Existence of solution to a periodic boundary value problem for a nonlinear impulsive fractional differential equation

We study the existence of solution to a periodic boundary value problem for nonlinear impulsive fractional differential equations by using Schaeffer’s fixed point theorem.


Introduction
In this work, we consider the following periodic boundary value problem for a nonlinear impulsive fractional differential equation where 0 < δ < 1, 0 = t 0 < t 1 < t 2 < • • • < t p < t p+1 = 1, D δ t k + represent the standard Riemann-Liouville fractional derivatives, I k ∈ C(R, R), k = 1, . . ., p, λ ∈ R, λ = 0, f is continuous at every point (t, u) ∈ (t k , t k+1 ] × R, k = 0, . . ., p, and satisfies the following restrictions concerning its behavior on the limit at t = t 0 and the impulse instants: for every k = 0, . . ., p 2 M. Belmekki, J. J. Nieto and R. Rodríguez-López and every function v ∈ C(t k , t k+1 ] such that the limit lim t→t + k v(t) exists and is finite, then there exists the (finite) limit lim Note that condition (1.2) is equivalent to since lim t→t + k (t − t k ) 1−δ u(t k ) = 0. Thus the limit in (1.2) exists if and only if the limit in (1.4) exists and the value is the same.
The theory of impulsive differential equations has been emerging as an important area of investigations in recent years.For some general aspects of impulsive differential equations, see the classical monographs [14,21], and Chapter 15 of [18].From a mathematical point of view, the reader can see, for instance, [6,8].Differential equations involving impulsive effects occur in many applications: control theory [2,9,10], population dynamics [17], or chemotherapeutic treatment in medicine [13].
Fractional order models are, in some cases, more accurate than integer-order models, i.e., there are more degrees of freedom in the fractional order models.Furthermore, fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes due to the existence of a 'memory' term in a model.This memory term insures the history and its impact to the present and future.For more details, see [15].
Recently Belmekki et al. [5] investigated the existence and uniqueness of solution to the (nonimpulsive) problem D δ u(t) − λu(t) = f (t, u(t)), t ∈ J := (0, 1], 0 < δ < 1, lim t→0 + t 1−δ u(t) = u(1), by using the fixed point theorem of Schaeffer and the Banach contraction principle.In [23], the authors consider a different impulsive problem and try to obtain existence and uniqueness results by using the Banach contraction principle.For δ = 1, we refer the reader to the paper by Nieto et al. [16].We cite [1,7,22] for some considerations on the concept and existence of solutions to fractional differential equations with impulses.The purpose of this paper is to study the existence of solution to the problem (1.1)-(1.3)by using Schaeffer's fixed point theorem.The results obtained extend in some sense those in [5,16] and allow some conclusions about the problem studied in [23].

Preliminary results
In this section, we introduce the notations, definitions, and preliminary facts which are used throughout this paper.
Let C(J) be the Banach space of all continuous real functions defined on J with the norm f := sup{| f (t)| : t ∈ J}.
Periodic BVP for nonlinear impulsive fractional DE 3 We also introduce the space PC r [a, b] for a general interval [a, b], a sequence a = t 0 < t 1 < t 2 < • • • < t p < t p+1 = b and a constant 0 < r < 1, as follows: k = 1, . . ., p, and there exists lim t→t + k (t − t k ) r f (t), k = 1, . . ., p , which obviously coincides with the set of functions f : The definition of the space clearly depends on the sequence {t k }, but we omit it in the notation for simplicity.Note also that f (t turns out to be a Banach space when it is endowed with the norm [4,14].Definition 2.1.([19, 20]).The Riemann-Liouville fractional primitive of order δ > 0 of a function f : (0, 1] → R is given by provided that the right-hand side is pointwise defined on (0, 1].Here, Γ is the classical Gamma function.
We have has as solution u(t) = ct δ−1 , where c is a real constant.
From this lemma, we deduce the following law of composition.
for any c ∈ R.
In this work, we also need the following concepts and properties of fractional primitives and derivatives.Definition 2.5.([12, 20]).The Riemann-Liouville fractional primitive of order δ > 0 of a function f : (0, 1] → R, I δ a+ f , where 0 ≤ a < 1, is given by provided that the right-hand side is pointwise defined on (a, 1].Definition 2.6.( [12,20]).The Riemann-Liouville fractional derivative of order 0 provided that the right-hand side is pointwise defined on (a, 1]. An issue which is interesting to our study is the behavior of the fractional primitives and derivatives over polynomials, deduced from the following properties.Proposition 2.7.([12, 20]).If δ ≥ 0 and β > 0, then In particular, the fractional derivative of a constant function is not zero: Concerning the impulsive problem of interest, the authors of [23] study the existence of solution to the problem where D δ is the standard Riemann-Liouville fractional derivative, f is continuous at every point (t, u) for a certain Green's function G λ,δ , and derive sufficient conditions for the existence of a unique solution.Although the approach of using integral formulations is important to the solvability of impulsive problems for fractional differential equations, the difficulty is that the solution to (2.1)-(2.3) is expected to be in the space PC 1−δ ([0, 1]) and, hence, the use of the fractional derivative of u, D δ 0 u, is combined with the possible existence of an 'infinite' jump of u at a point located inside the interval of interest.This produces that, for x ∈ PC 1−δ ([0, 1]), the function s → f (s, x(s)) is not necessarily continuous on (0, 1], due to the assumptions on the nonlinearity f . In this paper, we propose a new formulation for the impulsive problem for fractional differential equations of Riemann-Liouville type, in terms of problem (1.1)-(1.3)and study, through a different procedure, the existence of solution to this new problem.
We remark that the assumptions imposed in this paper on function f , namely the continuity of f on (t k , t k+1 ] × R, k = 0, . . ., p, and hypothesis (H) for every k = 0, . . ., p and v ∈ C(t k , t k+1 ] such that the limit lim t→t + k v(t) exists and is finite, then there exists the (finite) limit lim t→t guarantee the validity of the following property: Indeed, for a fixed u ∈ PC 1−δ [0, 1], the function t 1−δ u(t) is continuous on (0, t 1 ], so that u(t) is also continuous on (0, t 1 ], thus the continuity of f (t, u(t)) on (0, t 1 ] follows.On the other hand, for k = 1, . . ., p, (t − t k ) 1−δ u(t) is continuous on (t k , t k+1 ], hence u(t) and f (t, u(t)) are continuous on (t k , t k+1 ], by the continuity properties on f .Besides, for k = 0, . . ., p, the limit exists and it is finite, due to the hypotheses on f and the finiteness of the limit It is obvious that these restrictions on f are fulfilled for the nonlinearity in Example 4.1 [23].

