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Nonlocal Cauchy problem for abstract fractional semilinear evolution equations
Nonlinear Analysis 71 (2009) 4471–4475
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Nonlocal Cauchy problem for abstract fractional semilinear evolution equations K. Balachandran a,∗ , J.Y. Park b a
Department of Mathematics, Bharathiar University, Coimbatore-641 046, India
b
Department of Mathematics, Pusan National University, Pusan 609-735, South Korea
article
info
Article history: Received 13 January 2009 Accepted 3 March 2009 MSC: 34G10 34G20
abstract In this paper we prove the existence of solutions of fractional semilinear evolution equations in Banach spaces. Further nonlocal Cauchy problem is discussed for the evolution equations. The results are obtained by using fractional calculus and fixed point theorems. © 2009 Elsevier Ltd. All rights reserved.
Keywords: Existence of solution Evolution equation Nonlocal condition Fractional calculus Fixed point theorems
1. Introduction Recently fractional differential equations emerged as a new branch of applied mathematics which have been used for many mathematical models in science and engineering. In fact fractional differential equations are considered as an alternative model to nonlinear differential equations [1]. Theory of fractional differential equations has been extensively studied by Delbosco and Rodino [2] and Lakshmikantham et al. [3–6]. In [7,8] the authors have proved the existence of solutions of abstract differential equations by using semigroup theory and fixed point theorem. Many partial fractional differential equations can be expressed as fractional differential equations in some Banach spaces [9]. The nonlocal Cauchy problem for abstract evolution differential equation was first studied by Byszewski [10]. Subsequently several authors have investigated the problem for different types of nonlinear differential equations and integrodifferential equations including functional differential equations in Banach spaces [11–15]. Very recently N’Guerekata [16,17] discussed the existence of solutions of fractional abstract differential equations with a nonlocal initial condition. Motivated by this work we study in this paper the existence of solutions of fractional semilinear evolution equations in Banach spaces by using fractional calculus and fixed point theorems. 2. Preliminaries We need some basic definitions and properties of fractional calculus which are used in this paper. Definition 2.1. A real function f (t ) is said to be in the space Cα , α ∈ R if there exists a real number p > α , such that f (t ) = t p g (t ), where g ∈ C [0, ∞) and it is said to be in the space Cαm iff f (m) ∈ Cα , m ∈ N.
∗
Corresponding author. E-mail addresses: [email protected] (K. Balachandran), [email protected] (J.Y. Park).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.03.005
4472
K. Balachandran, J.Y. Park / Nonlinear Analysis 71 (2009) 4471–4475
Definition 2.2. The Riemann–Liouville fractional integral operator of order β > 0 of function f ∈ Cα , α ≥ −1 is defined as I β f (t ) =
t
Z
1
0 (β)
(t − s)β−1 f (s)ds
0
where 0 (·) is the Euler gamma function. m Definition 2.3. If the function f ∈ C− 1 and m is a positive integer then we can define the fractional derivative of f (t ) in the Caputo sense as
dα f (t ) dt α
=
t
Z
1
0 (m − α)
(t − s)m−α−1 f m (s)ds,
m − 1 < α ≤ m.
0
If 0 < α ≤ 1, then dα f (t ) dt α
=
0 (1 − α)
f 0 (s)
t
Z
1
(t − s)α
0
ds,
df (s)
where f 0 (s) = ds and f is an abstract function with values in X . Let C (J , X ) be the Banach space of continuous functions x(t ) with x(t ) ∈ X for t ∈ J, a compact interval in R and kxkC (J ,X ) = maxt ∈J kx(t )k. For basic facts about fractional derivative and fractional calculus one can refer the books [18–20]. Consider the linear fractional evolution equation dq u(t )
= A(t )u(t ), dt q u(0) = u0 ,
0 ≤ t ≤ T,
(1)
where 0 < q < 1 and A(t ) is a bounded linear operator on a Banach space X and u0 ∈ X . Let J = [0, T ] and assume the following condition. (HA) A(t ) is a bounded linear operator on X for each t ∈ J. The function t → A(t ) is continuous in the uniform operator topology. Eq. (1) is equivalent to the integral equation u(t ) = u0 +
t
Z
1
0 (q)
(t − s)q−1 A(s)u(s)ds.
