Existence of solutions of infinite system of nonlinear sequential fractional differential equations

In a recent paper (Filomat 32:4577–4586, 2018) the authors have investigated the existence and uniqueness of a solution for a nonlinear sequential fractional differential equation. To present an analytical improvement for Fazli–Nieto’s results with some conditions removed based on a new technique is the main objective of this paper. In addition, we introduce an infinite system of nonlinear sequential fractional differential equations and discuss the existence of a solution for them in the classical Banach sequence spaces c0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c_{0}$\end{document} and 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} by applying the Darbo fixed point theorem. Moreover, the proposed method is applied to several examples to show the clarity and effectiveness.


Introduction and preliminaries
As is well known, the fractional differential equations (FDEs) is a fundamental topic that considered as a powerful tool in many fields, for example, dynamic systems, rheology, blood flow phenomena, biophysics, electrical networks, modeled by different fractional order derivatives equations; see for details [2][3][4][5] and the references therein. Also, in the last two decades, FDEs have been used to model various stable physical phenomena [6][7][8]. For example, when the random oscillation force is assumed to be white noise, Brown's motion is well described by some fractional differential equations. On the other hand, during the last years, many studies have been done on the existence and uniqueness of solution of nonlinear initial fractional differential equations by the use of some fixed point theorems; see [9][10][11][12][13][14][15][16][17][18][19][20].
Recently, Fazli and Nieto [1] investigated the existence and uniqueness of the following interesting problem, which is a model of physical phenomena: where 0 < α ≤ 1, 0 < T < ∞. The term D 2α is for the sequence fractional derivative presented by Miller and Ross [21], where D α is the classical Riemann-Liouville fractional derivative of order α. Before giving the weighted Cauchy type problem obtained in [1], let us recall some notions introduced in that work. Let be the weighted spaces of continuous functions with the following norm: We define the following spaces of functions: Let f : [0, T] × R 2 → R be a function satisfying the following axioms: f is non-decreasing in all its arguments except for the first argument and for some L 1 , L 2 > 0 whenever x ∈ (0, T] and u ≥ũ, v ≥ṽ. The weighted Cauchy type problem presented in [1] is given by the following result. Moreover, the authors in [1] defined the generalization of (1) and obtained some results for it as follows: where 0 < α ≤ 1.
To prove the main results, we need the following assumptions: for some 0 ≤ γ < 1. (H 4 ) f is non-decreasing in all its arguments except for the first argument and there exists L > 0 such that In present paper, we address the following questions. (Q 1 ) Is it possible to remove the non-decreasing conditions of the mappings f in Theorem 1.1 and Theorem 1.2? (Q 2 ) Is it possible to remove assumption of the existence of a lower solution of the problems (1) and (4)? (Q 3 ) Is it possible to define the problem (1) as an infinite system and discuss the existence results of the solution to it in spaces c 0 and p ? In the sequel, we prove that the non-decreasing condition of function f in Theorem 1.1 and Theorem 1.2 is not necessary. Also, in Theorem 1.1 and Theorem 1.2, we need to find a lower solution of (1) and (4), respectively, while we show that do not need to this assumptions. In fact, by removing some of the assumptions and even with the weakening of other conditions of the main results of [1], using the new technique, we get the same results. Moreover, we present some remarks and examples to support the results herein and we compare the main results of Fazli and Nieto [1] and our results. In addition, since the theory of infinite systems of differential equations is an attractive research topic of the theory of differential equations in Banach spaces (for details, see [22][23][24]), we consider the problem (1) as an infinite system as follows: where 0 < T < ∞, α and D 2α are defined in (2), and also f n (x, u(x), D α u(x)), i = 1, 2, . . . , are real valued functions. Actually, we study the existence of the solution for the infinite system (5) in the spaces c 0 and p , 1 ≤ p < ∞, which c 0 is the space of sequences tends to zero. For this purpose, we use the Darbo fixed point theorem. Finally, illustrative examples are presented to evaluate the realization and effectiveness of our results. At first, we recall some important definitions, lemmas and theorems that we use in our proofs of the main results. For details see [25,26].

Definition 1.2
The Riemann-Liouville fractional integral of order γ of a function u ∈ C[0, T] is defined as provided the right-hand side is defined for almost every x ∈ (0, T). Herein, Γ (·) represents the classical Gamma function.
Lemma 1.5 Assume that u ∈ C(0, T] ∩ L 1 (0, T) with a fractional derivative of order 0 < α ≤ 1 that belongs to C(0, T] ∩ L 1 (0, T). Then Throughout this paper (X, · ) indicates a Banach space, for every E ⊂ X,Ē indicates the closure of E, and conv(E) indicates the closed convex hull of X. Also, note that M X is the family of non-empty bounded subsets of X and N X is the family of non-empty and relatively compact subsets of X. The use of the measure of noncompactness(MNC) concepts was first proposed by Kuratowski [27]. Here, we will give a brief overview of this notion, which is used in Sect. 3.

