Periodic BVPs for fractional order impulsive evolution equations

Yu and Wang Boundary Value Problems 2014, 2014:35 http://www.boundaryvalueproblems.com/content/2014/1/35 RESEARCH Open Access Periodic BVPs for fractional order impulsive evolution equations Xiulan Yu1 and JinRong Wang2,3* * Correspondence: 2 Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, P.R. China 3 School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou 550018, P.R. China Full list of author information is available at the end of the article Abstract In this paper, we study periodic BVPs for fractional order impulsive evolution equations. The existence and boundedness of piecewise continuous mild solutions and design parameter drift for periodic motion of linear problems are presented. Furthermore, existence results of piecewise continuous mild solutions for semilinear impulsive periodic problems are showed. Finally, an example is given to illustrate the results. MSC: 34B05; 34G10; 47D06 Keywords: fractional order; impulsive evolution equations; periodic BVPs 1 Introduction In order to describe dynamics of populations subject to abrupt changes as well as other evolution processes such as harvesting, diseases, and so forth, many researchers have used impulsive differential systems to describe the model since the last century. For a wideranging bibliography and exposition on this important object see for instance the monographs of [–] and the papers [–]. Fractional differential equations appear naturally in fields such as viscoelasticity, electrical circuits, nonlinear oscillation of earthquakes etc. In particular, impulsive fractional evolution equations are used to describe many practical dynamical systems in many evolutionary processes models. Recently, Wang et al. [] discussed Cauchy problems and nonlocal problems for impulsive fractional evolution equations involving the Caputo fractional derivative. However, periodic boundary value problems (BVPs for short) for impulsive fractional evolution equations have not been studied extensively. In this paper we study the periodic BVPs for impulsive fractional evolution equations. Firstly, we discuss periodic BVPs for impulsive fractional evolution equations: ⎧ q c ⎪ ⎨ D,t x(t) = Ax(t) + f (t), q ∈ (, ), t ∈ J = [, T], t = tk , x() = x(T), ⎪ ⎩ + x(tk ) = x(tk– ) + yk , k = , , . . . , δ, () q in Banach space X, where c D,t is the Caputo fractional derivative of order q with the lower limit zero, A : D(A) ⊆ X → X is the generator of a C -semigroup {S(t), t ≥ } on a Banach space X, f : J → X is continuous, x() and yk are the elements of X,  = t < t < t < · · · < tδ < tδ+ = T, and x(tk+ ) = limh→+ = x(tk + h) and x(tk– ) = x(tk ) represent respectively the right and left limits of x(t) at t = tk . ©2014 Yu and Wang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Yu and Wang Boundary Value Problems 2014, 2014:35 http://www.boundaryvalueproblems.com/content/2014/1/35 Page 2 of 11 Secondly, we design parameter drift for the above periodic motion. We study the following impulsive periodic BVPs with parameter perturbations: ⎧ q c ⎪ ⎨ D,t x(t) = Ax(t) + f (t) + p(t, x(t), ξ ), x() = x(T), ⎪ ⎩ + x(tk ) = x(tk– ) + yk , k = , , . . . , δ, t ∈ J, t = tk , () where p is a given function and ξ ∈  = (–ξ̃ , ξ̃ ) is a small parameter perturbation that may be caused by some adaptive control algorithms or parameter drift. Finally, we consider semilinear impulsive periodic problems: ⎧ q c ⎪ ⎨ D,t x(t) = Ax(t) + f (t, x(t)), t ∈ J, t = tk , x() = x(T), ⎪ ⎩ + x(tk ) = x(tk– ) + Ik (x(tk– )), k = , , . . . , δ, () where f : J × X → X is continuous and Ik : X → X is continuous. The rest of this paper is organized as follows. In Section , the existence and boundedness of the operator B = [I – T (T)]– are given. In Section , the existence and boundedness of PC-mild solutions and the design parameter drift for such a periodic motion are presented. In Section , existence results of PC-mild solutions for impulsive periodic problems are showed. Finally, an example is presented to illustrate the theory. 2 Existence and boundedness of operator B = [I – T (T)]–1 Suppose T > , let J = [, T]. Let M = supt≥ S(t) < ∞. We denote by C(J, X) the Banach space of all continuous functions from J into X with the norm xC = sup{x(t) : t ∈ J}. We also introduce the set of functions PC(J, X) = {x : J → X : x is continuous at t ∈ J \ {t , t , . . . , tδ }, and x is continuous from the left and has right-hand limits at t ∈ {t , t , . . . , tδ }}. Endowed with the norm      xPC = max supx(t + ), supx(t – ) , t∈J t∈J it is easy to see (PC(J, X),  · PC ) is a Banach space.  For measurable functions l : J → R, define the norm lLp (J,R) = ( J |l(t)|p dt) p ,  ≤ p < ∞. We denote by Lp (J, R) the Banach space of all Lebesgue measurable functions l with lLp (J,R) < ∞. Definition . ([]) The fractional integral of order γ with the lower limit zero for a function f is defined as γ I,t f (t) =  (γ ) t f (s) ds, (t – s)–γ  t > , γ > , provided the right side is point-wise defined on [, ∞), where (·) is the gamma function. Definition . ([]) The Riemann-Liouville derivative of order γ with the lower limit zero for a function f : [, ∞) → R can be written as L γ D,t f (t) = dn  (n – γ ) dt n t f (s) ds, (t – s)γ +–n  t > , n –  < γ < n. Yu and Wang Boundary Value Problems 2014, 2014:35 http://www.boundaryvalueproblems.com/content/2014/1/35 Page 3 of 11 Definition . ([]) The Caputo derivative of order γ for a function f : [, ∞) → R can be written as n– c γ γ D,t f (t) =L D,t f (t) – k= t k (k) f () , k! t > , n –  < γ < n. Remark . If f is an abstract function with values in X, then the integrals which appear in Definitions . and . are taken in Bochner’s sense. As in our previous work [], by a PC-mild solution of () we mean the function x ∈ PC(J, X) satisfying k x(t) = T (t)x() + T (t – ti )yi i= t (t – s)q– S(t – s)f (s) ds, + t ∈ (ti , ti+ ], k = , , . . . , δ,  and x() = x(T), where ∞ T (t) =   ξq (θ)S t q θ dθ , ∞ S(t) = q    θ ξq (θ )S t q θ dθ  and     ξq (θ) = θ –– q ϑq θ – q ≥ , q ∞ ϑq (θ) = (nq + )  sin(nπq), (–)n– θ –qn– π n= n! θ ∈ (, ∞), here ξq is a probability density function defined on (, ∞), that is, ξq (θ) ≥ , ∞ θ ∈ (, ∞) and ξq (θ ) dθ = .  Lemma . (see Lemma . []) The operator T has the following properties: (i) For any fixed t ≥ , T (t) and S(t) are linear and bounded operators, i.e., for any M x ∈ X, T (t)x ≤ Mx and S(t)x ≤ (q) x. (ii) Both {S(t), t ≥ } and {T (t), t ≥ } are strongly continuous. (iii) For every t > , T (t) and S(t) are compact op (...truncated)