1 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 [2–5] and the references therein. Also, in the last two decades, FDEs have been used to model various stable physical phenomena [6–8]. 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 [9–20].

Recently, Fazli and Nieto [1] investigated the existence and uniqueness of the following interesting problem, which is a model of physical phenomena:

$$ \left \{ \textstyle\begin{array}{ll} \mathcal{D}^{2\alpha}u(x) = f(x,u(x),\mathcal{D}^{\alpha}u(x)),\quad x \in(0,T], \\ \lim_{x \rightarrow0}x^{1-\alpha}u(x)=u_{0}, \qquad\lim_{x \rightarrow 0}x^{1-\alpha}\mathcal{D}^{\alpha}u(x)= u_{1}, \end{array}\displaystyle \right . $$
(1)

where \(0 < \alpha\leq1\), \(0 < T < \infty\). The term \(\mathcal {D}^{2\alpha}\) is for the sequence fractional derivative presented by Miller and Ross [21],

$$ \left \{ \textstyle\begin{array}{ll} \mathcal{D}^{\alpha}u = D^{\alpha}u, \\ \mathcal{D}^{k\alpha}u = \mathcal{D}^{\alpha}\mathcal {D}^{(k-1)\alpha}u\quad (k=2,3,\ldots), \end{array}\displaystyle \right . $$
(2)

where \(\mathcal{D}^{\alpha}\) 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

$$ C_{1-\alpha}[0,T] = \bigl\{ u\in C[0,T]:x^{1-\alpha}u \in C[0,T]\bigr\} $$
(3)

be the weighted spaces of continuous functions with the following norm:

$$\Vert u \Vert _{C_{1-\alpha}[0,T]} = \max_{0\leq x \leq T} \bigl\vert x^{1-\alpha}u(x) \bigr\vert . $$

We define the following spaces of functions:

$$C_{1-\alpha}^{\alpha}[0,T] = \bigl\{ u\in C[0,T]:x^{1-\alpha}u \in C_{1-\alpha}[0,T], \mathcal{D}^{\alpha}u \in C_{1-\alpha}[0,T] \bigr\} , $$

with the norm

$$\Vert u \Vert _{C_{1-\alpha}^{\alpha}[0,T]} = \Vert u \Vert _{C_{1-\alpha}[0,T]} + \bigl\Vert \mathcal{D}^{\alpha}u \bigr\Vert _{C_{1-\alpha}[0,T]}, $$

which are Banach spaces.

A function \(\underline{u} \in C_{1-\alpha}^{\alpha}[0,T]\) is called a lower solution of the initial value problem (1), if \(\mathcal{D}^{2\alpha}\underline{u}(x) \leq f(x,\underline {u}(x),\mathcal{D}^{\alpha}\underline{u}(x))\) for every \(x \in (0,T]\) and

$$\lim_{x \rightarrow0}x^{1-\alpha} \underline{u}(x) \leq u_{0},\qquad \lim_{x \rightarrow0}x^{1-\alpha} \mathcal{D}^{\alpha}\underline {u}(x) \leq u_{1}. $$

Let \(f : [0,T]\times\mathbb{R}^{2} \rightarrow\mathbb{R}\) be a function satisfying the following axioms:

\((H_{1})\):

for every \(u \in C_{1-\alpha}^{\alpha}[0,T]\), \(f(x,u(x),\mathcal{D}^{\alpha}u(x)) \in C_{\gamma}[0,T]\) for some \(0 \leq\gamma< 1\).

\((H_{2})\):

f is non-decreasing in all its arguments except for the first argument and

$$f(x,u,v) - f(x,\tilde{u},\tilde{v}) \leq L_{1} (u-\tilde{u}) + L_{2}(v - \tilde{v}) $$

for some \(L_{1},L_{2} > 0\) whenever \(x \in(0,T]\) and \(u \geq\tilde{u}\), \(v \geq \tilde{v}\).

The weighted Cauchy type problem presented in [1] is given by the following result.

Theorem 1.1

Assume that\((H_{1})\)–\((H_{2})\)hold. Then there exists\(0 <\delta\leq T\)such that the existence of a lower solution for (1)–(2) in\(C_{1-\alpha}^{\alpha}[0,\delta]\)provides the existence of a unique solution\(u \in C_{1-\alpha}^{\alpha}[0,\delta]\)for (1).