(2)
0
By a solution of the abstract Cauchy problem (1), we mean an abstract function u such that the following conditions are satisfied: (i) u ∈ C (J , X ) and u ∈ D(A(t )) for all t ∈ J; q (ii) ddt qu exists and continuous on J, where 0 < q < 1; (iii) u satisfies Eq. (1) on J with the initial condition u(0) = u0 ∈ X . Theorem 2.1. If the hypothesis (HA) is satisfied, then Eq. (1) has a unique solution. Proof. The proof is based on the application of Picard’s iteration method. Let M = max0≤t ≤T kA(t )k (see [21]) and define a mapping P : C ([0, T ] : X ) → C ([0, T ] : X ) by
(Pu)(t ) = u0 +
1
0 (q)
t
Z
(t − s)q−1 A(s)u(s)ds.
(3)
0
From (3) we have
kPu(t ) − P v(t )k ≤ (MT q /0 (q + 1))ku − vk and by induction
kP n u(t ) − P n v(t )k ≤
(MT q /0 (q + 1))n ku − vk n!
and therefore
kP n u − P n vk ≤ (MT q /0 (q+1))n
(MT q /0 (q + 1))n ku − vk. n!
Since < 1 for large n then by the well-known generalization of the Banach contraction principle P has a unique n! fixed point u ∈ C ([0, T ] : X ). This fixed point is the solution of Eq. (1).
K. Balachandran, J.Y. Park / Nonlinear Analysis 71 (2009) 4471–4475
4473
3. Semilinear evolution equation Now consider the semilinear fractional evolution equation d q u( t )
= A(t )u(t ) + f (t , u(t )), dt q u(0) = u0 ,
0 ≤ t ≤ T,
(4)
where A(t ) is a bounded linear operator and f : J × X → X is continuous. The equation is equivalent to the integral equation u( t ) = u0 +
t
Z
1
0 (q)
(t − s)
q −1
A(s)u(s)ds +
0
t
Z
1
0 (q)
(t − s)q−1 f (s, u(s))ds.
(5)
0
We need the following additional assumption to prove the existence of solution of Eq. (4). (Hf) f : J × X → X is continuous and there exists a constant L > 0 such that
kf (t , u) − f (t , v)k ≤ Lku − vk,
for all u, v ∈ X .
q
For brevity let us take γ = 0 (qT+1) and N = maxt ∈J kf (t , 0)k. Theorem 3.1. If the hypotheses (HA)–(Hf) are satisfied and if γ (M + L) < unique solution.
1 , 2
then the fractional evolution equation (4) has a
Proof. Let Z = C ([0, T ] : X ). Define the mapping Φ : Z → Z by
Φ u(t ) = u0 +
1
0 (q)
t
Z
(t − s)q−1 A(s)u(s)ds + 0
1
t
Z
0 (q)
(t − s)q−1 f (s, u(s))ds
(6)
0
and we have to show that Φ has a fixed point. This fixed point is then a solution of Eq. (4). Choose r ≥ 2(ku0 k + N γ ). Then we can show that Φ Br ⊂ Br where Br := {x ∈ Z : kxk ≤ r }. From the assumptions we have
kΦ u(t )k ≤ ku0 k + ≤ ku0 k +
t
Z
1
0 (q)
(t − s)q−1 kA(s)kku(s)kds + 0 t
Z
1
0 (q)
( t − s)
q−1
kA(s)kku(s)kds +
0
Tq
≤ ku0 k + Mr
+ (Lr + N )
0 (q + 1) = ku0 k + Mr γ + γ (Lr + N ) ≤ r.