Definition 1.4 ([28])
A mapping μ : M X − → R + is said to be a measure of the noncompactness in E if it satisfies the following conditions: In addition, the definition of the Hausdorff measure of noncompactnesss χ which can be found in [27] is expressed as follows: is a Schauder basis for c 0 , the Hausdorff MNC χ for Q is given by is a Schauder basis for p , the Hausdorff MNC χ for Q is given by

An improvement of the existence and uniqueness of solutions to the initial value problem
In the following theorems, we remove some of the hypotheses of Theorems 1.1 and 1.2. Moreover, we show that under our assumptions (1) and (4) have a unique solution. This gives a partial answer to (Q 1 ) and (Q 2 ).
Proof Fix δ > 0 such that Consider the operator A defined on C α 1-α [0, δ] by It is obvious that u is a solution of the problem (1) if and only if u is fixed point of the operator A. By the same arguments as given in the proof of Theorem 4.2 of [1], we draw the conclusion that the operator A is well defined. Now, we only need to show that A is a contraction mapping. For each u, Since the space C α 1-α [0, δ] is a complete metric space, applying the Banach contraction, the operator A has a unique fixed point and this fixed point is the unique solution of the problem (1).  Example 2.1 The linear initial value problem is given as follows: This problem is a special case of (1) with α = 1 2 , T, a, b, ν 1 , ν 2 > 0, max{ν 1 , . It is easy to see that Applying Theorem 2.1 the linear initial value problem (6) possesses a unique solution in C α 1-α [0, γ ]. It is simple to verify that Theorem 1.1 cannot be applied to our example. Because f is not increasing in all its arguments except for the first argument, that is, the condition (H 2 ) of Theorem 1.1 is not satisfied.
If max 1≤i≤4 {ζ i } < e 4 5.86 , then, by applying Theorem (4), the problem where x ∈ (0, T], possesses a unique solution. On the other hand, since f is not increasing, Theorem 2.2 is not applicable here.

Solution of infinite system (5)
In this section, we give a partial answer to (Q 3 ). For this purpose, firstly, we present some weighted continuous spaces. Then we discuss the existence of solution of infinite system (5) in the Banach space c 0 and p in Sects. 3.1 and 3.2, respectively. Definition 3.1 Let X be a norm space and C(I, X) be the family of all continuous functions on I to X. We define a weighted spaces of continuous functions as follows:
Throughout this section, we define I = (0, T], , which belongs to some Banach space (X, · ). Therefore, one has system (5) as follows: where u n (x), n = 1, 2, 3, . . . , are continuous on I, f is defined on I × X × X − → X and f i is a real valued function.

Solution in space c 0
In this subsection, let X = c 0 . We intend to show the existence of a solution of the infinite system (5) in the Banach space c 0 with the norm u = sup{|u i | : i = 1, 2, 3, . . .}. Suppose that the following conditions are satisfied: (C 1 ) {u n 0 } ∞ n=1 and {u n 1 } ∞ n=1 belong to c 0 ; (C 2 ) for any fixed u, f (x, u(x), D α u(x)) is measurable; (C 3 ) for each x ∈ I, u(x) ∈ c 0 and i = 1, 2, . . . , we have where j i (x) and k i (x) are continuous real valued functions on I such that the sequence {k i (x)} ∞ i=1 is equibounded on I and the sequence (j i (x)) converges uniformly on I to the zero function identically; D α u(x)) is equicontinuous at any point of the space c 0 .

Theorem 3.1 Under the conditions
satisfies the boundary conditions of the infinite system (8). We define the operator A : C α 1-α (I, c 0 ) − → C α 1-α (I, c 0 ) by Applying (C 2 ), A is well defined. We show that A is bounded on C α 1-α (I, c 0 ), Using (C 3 ), there exists J = max i sup x∈I |j i (x)|, therefore and so Then we conclude that is the optimal solution of the inequality Define the closed, bounded and convex set where n = 1, 2, . . . . Clearly, A is bounded on B r . In the following, we show that A is continuous on B r . We can write is equicontinuous on c 0 . Bearing (C 4 ) in mind, we have ∀v, u ∈ B r and ∀ > 0, ∃δ > 0: where z = T 2 Γ (2α+1) . Therefore, we conclude that which means that A is continuous. Without loss of generality, we can suppose x 1 > x 2 . There exist m 1 , m 2 and m 3 in R + such that Applying (9), for any u ∈ B r , we have which tends to zero when x 1 − → x 2 . Thus, we deduce that A is equicontinuous on B r . SettingB = conv(A(B r )), clearlyB ⊂ B r . Let Y ⊂B, then A is continuous on Y and the functions from the set of Y are equicontinuous on I. In view of the definition of the Hausdorff MNC χ on the space C α 1-α (I, c 0 ), Proposition 1.7 and Theorem 1.8, we have Recalling Theorem 1.8, for any u ∈ Y , we observe and As KT 2α Γ (2α+1) < 1, applying Lemma 1.6, A possesses at least one fixed point in A, which is a solution for (5) in the space C α 1-α (I, c 0 ). Now, with the following example, we clarify the main result of this subsection.

Solution in space l p
In this subsection, let X = p . For a real number p ≥ 1, the space denoted by p is the Banach sequence space, when equipped with the following norm: In the following, we show that the infinite system (5) has at least on solution in the space p , when the following conditions are satisfied: (C 1 ) u 0 and u 1 belong to p ; (C 2 ) f : I × p − → p is continuous; (C 3 ) for each x ∈ [0, t], u(x) ∈ p and i = 1, 2, . . . , we have