Mild solutions of Riemann–Liouville fractional differential equations with fractional impulses

. We consider Riemann–Liouville fractional differential equations with fractional-order derivative in the impulsive conditions. We study the existence of the mild solution by applying the Laplace transform method and ( a, k ) -regularized resolvent operator. We use the contraction mapping principle and ﬁxed point theorem for condensing map to prove our existence results.


Introduction
In recent years, fractional differential equations have been considered to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Since the definition of fractional derivative exhibits the past history of the functions, fractional-order derivative has been found as an excellent mathematical tool for characterizing memory and hereditary properties of complex systems such as viscoelastic deformation, anomalous diffusion, signal processing, stock market, and so on. This is the main advantage of using fractional-order derivatives to real world problems in comparison with integer-order derivatives. For more detailed work, see [5,7,17,21,23] and the references therein.
Impulsive differential equations have more consideration because of its extensive applications in physics, biology, medical, engineering, and technology [1,2,11]. Impulsive differential equations are suitable model for describing processes, which change their state abruptly at a certain moment. In most of the study on fractional impulsive differential equations, the Caputo derivative is used to define the system, and integerorder derivative is used in the impulsive conditions [4,6,18,20,22]. In [9], Podulbny shown that it is possible to attribute physical meaning to initial conditions defined in the form of Riemann-Liouville fractional derivatives in the field of viscoelasticity. In [10], Kosmatov introduced the fractional-order derivative in the impulsive conditions to study the existence of solutions of fractional impulsive differential equations using both Caputo and the Riemann-Liouville derivatives.
In [4], Feckan et al. cited some papers in which the mild solutions are defined inappropriatly. But in [19], Wang et al. refuted Feckan's argument and shown that the example framed by Feckan in [4] was wrong. But we observed that Feckan's arguments are correct if the lower limit is taken as zero in equation (1.14) of [19]. The above dispute will not be arised if the lower bound is taken as same in the statement of the problem, definition of fractional derivative and the definition of solution. These comments are also justified by Feckan et al. in [3] and Liu et al. in [12].
In [8], Hernández et al. have shown that the definition of mild solutions is not appropriate in some recent papers. To make it more appropriate, he introduced the resolvent operator for integral equations to define the mild solutions in [8].
But in [14], Lizama et al. studied that not all the fractional equation can be formulated as an integral equation, so that the concept proposed in [8] fails in the general case. He also derived a suitable variation of constant formula for a large class of fractional differential equations using Laplace transform and with the help of (a, k)-regularized resolvent families. The advantage of this approach is that the domain of A is not necessarily dense in Z [13].
Here we derive the mild solution of the above problem (1)-(3) by using the Laplace transform method and (t α−1 /Γ(α), 1)-regularized resolvent operator. The existence results of Riemann-Liouville fractional differential equations with fractional-order impulsive conditions is established by means of a fixed point theorem for condensing map, and the uniqueness result is verified via contraction mapping principle.