1
0 (q) 1
0 (q)
Z
t
(t − s)q−1 kf (s, u(s))kds
0
Z
t
(t − s)q−1 [kf (s, u(s)) − f (s, 0)k + kf (s, 0)k]ds
0
Tq
0 (q + 1)
Thus, Φ maps Br into itself. Now, for u1 , u2 ∈ Z , we have
kΦ u1 (t ) − Φ u2 (t )k ≤ ≤ ≤
t
Z
1
0 (q)
(t − s)q−1 kA(s)(u1 (s) − u2 (s))kds + 0
Tq
0 (q + 1) 1 2
1
0 (q)
t
Z
(t − s)q−1 kf (s, u1 (s)) − f (s, u2 (s))kds 0
(M + L)ku1 (t ) − u2 (t )k
ku1 (t ) − u2 (t )k.
Hence Φ is a contraction mapping and therefore there exists a unique fixed point u ∈ Br such that Φ u(t ) = u(t ). Any fixed point of Φ is the solution of (4). 4. Nonlocal problem In this section we discuss the existence of solution of evolution equation (4) with nonlocal condition of the form u(0) + g (u) = u0
(7)
where g : Z → X is a given function which satisfies the following condition (Hg) g : Z → X is continuous and there exists a constant G > 0 such that
kg (u) − g (v)k ≤ Gku − vk,
for u, v ∈ C (J , X ).
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K. Balachandran, J.Y. Park / Nonlinear Analysis 71 (2009) 4471–4475
Theorem 4.1. If the hypotheses (HA), (Hf), (Hg) are satisfied and if 2[G +(M + L)γ ] < 1, then the fractional evolution equation (4) with nonlocal condition (7) has a unique solution. Proof. We want to prove that the operator Ψ : Z → Z defined by
Ψ u(t ) = u0 − g (u) +
t
Z
1
0 (q)
(t − s)q−1 A(s)u(s)ds + 0
1
0 (q)
t
Z
(t − s)q−1 f (s, u(s))ds
(8)
0
has a fixed point. This fixed point is then a solution of Eqs. (4) and (7). Choose r ≥ 2(ku0 k + kg (0)k + N γ ). Then we can show that Φ Br ⊂ Br .
Z t 1 kΨ u(t )k ≤ ku0 k + kg (u) − g (0)k + kg (0)k + (t − s)q−1 kA(s)u(s)kds 0 (q) 0 Z t 1 (t − s)q−1 [kf (s, u(s)) − f (s, 0)k + kf (s, 0)k]ds + 0 (q) 0 ≤ ku0 k + Gr + kg (0)k + Mr γ + Lr γ + N γ ≤ r. Now, for u1 , u2 ∈ Z , we have 1
Z
t
kΨ u1 (t ) − Ψ u2 (t )k ≤ kg (u1 ) − g (u2 )k + (t − s)q−1 kA(s)(u1 (s) − u2 (s))kds 0 (q) 0 Z t 1 + (t − s)q−1 kf (s, u1 (s)) − f (s, u2 (s))kds 0 (q) 0 ≤ [G + γ (M + L)]ku1 (t ) − u2 (t )k ≤
1 2
ku1 (t ) − u2 (t )k.
The result follows by the application of contraction mapping principle. Our next result is based on the following well-known fixed point theorem. Krasnoselkii Theorem. Let S be a closed convex and nonempty subset of a Banach space X . Let P , Q be two operators such that (i) Px + Qy ∈ S whenever x, y ∈ S; (ii) P is a contraction mapping; (iii) Q is compact and continuous. Then there exists z ∈ S such that z = Pz + Qz. Now we assume the following condition instead of (Hf) and apply the above fixed point theorem. 0
(Hf) f : J × X → X is continuous and there exists a function µ ∈ L1 (J ) such that sup kf (t , u)k ≤ µ(t ),
for all (t , u) ∈ J × X 0
Theorem 4.2. Assume that (HA), (Hf) , (Hg) hold. If G + γ M < 1, then the fractional evolution equation (4) with nonlocal condition (7) has a solution. Proof. Choose r ≥
ku0 k+kg (0)k+γ kµkL1 1−(G+γ M )
Pu(t ) = u0 − g (u) +
1
0 (q)
Z
and define the operators P and Q on Br as
t
(t − s)q−1 A(s)u(s)ds
0
and Qu(t ) =
1
0 (q)
t
Z
(t − s)q−1 f (s, u(s))ds. 0
For any u, v ∈ Br we have kPu + Q vk ≤ ku0 k + Gr + kg (0)k + Mr γ + γ kµkL1 ≤ r and so Pu + Q v ∈ Br . Obviously P is a contraction. The equicontinuity of (Qu)(t ) is already proved in [17] and so Q (Br ) is relatively compact. By the Arzela–Ascoli Theorem, Q is compact. Hence by the Krasnoselkii theorem there exists a solution to the problem (4) and (7). Acknowledgement The work of the first author is supported by Korea Brain Pool Program of 2008.
