Analysis of Caputo impulsive fractional order differential equations with applications

We use Sadavoskii's fixed point method to investigate the existence and uniqueness of solutions of Caputo impulsive fractional differential equations of order \alpha between 0 and 1 with one example of impulsive logistic model and few other examples as well. We also discuss Caputo impulsive fractional differential equations with finite delay. The results proven are new and complement the existing one.


Introduction
Dynamics of many evolutionary processes from various field as population dynamics, control theory, physics, biology, medicine etc. undergo abrupt changes at certain moment of times like earthquake, harvesting, shock etc. These perturbations can be well-approximated as instantaneous change of state or impulses. These processes are modelled by impulsive differential equations. In 1960, Milman and Myshkis introduced impulsive differential equations in their paper [22]. Based on their work, several monographs have been published by many authors like Semoilenko and Perestyuk [28], Lakshmikantham et. al. [18], Bainov and Semoinov [4,5], Bainov and Covachev [6] and Benchohra et. al. [7]. All the authors mentioned above have considered impulsive differential equations as ordinary differential equations coupled with impulsive effects. They considered the impulsive effects as difference equations being satisfied at impulse times. So, the solutions are piecewise continuous with discontinuities at impulse times. In the field like biology, population dynamics, etc., problems with hereditary are best modelled by delay differential equations [15]. Problems associated with impulsive effects and hereditary property are modelled by impulsive delay differential equations.
The origin of fractional calculus (derivatives d α dt α f and integrals I α f of arbitrary order α > 0) goes back to Newton and Leibniz in the 17th century. In a letter correspondence with Leibniz, l'Hospital asked "What if the order of the derivative is 1 2 "? Leibniz replied, "Thus it follows that will be equal to x √ dx : x, an apparent paradox, from which one day useful consequences will be drawn." This letter of Leibniz was in September, 1695. So 1695 is considered as the birthday of fractional calculus. Fractional order differential equations are generalizations of classical integer order differential equations and are increasingly used to model problems in fluid dynamics, finance and other areas of application. Recent investigations have shown that often times physical systems can be modelled more accurately using fractional derivative formulations [20]. There are several excellent monographs available on this field [10,17,21,23,25,27]. In [17], the authors give a recent and up-to-date description of the developments of fractional differential and fractional integro-differential equations including applications. The existence and uniqueness of solutions to fractional differential equations has been considered by many authors [1,2,3,14,16,19]. Impulsive fractional differential equations represent a real framework for mathematical modelling to real world problems. Significant progress has been made in the theory of impulsive fractional differential equations [7,8,9,11]. Xu et. al. in his paper [29] has described an impulsive delay fishing model. Fractional derivatives arise naturally in mathematical problems, for α > 0 and a function f : provided the right hand sides exist point-wise on [0, T ] (Γ denotes the gamma function). Using the Riemann-Liouville fractional integral [10, Definition , moreover I α 0 f (0) = 0. One can see that the both the fractional derivatives are actually non-local operator because integral is a non-local operator. Moreover, calculating time fractional derivative of a function at some time requires all the past history and hence fractional derivatives can be used for modelling systems with memory. Fractional differential equations can be formulated using both Caputo or Riemann-Liouville fractional derivatives. A Riemann-Liouville initial value problem can be stated as follows [17,Theorem 3.24]. For a physical interpretation of the initial conditions in (1.1) see [12,13,24]. If derivatives of Caputo type are used instead of Riemann-Liouville type then initial conditions for the corresponding Caputo fractional differential equations can be formulated as for classical ordinary differential equations, namely x(0) = x 0 .
Our main objective is to discuss existence and uniqueness of solutions of the following impulsive fractional differential equation of Caputo type in a Banach space X with norm |.|, We break our function f into two components which satisfy different conditions. We observed that these kind of functions occurs in ecological modelling. We have given the example of logistic equation in the last section. Our main tool is Sadovskii's fixed point theorem. Now, we define some important spaces and norm which will encounter frequently: with sup-norm . , defined by x = sup{|x(t)| : t ∈ I}.
and hence Φ is a condensing map.
The structure of the paper is as follows. In section 2, we prove existence (Theorem 2.2) and uniqueness of solutions to (1.2). We show in section 3, the existence and uniqueness of solutions for a general class of impulsive functional differential equations of fractional order α ∈ (0, 1). In the last section 4, we give some examples in favor of our sufficient conditions.
