Study of impulsive fractional differential equation under Robin boundary conditions by topological degree method

This research work is dedicated to investigating a class of impulsive fractional order differential equations under the Robin boundary conditions via the application of topological degree theory (TDT). We establish some adequate results for the existence of at most one solution for the consider problem. Further, the whole analysis is illustrated by providing a pertinent example. We keep in mind that the conditions we develop by using TDT are much weaker than using ordinary fixed point theory. Hence TDT can be used as powerful tool for the theoretical analysis of many linear and nonlinear problems.


Introduction
In previous few decades, the area devoted to studying fractional calculus and derivatives and integrals of real or complex order has got proper attention. Fractional ordinary differential equations (FODEs) have numerous applications in the fields of physical science, biological sciences and engineering disciplines (for further details, see [1][2][3][4][5][6][7] and the references therein). The concerned area has the ability to model hereditary and memory process of many real world problems more comprehensively than classical derivatives and integral. Recently, some new differential operators of fractional order of nonsingular type have also attracted attention. Various researchers are working on it, it having also many applications in modeling real world problems. In this regard, some authors recently published very good work [8][9][10].
In real world problems, many phenomena face abrupt or sudden changes in their state of motion or rest. These abrupt changes are modeled by using impulsive differential equations. The area which addresses the aforementioned problems has been established very well in respect of ordinary derivatives and integrals. For the concerned investigations, researchers have used fixed point theory and nonlinear analysis tools. Further some results regarding numerical analysis corresponding to impulsive problems have also established [11,12]. On the other hand the investigation of impulsive problems under the concept of fractional calculus has also developed properly. The problem related to the moments of sudden changes like heartbeat, earthquake, temperature, shock, and so forth occurs in our daily life problems. All of these sudden changes are concerned with the impulsive differential equations, for these types of sudden changes the investigation of impulsive differential equations is an important tool for analysis. Large numbers of articles can be found in this regard [13][14][15][16][17].
An important aspect to be investigated of the physical problems is whether the problem physically exists or not. For the existence theory, fixed point theory has been utilized in the last many years. But one thing which restricted the application of fixed point theory is the requirement of strong conditions. Because when we deal with problems of integral or differential equations by fixed point theory, we first convert it to operator equations for which we search fixed points. To achieve this goal often we impose strong compact conditions on the concerned operator, which restricts the applicability of the area to a limited class of problems. Therefore it is necessary to search some more sophisticated methods and theories to relax the criteria. Therefore Mawhin [18] used TDT for the first time to deal with integral equations. Later on Isaia [19] used the same theory to investigate some integral equations. During 2012 Wang [20] investigated some classes of initial value problems of FODEs by using the TDT including the following problem of impulsive FODEs: where i = 1, 2, . . . , l, h is a nonlinear function and I i : R → R is a nonlinear map which is the size of the jump at t i . As from present literature, the mentioned method has been used to investigate the usual initial and "boundary value problems" (BVPs) of FODEs (for details, see [21][22][23]).
On the other hand BVPs have many applications in engineering disciplines; see [24,25]. The above-mentioned problems involve different kinds of boundary conditions like "Neumann boundary conditions and Dirichlet boundary conditions". When the aforesaid boundary conditions are mixed then for the problem the mixed boundary conditions are called Robin boundary conditions (RBCs) or impedance BVPs. The concerned problems have many applications in heat transfer phenomena and electromagnetic theory. Further problems under RBCs are increasingly used in solving Sturm-Liouville equations which are largely used in engineering disciplines (see [26,27]). Keeping in mind the mentioned literature, we investigate the following nonlinear problem of impulsive FODEs under RBCs for t ∈ [0, T]: , where j = 1, 2, . . . , l, 0 < γ ≤ 2 and the nonlinear function h is continuous and I j , J j are nonlinear maps which determine the size of the jump at t i , where 0 < t 1 < t 0 < t 2 < t 3 · · · < t l and ) are right and left limits, respectively, and D γ represents the Caputo derivative of various orders where 1 < γ ≤ 2. We use TDT to establish some adequate results that ensure the existence of a solution to the considered problem. A pertinent example is given to demonstrate the main contribution.

Background materials
Some basic notions and results [1,2] which we need for our analysis are given as follows.

Proposition 2.1 The following statements hold:
(1) ψ(H) = 0, iff H is relatively compact; (2) ψ is of semi-norm, that is, ψ(λH) = |λ|ψ(H) and ψ( Then F has at least one fixed point and the set of fixed points of F lies in B r (0).

Lemma 2.1
The FODE wih γ > 0, has a unique solution of the form Ψ (t) = m n=0 c i t m-1 , and c i are real constants and n = [γ ] + 1.

The existence results for the considered problem
Here we define some hypotheses for the existence and uniqueness theory for the problem under consideration and we name the space PC(J, R) = Υ ; is needed throughout this work. ( for all Ψ 1 , Ψ 2 ∈ R and i = 1, 2, 3, . . . .

