Existence results for infinite systems of the Hilfer fractional boundary value problems in Banach sequence spaces

The main aim of this paper is to present some existence criteria for an infinite system of Hilfer fractional boundary value problems of the form D a + α , β u i = − F i ( t , u ) , u i ( a ) = u i ( b ) = 0 , a < t < b , i = 1 , 2 , … , $$ \mathcal{D}_{a^{+}}^{\alpha,\beta }u_{i}=-F_{i}(t,u),\quad u_{i}(a)=u_{i}(b)=0, a< t< b,i=1,2,\ldots, $$ in Banach sequence spaces of c 0 $c_{0}$ and l p , p ≥ 1 $l_{p},p\geq 1$ types. Our approach is based on the Darbo-type fixed point theorems acting on the condensing operators. The obtained existence results in each of the above sequence spaces are illustrated by presenting some numerical examples.


Introduction
Nowadays it is a well-known fact that one may take the integration and differentiation of arbitrary order in the appropriate functional spaces. The governing theory of such operations is known as fractional calculus, and consequently differential equations reconstructed in this theory are called fractional differential equations. So, it is a natural expectation that we try to re-obtain the fractional-order approaches of the current results belonging to the ordinary differential calculus. Between various versions of the fractionalorder differentiation the Hilfer fractional derivatives have an interesting technical property, which makes these derivatives much more important than other fractional-order derivatives, that is, the Hilfer fractional derivatives unify the Riemann-Liouville and Caputo fractional derivatives. This property makes the Hilfer fractional derivatives a more powerful mathematical tool in studying real-life phenomena and their engineering refinements. In this way, we suggest some resources for fractional differential equations such as [1, 5, 7, 8, 11-15, 19-22, 25, 27-29, 31, 32, 34, 36-40] and the references cited therein, for more consultation on this topic.
In this paper we deal with one of the above-mentioned approaches, that is, the solvability assessment of an infinite system of the boundary value problems equipped with the Hilfer fractional derivatives. This fractional-order system can be formulated as D α,β a + u i = -F i (t, u), u i (a) = u i (b) = 0, a < t < b, i = 1, 2, . . . , (1.1) where (A 1 ) D α,β a + denotes the Hilfer fractional derivative of order 1 < α ≤ 2 and type 0 ≤ β ≤ 1. As special cases of the infinite systems of the Hilfer fractional differential equations (1.1), one may consider the following instances: 1. The Hilfer fractional relaxation: 2. The Hilfer fractional diffusion: with the Laplacian . The importance of the infinite systems of differential equations arises naturally in solving of partial differential equations. For instance, in the field of statistical physics infinite systems of ordinary differential equations appear as moment problems associated with the Fokker-Planck partial differential equations as described in [30]. As another application for the infinite systems of ordinary differential equations, one can consider the secondorder nonlinear partial differential equation of the form such that solving it can be reduced to solving the infinite system of the ordinary differential equations More details can be found in [18] and the cited bibliography. Also, by the use of the longitudinal method of lines acting on the infinite systems of ordinary differential equations corresponding to some parabolic partial differential equations, one may obtain the approximate solutions of the partial differential equations as has been presented in detail in [35]. Besides, studying the infinite systems of differential equations leads to some significant problems in engineering, mechanics, theory of the branching processes, theory of the neural networks, and the theory of the dissociation of polymers (cf. [18,23] and [24]). On the other hand, studying the infinite systems of differential equations on appropriate Banach spaces by themselves may occupy a distinguished place from the point of view of their solvability processes. As a consequence, one can consider the measures of noncompactness and constructed fixed point theory based on them as a pioneering technique to investigate the existence of solutions for the infinite systems of differential equations. Actually, measures of noncompactness from the point of view of applicability have a wide range of applications in providing solvability criteria for nonlinear differential equations via various refinements of the Darbo fixed point theorem. The history of measures of noncompactness originated with the Polish mathematician Kuratowski [17], and the first measure of noncompactness introduced by him was as follows: for the bounded subset S of the metric space X. Besides, diam(T) denotes the diameter of the set T ⊂ X, i.e., diam(T) := sup{d(x, y)|x, y ∈ T}. However, from the point of view of applicability there is an another measure of noncompactness, which can be considered as the one most convenient and useful in applications. This measure is called the Hausdorff measure of noncompactness and is defined by As a useful and compact collection of resources for more detailed consultation we refer to [4,6,9] and [10] and the references therein. From now on, for the sake of convenience we will use the abbreviation MNC for the measure of noncompactness.
Having the measures of noncompactness and related fixed point theory as our main technical tool, we now mention the following papers that are the inspiring resources of this paper, indeed. The authors in [23], studied the solvability of the infinite system of second-order periodic boundary value problem (1.2) by the use of the measures of noncompactness and Darbo-type fixed point theorems on condensing operators. Motivated by this problem, and essentially following the applied technique in [23], we initiate studying the solvability of the infinite system of the Hilfer fractional boundary value problem (1.1). At the end of this section, we state the organization of this investigation as follows. In Sect. 2, the Hilfer fractional derivatives and their importance in fractional calculus will be summarized. Besides, the concept of MNC and related Darbo-type fixed point theory will be presented. Section 3, is devoted to the some solvability criteria for the Hilfer fractional infinite system (1.1). We divide this section into the two parts. First, we define the Banach sequence space c 0 , and consequently we provide some conditions on nonlinearities to obtain at least one solution for the infinite system (1.1). In the second part, presenting the Banach sequence space l p , p ≥ 1, we give some another sufficient conditions that lead us to the existence of at least one solution of the infinite system (1.1). Applicability of the obtained theoretical results in Sect. 3 will be illustrated in Sect. 4 by presenting some numerical examples for each existence criterion in c 0 and l p .

