Basic Concepts of Riemann – Liouville Fractional Differential Equations with Non-Instantaneous Impulses

In this paper a nonlinear system of Riemann–Liouville (RL) fractional differential equations with non-instantaneous impulses is studied. The presence of non-instantaneous impulses require appropriate definitions of impulsive conditions and initial conditions. In the paper several types of initial value problems are considered and their mild solutions are given via integral representations. In the linear case the equivalence of the solution and mild solutions is established. Conditions for existence and uniqueness of initial value problems are presented. Several examples are provided to illustrate the influence of impulsive functions and the interpretation of impulses in the RL fractional case.


Introduction
Fractional differential equations model nonlocal phenomena in time.One of the basic fractional derivatives is the Riemann-Liouville one which arises naturally in real world phenomena.For example, Heymans and Podlubny [1] provide several examples from the field of viscoelasticity.Several applications of fractional calculus to control theory, electrical circuits, fractional-order multipoles in electromagnetism, electrochemistry, and the neurons in biology are provided in [2][3][4][5].In particular, several applications of fractional derivatives in physics are given in the book [6].
Many physical phenomena have short-term perturbations at some points caused by external interventions during their evolution.Adequate models for this kind of phenomena are impulsive differential equations.Two types of impulses are popular in the literature: instantaneous impulses (whose duration is negligible small) and non-instantaneous impulses (these changes start impulsively and remain active on finite initially given time intervals).There are mainly two approaches for the interpretation of the solutions of impulsive fractional differential equations: one by keeping the lower bound of the fractional derivative at the fixed initial time and the other by switching the lower limit of the fractional derivative at the impulsive points.The statement of the problem depends significantly on the type of fractional derivative.Caputo fractional derivatives have some properties similar to ordinary derivatives (such as the derivative of a constant) which lead to similar initial value problems as well as similar impulsive conditions (instantaneous and non-instantaneous).In the literature many types of initial value problems and boundary value problems for Caputo fractional differential equations with instantaneous and non-instantaneous impulses are studied (see, for example, [7][8][9]).For Riemann-Liouville (RL) fractional differential equations with instantaneous impulses several results are obtained in [8,[10][11][12].However for the RL fractional derivative, the physical interpretation of the initial condition (see [1]) requires different impulsive conditions, so as a result the statement of the problem considered is crucial.To the best of our knowledge this is the first paper concerning Riemann-Liouville fractional differential equations with non-instantaneous impulses.
In this paper the basic ideas in introducing non-instantaneous impulses for RL fractional differential equations are presented.The appropriate definition of both the initial conditions and the impulsive conditions for RL fractional differential equations is an important starting point for their qualitative investigation.We set up and discuss several types of initial conditions and impulsive conditions which are deeply connected with the RL fractional derivative.We use both the RL integral and the weighted limit to present the initial condition and the impulsive conditions.We consider both approaches in the literature in the fractional case with the presence of impulses in the equations: when the lower bound of the fractional derivative is fixed at the initial time and when the lower bound of the fractional derivative is changed at any point of impulse.In all cases mild solutions are defined by appropriate Volterra integro-algebraic representations and some conditions for existence and uniqueness are given.Note since instantaneous impulses are a special case of non-instantaneous ones, a brief overview of impulsive conditions in the instantaneous case is given.

Some Preliminary Results From Fractional Calculus
In this paper we will use the following definitions for fractional derivatives and integrals: -Riemann-Liouville fractional integral of order q ∈ (0, 1) ([13]) where m ∈ L loc 1 ([t 0 , ∞), R) and Γ(.) is the Gamma function.This is called by some authors the left Riemann-Liouville fractional integral of order q.

Note sometimes the notation
).This is also called the left Riemann-Liouville fractional derivative.

