On Hilfer fractional difference operator

In this article, a new definition of fractional Hilfer difference operator is introduced. Definition based properties are developed and utilized to construct fixed point operator for fractional order Hilfer difference equations with initial condition. We acquire some conditions for existence, uniqueness, Ulam–Hyers, and Ulam–Hyers–Rassias stability. Modified Gronwall’s inequality is presented for discrete calculus with the delta difference operator.


Introduction
In the topics of discrete fractional calculus a variety of results can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], which has helped to construct theory of the subject. A rigorous intrigue in fractional calculus of differences has been exhibited by Atici and Eloe [3,5]. They explored characteristics of falling function, a new power law for difference operators, and the composition of sums and differences of arbitrary order. Holm presented advance composition formulas for sums and differences in his dissertation [12].
Hilfer fractional order derivative was introduced in [17]. Hilfer's definition is illustrated as follows: the fractional derivative of order 0 < μ < 1 and type 0 ≤ ν ≤ 1 is The special cases are Riemann-Liouville fractional derivative for ν = 0 and Caputo fractional derivative for ν = 1. Furati et al. [18,19] primarily studied the existence theory of Hilfer fractional derivative and also explained the type parameter ν as interpolation between the Riemann-Liouville and the Caputo derivatives. It generates more types of stationary states and gives an extra degree of freedom on the initial condition. Hilfer fractional calculus has been examined broadly by a lot of researchers. Some recent studies involving Hilfer fractional derivatives can be found in [20][21][22][23][24][25][26][27][28]. The majority of the work in discrete fractional calculus is developed as analogues of continuous fractional calculus. Extensive work on Hilfer fractional derivative and on its extensions has been done, namely: Hilfer-Hadamard [29][30][31][32], K-fractional Hilfer [33], Hilfer-Prabhakar [34], Hilfer-Katugampola [35], and ψ-Hilfer [36] fractional operator. However, to the best of our knowledge no work is available for Hilfer fractional difference operator in the delta fractional setting. Also formation of fractional difference operator is an important aspect of mathematical interest and numerical formulae as well as the applications. It motivated us to generalize the two existing fractional difference operators namely, Riemann-Liouville and Caputo difference operator in Hilfer's sense.
We started by introducing a generalized difference operator analogous to Hilfer fractional derivative [17]. To keep the interpolative property of Hilfer fractional difference operators, we carefully chose the starting points of fractional sums. Some important composition properties were developed and utilized to construct fixed point operator for a new class of Hilfer fractional nonlinear difference equations with initial condition involving Riemann-Liouville fractional sum. An application of Brouwer's fixed point theorem gave us conditions for the existence of solution for a new class of Hilfer fractional nonlinear difference equations. For the uniqueness of solution, we applied the Banach contraction principle. To solve linear fractional Hilfer difference equation, we used successive approximation method and then defined the discrete Mittag-Leffler function in the delta difference setting. Gronwall's inequality for discrete calculus with the delta difference operator has been modified. An application of Gronwall's inequality has been given for the stability of solution to fractional order Hilfer difference equation with different initial conditions.
In the continuous setting extensive work on Ulam-Hyers-Rassias stability for noninteger order differential equation has been done. The idea of Ulam-Hyers type stability is important for both pure and applied problems; especially in biology, economics, and numerical analysis. Rassias [37] introduced the continuity condition, which produced acceptable stronger results. However, in discrete fractional setting a limited work can be found [38][39][40]. For Hilfer delta difference equation, conditions have been acquired for Ulam-Hyers and Ulam-Hyers-Rassias stability with illustrative example. Interested reader may find some details on Ulam-Hyers-Rassias stability in [37,[41][42][43].
In this article, we shall study initial value problem (IVP) for the following Hilfer fractional difference equation. Let η = μ + νμν, then for 0 < μ < 1 and 0 ≤ ν ≤ 1, we have In Sect. 2, we state a few basic but important results from discrete calculus. In the third section, a new fractional Hilfer difference operator is introduced which interpolates Riemann-Liouville and Caputo fractional differences; we also develop some important properties of a newly defined operator. Conditions for existence, uniqueness, and Ulam-Hyers stability are obtained in Sect. 4. The last section comprises modification and application of discrete Gronwall's inequality in delta setting.