Moreover, the authors in [1] defined the generalization of (1) and obtained some results for it as follows:

$$ \left \{ \textstyle\begin{array}{ll} \mathcal{D}^{n\alpha}u(x) = f(x,u(x),\mathcal{D}^{\alpha }u(x),\mathcal{D}^{2\alpha}u(x), \ldots, \mathcal{D}^{(n-1)\alpha }u(x)),\quad x \in(0,T], \\ \lim_{x \rightarrow0}x^{1-\alpha}\mathcal{D}^{k\alpha}u(x)= u_{k}\quad (k=0,1, \ldots, n-1), \end{array}\displaystyle \right . $$
(4)

where \(0 < \alpha\leq1\).

Definition 1.1

For \(0 < \alpha\leq1\), we define the space

$$C_{1-\alpha}^{n\alpha}[0,T] = \bigl\{ u\in C_{1-\alpha}[0,T]: \mathcal {D}^{k \alpha}u \in C_{1-\alpha}[0,T], k = 1,2, \ldots,n-1 \bigr\} , $$

equipped with the norm

$$\Vert u \Vert _{C_{1-\alpha}^{n\alpha[0,T]}} = \sum_{k=0}^{n-1} \bigl\Vert \mathcal {D}^{k\alpha}u \bigr\Vert _{C_{1-\alpha}[0,T]}. $$

To prove the main results, we need the following assumptions:

\((H_{3})\):

\(f : [0,T]\times\mathbb{R}^{n} \rightarrow\mathbb{R}\) be a function such that, for every \(u \in C_{1-\alpha}^{n\alpha} [0,T]\),

$$f\bigl(\cdot,u(\cdot),\mathcal{D}^{\alpha}u(x),\mathcal{D}^{2\alpha}u(x), \ldots ,\mathcal{D}^{(n-1)\alpha}u(x)\bigr)\in C_{\gamma}[0,T] $$

for some \(0 \leq\gamma< 1\).

\((H_{4})\):

f is non-decreasing in all its arguments except for the first argument and there exists \(L> 0\) such that

$$f(x,u_{1},\ldots,u_{n}) - f(x,\tilde{u}_{1}, \ldots,\tilde{u}_{n}) \leq L \sum_{i=1}^{n}(u_{i} - \tilde{u}_{i}), u_{i} \geq\tilde{u}_{i},\quad i=1,2,\ldots, n. $$

Theorem 1.2

Assume that\((H_{3})\)–\((H_{4})\)hold. Then there exists\(0 <\delta\leq T\)such that the existence of a lower solution for (4) in\(C_{1-\alpha}^{\alpha}[0,\delta]\)provides the existence of a unique solution\(u \in C_{1-\alpha}^{\alpha}[0,\delta]\)for (4).

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 \(\ell_{p}\)?

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 [22–24]), we consider the problem (1) as an infinite system as follows:

$$ \left \{ \textstyle\begin{array}{ll} \mathcal{D}^{2\alpha}u_{n}(x) = f_{n}(x,u_{n}(x),\mathcal{D}^{\alpha }u_{n}(x)),\quad x \in(0,T], \\ \lim_{x \rightarrow0}x^{1-\alpha}u_{n}(x)=u_{n}^{0}, \qquad\lim_{x \rightarrow 0}x^{1-\alpha}\mathcal{D}^{\alpha}u_{n}(x)= u_{n}^{1}, \quad n=1,2,\ldots, \end{array}\displaystyle \right . $$
(5)

where \(0 < T < \infty\), α and \(\mathcal{D}^{2\alpha}\) are defined in (2), and also \(f_{n}(x,u(x),\mathcal{D}^{\alpha }u(x))\), \(i = 1,2, \dots\), are real valued functions. Actually, we study the existence of the solution for the infinite system (5) in the spaces \(c_{0}\) and \(\ell_{p}\), \(1 \leq p < \infty\), 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 \(u \in C[0,T]\) is defined as

$$I^{\gamma} u(x) = \frac{1}{\varGamma(\gamma)} \int_{0}^{x} \frac {u(s)}{(x - s)^{1 - \gamma}}\,ds, \quad0 \leq x \leq T. $$

Definition 1.3

The Riemann–Liouville fractional derivative \(D^{\gamma}\) of order \(0 < \gamma\leq1\) of a function \(u : [0,T) \longrightarrow\mathbb{R}\) is defined by

$$\mathcal{D}^{\gamma}u(x) = \frac{1}{\varGamma(1 - \gamma)} \frac {d}{dx} \int_{0}^{x} (x - s)^{ - \gamma}u(s)\,ds, $$

provided the right-hand side is defined for almost every \(x \in(0, T)\). Herein, \(\varGamma(\cdot)\) represents the classical Gamma function.