Technical background
In this section we first define the Hilfer fractional derivatives and their essential properties. So, we have the following definition. Definition 2.1 ([15, 33]) Assume n -1 < α ≤ n, n ∈ N, 0 ≤ β ≤ 1 and f ∈ AC n [a, b]. Then the left and right sided Hilfer fractional derivatives of order α and type β, are defined as follows: (2.1) We note that in this definition, I α a + (b -) stands for the left (right) sided Riemann-Liouville fractional integral of order α that acts on f ∈ L(a, b) as follows [16]: In addition, D α a + (b -) stands for the Riemann-Liouville fractional derivative of order α that acts on f ∈ AC n [a, b] as follows [16]: Remark 2.2 Concentrating on the Hilfer fractional derivative D α,β a + (b -) , it follows that if we take β := 0, then this fractional derivative reduces to the Riemann-Liouville fractional derivatives, while taking β := 1, one obtains the left (right) sided Caputo fractional derivatives In the following lemma, the inversion formula for the Hilfer fractional derivatives is given.

Lemma 2.3 ([33])
Assume n -1 < α ≤ n, n ∈ N and 0 ≤ β ≤ 1. If f ∈ L(a, b) and Now, we can represent the Green function corresponding to the infinite system of the Hilfer fractional boundary value problems (1.1).

Lemma 2.4 ([26]) Assume h ∈ L(a, b). Then the Hilfer boundary value problem
is equivalent to the integral equation (2.5) In this position, we are going to show the importance of the Hilfer fractional differential equations as a consequence of Remark 2.2.
Remark 2.6 Let us consider the Hilfer fractional boundary value problem (2.3). Table 1 illustrates that in view point of comprehensiveness, the Hilfer fractional differential equations include not only their related integer-order differential equations but also the Riemann-Liouville and Caputo versions of the corresponding fractional differential equations.
Consequently, the following table shows that the corresponding Green functions depend on the special values of the parameters α and β and their maximums.
The first and second rows in Table 2 generalize the third row and the Hilfer-Green function (2.4) and its maximum, (2.5), generalizes the first and second rows. So, we arrive at the conclusion that on the most elementary level, the infinite system of Hilfer fractional boundary value problems (1.1) generalizes the research work [23], which was mentioned in the previous section.
It is time to have a quick overview on MNCs and the related fixed point theory. To this aim, we first state the following notations. Let (E, · ) be a Banach space, R + := [0, ∞). If X is a subset of E, then the symbols X and Conv X denote the closure and the convex closure of X, respectively. Let M E and N E denote the families of the nonempty bounded and nonempty relatively compact subsets of E, respectively. We are now ready to define the concept of MNC.
If an MNC satisfies the following additional conditions, we call it a regular MNC: Since our solvability approach will be constructed by the Darbo-type fixed point theorem on the condensing operators, we first have to present the classic Darbo fixed point theorem and definition of a condensing operator.  We note that if E 1 = E 2 and μ 1 = μ 2 , then T is called a μ-condensing operator.
As stated above we are interested in the solvability processes represented in [23]. Thus, we shall consider a particular class of the condensing operators, the so-called the Meirkeeler condensing operators that are defined as follows.
Definition 2.10 ( [3,23]) Let C be a nonempty subset of a Banach space E and let μ be an arbitrary MNC on E. We say that an operator T : C → C is a Meir-Keeler condensing operator if for any > 0, there exists δ > 0 such that for any bounded subset X of C.

