Measure of noncompactness for an infinite system of fractional Langevin equation in a sequence space

A new sequence space related to the space ℓp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell _{p}$\end{document}, 1≤p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\leq p<\infty $\end{document} (the space of all absolutely p-summable sequences) is established in the present paper. It turns out that it is Banach and a BK space with Schauder basis. The Hausdorff measure of noncompactness of this space is presented and proven. This formula with the aid the Darbo’s fixed point theorem is used to investigate the existence results for an infinite system of Langevin equations involving generalized derivative of two distinct fractional orders with three-point boundary condition.


Introduction
Infinite systems of differential equations play a significant role in many subjects of nonlinear analysis. The infinite systems of ODEs represent some problems faced within the theories of neural nets, branching processes, and dissociation of polymers. Therefore, the notion of infinite systems of differential equations is considered a substantial part of the theory of differential equations, especially in Banach spaces [1,2]. To date, many existence results have been obtained for the infinite systems of ODEs in Banach spaces. Banas and Lecko [3] discussed the existence of solutions for an infinite system The authors in [6,7] considered the second-order infinite system with the initial conditions x i (0) = x i (T), i ∈ N in the space p , p > 1.
Mursaleen [8] discussed the fractional infinite system where u(t) = {u i (t)} ∞ i=0 , with the initial conditions in Banach spaces, where D α is the R-L fractional derivative of order α.
Fractional calculus, differentiation and integration, appears naturally in several fields of science and engineering; see, for instance, [10] and more recent [11,12], and the references given therein. A large number of existence results of differential and fractional differential equations have been formulated in terms of measures of noncompactness. Measures of noncompactness provide helpful information, which is extensively used in the theory of integral and integro-differential equations. Especially, the measure of noncompactness has been used extensively by many authors when studying infinite systems of differential and fractional differential equations.
Motivated by the former contributions, we consider the infinite system of the generalized Langevin equations c D α is the generalized Liouville-Caputo derivative as in [13], and the space p , 1 ≤ p < ∞ is the space of all absolutely p-summable sequences.
This infinite system is subject to the boundary conditions where a i ∈ R, 0 < η i < 1 and a i η By using the measure of noncompactness technique and applying the Darbo's fixed point theorem, we investigate the existence of solutions for the infinite system (1.1)-(1.2) in the Banach spaces p , p ≥ 1.

Preliminaries
This section is divided into three subsections: The first introduces the concepts and the main results in fractional calculus that the paper needs. The second subsection represents a brief review on the measure of noncompactness and its applications when investigating the existence of solutions for differential equations. The last subsection presents a brief review about the sequence spaces used in this paper.

Fractional calculus
In our paper, we deal with the generalized Liouville-Caputo derivative which is considered a generalization for many known fractional derivatives [15]. Historically, in 2011, Katugampola [30] introduced a new version of fractional integral given by where ρ and ν are positive real numbers, while the function f ∈ X p c (a, b) (the space of Lebesgue measurable functions). He proved that this fractional operator satisfies the semigroup property: 3) The importance of this approach comes from the fact that it is a generalization of Riemann-Liouville and Hadamard fractional integrals. It is easy to show that where RL I ν a and H I ν a are the Riemann-Liouville and Hadamard fractional integral operators, respectively. Also, it is considered a special case of Erdélyi-Kober fractional integral operator [31, formula (1.1.17)] Jarad et al. [32] introduced a new version of a fractional derivative in the Caputo sense which later became known as the generalized Liouville-Caputo derivative, given by It is worth mentioning that the previous formula tends to Caputo derivative as ρ → 1 and tends to Caputo-Hadamard derivative as ρ → 0. Some of its semigroup properties are presented as follows.
From here onward, we replace ρ I λ 0 and ρ c D λ 0 by ρ I λ and ρ c D λ , respectively.
has the unique solution where and F k (t, s, λ, ν), k = 1, 2 are given by Proof Apply ρ I λ on both sides of (2.5) twice in succession using the semigroup property (2.2) and the relation ( Inserting the first and second conditions into (2.9) and (2.10) gives c 0 = c 2 = 0. By applying the last condition, we find that Substituting into (2.10) gives which gives the kernels in (2.7) and (2.8). Conversely, inserting (2.7) into the left-hand side of (2.5) and using the first three relations of Lemma 2.1, we obtain the right-hand side of (2.5). Also, it is not difficult to see that the solution (2.7) satisfies all conditions of (2.6). The proof is complete.

Hausdorff measure of noncompactness
There are many measures of noncompactness. The three main and most frequently used measures are Kuratowski, Istratescu, and Hausdorff measures of noncompactness [1]. Here, we deal with Hausdorff measure of noncompactness that needs the following notions and definitions. In order to render some essential identities of the Hausdorff measure of noncompactness, let us consider subsets N , N n ⊂ N E , n ∈ N. Then we have: (a) β(N ) = 0 for a relatively compact subset of E,

Sequence spaces
Let ω be the set of all real sequences where a = lim n→∞ sup I -P n ω .
Remark 2.1 According to the proof of Theorem 5.16 in [1], if the space ω is a BK space with AK property and monotone norm · ω , then a = lim n→∞ sup I -P n ω = 1.