K. Balachandran, J.Y. Park / Nonlinear Analysis 71 (2009) 4471–4475
4475
References [1] B. Bonilla, M. Rivero, L. Rodriguez-Germa, J.J. Trujillo, Fractional differential equations as alternative models to nonlinear differential equations, Applied Mathematics and Computation 187 (2007) 79–88. [2] D. Delbosco, L. Rodino, Existence and uniqueness for a fractional differential equation, Journal of Mathematical Analysis and Applications 204 (1996) 609–625. [3] V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Analysis 69 (2008) 3337–3343. [4] V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Analysis 69 (2008) 2677–2682. [5] V. Lakshmikantham, A.S. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations, Applied Mathematics Letters 21 (2008) 828–834. [6] V. Lakshmikantham, J. Vasundhara devi, Theory of fractional differential equations in Banach spaces, European Journal of Pure and Applied Mathematics 1 (2008) 38–45. [7] M.M. El-Borai, Semigroups and some nonlinear fractional differential equations, Applied Mathematics and Computation 149 (2004) 823–831. [8] O.K. Jaradat, A. Al-Omari, S. Momani, Existence of the mild solution for fractional semilinear initial value problem, Nonlinear Analysis 69 (2008) 3153–3159. [9] M.A.A. El-Sayeed, Fractional order diffusion wave equation, International Journal of Theoretical Physics 35 (1996) 311–322. [10] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, Journal of Mathematical Analysis and Applications 162 (1991) 494–505. [11] K. Balachandran, Existence and uniqueness of mild and strong solutions of nonlinear integrodifferential equations with nonlocal condition, Differential Equations and Dynamical Systems 6 (1998) 159–165. [12] K. Balachandran, J.Y. Park, Existence of a mild solution of a functional integrodifferential equation with nonlocal condition, Bulletin of the Korean Mathematical Society 38 (2001) 175–182. [13] K. Balachandran, J.Y. Park, Existence of solution of second order nonlinear differential equations with nonlocal conditions in Banach spaces, Indian Journal of Pure and Applied Mathematics 32 (2001) 1883–1892. [14] L. Byszewski, H. Acka, Existence of solutions of a semilinear functional differential evolution nonlocal problems, Nonlinear Analysis 34 (1998) 65–72. [15] J. Liang, J.H. Liu, T.J. Xiao, Nonlocal Cauchy problems governed by compact operator families, Nonlinear Analysis 57 (2004) 183–189. [16] G.M. Mophou, G.M. N’guerekata, Existence of the mild solution for some fractional differential equations with nonlocal conditions, Semigroup Forum (in press). [17] G.M. N’guerekata, A Cauchy problem for some fractional abstract differential equation with nonlocal condition, Nonlinear Analysis 70 (2009) 1873–1876. [18] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [19] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. [20] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Nonlocal Cauchy problem for abstract fractional semilinear evolution equations K. Balachandran a,∗ , J.Y. Park b a
Department of Mathematics, Bharathiar University, Coimbatore-641 046, India
b
Department of Mathematics, Pusan National University, Pusan 609-735, South Korea
article
info
Article history: Received 13 January 2009 Accepted 3 March 2009 MSC: 34G10 34G20
abstract In this paper we prove the existence of solutions of fractional semilinear evolution equations in Banach spaces. Further nonlocal Cauchy problem is discussed for the evolution equations. The results are obtained by using fractional calculus and fixed point theorems. © 2009 Elsevier Ltd. All rights reserved.