Theorem 2.11 ([3, 23]) Let C be a nonempty, bounded, closed and convex subset of a Banach space E and let μ be an arbitrary MNC on E. If T : C → C is a continuous and condensing operator, then T has a fixed point and the set of all of fixed points of T in C is compact.
We finalize this section with formulating the Hausdorff MNC in two Banach sequence spaces c 0 and l 1 . More details can be found in ( [10], pp. 162-163, Th. 5.18, (a), (b)).
In the Banach sequence space (c 0 , · c 0 ), the Hausdorff MNC χ can be formulated as . Furthermore, in the Banach sequence space l p , p ≥ 1, the Hausdorff MNC χ can be formulated as

Existence criteria via measures of noncompactness in Banach sequence spaces
As discussed above, depending on the Banach sequence spaces c 0 and l p we divide this section into two parts and then we attempt to present solvability criterion in each case for the infinite system of the Hilfer fractional boundary value problems (1.1). From now on by a MNC, we mean the Hausdorff measure of noncompactness χ defined by (2.8) and (2.9).

Existence criterion in c 0
To begin with, we have to consider the following hypotheses: has the same conditions as asserted in (C.2) and q i (t) ∈ C([a, b]; R + ) and the sequence . Before presenting the first main result, for the sake of convenience we set the following notation: it is clear that In other mean, the integral operators F i u, i = 1, 2, . . . , satisfy the boundary conditions of (1.1).
Having this fixed point problem in hand, we now able to prove that all conditions of Theorem 2.11 hold. So, let us begin as follows.
According to the (3.1), we get So, we have To proceed, we recall these basic facts; as a result of the hypothesis (C.1), the operator T defined by (3.5) belongs to C ([a, b], B). Besides, according to (3.2) and (3.4), it is clear that Thus, T is self-mapping on B. Furthermore, because of the (3.3), T satisfies the boundary conditions of the infinite system of the Hilfer fractional boundary value problems (1.1) that is (Tu)(a) = 0 and (Tu)(b) = 0.
Looking back into the proof and compare it with the proof of Theorem 2.11, it just remains to prove that the operator T is a Meir-Keeler condensing operator. To this aim, for each arbitrary > 0 we have to find an δ > 0 such that Here we recall the c 0 -definition of the Hausdorff MNC presented in (2.8), and apply it as follows:

Now, it is enough to choose
to obtain χ(T(B)) < . This means that the operator T is a Meir-Keeler condensing operator on B ⊂ c 0 . So, T has at least one fixed point in B that is a solution of the infinite system of the Hilfer fractional boundary value problems (1.1). In addition, if we consider the fixed point set clearly K is closed and nonempty (u 0 ∈ K). Moreover, since K ⊂ Conv(TK ∪ {u 0 }), it follows that So, we arrive at the conclusion that K ⊂ ker χ . This completes the proof.