Problem with a single impulse point
For simplicity, we focus our attention on the study of the problem where 0 represent the standard Riemann-Liouville fractional derivatives, I ∈ C(R, R), λ ∈ R, λ = 0 and f is continuous at every point (t, u) ∈ (t k , t k+1 ] × R, k = 0, 1, and satisfying the restriction (H) concerning its behavior on the limit at the instants t = 0 and t = t 1 , that is: • for every function v ∈ C(0, t 1 ] such that the limit lim t→0 + v(t) exists and it is finite, then there exists the (finite) limit lim t→0 + f (t, t δ−1 v(t)); and • for every function v ∈ C(t 1 , 1] such that the limit lim t→t + 1 v(t) exists and it is finite, then there exists the (finite) limit lim t→t The space of solutions will be the set where it is left-continuous and has finite right-hand limit.

Some existence and characterization results
The following lemma is useful for the study of the solutions to (3.1)- (3.4).
is given, for t ∈ (a, b], by Proof.Similar to the results in [5].Obviously, these considerations provide the existence of solution to problem (3.5)- (3.6).Concerning the uniqueness of solution to (3.5)-(3.6),we refer to the results in [12], where the following general nonlinear fractional differential equation is considered (D δ a + y)(x) = f (x, y(x)), (δ) > 0, x > a, which admits the particular case f : (a, b] × R −→ R, f (x, y) = λy + σ(x).In this reference [12], the equivalence between the Cauchy type problem for the above-mentioned nonlinear differential equation and a Volterra integral equation is proved (see Theorems 3.1 and 3.10 [12] and also Theorems 1 and 2 [11]).This equivalence is used to prove the uniqueness of solution to the Cauchy problem by adding the Lipschitz type condition (see (3.2.15) [12]) where A > 0 and G is an open set in R, condition which is trivially fulfilled by f (x, y) = λy + σ(x).We refer to Theorems 3.3 and 3.11 [12] for these existence and uniqueness results.On the other hand, in [12,Section 3.3.3],the weighted Cauchy problem is considered for the case 0 < δ < 1, proving the existence and uniqueness of solution to the weighted Cauchy problem accordingly by using the Lipschitz condition (see [12,Theorem 3.12]).
We also mention the monograph [19], where the fractional Green's function for a differential equation with fractional order and constant coefficients is obtained, getting an expression close to the solution to the homogeneous linear differential equation studied in [5] (see [5,Eq. (3.16) and, for t ∈ (t 1 , 1], by Proof.Obvious from Lemma 3.1. Next, we consider the existence of solution to problem (3.1)-(3.4),for a function f which is independent of the second variable, that is, f (t, u) = σ(t), as follows: where, for the rest of the paper, σ ∈ PC[0, 1] is piecewise continuous on [0, 1], and thus, allowing perhaps finite jump discontinuities at the impulse instants, in this case t 1 .
has a unique solution u(t) given by M. Belmekki, J. J. Nieto and R. Rodríguez-López Proof.From the study in [5] (also Lemma 3.1 or Lemma 3.2), the solution to (3.14) joint to condition (3.18) is given by The integral characterization of the solution to the impulsive equation subject to an 'initial condition' given in Lemma 3.3 allows to obtain some conclusions for the periodic boundary value problem (3.14)- (3.17).In this sense, taking an appropriate 'initial value' c 0 for its replacement in expression (3.19), we can derive some immediate consequences concerning existence and uniqueness results for problem (3.14)-(3.17).The idea is to find which are the adequate numbers c 0 ∈ R for which the solution to problem (3.14)-(3.16)subject to the initial condition lim t→0 + t 1−δ u(t) = c 0 satisfies that u(1) = c 0 .These appropriate choices for c 0 are those which would make true the periodic boundary condition (3.17) and the corresponding solution can also be calculated by using (3.19).Note that, in the previous lemma, φ(c 0 ) coincides with u(1) for the solution u in (3.19).This way, the fixed points of φ are those 'initial conditions' for which u(1) = c 0 .This way, to solve the periodic boundary value problem, we just write u(1) as a function of c 0 , which is possible by using the composition of several functions, u(1) = ψ(I(ϕ(c 0 ))), being ψ, ϕ linear and I the impulse function.Proposition 3.5.If the impulse function I is linear, I(x) = µx, for some µ ∈ R, and