Preliminaries
Let L(Z ) be the space of bounded linear operators from Banach space Z into Z with norm · L(Z ) . The domain of A is provided with the graph norm · D(A) = u + Au , and denote by B r (u, Z ) the closed ball with center at u and distance r in Z . The space We consider the space PC(Z ), which is formed by all the functions u : Here we use the (a, k)-regularized resolvent family, which was initialized in [13] and analyzed in some recent papers [14].
In [13,Prop. 3.1], Lizama established that the Laplace transform of strongly continuous familyR α (λ) in L(Z ) exists for λ > ω. It follows that the Laplace transform of see [14]. The Riemann-Liouville fractional integral of a function u ∈ L p (0, 1), 1 p < ∞, of order α > 0 is The fractional derivative of order α > 0 is defined in the Riemann-Liouville sense as In general, Riemann-Liouville derivative is a left inverse of the operator I α 0 + but not a right inverse. That is, we have D α We also recollect the formula for the Laplace transform of Riemann-Liouville derivative given byD Now we derive the mild solution of equations (1)-(3). First, we split u(·) as p(·) + q(·) in problem (1)- (3), where p is the continuous mild solution for and q is the PC-mild solution for . . , m, and any solution of (1)-(3) can be decomposed to the solutions of (4)- (5) and (6)-(8).
Applying Laplace transform and inverse Laplace transform technique to (4), the mild solution of (4)-(5) can be written as on I. Next, we rewrite (6)-(8) into the equivalent integral equation. For that, we operate fractional integral operator I α 0 + on each side of (6), which gives Since the impulsive condition (7) is a discontinuity condition, we have Then, for t ∈ (t 1 , t 2 ], we get So that From (7) we obtain Hence, Proceeding in this way, we can derive for the remaining interval I, and we get the integral equation as Taking Laplace transform on both sides, we get Taking the inverse Laplace transform on each side of the above equation, we get the mild solution of (6)-(8) as Finally, from (9) and (11) we can obtain the mild solution of problem (1)-(3) as is satisfied.

Existence and uniqueness results
We assume the hypotheses given below: (H1) f : I × R → R is continuous, and there exists a function Then there exists a unique mild solution of (1)-(3).
Proof. Define the fixed point operator T : https://www.mii.vu.lt/NA From the assumption we easily conclude that T is well defined. Now we prove that T is contraction.
Let u and v in PC(Z ) and t ∈ I, we get which shows that T is contraction and there exists a unique mild solution of (1)-(3).
The accompanying theorem provides the existence results by means of fixed point criterion for condensing map. Proof. Take r > 0 such that for all s r.
Here we prove that the operator T introduced in the previous Theorem 1 is a condensing map from B r (0, PC(Z )) into B r (0, PC(Z )).
We first show that T has values in B r (0, PC(Z )), that is T B r (0, PC(Z )) ⊂ B r (0, PC(Z )).
For u ∈ B r (0, PC(Z )) and t ∈ I, we get which implies that T u(t) r and T B r (0, PC(Z )) ⊂ B r (0, PC(Z )). To continue the remainder of the proof, we introduce the decomposition operator T by T 1 + T 2 , where Step 1. The map T 1 is contraction on B r (0, PC(Z )). For u, v ∈ B r (0, PC(Z )), t ∈ I, from Theorem 1 it is easy to see that https://www.mii.vu.lt/NA Step 2. The map T 2 is completely continuous on B r (0, PC(Z )).
First, we show that T 2 is a continuous operator. Let the sequence (u n ) in B r (0, PC(Z )) and u ∈ B r (0, PC(Z )). Assume u n → u as n → ∞. Then we have Since f is continuous, so T 2 u n − T 2 u → 0 as n → ∞. From this we know T 2 is continuous.
Further, we prove that T 2 is a compact operator.
In continuation to this, we show that {T 2 u(t): u ∈ B r (0, PC(Z ))} is relatively compact in Z for every t ∈ I.
The operator A : D(A) ⊂ Z → Z is determined by Ax = x with domain D(A) := {x ∈ Z : x ∈ Z , x(0) = x(π) = 0}. The class of operator {T (t)} t 0 is an analytic semigroup in Z with the infinitesimal generator A. Here A has a discrete spectrum with eigenvalues −n 2 , n ∈ N, and the corresponding normalized eigenfunction is defined by u n (ζ) = (2/π) 1/2 sin(nζ). The value T (t)x = ∞ n=1 e −n 2 t x, u n u n for x ∈ Z , and the set {u n : n ∈ N} is an orthonormal basis for Z . From these expressions it follows that {T (t)} t 0 is a compact, so that (λ − A) −1 is a compact operator for all λ ∈ ρ(A).
If the subsequent condition is satisfied then, as per Theorem 2, problem (14)- (17) has at least one mild solution on B r (0, PC(Z )).