2 Impulsive fractional differential equation x ∈ B(0, r)} for some fixed T > 0, r > 0, and assume that there exist p ∈ (0, α), α ∈ (0, 1), M 1 , M 2 , L 1 ∈ L 1/p ([0, T ], R + ) and functions f 1 , f 2 ∈ C(R, X) such that f = f 1 + f 2 and the following assumptions are satisfied is equivalent to the non-linear integral equation In other words, every solution of the integral equation (2.1) is also solution of our original initial value problem (1.2) and conversely.
Proof. Let B λ be the closed bounded and convex subset of Let us consider Step:2 F 1 is continuous and γ− contraction.
To prove the continuity of F 1 for t ∈ [0, T ], let us consider a sequence x n converging to x. Taking the norm of From the above analysis we obtain To prove that F 1 is γ 1 − a contraction, let us consider for x, y ∈ B r , Thus for Step:3 F 2 is compact. For 0 ≤ τ 1 ≤ τ 2 ≤ T , we have The right-hand side of the above expression does not depend on x. Thus using Arzel-Ascoli theorem for equi-continuous functions (Diethelm, Theorem D.10 [10]), we conclude that F 2 (B r ) is relatively compact and hence F 2 is completely continuous on I − {t 1 , t 2 · · · t m }. In similar way it can be prove the equi-continuity of F on t = t − k and t = t + k , k = 1, 2, · · · , m. And thus F 2 is compact on [0, T ].
Step:4 F is condensing. As F = F 1 + F 2 , F 1 is continuous, contraction and F 2 is compact, so using the Lemma 1.5, F is condensing map on B r . And hence using the Theorem 1.4 we conclude that (1.2) has a solution in B r . Theorem 2.3. If f is bounded and Lipschitz, in particular, |f (t, x)−f (t, y)| ≤ L * 1 (t)|x−y| for all (t, x), (t, y) ∈ R and L * 1 ∈ L 1/p ([0, T ], R + ),then the problem (1.2) has unique solution in B λ , provided that 3 Impulsive fractional differential equations with finite delay In this section, we discuss existence and uniqueness of solutions of the following impulsive fractional differential equations of Caputo type with finite delay in a Banach space X with norm |.| with sup-norm . , defined by x = sup{|x(t)| : t ∈ [−r, T ]}. Consider the initial value problem (3.1) on I × C for some fixed T > 0, and assume that there exist p ∈ (0, α), α ∈ (0, 1), M 3 , M 4 , L 2 ∈ L 1/p ([0, T ], R + ) such that the following assumptions are satisfied
In other words, every solution of the integral equation ( (3.2)) is also solution of our original initial value problem (3.1) and conversely.
Proof. In this case we define the operator F : P C(I, X) → P C(I, X) by Step:1 To prove that F is self mapping, we need to prove that for each x ∈ P C(I, X), F x ∈ P C(I, X). We can see that the proof is similar to the proof of continuity of F 1 in Step:2 of the Theorem 2.2 and hence we omit it.
Step:2 F is continuous and γ 2 − contraction. The proof of this step is also similar to the proof of continuous and γ 1 − contraction of F 1 in Step:2 of the Theorem 2.2. Now by applying Banach's fixed point theorem, we get that the operator F has an unique fixed point in P C(I, X) and hence the problem (3.1) has an unique solution in P C([−r, T ], X).
Our next result is based on Schaefer's fixed point theorem. In this case we replace assumption A.3 with the followings linear growth condition: Proof. We transform the problem into a fixed point problem. For this purpose, consider the operator F : P C(I, X) → P C(I, X) defined by Step:1 F is continuous. Let {x n } be a sequence such that x n → x in P C(I, X). Then for every t ∈ I, we have We can see that if n → ∞, F x n → F x, as x n → x and I k and f both are continuous. Hence F is continuous.
Step:2 F maps bounded sets into bounded sets in P C(I, X).
It is enough to show that for any δ > 0, there exists a l > 0 such that Step:3 F maps bounded sets into equi-continuous sets in P C(I, X). The proof of this step is similar to the proof of compactness of F 2 in Step:3 of the Theorem (2.2). As a consequence of steps 1-3 together with PC-type Arzela-Ascoli theorem (Fecken et al, Theorem 2.11 [11]) the map F : P C(I, X) → P C(I, X) is completely continuous.
This shows that E(F ) is bounded. As a consequence of Schaefer's fixed point theorem, the problem (3.1) has at least one solution in P C([−r, T ], X).
Theorem 3.6. If f is bounded and Lipschitz, in particular, |f (t, φ)−f (t, ψ)| ≤ L * 2 (t) φ − ψ , for all (t, φ), (t, ψ) ∈ I × C and L * 2 ∈ L 1/p ([0, T ], R + ), then problem (3.1) has an unique solution in P C([−r, T ], X), provided that Further, we consider the following more general Caputo fractional differential equation where f : I × X × C → X and C = C([−r, 0], X), I k ∈ C(X, X), (k = 1, 2, ..., m) and X is a separable real Banach space with the norm |.|. Here, our tools will be Banach and Schaefer fixed point theorems. Consider the initial value problem (3.3) on I × C × X for some fixed T > 0 and assume that there exist p ∈ (0, α), M 5 , M 6 , L 3 , L 4 ∈ L 1/p ([0, T ], R + ) such that the following assumptions are satisfied: is equivalent to the following non-linear integral equation In other words, every solution of the integral equation  has an unique solution, provided that Proof. The proof is similar to the proof of Theorem 3.4. Proof. The proof is similar to the proof of Theorem 3.5.
Theorem 3.11. If f is bounded and Lipschitz, in particular, , then the problem (3.3) has an unique solution in P C([−r, T ], X), provided that

Examples
Example 4.1. Fractional impulsive logistic equation: Consider the following class of fractional logistic equations where a(t) ∈ [a * , a * ] and b(t) ∈ [b * , b * ] with a * , b * > 0. We give some more examples which are inspired by [11].