Lemma 3.4. Consider the function φ defined by
Proof.It is deduced from the identity φ(c 0 ) = c 0 , where φ is given in Lemma 3.4, that is, Periodic BVP for nonlinear impulsive fractional DE 9 which is equivalent to and, under the hypotheses imposed, the solution to the boundary value problem is uniquely determined.
However, if K = 0, then the boundary value problem is solvable if and only if the right-hand side in the previous expression is null, obtaining an infinite number of solutions corresponding to any value of c 0 ∈ R.This is a problem at resonance and will be considered in the future.
Remark 3.6.The case µ = 1 (I(x) = x, for every x ∈ R) corresponds, in the ordinary case δ = 1, to a nonimpulsive problem u(t 1 +) = u(t 1 ).The peculiarities of fractional differential equations force the non-continuous behavior of the solution at t = t 1 , even for µ = 1, since Remark 3.7.For I nonlinear, problem (3.14)-(3.17) is also nonlinear and, to deduce the existence of solution, we prove the existence of fixed points for function φ defined in Lemma 3.4 without obtaining their explicit expression.Proof.For b 0 , c 0 ∈ R, we get, from the definitions of φ, R in Lemma 3.4, Proof.Note that φ in Lemma 3.4 is a continuous mapping.Let m > 0 be such that Then, the restriction of φ to the nonempty compact and convex set In consequence, by Schauder's theorem, there exists a fixed point c 0 of φ in [−A, A], which gives a solution to (3.14)-(3.17)through (3.19).

Lemma 3.10. If I is continuous and there exists
However, the approach followed previously is not useful for equations with a general righthand side depending on u, since it is not possible to avoid this dependence in the definition of the mapping φ.
For a different approach to the problem which will allow to deal with a nonlinearity f , we first consider the periodic boundary value problem whose solution is given by where where G λ,δ (t, s) is defined, for (t, s) ∈ (0 (3.30) Proof.It is deduced from (3.27) and (3.28), taking into account that so that they are the fixed points of the mapping B defined as where G λ,δ is given by (3.30).
Remark 3.14.The mapping B has an expression similar to the operator A defined in equation (3.2) [23], but the Green's function G λ,δ is different.
Remark 3.15.From (3.27) and (3.28), it is clear that the expression of the mapping B can be expanded as [Bu](t

Analysis of the nonlinear problem
In this section, we shall be concerned with the existence and uniqueness of solution to the nonlinear impulsive boundary value problem (3.1)-(3.4).To this end, we use the following fixed point theorem of Schaeffer.
Theorem 3.16.Assume X to be a normed linear space, and let the operator F : X → X be compact.Then either i) the operator F has a fixed point in X, or ii) the set E = {u ∈ X : u = µF(u), µ ∈ (0, 1)} is unbounded.
We define the operator B :