1) if and only is Ψ is the solution of the fractional impulsive integral equation as
, and using Lemma 2.2 for the given problem, we have two constants f 0 , f 1 , that is, , Differentiating we get , Differentiating, we get and (3.8) Next applying the impulsive conditions after simplification we get the following: Putting these values in the given equation we get (3.10) Reiterating the same procedure generally for t ∈ (t j-1 , t j ], we have (3.11) In view of the RBCs and after simplification, we have the subsequent values for the constant f 0 , f 1 , Putting these values in (3.11), we get the required solution given in (3.2) as Rearranging terms so we can achieve the solution given in (3.2). On the other hand, we presume that Ψ is a solution of the given equation, it being straightforward we give the solution given by (3.2), by direct computation which satisfies (3.12).

Corollary 3.1 In view of Lemma 3.1, the solution of the considered problem (1.1) is given by
With the help of Corollary 3.1, the problem (3.12) is reduced to the fixed point problem as Ψ = T(Ψ ). Keeping in mind this operator problem we develop results regarding the solution of the considered problem.
Next we define five operators to establish the main results as regards the existence and uniqueness as follows: For the integral part we define the operators by Let T : Υ → Υ , then T is defined by Therefore, investigating the solution to the given (3.12) the problem is just like investigating the fixed point for the operator T. 1). Further F 0 verifies the following relation: Further simplification yields Hence one has (3.16) Using Then (3.16) becomes Thus F 0 is Lipschitz with constant K ∈ [0, 1). For the growth relation using (A 2 ) and (A 5 ), we get the following: (3.17) Then (3.17) becomes where A and B are given in (3.14) and (3.15).

Lemma 3.2 The operator
is continuous and satisfies the following growth condition: Using the Lebesgue dominated convergence theorem, F 1 (Ψ n ) -F 1 (Ψ ) → 0 as n → ∞, this implies that F 1 is continuous. For the growth relation, using (A 3 ), we then find

Theorem 3.2
The operator defined as F 1 : Y → Y is compact and ψ-Lipschitz with constant zero.
Proof Clearly F 1 satisfies the growth condition so F 1 is bounded on B k = {Ψ ∈ Y : Ψ ≤ r}.
Let Ψ ∈ B k , we have Hence F 1 is bounded. Further for 0 ≤ t 1 ≤ t 2 ≤ T, we show F 1 is equi-continuous.

Lemma 3.3 The operator
is continuous and satisfies the following growth condition: Using the Lebesgue dominated convergence theorem, we see that This implies that F 2 is continuous. Further for the growth relation using (A 3 ), then one has Thus one has This is the required relation.

Lemma 3.4
The operator defined as F 2 : Y → Y is compact and F 2 is ψ-Lipschitz with constant zero.
Proof It can be easily derived similarly to Lemma 3.3.

Lemma 3.5 The operator
is continuous and satisfies the following growth condition: This implies that Using the Lebesgue dominated convergence theorem This implies that F 3 is continuous. For the growth relation using (A 3 ), then Hence one has This is the required relation.

Lemma 3.6
The operator defined by F 3 : Y → Y is compact and F 3 is ψ-Lipschitz with constant zero.
Proof The proof is so easy that we omit it.

Lemma 3.7 The operator
is continuous and satisfies the following growth condition: Proof Let Ψ n be a sequence in B k = {Ψ ∈ Y : Ψ ≤ r} such that Ψ n → Ψ as n → ∞. This implies that Using the Lebesgue dominated convergence theorem, F 4 (Ψ n ) -F 4 (Ψ ) → 0 as n → ∞. This implies that F 4 is continuous.
For the growth relation using (A 3 ), then This is the required relation.

Lemma 3.8 The operator defined by F 4 : Y → Y is compact and F 4 is ψ-Lipschitz with constant zero.
Proof The proof is so easy that we omit it.
Next we show in combined form thta the four operators F 1 , F 2 , F 3 and F 4 , satisfy the growth condition and are continuous, then we show the operators F 1 , F 2 , F 3 and F 4 are ψ-Lipschitz with constants zero and co-mpact.

Theorem 3.3
The operators F 1 , F 2 , F 3 and F 4 : Υ → Υ , are continuous and satisfy the following relation: Thus T is a contraction mapping and by the Banach contraction theorem T has a unique fixed point. Thus the considered problem has a unique solution.  As regards Theorem 3.4, it is not difficult to see that all the hypotheses are satisfied. Thus, the said problem has a solution in Υ . For the uniqueness we presume K Ψ = 1 16 , K g 1 = 1 10 , K g 2 = 1 20 , K

Conclusion
With the support of a degree method of topology, we found sufficient conditions for the impulsive problem (1.1) and by a well-known Banach contraction theorem the uniqueness conditions have been obtained. A test example is presented and it is observed that our results are effective under some conditions.