Keywords: Existence of solution Evolution equation Nonlocal condition Fractional calculus Fixed point theorems
1. Introduction Recently fractional differential equations emerged as a new branch of applied mathematics which have been used for many mathematical models in science and engineering. In fact fractional differential equations are considered as an alternative model to nonlinear differential equations [1]. Theory of fractional differential equations has been extensively studied by Delbosco and Rodino [2] and Lakshmikantham et al. [3–6]. In [7,8] the authors have proved the existence of solutions of abstract differential equations by using semigroup theory and fixed point theorem. Many partial fractional differential equations can be expressed as fractional differential equations in some Banach spaces [9]. The nonlocal Cauchy problem for abstract evolution differential equation was first studied by Byszewski [10]. Subsequently several authors have investigated the problem for different types of nonlinear differential equations and integrodifferential equations including functional differential equations in Banach spaces [11–15]. Very recently N’Guerekata [16,17] discussed the existence of solutions of fractional abstract differential equations with a nonlocal initial condition. Motivated by this work we study in this paper the existence of solutions of fractional semilinear evolution equations in Banach spaces by using fractional calculus and fixed point theorems. 2. Preliminaries We need some basic definitions and properties of fractional calculus which are used in this paper. Definition 2.1. A real function f (t ) is said to be in the space Cα , α ∈ R if there exists a real number p > α , such that f (t ) = t p g (t ), where g ∈ C [0, ∞) and it is said to be in the space Cαm iff f (m) ∈ Cα , m ∈ N.
∗
Corresponding author. E-mail addresses: [email protected] (K. Balachandran), [email protected] (J.Y. Park).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.03.005
4472
K. Balachandran, J.Y. Park / Nonlinear Analysis 71 (2009) 4471–4475
Definition 2.2. The Riemann–Liouville fractional integral operator of order β > 0 of function f ∈ Cα , α ≥ −1 is defined as I β f (t ) =
t
Z
1
0 (β)
(t − s)β−1 f (s)ds
0
where 0 (·) is the Euler gamma function. m Definition 2.3. If the function f ∈ C− 1 and m is a positive integer then we can define the fractional derivative of f (t ) in the Caputo sense as
dα f (t ) dt α
=
t
Z
1
0 (m − α)
(t − s)m−α−1 f m (s)ds,
m − 1 < α ≤ m.
0
If 0 < α ≤ 1, then dα f (t ) dt α
=
0 (1 − α)
f 0 (s)
t
Z
1
(t − s)α
0
ds,
df (s)
where f 0 (s) = ds and f is an abstract function with values in X . Let C (J , X ) be the Banach space of continuous functions x(t ) with x(t ) ∈ X for t ∈ J, a compact interval in R and kxkC (J ,X ) = maxt ∈J kx(t )k. For basic facts about fractional derivative and fractional calculus one can refer the books [18–20]. Consider the linear fractional evolution equation dq u(t )
= A(t )u(t ), dt q u(0) = u0 ,
0 ≤ t ≤ T,
(1)
where 0 < q < 1 and A(t ) is a bounded linear operator on a Banach space X and u0 ∈ X . Let J = [0, T ] and assume the following condition. (HA) A(t ) is a bounded linear operator on X for each t ∈ J. The function t → A(t ) is continuous in the uniform operator topology. Eq. (1) is equivalent to the integral equation u(t ) = u0 +
t
Z
1
0 (q)
(t − s)q−1 A(s)u(s)ds.
(2)
0
By a solution of the abstract Cauchy problem (1), we mean an abstract function u such that the following conditions are satisfied: (i) u ∈ C (J , X ) and u ∈ D(A(t )) for all t ∈ J; q (ii) ddt qu exists and continuous on J, where 0 < q < 1; (iii) u satisfies Eq. (1) on J with the initial condition u(0) = u0 ∈ X . Theorem 2.1. If the hypothesis (HA) is satisfied, then Eq. (1) has a unique solution. Proof. The proof is based on the application of Picard’s iteration method. Let M = max0≤t ≤T kA(t )k (see [21]) and define a mapping P : C ([0, T ] : X ) → C ([0, T ] : X ) by
(Pu)(t ) = u0 +
1
0 (q)
t
Z
(t − s)q−1 A(s)u(s)ds.