(a)
If there exists a limit lim t→t 0 + [(t − t 0 ) q−1 m(t)] = c ∈ R, then there also exists a limit (b) If there exists a limit t 0 I 1−q t m(t)| t=t 0 = b ∈ R, and if the limit lim t→t 0 + [(t − t 0 ) 1−q m(t)] exists, then Let a < b ≤ ∞ be real numbers and consider the nonlinear RL fractional differential equation (RLFrDE) Note that according to [14] the initial conditions to (1) could be: the integral form (see (3.1.6)[14]) a weighted Cauchy type problem (see (3.1.7)[14]) the initial condition at the inner point of a finite interval (see (3.4.71) [14]) Remark 1.According to Lemma 1 if the function x(t) satisfies the initial condition (3), then, x(t) also satisfies the condition (2) with B = C Γ(q).Any of the above initial value problem (IVP)'s of RLFrDE (1) has an equivalent integral representation.
Then there exists a unique solution to the integral type initial value problem for RLFrDE (1), (2) in the space C 1−q [a, b].

Lemma 4. ([14]) (i)
If g ∈ C(t 0 , T], then for any point t ∈ (t 0 , T] (ii) If g ∈ C(t 0 , T] and I 1−q t 0 g(t) ∈ C(t 0 , T], then for any point t ∈ (t 0 , T] Remark 3. In the vector case, the fractional derivatives with the same fractional order are taken for all components.

Non-Instantaneous Impulses in RL Fractional Differential Equations
The appropriate definition of both the initial conditions and the impulsive conditions for RL fractional differential equations is an important starting point.We will discuss the initial value problem for nonlinear Riemann-Liouville fractional differential equations with non-instantaneous impulses (NIRLFrDE).We will set up in an appropriate way both the initial conditions and the impulsive conditions.
Note RL fractional functional differential equations with non-instantaneous impulses were studied in [15] but the impulsive conditions as well as the initial condition do not depend on the fractional order.
We consider the following sets: When impulses are involved in fractional differential equations there are two main approaches for interpretation of the solutions.We will consider both approaches.

Fixed Lower Bound of the RL Fractional Derivative at the Given Initial Time
For our considerations in this case we need the function f (t, x) to be defined for all As is mentioned in Section 2 there are two types of initial conditions to RL fractional differential equations: the integral form (2) and the weighted form (3). Following this idea we will define two different types of initial value problems for NIRLFrDE.

Integral Form of the Initial Value Problem
Consider the nonlinear non-instantaneous impulsive Riemann-Liouville fractional differential equation (NIRLFrDE) with impulsive conditions and the initial condition in integral form where The impulses in problem ( 8), ( 9) start abruptly at the points s k , k = 0, 1, 2, . . ., p − 1 and their action continues on the interval (s k , t k+1 ].The function x takes an impulse at s k , k = 0, 1, 2, . . ., p − 1 and it follows different rules in the two consecutive intervals (s k , t k+1 ] and (t k+1 , s k+1 ].At the point t k+1 , k = 0, 1, 2, . . ., p − 1, the function x is continuous. We will define a mild solution of the initial value problem ( 8), ( 9) by properly handling the RL fractional derivative and both the initial condition and the impulsive conditions.
Next we prove some auxiliary results. Then: (i) the function x satisfies the linear problem The function x is continuous at t i , i = 1, 2, . . ., p.
. ., p (the proof in the case t ∈ (t 0 , s 0 ] is obvious and we omit it).
According to Lemma 4i and Proposition 1 the derivative RL t 0 D q t x(t) exists.Apply the operator RL t 0 D q t to both sides of (11), use Proposition 1, Lemma 4i and we obtain Therefore the function x(t) satisfies the first equation in (12).
Applying the operator t 0 I 1−q t x(t) to the first equation of (11), and using Proposition 1 and the equality t 0 I 1−q t t 0 I q t = t 0 I 1 t (see p. 10, (1.10) [16]), we get Therefore, lim t→t 0 t 0 I 1−q t x(t) = x 0 and the function x(t) satisfies the initial condition in (12).(ii) Let k = 1, 2, . . ., p. Taking the limit in (11), we obtain lim T] and satisfies (12), then x satisfies (11).
Then according to Lemma 4ii we get i.e., the function x(t) satisfies the first equation of (11).
i.e., the last equality in ( 11) is satisfied.Similarly, we can prove that x satisfies (11) for the other subintervals.Now, we introduce the concept of a mild solution for IVP for NIRLFrDE ( 8)- (10).
Definition 2. A function x : [t 0 , T] → R n is called a mild solution of the IVP for NIRLFrDE ( 8)- (10) if it satisfies the following Volterra integral-algebraic equation Remark 5. Note in formula (16) the function f (t, x) has to be defined on the whole interval [t 0 , T].Now we will establish existence results for mild solutions to the integral form of IVP for NIRLFrDE ( 8)-( 10) on a finite interval.Theorem 1. (Existence and uniqueness).Let the following assumptions be satisfied: 1.

