Perturbation Results and Monotone Iterative Technique for Fractional Evolution Equations

and Applied Analysis 3 system. Secondly, do the solution operators for fractional evolution equations have the perturbation properties analogous to those for the C0-semigroup? For evolution equations of integer order, perturbation properties play a significant role in monotone iterative technique; see 24 . Our paper copes with the above difficulties, and the new features of this paper mainly include the following aspects. We firstly introduce a new concept of a mild solution based on the well-known theory of Laplace transform, and the form is very easy. Secondly, we discuss the perturbation properties for the corresponding solution operators. Thirdly, by the monotone iterative technique based on lower and upper solutions, we obtain results on the existence and uniqueness of mild solutions for problem 1.1 and 1.3 . 2. Preliminaries In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. Definition 2.1 see 5 . The Riemann-Liouville fractional integral operator of order α > 0 of function f ∈ L1 R is defined as I 0 f t 1 Γ α ∫ t 0 t − s α−1f s ds, 2.1 where Γ · is the Euler gamma function. Definition 2.2 see 5 . The Caputo fractional derivative of order α > 0, n − 1 < α < n, is defined as D 0 f t 1 Γ n − α ∫ t 0 t − s n−α−1f s ds, 2.2 where the function f t has absolutely continuous derivatives up to order n − 1. If f is an abstract function with values in X, then the integrals and derivatives which appear in 2.1 and 2.2 are taken in Bochner sense. Proposition 2.3. For α, β > 0 and f as a suitable function (e.g., [5]) one has the following: i I 0 I β 0 f t I α β 0 f t ; ii I 0 I β 0 f t I β 0 I α 0 f t ; iii I 0 f t g t I α 0 f t I α 0 g t ; iv CD 0 I α 0 f t f t ; v CD 0 D β 0 f t / D α β 0 f t ; vi CD 0 D β 0 f t / D β 0 C D 0 f t . We observe from the above that the Caputo fractional differential operators do not possess neither semigroup nor commutative properties, which are inherent to the derivatives 4 Abstract and Applied Analysis on integer order. For basic facts about fractional integrals and fractional derivatives one can refer to the books 5, 32–34 . Let X be an ordered Banach space with norm ‖ · ‖ and partial order ≤, whose positive cone P {y ∈ X | y ≥ θ} θ is the zero element of X is normal with normal constant N. Let C I, X be the Banach space of all continuous X-value functions on interval I with norm ‖u‖c maxt∈I‖ t ‖. For u, v ∈ C I, X , u ≤ v ⇔ u t ≤ v t for all t ∈ I. For v,w ∈ C I, X , denote the ordered interval v,w {u ∈ C I, X | v ≤ u ≤ w}, and v t , w t {y ∈ X | v t ≤ y ≤ w t }, t ∈ I. By B X we denote the space of all bounded linear operators from X to X. Definition 2.4. If CD 0 v0, Av0, v ′ 0 ∈ C I, X , and v0 satisfies D 0 v0 t Av0 t ≤ f t, v0 t , Gv0 t , t ∈ I, v0 ≤ x ∈ X, v′ 0 0 ≤ θ, 2.3 then ṽ0 is called a lower solution of problem 1.1 ; if all inequalities of 2.3 are inverse, we call it an upper solution of problem 1.1 . Similarly, we give