(3)
0
From (3) we have
kPu(t ) − P v(t )k ≤ (MT q /0 (q + 1))ku − vk and by induction
kP n u(t ) − P n v(t )k ≤
(MT q /0 (q + 1))n ku − vk n!
and therefore
kP n u − P n vk ≤ (MT q /0 (q+1))n
(MT q /0 (q + 1))n ku − vk. n!
Since < 1 for large n then by the well-known generalization of the Banach contraction principle P has a unique n! fixed point u ∈ C ([0, T ] : X ). This fixed point is the solution of Eq. (1).
K. Balachandran, J.Y. Park / Nonlinear Analysis 71 (2009) 4471–4475
4473
3. Semilinear evolution equation Now consider the semilinear fractional evolution equation d q u( t )
= A(t )u(t ) + f (t , u(t )), dt q u(0) = u0 ,
0 ≤ t ≤ T,
(4)
where A(t ) is a bounded linear operator and f : J × X → X is continuous. The equation is equivalent to the integral equation u( t ) = u0 +
t
Z
1
0 (q)
(t − s)
q −1
A(s)u(s)ds +
0
t
Z
1
0 (q)
(t − s)q−1 f (s, u(s))ds.
(5)
0
We need the following additional assumption to prove the existence of solution of Eq. (4). (Hf) f : J × X → X is continuous and there exists a constant L > 0 such that
kf (t , u) − f (t , v)k ≤ Lku − vk,
for all u, v ∈ X .
q
For brevity let us take γ = 0 (qT+1) and N = maxt ∈J kf (t , 0)k. Theorem 3.1. If the hypotheses (HA)–(Hf) are satisfied and if γ (M + L) < unique solution.
1 , 2
then the fractional evolution equation (4) has a
Proof. Let Z = C ([0, T ] : X ). Define the mapping Φ : Z → Z by
Φ u(t ) = u0 +
1
0 (q)
t
Z
(t − s)q−1 A(s)u(s)ds + 0
1
t
Z
0 (q)
(t − s)q−1 f (s, u(s))ds
(6)
0
and we have to show that Φ has a fixed point. This fixed point is then a solution of Eq. (4). Choose r ≥ 2(ku0 k + N γ ). Then we can show that Φ Br ⊂ Br where Br := {x ∈ Z : kxk ≤ r }. From the assumptions we have
kΦ u(t )k ≤ ku0 k + ≤ ku0 k +
t
Z
1
0 (q)
(t − s)q−1 kA(s)kku(s)kds + 0 t
Z
1
0 (q)
( t − s)
q−1
kA(s)kku(s)kds +
0
Tq
≤ ku0 k + Mr
+ (Lr + N )
0 (q + 1) = ku0 k + Mr γ + γ (Lr + N ) ≤ r.
1
0 (q) 1
0 (q)
Z
t
(t − s)q−1 kf (s, u(s))kds
0
Z
t
(t − s)q−1 [kf (s, u(s)) − f (s, 0)k + kf (s, 0)k]ds
0
Tq
0 (q + 1)
Thus, Φ maps Br into itself. Now, for u1 , u2 ∈ Z , we have
kΦ u1 (t ) − Φ u2 (t )k ≤ ≤ ≤
t
Z
1
0 (q)
(t − s)q−1 kA(s)(u1 (s) − u2 (s))kds + 0
Tq
0 (q + 1) 1 2
1
0 (q)
t
Z
(t − s)q−1 kf (s, u1 (s)) − f (s, u2 (s))kds 0
(M + L)ku1 (t ) − u2 (t )k
ku1 (t ) − u2 (t )k.