Lemma 1.3

Let\(\alpha,\beta\geq0\). If\(u \in L^{1}(0,T)\), then\(I^{\alpha}I^{\beta}u = I^{\alpha+ \beta}u\)almost everywhere on\((0,T)\).

Lemma 1.4

Let\(\alpha\geq0\). If\(u \in L^{1}(0,T)\), then\(D^{\alpha}I^{\alpha}u = u\)almost everywhere on\((0,T)\).

Lemma 1.5

Assume that\(u \in C(0,T] \cap L^{1}(0,T)\)with a fractional derivative of order\(0 < \alpha\leq1\)that belongs to\(C(0,T] \cap L^{1}(0,T)\). Then

$$I^{\alpha}D^{\alpha}u(x) = u(x)+ cx^{\alpha-1} $$

for some\(c \in\mathbb{R}\).

Throughout this paper \((X, \Vert\cdot \Vert)\) indicates a Banach space, for every \(E \subset X\), Ē indicates the closure of E, and \(\operatorname{conv}(E)\) indicates the closed convex hull of X. Also, note that \(\mathcal{M}_{X}\) is the family of non-empty bounded subsets of X and \(\mathcal{N}_{X}\) 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 \(\mu: \mathcal{M}_{X} \longrightarrow\mathbb{R}^{+}\) is said to be a measure of the noncompactness in E if it satisfies the following conditions:

\((A_{1})\):

The family \(\operatorname{Ker}\mu= \lbrace X \in\mathcal{M}_{X} : \mu (E) = 0 \rbrace\) is non-empty and \(\operatorname{Ker}\mu\subset\mathcal{N}_{X}\);

\((A_{2})\):

\(X_{1} \subset E_{2} \Rightarrow\mu(E_{1}) \leq\mu(E_{2})\);

\((A_{3})\):

\(\mu(\bar{E}) = \mu(E)\);

\((A_{4})\):

\(\mu(\operatorname{conv}E) = \mu(E)\);

\((A_{5})\):

\(\mu(\gamma E_{1} + (1-\gamma)E_{2}) \leq\gamma\mu (E_{1}) + (1-\gamma)\mu(E_{2})\) for \(0 \leq\gamma\leq1\);

\((A_{6})\):

if \((E_{n})\) is a sequence of closed sets from \(\mathcal {M}_{E}\) such that \(E_{n +1} \subset E_{n}\) and \(\lim_{n \rightarrow \infty}\mu(E_{n}) = 0\), then the intersection set \(E_{\infty} = \bigcap_{n=1}^{\infty}E_{n}\) is non-empty.

In addition, the definition of the Hausdorff measure of noncompactnesss χ which can be found in [27] is expressed as follows:

$$\chi(s) = \inf\lbrace\epsilon> 0: S \textit{ has finite } \epsilon \textit{-net in } X \rbrace. $$

Lemma 1.6

([29])

LetΛbe a non-empty, closed, bounded and convex subset of a Banach spaceXand let\(H : \varLambda\longrightarrow\varLambda\)be a continuous mapping such that there exists a constant\(L\in [0,1)\)with the property\(\mu(H(\varLambda)) \leq L\mu(\varLambda)\). ThenHhas a fixed point inΛ.

Proposition 1.7

([30])

If\(W \subset C(I,X)\)for all continuous functions onItoEis bounded and equicontinuous, then the set\(\mu(W(x))\)is continuous onIand

$$\mu(W) = \sup_{x \in I} \mu\bigl(W(x)\bigr),\qquad \mu \biggl( \int_{0}^{x} W(\eta)\,d\eta \biggr) \leq \int_{0}^{x} \mu\bigl(W(\eta)\bigr)\,d\eta. $$

Theorem 1.8

([30])

LetQbe a bounded subset of the Banach space\(X = c_{0}\). As\((e^{(1)},e^{(2)},\ldots)\)is a Schauder basis for\(c_{0}\), the HausdorffMNCχforQis given by

$$\chi_{c_{0}} = \lim_{n \rightarrow\infty} \Bigl\lbrace \sup _{x \in Q}\Bigl(\max_{k\geq n} \vert x_{k} \vert \Bigr) \Bigr\rbrace . $$

Theorem 1.9

([30])

LetQbe a bounded subset of the Banach space\(X = \ell_{p}\). As\((e^{(1)},e^{(2)},\ldots)\)is a Schauder basis for\(\ell_{p}\), the Hausdorff\(MNC\)χforQis given by

$$\chi_{\ell_{p}} = \lim_{n \rightarrow\infty} \biggl\lbrace \sup _{x \in Q} \biggl(\sum_{k\geq n} \vert x_{k} \vert ^{p} \biggr)^{\frac{1}{p}} \biggr\rbrace . $$

2 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 \((Q_{2})\).