.34)
(H2) There exist positive constants k and l such that Then the operator B defined in Lemma 3.13 is well-defined, continuous and compact.
Proof.(a) First, we prove that the mapping B is well-defined, that is, Bu ∈ PC 1−δ [0, 1], for every u ∈ PC 1−δ [0, 1].We take u ∈ PC 1−δ [0, 1] and prove that t 1−δ B(u ] and the existence of the limit Indeed, for any 0 < τ 1 < τ 2 ≤ t 1 , we have which is derived from (H1) and the inequality It is obvious that the first term on the right-hand side of the previous inequality tends to zero as |τ 1 − τ 2 | → 0. From the calculations in section (a) of Lemma 4.1 proof in [5], the second term tends to zero as |τ 1 − τ 2 | → 0 and, similarly to the calculations in (a) (proof of Lemma 4.1 in [5]) and those in [23], the two other terms also tend to zero as tends to 0 as |τ 1 − τ 2 | → 0. Indeed, for t 1 < τ 1 < τ 2 ≤ 1, from (H1) and following the calculations in Lemma 3.1 in [23], we get At this point, the calculations differ from [23], since the Green's function is defined in (3.30), then Again, the first term in the right-hand side of the previous inequality tends to zero as |τ 1 − τ 2 | → 0 and analogously with the last term due to (3.37).Finally, the integral term (multiplying M) is bounded by It is obvious that (τ 1 − t 1 ) 1−δ − (τ 2 − t 1 ) 1−δ tends to zero as |τ 1 − τ 2 | → 0 and, similarly to (3.37), On the other hand, as justified in [23], estimate which is of the type of the last term in equation (4.8) [5] and, in consequence, it has limit zero as |τ 1 − τ 2 | → 0 due to inequalities (4.13) and (4.14) in [5] (analogously to the reasoning in [23]).
Let u be any element of E , then u = µB(u) for some µ ∈ (0, 1).Thus, for each t ∈ (0, t 1 ], we have by (3.42) and, using (3.43), we get, for t ∈ (t 1 , 1], and the set E is bounded independently of µ ∈ (0, 1).Using Lemma 3.17 and Theorem 3.16, we obtain that the operator B has at least one fixed point.
Proof.We prove, by induction, that for b 0 , c 0 ∈ R and j = 1, . . ., p, Indeed, take b 0 , c 0 ∈ R. For j = 1, we have which coincides with (4.9) for j = 1.Next, we suppose that (4.9) is true for j − 1, that is, and we deduce that Taking into account that t p+1 = 1, we write Thus, the restriction of φ to the nonempty compact and convex set The result is a consequence of Schauder's theorem.
Lemma 4.9.The solutions to problem (1.1)-(1.3)are the fixed points of the mapping A defined as where G λ,δ is given by (4.16).
We remark that the operator A is an extension of (3.31).Due to the similarities of the expression of the mapping A with respect to the case p = 1 and analogously to the procedure followed in Lemma 3.17 and Theorem 3.18, it is possible to deduce the continuity and compactness of the mapping A and the existence of at least one solution to problem (1.1)-(1.3) in the space PC 1−δ [0, 1], just by imposing the boundedness and the Lipschitzian character of the function f and the impulse functions I 1 , . . ., I p and, of course, assuming (H).Then the operator A defined in Lemma 4.9 is well-defined, continuous and compact.
Proof.The expression of the mapping A can be written as [Au](t) = I p (u(t p ))(Γ(δ)) Comparing this with the expression of the mapping B given in Remark 3.15, it is clear that the proof can be completed similarly to the proof of Lemma 3.17.If u = µA(u) for some µ ∈ (0, 1), then, for each t ∈ (0, ds .Next, the solution to the equation (3.15)-(3.16) is obtained by applying Lemma 3.1 (or Lemma 3.2).
then the periodic boundary value problem (3.14)-(3.17)has a unique solution given by(3.19),where

. 21 )
then problem (3.14)-(3.17)has a unique solution given by (3.19), for c 0 ∈ R the unique fixed point of the mapping φ defined in Lemma 3.4.
and the conclusion follows.Lemma 3.9.If I is continuous and bounded, then problem (3.14)-(3.17)has at least one solution.
given in Lemma 3.4), then problem (3.14)-(3.17)has at least one solution.Proof.Note that, from the continuity of I, R([−B, B]) is a compact set in R, for every B ∈ R. To deal with a problem where the nonlinearity depends on the second variable, we could extend Lemma 3.3 to the context of the more general type of right-hand side f (t, u(t)) in the equation, as follows: Lemma 3.11.Solutions to problem (3.1)-(3.3)joint to the condition (3.18) are the solutions of the integral equation

1 ]
by expression (3.31) in such a way that problem (3.1)-(3.4)has solutions if and only if the operator equation Bu = u has fixed points.Lemma 3.17.Suppose that the following conditions hold: (H1) There exist positive constants M and m such that

Lemma 4 . 3 .
The solutions to the problem (4.4)-(4.5)subject to the boundary condition (3.17) are given by (4.7), for c 0 ∈ R a fixed point of the mapping φ