the definitions of lower and upper solutions of problem 1.3 . Definition 2.5. If CD 0 ṽ0, Aṽ0, ṽ ′ 0 ∈ C I, X , and ṽ0 satisfy D 0 ṽ0 t Aṽ0 t ≤ f t, ṽ0 t , t ∈ I, ṽ0 0 ≤ x ∈ X, ṽ′ 0 0 ≤ θ, 2.4 then ṽ0 is called a lower solution of problem 1.3 ; if all inequalities of 2.4 are inverse, we call it an upper solution of problem 1.3 . Consider the following problem: D 0 u t Au t θ, t ∈ I, u 0 x, u′ 0 θ. 2.5 Definition 2.6 see 9 . A family {Sα t }t≥0 ⊂ B X is called a solution operator for 2.5 if the following conditions are satisfied: 1 Sα t is strongly continuous for t ≥ 0 and Sα 0 I; 2 Sα t D A ⊂ D A and ASα t x Sα t Ax for all x ∈ D A , t ≥ 0; 3 Sα t is a solution of u t x − 1 Γ α ∫ t 0 t − s α−1Au s ds, 2.6 for all x ∈ D A , t ≥ 0. In this case, −A is called the generator of the solution operator Sα t and Sα t is called the solution operator generated by −A. Abstract and Applied Analysis 5 Definition 2.7 see 9 . The solution operator Sα t is called exponentially bounded if there are constants M ≥ 1 and ω ≥ 0 such that ‖Sα t ‖ ≤ Me, t ≥ 0. 2.7 An operator −A is said to belong to C X;M,ω , or Cα M,ω for short, if problem 2.5 has a solution operator Sα t satisfying 2.7 . Denote Cα ω ∪{Cα M,ω | M ≥ 1}, Cα ∪{Cα ω | ω ≥ 0}. In these notations C1 and C2 are the sets of all infinitesimal generators ofC0emigroups and cosine operator families COF , respectively. Next, we give a characterization of Cα M,ω . Lemma 2.8 see 9 . Let 1 < α < 2, −A ∈ C M,ω and let Sα t be the corresponding solution operator. Then for λ > ω, one has λ ∈ ρ −A and λα−1R λα,−A x ∫ ∞and Applied Analysis 5 Definition 2.7 see 9 . The solution operator Sα t is called exponentially bounded if there are constants M ≥ 1 and ω ≥ 0 such that ‖Sα t ‖ ≤ Me, t ≥ 0. 2.7 An operator −A is said to belong to C X;M,ω , or Cα M,ω for short, if problem 2.5 has a solution operator Sα t satisfying 2.7 . Denote Cα ω ∪{Cα M,ω | M ≥ 1}, Cα ∪{Cα ω | ω ≥ 0}. In these notations C1 and C2 are the sets of all infinitesimal generators ofC0emigroups and cosine operator families COF , respectively. Next, we give a characterization of Cα M,ω . Lemma 2.8 see 9 . Let 1 < α < 2, −A ∈ C M,ω and let Sα t be the corresponding solution operator. Then for λ > ω, one has λ ∈ ρ −A and λα−1R λα,−A x ∫ ∞ 0 e−λtSα t xdt, x ∈ X. 2.8 Lemma 2.9 see 9 . Let 1 < α < 2 and −A ∈ C. Then the corresponding solution operator is given by Sα t x lim n→∞ 1 n! n 1 ∑ k 1 b k,n 1 ( I ( t n )α A )−k x lim n→∞ 1 n! n 1 ∑ k 1 b k,n 1 [(n t )α R ((n t )α ,−A )]k x, 2.9 where b k,n are given by the recurrence relations: b 1,1 1, b k,n n − 1 − ka b k,n−1 α k − 1 b k−1,n−1, 1 ≤ k ≤ n, n 2, 3, . . ., b k,n 0, k > n, n 1, 2, . . .. The convergence is uniform on bounded subsets of 0, ∞ for any fixed x ∈ X. Lemma 2.10 see 9 . Let 1 < α < 2. Then −A ∈ C M,ω if and only if ωα,∞ ⊂ ρ −A and ∥∥∥ ∂ n ∂λn ( λα−1R λα,−A )∥∥∥ ≤ Mn! λ −ω n 1 λ > ω, n 0, 1, . . . . 2.10 Lemma 2.11. Assume h ∈ C I, X . For the linear Cauchy problem D 0 u t Au t h t , t ∈ I, u 0 x ∈ X, u′ 0 θ, 2.11