Existence criterion in l p
Guaranteeing the existence at least one solution for the infinite system of the Hilfer fractional boundary value problems (1.1) in Banach sequence space l p requires the following general assumptions: such that the class of all functions ((Su)(t)) t∈ [a,b] is equicontinuous at every point of the sequence space l p . L.2 a i (t) ∈ C([a, b]; R + ), i = 1, 2, . . . , and the function series ∞ i=1 a i (t) is uniformly convergent. C([a, b]; R + ) such that

L.3 For any natural i, there exists a function q i (t) ∈
holds for t ∈ [a, b] and u = (u i ) ∈ l p for i = 1, 2, . . . , the sequence {b i (t)} ∞ i=1 behaves as mentioned in the hypothesis (L.2) and the sequence {q Similar to the previous section, we introduce here the following finite constants: is a continuous and uniformly convergent series function on [a, b].
Here we are ready to state and prove the second existence criterion for the infinite system of the Hilfer fractional boundary value problems (1.1). So, we have the following theorem.
Proof Since in Sects. 3.1 and 3.2 the infinite system (1.1) must be studied, in the proof of this theorem we try to skip those parts that are the same as those in Theorem 3.1. So, we do not repeat the first part of the proof of Theorem, (3.1) up to (3.3), here. Therefore, we start as follows.
where q > 1 is the conjugate number of p, that is, 1 p + 1 q = 1. So, we get the following: This inequality ensures that Lu is bounded on the interval [a, b] that guarantees that the operator L transforms the space C([a, b]; l p ) into itself. Besides, considering the fixed point problem (3.1)-(3.2), we obtain the following inequality: which leads us to the following straightforward inequality: where q is the conjugate number of p. So, we have Equivalently, it follows that Therefore, choosing

Numerical examples
Example 4.1 We start this section with the following infinite system of the Hilfer fractional boundary value problems: subject to the following setting: Note that by E α,β (t) we mean the two-parametric Mittag-Leffler function , α > 0, β ∈ C.
For the sake of better applicability, we choose p := 2, that is, we are migrating into the Banach sequence space l 2 . Here, we begin by proving that all of the hypotheses (L.1)-(L.3) are satisfied. But formerly we paid attention to the key point that, for an arbitrary positive constant ρ, it follows that max ω>0 ω 1 + ρ 4 ω 4 := 3 3 4 4ρ . (4.4) We fix the arbitrary natural number i to get the following: and fix arbitrary t ∈ [a, b] and u = (u i ) := (u 1 , u 2 , . . .) ∈ l 2 . So, we get This proves that the operator S = (F i ) transforms the space [a, b] × l 2 into l 2 . Next, we shall take a fixed arbitrary positive constant and an arbitrary point u = (u i ) ∈ l 2 . So, taking the To complete this estimation, we need to apply the Cauchy-Schwarz inequality to obtain the following: 540 E 2 α,β (1).
If we summarize the above process, we conclude that Therefore, since all conditions of Theorem 3.2 hold, it has been proved that the infinite system of the Hilfer fractional boundary value problems (4.1) has at least one solution u(t) = (u i (t)) in the sequence space C([a, b]; l 2 ).
Example 4.2 In order to illustrate the applicability of the obtained theoretical existence criterion in Theorem 3.1, we consider the following infinite system of the Hilfer fractional boundary value problems:  We assume that all of the parameters in this application are the same as those in Example 4.1. To see that the infinite system (4.9) has at least one solution in the sequence space c 0 , we have to show that all conditions of Theorem 3.1 hold. To this aim, first one can observe that max ω>0 ω 1 + ρ 4 ω 2 := 1 2ρ 2 , ρ > 0.
So, if u(t) = (u i (t)) ∈ c 0 , then, by the use of this fact, we get the following: is similar to that of in Example 4.1, so we omit it here. Hence, we begin with the hypothesis (C.2) as follows. According to (4.10), it is easy to check that a i (t) ∈ C([a, b]; R + ) for i = 1, 2, . . . . Also, based on Fig. 2 and its corresponding numerical data Table 4, as represented follows that (ba)Q ≈ 0.434314 < 1.
Therefore, Theorem 3.2 ensures that the infinite system of Hilfer fractional boundary value problems (4.9) has at least one solution u(t) := (u i (t)) in c 0 .