2.
For all k = 1, 2, . . ., p the functions φ k (t, x, y) ∈ C[s k−1 , t k ] for any x, y ∈ R n and the inequality ||φ k (t, The inequality K < 1 holds where Then there exists a unique mild solution to the integral form of IVP for NIRLFrDE (8)- (10) in the space PC 1−q [t 0 , T].
We will apply the Banach contraction principle.For any function x ∈ PC 1−q [t 0 , T] we define the operator From condition 1 it follows that the operator T is well defined.
Step 1.We prove that Tx(t) ∈ PC 1−q [t 0 , T] for x ∈ PC 1−q [t 0 , T].From (17) and lim t→t k +0 Tx(t) = lim t→t k −0 Tx(t) it follows that Tx(t) ∈ C(t 0 , T]/ ∪ {s k }.Let t ∈ (t 0 , s 0 ].Then from condition 1 we get Step 2. The operator T is a contraction in Next let t ∈ (s 0 , Continuing this procedure.For example let t ∈ (t k , s k ].Then we get sup where the inequalities T] which proves the Theorem.

Weighted Initial Value Problem
Consider the NIRLFrDE (8) with impulsive conditions (9) and the weighted Cauchy type condition lim t→t 0 Then the following auxiliary results hold.

Lemma 7. Let h
Then: (i) the function x satisfies the linear problem The function x is continuous at t i , i = 1, 2, . . ., p.
The proof follows from Lemma 5 and Remark 1 so we omit it.

Changed Lower Bounds of the RL Fractional Derivative at the Impulsive Points
Here our function f (t, x) is defined only for t ∈ ∪ p k=0 [t k , s k ] and x ∈ R n .In this case, we will consider the RL fractional derivative with a changeable lower bound at any point of jump, i.e., instead of RL t 0 D q t for all t > t 0 we consider RL t k D q t on (t k , s k ], k = 0, 1, . . . .It does not seem to be possible to consider the usual Cauchy conditions at the point t k to the RLFrDE RL t k D q t x(t) = f (t, x(t)), because the solution to this problem, in general, has a singularity at t k and therefore it is not bounded and continuous at the point t k (see Section 3.4.2[14]).There are two types of initial conditions to RL fractional differential equations: the integral form (2) and the weighted form (3). Following this idea we will define four different types of initial value problems for NIRLFrDE.