Remark 2.2
The space of all absolutely p-summable sequences p , 1 ≤ p < ∞ is a Banach space equipped with the norm and it is a BK space with AK property and monotone norm · p . Therefore, a nonempty bounded subset N ∈ p satisfies the inequality in Theorem 2.2 with a = lim n→∞ sup I -P n p = 1.

Basic constructions
Since ρ c D ν u j (t) → y(t) as j → ∞ uniformly on [0, 1], we find that ρ I ν ρ c D ν u j (t) → ρ I ν y(t) as j → ∞ uniformly on [0, 1]. Hence, by using the last relation in Lemma 2.1, we find that u j (t)u j (0) → ρ I ν y(t) as j → ∞, which leads to x(t)c = ρ I ν y(t) where c is a constant. Applying ρ c D ν on both sides and using the first and third relations in Lemma 2.1, we obtain ρ c D ν x(t) = y(t). Hence, φ( ρ c D ν x(t)) = φ(y(t)). Proof Let u ∈ E p , then u ∈ p . In view of Theorem 3.1 and the fact that p , 1 ≤ p < ∞ is a BK space with Schauder basis, the space E p is a BK space with the same Schauder basis. Then, according to Theorem 2.2, we get where a = lim n→∞ sup I -P n . The left-and right-hand sides of the inequality above show that a ≥ 1. So, it is suffices to prove that a ≤ 1. Since · p , 1 ≤ p < ∞ is monotone, (I -P n )(u) p ≤ u p for all u ∈ p and all n ∈ N (see the proof of Theorem 5.16 [1]).

Theorem 3.2 Let N E p be a nonempty bounded subset of
Hence, for all n ∈ N, it is easy to see that This means that a = lim n→∞ sup I -P n ≤ 1, which is the desired result.
Case I. If 0 ≤ t 1 < t 2 ≤ η < 1, then we find that . By using the well-known fact that (xy) q ≤ x qy q for all x ≥ y > 0 and q ≥ 1, we can deduce by Lemma 3.1, when 1 < λ ≤ 2 (α ν > 1), that and, when λ = 0 (α ν < 1), we find that By applying the result of Lemma 3.1, we can deduce that Similarly, the upper bound of the integral I 3 can be evaluated as It is obvious that the integrals I 1 , I 2 , and I 3 approach uniformly zero as t 1 → t 2 , which implies the desired results.

Main results
By using the Hausdorff measure of noncompactness and applying Theorem 3.2 together with the Darbo's fixed point Theorem 2.1, we obtain the existence of solution for the infinite system of fractional Langevin equations (1.1) subject to the boundary conditions (1.2) with the same constraints mentioned in the first section. It is obvious, due to Lemma 2.2, that the solution of the infinite system (1.1)-(1.2) u(t) ∈ E p satisfies the single integral equation Here, F i , i ∈ N are defined as and F i1 = F 1 and F i2 = F 2 where F 1 and F 2 are defined in (2.8) by replacing the symbols λ, ν, μ, ρ, a, and η by the indexed symbols λ i , ν i , μ i , ρ i , a i , and η i , respectively. Define the sequence operators P, P 1 , P 2 : E p → E p by (Pu i )(t) = (P 1 u i )(t) -(P 2 u i )(t), (4.2) (4.4) for all i ∈ N. Their fractional derivatives of order 0 < ν < μ, by using the third identity in Lemma 2.1, can be computed as The investigation of the existence of solutions for the infinite system (1.1)-(1.2) will be discussed under the following assumptions: (M 2 ) There exist nonnegative sequence functions x i (t) and y i (t), satisfying, for all i ∈ N, t ∈ [0, 1] and u, v ∈ C([0, 1], p ), the inequality 1]. This means that lim n→∞ (M 6 ) The functions φ i : R → R are continuous and additive for all i ∈ N. That is, they satisfy Cauchy's functional equation By the same technique, we arrive at which implies that the operator P is continuous on the set N . To show it is continuous uniformly on the interval [0, 1], let t 0 ∈ [0, 1]. Then, we find that which approaches zero uniformly as t → t 0 due to Lemma 3.2. In the same way, we can see that |(Pu)(t) -(Pu)(t 0 )| → 0 uniformly as t → t 0 , which implies that the operator P is continuous on [0, 1]. Now, let us use the Hölder integral inequality with 1/p + 1/q = 1 as follows: (M 2 ) We choose x i (t) = t 4 (t + 1) 4 e -it i! and y i (t) = te -it 2(5t) 4 i 2 2 2(p-1) ζ p-1 (9q).

Conclusion
In the present research, we studied an infinite system of Langevin equations of fractional order. The fractional derivative used in our model is the so-called generalized Liouville-Caputo derivative, which associates with many well-known fractional derivatives. By applying the measure of noncompactness technique and using the Darbo's fixed point theorem, we examined the existence of solution to this infinite system. This investigation has been performed in a new sequence space related to the p , 1 ≤ p < ∞ space. A numerical example is presented to illustrate our idea by investigating a function satisfying all the proposed assumptions.