Preliminaries
Some basics from discrete fractional calculus are given for later use in the following sections. The functions we consider are usually defined on the set N a := {a, a + 1, a + 2, . . .}, where a ∈ R is fixed. Sometimes the set N a is called isolated time scale. Similarly, the sets N b a := {a, a + 1, a + 2, . . . , b} and [a, b] N a := [a, b] ∩ N a [44] for b = a + k, k ∈ N 0 . The jump operators σ (t) = t + 1 and ρ(t) = t -1 are forward and backward, respectively, for t ∈ N a . Furthermore, the set R = {p i : 1 + p i (x) = 0} contains regressive functions.
Definition 2.1 ([45]) Assume that f : N a → R and b ≤ c are in N a , then the delta definite integral is defined by Note that the value of integral is μth fractional Taylor monomial based at s and t μ is the generalized falling function.
Definition 2.6 ([9]) Assume p ∈ R and x, y ∈ N a . Then the delta exponential function is given by By empty product convention y-1 t=y [h(t)] := 1 for any function h. for all complex numbers y = -1 such that this improper integral converges.

Hilfer-like fractional difference
In this section, we generalize the definition of fractional difference operators. Motivated by the concept of Hilfer fractional derivative [17], and to keep the interpolative property, we introduce the following definition. Assume f : N a → R, then the fractional difference of order m -1 < μ < m for m ∈ N 1 is given by whereas integer order differences keep the same domain [12]. The starting point of the last sum is compatible with the starting point for the domain of the function m -(1-ν)(m-μ) a f (x), which is a + (1ν)(mμ). This allows us the successive composition of operators in the above expression, and the final domain of To get some nice properties, we restrict 0 < μ < 1 throughout the article.
First of all we will develop some composition properties to use them in the next section and to construct a fixed point operator for a new class of Hilfer fractional nonlinear difference equations with initial condition involving Riemann-Liouville fractional sum. Also we will present the delta Laplace transform for newly defined Hilfer fractional difference operator.
Proof (i) On the left-hand side we use Definition 3.1 and (Theorem 5 [12]) to obtain (ii) On the left-hand side, use (i) and the first part of (Lemma 6 [12]) to get (iii) Using Definition 3.1 and (Theorem 5 [12]), we obtain In the preceding step we also used the first part of (Lemma 6 [12]).
(iv) Consider the left-hand side, use (iii) and the second part of (Theorem 8 [12]), For a nonempty set N T a , the set of all real-valued bounded functions The weighted space of bounded functions is considered for finding left inverse property, however analysis in the following sections is not influenced by this space.
In the preceding step we used the fact -μ The desired result is achieved by applying limit process x → a + μ.
Next we will state the left inverse property.
Hence the result follows from part (iv) of Lemma 3.2.
Then, for |y + 1| > r, we have the delta Laplace transform given as Proof Considering the left-hand side and using Lemmas 2.8 and 2.9, we have Remark 1 Notice that, if in Theorem 3.5 we set ν = 0, then we recover Theorem 2.70 in [9]. Further, if we set ν = 1, we obtain the delta Laplace transform for the Caputo fractional difference.