Integral form of the Initial Conditions and Impulses
Consider the IVP for NIRLFrDE with impulsive conditions in the form of RL integrals and initial conditions where Then: (i) the function x satisfies the linear problem The function x satisfies lim t→t k + t k I 1−q t x(t) = lim t→t k − x(t) for k = 1, 2, . . ., p.
Proof.(i).Let t ∈ (t k , s k ), k = 1, 2, . . ., p.According to Lemma 4i and Proposition 1 the derivative RL t k D q t x(t) exists.Applying the operator RL t k D q t to both sides of the last equality of (28), and useing Corollary 1 and Lemma 4i, we obtain Therefore the function x(t) satisfies the first equation in (29).A similar argument is needed for t ∈ (t 0 , s 0 ]. Let t ∈ (t k , s k ), k = 1, 2, . . ., p and apply the operator t k I 1−q t x(t) to both sides of the last equality of (28), use Corollary 1 and the equality t k I 1−q t t k I q t = t k I 1 t (see p. 10, (1.10) [16]), to get ) and the function x(t) satisfies the impulsive condition in (29).
Apply the operator t 0 I 1−q t x(t) to the first equation of (28), use Corollary 1 and the equality 1−q t t k I q t = t 0 I 1 t (see p. 10, (1.10) [16]) and we get Therefore, lim t→t 0 t 0 I 1−q t x(t) = x 0 and the function x(t) satisfies the initial condition in (29).(ii) Let k = 1, 2, . . ., p.According to (31) we obtain lim T] and satisfies (29), then x satisfies (28).
Proof.Let t ∈ (t 0 , s 0 ] and RL t 0 D q t x(t) = h(t).Then according to Lemma 4ii we get i.e., the function x(t) satisfies the first equation of (28).Let t ∈ (t 1 , s 1 ] and RL t 1 D q t x(t) = h(t).According to Lemma 2 and ( 6) i.e., the last equality in (28) is satisfied for k = 1.Similarly, we can prove that x satisfies (28) for the other subintervals.Now, we define a mild solution for IVP for NIRLFrDE (25)-( 27) following the idea with the presence of impulses in differential equations ( [17]) and Lemma 2 with We give conditions for the existence of a mild solution to the integral type IVP for NIRLFrDE (25)-( 27).
The function f (t, x) : Then there exists a unique mild solution to the integral form of IVP for NIRLFrDE (8)- (10) in the space P C 1−q [t 0 , T].
We will apply the Banach contraction principle.For any function x ∈ P C 1−q [t 0 , T] we define the operator From condition 1 it follows that the operator T is well defined.
Step 1.We prove that Tx(t) 2, . . ., p. Then from (34) and condition 1 we get From (34), condition 1, Corollary 1 and t k I 1−q t t k I q t = t k I 1 t we get Therefore, lim Step 2. The operator T is a contraction in P C 1−q [t 0 , T].Let x 1 , x 2 ∈ P C 1−q [t 0 , T].First let t ∈ (t 0 , s 0 ].Then similar to (18) we get sup where the equality is applied.Next let t ∈ (s 0 , t 1 ].As in (19) we get sup Continuing this procedure.For example let t ∈ (t k , s k ].Then from (34) we get sup where the inequality (36) with a = t k and are used.

Weighted Form of the Initial Conditions and Impulses
Consider the NIRLFrDE (25) with impulsive conditions in weighted form and initial conditions in the weighted form: Applying Lemmas 2, 9 and 10 and ( 6) we could define a mild solution of IVP for NIRLFrDE (25), (39), (40): Definition 5. A function x : [t 0 , T] → R n is called a mild solution of the IVP for NIRLFrDE (25), (39), (40) if it satisfies the following Volterra integral-algebraic equations: Note Theorem 2 for existence and uniqueness could be easily converted to the mild solution of the IVP for NIRLFrDE (25), (39), (40) given by (41).
The integral representation of the solution of instantaneous impulsive RL fractional derivative of order q ∈ (0, 1) is given in [8] in the case when the impulsive conditions are lim t→t i +0 (t − t i ) 1+β−q x(t) = x i , i = 1, 2, . . ., p, and the initial condition is RL 0 I 1−q t x(t)| t=0 = x 0 where β ∈ (0, q).

Definition 4 .
A function x : [t 0 , T] → R n is called a mild solution of the IVP for NIRLFrDE (25)-(27) if it satisfies the following Volterra integral-algebraic equations: