, y ds ≤ t 0 10e t 1 x − y , h1 t, s, x − h1 t, s, y ds ≤ t 0 10e f t, x1 y1 , z1 − f t, x2 , y2 , z2 ≤
1 x1 − x2 y1 − y2 z1 − z2 , 10 t ft, 0, 0, 0
1 . 10 3.16
International Journal of Differential Equations
15
Hence, the conditions H1–H5 hold with G1 G2 0.1 , b1 b2 0.01, M1∗ 0.12, w 0.1, Co 6, N1 0.1, M∗ 0.12, and C1 6, thus b1 1 T b2 T − 1
wM∗ C1 1 T 0.10.1262 < 1. < 1 ⇐⇒ 0.012 2−α Γ2.5 Γα 1T
3.17
We conclude from the above example that the integrodifferential equation has unique solution.
References 1 M. Amairi, M. Aoun, S. Najar, and M. N. Abdelkrim, “A constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 2162–2168, 2010. 2 Z. Drici, F. A. McRae, and J. V. Devi, “Fractional differential equations involving causal operators,” Communications in Applied Analysis., vol. 14, no. 1, pp. 81–88, 2010. 3 S. B. Hadid, “Local and global existence theorems on differential equations of non-integer order,” Journal of Fractional Calculus, vol. 7, pp. 101–105, 1995. 4 R. W. Ibrahim, “Existence results for fractional boundary value problem,” International Journal of Contemporary Mathematical Sciences, vol. 3, no. 33-36, pp. 1767–1774, 2008. 5 S. M. Momani, “Local and global existence theorems on fractional integro-differential equations,” Journal of Fractional Calculus, vol. 18, pp. 81–86, 2000. 6 S. M. Momani and S. B. Hadid, “On the inequalities of integro-differential fractional equations,” International Journal of Applied Mathematics, vol. 12, no. 1, pp. 29–37, 2003. 7 B. Ahmad, A. Alsaedi, and B. S. Alghamdi, “Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions,” Nonlinear Analysis Real world Applications, vol. 9, no. 4, pp. 1727–1740, 2008. 8 H L. Tidke, “Existence of global solutions to nonlinear mixed Volterra-Fredholm integrodifferential equations with nonlocal conditions,” Electronic Journal of Differential Equations, vol. 2009, pp. No. 55–7, 2009. 9 B. Ahmad and J. J. Nieto, “Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions,” Boundary Value Problems, vol. 2009, Article ID 708576, 11 pages, 2009. 10 G. M. N’Gu´er´ekata, “A Cauchy problem for some fractional abstract differential equation with non local conditions,” Nonlinear Analysis: Theory , Method and Applications, vol. 70, no. 5, pp. 1873–1876, 2009. 11 A. Anguraj, P. Karthikeyan, and J. J. Trujillo, “Existence of solutions to fractional mixed integrodifferential equations with nonlocal initial condition,” Advances in Difference Equations, vol. 2011, Article ID 690653, 12 pages, 2011. 12 A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006. 13 M. A. Krasnosel’ski˘ı, “Two remarks on the method of successive approximations,” Uspekhi Matematicheskikh Nauk, vol. 10, no. 163, pp. 123–127, 1955. 14 I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. 15 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993.