Fixed point operators for initial value problem
To establish existence theory for Hilfer fractional difference equation with initial conditions, we transform the problem to an equivalent summation equation which in turn defines an appropriate fixed point operator.
The proof of the above lemma is an implication of Lemma 3.2 (i) and (ii) and the second part of Theorem 8 in [12]. In next result Brouwer's fixed point theorem [38] is utilized for establishing existence conditions. The set Z of all real sequences u = {u(x)} T x=a , with u = sup x∈N T a |u(x)| is a Banach space. Using Definition 2.2 and Lemma 4.1 we define an operator A : Z → Z by The fixed points of A coincide with the solutions of problem (1). where Proof For M > 0, define the set To prove this theorem we just have to show that A maps W into itself. For u ∈ W , we have We have Au ≤ M, which implies that A is a self map. Therefore, by Brouwer's fixed point theorem, A has at least one fixed point.
Proof Let u, v ∈ Z and x ∈ [a, T] N a , we have by assumption In the preceding step, we used τ h ν-1 (x, σ (τ )) = -h ν (x, τ ) and Lemma 2.10. Now taking supremum on both sides, we have Using inequality (8), we get Au -Av ≤ uv , which implies A is a contraction. Therefore, by Banach's fixed point theorem, A has a unique fixed point. Proof For simplicity the solution of IVP (1) can be rewritten by using equation (6) as follows: where w(x) = ζ h η-1 (x, a + 1η). Now, for [a, T] N a , it follows from inequality (2) that For [a, T] N a , making use of equation (9) and inequality (10) In the preceding step, we used assumption and the same argument used in Theorem 4.3. Now taking supremum on both sides and simplifying, we have Therefore by inequality (8) To illustrate the usefulness of Theorem 4.4, we present the following example.
Example 4.6 Consider the following fractional Hilfer difference equation with initial condition involving Riemann-Liouville fractional sum: Here, a = 0.3, T = 9.3, μ = 0.7, and ν = 0.5. Therefore η = 0.85. Thus, for K < 0.1974, the solution to the given problem with inequalities is Ulam-Hyers stable and Ulam-Hyers-Rassias stable with respect to function ψ : To solve the linear Hilfer fractional difference IVP, we use the successive approximation method.
Remark 2 If we set ν = 1 in Example 4.7 above (hence η = 1) and take a = μ -1, then we recover Example 17 in [1]. In fact, the solution of the initial Caputo difference equation will be given by Observe that the case a = μ -1 will result in (66) in [1]. That is, formula (66) in [1] represents E μ (λ, t -(α -1)). Also, one can see that the substitution μ = 1 will give the delta discrete Taylor expansion of the delta discrete exponential function.
The observations in Remark 2 suggest the following modified definitions which are different from those appearing in [1].

Modified Gronwall's inequality and its application in delta difference setting
First we develop a Gronwall's inequality for the delta difference operator. Then a simple utilization of Gronwall's inequality leads to stability for Hilfer difference equation. For this purpose, choose u and w such that Lemma 5.1 Assume u and w respectively satisfy (22) and (23).
Proof We give the proof by induction principle. Assume w(τ )u(τ ) ≥ 0 is valid for τ = a, a + 1, . . . , x -1. Then we have where the last summation is valid for x ∈ N a+μ . Now we shift the domain of summation to N a : By assumption, for τ = a, a + 1, . . . , x -1, we have This implies that (1φ(x))(w(x)u(x)) ≥ 0 and for |φ(x)| < 1, which is the desired result.
Following the approach for nabla fractional difference in [49], let . For constant φ one can use E v φ to express the Mittag-Leffler function.
For k = 1, Proceeding inductively, we obtain and let k → ∞, Next we derive a Gronwall's inequality in delta discrete setting.
For η = 1, a special case is obtained as follows.
where e v (x, a) is the delta exponential function.
Proof It follows from Theorem 5.3 that , a). To justify our claim, we utilize the uniqueness of solution of the following IVP: of IVP is given in [9] for regressive function v(x). Thus, we have to show that ∞ Also, by Definition 2.2 and empty sum convention, we have Then the result follows.

(24)
Theorem 5.5 Assume that the Lipschitz condition |g(x, u)g(x, v)| ≤ K|u -v| holds for function g. Then the solution to Hilfer fractional difference system is stable.

Conclusion
We finish by concluding the following: • A new definition of Hilfer-like fractional difference on discrete time scale has been introduced. • The delta Laplace transform has been developed for newly defined Hilfer fractional difference operator. • We have investigated a new class of Hilfer-like fractional nonlinear difference equations with initial condition involving Riemann-Liouville fractional sum. • In particular, conditions for the existence, uniqueness, and two types of stabilities, called Ulam-Hyers stability and Ulam-Hyers-Rassias stability, have been obtained. • The linear Hilfer fractional difference equation with initial conditions has been solved and alternative versions of discrete Mittag-Leffler functions are presented in comparison to [1]. • A Gronwall's inequality has been modified and applied for discrete calculus with the delta operator.