Hence Φ is a contraction mapping and therefore there exists a unique fixed point u ∈ Br such that Φ u(t ) = u(t ). Any fixed point of Φ is the solution of (4). 4. Nonlocal problem In this section we discuss the existence of solution of evolution equation (4) with nonlocal condition of the form u(0) + g (u) = u0
(7)
where g : Z → X is a given function which satisfies the following condition (Hg) g : Z → X is continuous and there exists a constant G > 0 such that
kg (u) − g (v)k ≤ Gku − vk,
for u, v ∈ C (J , X ).
4474
K. Balachandran, J.Y. Park / Nonlinear Analysis 71 (2009) 4471–4475
Theorem 4.1. If the hypotheses (HA), (Hf), (Hg) are satisfied and if 2[G +(M + L)γ ] < 1, then the fractional evolution equation (4) with nonlocal condition (7) has a unique solution. Proof. We want to prove that the operator Ψ : Z → Z defined by
Ψ u(t ) = u0 − g (u) +
t
Z
1
0 (q)
(t − s)q−1 A(s)u(s)ds + 0
1
0 (q)
t
Z
(t − s)q−1 f (s, u(s))ds
(8)
0
has a fixed point. This fixed point is then a solution of Eqs. (4) and (7). Choose r ≥ 2(ku0 k + kg (0)k + N γ ). Then we can show that Φ Br ⊂ Br .
Z t 1 kΨ u(t )k ≤ ku0 k + kg (u) − g (0)k + kg (0)k + (t − s)q−1 kA(s)u(s)kds 0 (q) 0 Z t 1 (t − s)q−1 [kf (s, u(s)) − f (s, 0)k + kf (s, 0)k]ds + 0 (q) 0 ≤ ku0 k + Gr + kg (0)k + Mr γ + Lr γ + N γ ≤ r. Now, for u1 , u2 ∈ Z , we have 1
Z
t
kΨ u1 (t ) − Ψ u2 (t )k ≤ kg (u1 ) − g (u2 )k + (t − s)q−1 kA(s)(u1 (s) − u2 (s))kds 0 (q) 0 Z t 1 + (t − s)q−1 kf (s, u1 (s)) − f (s, u2 (s))kds 0 (q) 0 ≤ [G + γ (M + L)]ku1 (t ) − u2 (t )k ≤
1 2
ku1 (t ) − u2 (t )k.
The result follows by the application of contraction mapping principle. Our next result is based on the following well-known fixed point theorem. Krasnoselkii Theorem. Let S be a closed convex and nonempty subset of a Banach space X . Let P , Q be two operators such that (i) Px + Qy ∈ S whenever x, y ∈ S; (ii) P is a contraction mapping; (iii) Q is compact and continuous. Then there exists z ∈ S such that z = Pz + Qz. Now we assume the following condition instead of (Hf) and apply the above fixed point theorem. 0
(Hf) f : J × X → X is continuous and there exists a function µ ∈ L1 (J ) such that sup kf (t , u)k ≤ µ(t ),
for all (t , u) ∈ J × X 0
Theorem 4.2. Assume that (HA), (Hf) , (Hg) hold. If G + γ M < 1, then the fractional evolution equation (4) with nonlocal condition (7) has a solution. Proof. Choose r ≥
ku0 k+kg (0)k+γ kµkL1 1−(G+γ M )
Pu(t ) = u0 − g (u) +
1
0 (q)
Z
and define the operators P and Q on Br as
t
(t − s)q−1 A(s)u(s)ds
0
and Qu(t ) =
1
0 (q)
t
Z
(t − s)q−1 f (s, u(s))ds. 0
For any u, v ∈ Br we have kPu + Q vk ≤ ku0 k + Gr + kg (0)k + Mr γ + γ kµkL1 ≤ r and so Pu + Q v ∈ Br . Obviously P is a contraction. The equicontinuity of (Qu)(t ) is already proved in [17] and so Q (Br ) is relatively compact. By the Arzela–Ascoli Theorem, Q is compact. Hence by the Krasnoselkii theorem there exists a solution to the problem (4) and (7). Acknowledgement The work of the first author is supported by Korea Brain Pool Program of 2008.
K. Balachandran, J.Y. Park / Nonlinear Analysis 71 (2009) 4471–4475
4475
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