Theorem 2.1

Let\(f : [0,T]\times\mathbb{R}^{2} \rightarrow\mathbb{R}\)be a function such that, for every\(u \in C_{1-\alpha}^{\alpha}[0,T]\), \(f(x,u(x),\mathcal{D}^{\alpha }u(x)) \in C_{\gamma}[0,T]\)for some\(0 \leq\gamma< 1\), and also

$$\bigl\vert f(x,u,v) - f(x,\tilde{u},\tilde{v}) \bigr\vert \leq L_{1} \vert u-\tilde{u} \vert + L_{2} \vert v - \tilde{v} \vert ,\quad \forall x \in(0,t], \forall(u,v),(\tilde{u},\tilde{v}) \in \mathbb{R}^{2} $$

for some\(L_{1},L_{2} > 0\). Then there exists\(0 < \delta\leq T\)such that the problem (1) possesses a unique solution in\(C_{1-\alpha}^{\alpha}[0,\delta]\).

Proof

Fix \(\delta>0\) such that

$$l = \max\{L_{1},L_{2}\} \biggl(\delta^{2\alpha} \frac{\varGamma(\alpha )}{\varGamma(3\alpha)} + \delta^{\alpha}\frac{\varGamma(\alpha )}{\varGamma(2\alpha)} \biggr)< 1. $$

Consider the operator A defined on \(C_{1-\alpha}^{\alpha}[0,\delta]\) by

$$Au(x) = u_{0}x^{\alpha-1} + u_{1} \frac{\varGamma(\alpha)}{\varGamma (2\alpha)}x^{2\alpha-1} + I^{2\alpha} f\bigl(x,u(x), \mathcal{D}^{\alpha}u(x)\bigr) $$

for each \(u \in C_{1-\alpha}^{\alpha}[0,\delta]\) and \(x \in[0,\delta ] \). 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 \(u, \tilde{u} \in C_{1-\alpha}^{\alpha}[0,\delta]\), we have

$$\begin{aligned} & \bigl\Vert Au(x) - A\tilde{u}(x) \bigr\Vert _{ C_{1-\alpha}^{\alpha}[0,\delta]} \\ &\quad= \bigl\Vert I^{2\alpha} \bigl[f\bigl(x,u(x),\mathcal{D}^{\alpha}u(x) \bigr)- f\bigl(x,\tilde{u}(x), \mathcal{D}^{\alpha}\tilde{u}(x)\bigr) \bigr] \bigr\Vert _{C_{1-\alpha}^{\alpha}[0,\delta]} \\ &\quad\leq L_{1} \bigl\Vert I^{2\alpha} \bigl\vert \bigl(u(x)-\tilde{u}(x)\bigr) \bigr\vert \bigr\Vert _{C_{1-\alpha }^{\alpha}[0,\delta]} + L_{2} \bigl\Vert I^{2\alpha} \bigl\vert {D}^{\alpha }\bigl(u(x)-\tilde{u}(x)\bigr) \bigr\vert \bigr\Vert _{C_{1-\alpha}^{\alpha}[0,\delta]} \\ &\quad= L_{1} \bigl\Vert I^{2\alpha} \bigl\vert \bigl(u(x)- \tilde{u}(x)\bigr) \bigr\vert \bigr\Vert _{C_{1-\alpha }[0,\delta]} + L_{1} \bigl\Vert I^{\alpha} \bigl\vert \bigl(u(x)- \tilde{u}(x)\bigr) \bigr\vert \bigr\Vert _{C_{1-\alpha}[0,\delta]} \\ &\qquad{}+ L_{2} \bigl\Vert I^{2\alpha} \bigl\vert {D}^{\alpha}\bigl(u(x)-\tilde{u}(x)\bigr) \bigr\vert \bigr\Vert _{C_{1-\alpha} [0,\delta]} + L_{2} \bigl\Vert I^{\alpha} \bigl\vert {D}^{\alpha }\bigl(u(x)-\tilde{u}(x)\bigr) \bigr\vert \bigr\Vert _{C_{1-\alpha} [0,\delta]} \\ &\quad\leq\max\{L_{1},L_{2}\} \biggl(\delta^{2\alpha} \frac{\varGamma (\alpha)}{\varGamma(3\alpha)} + \delta^{\alpha}\frac{\varGamma (\alpha)}{\varGamma(2\alpha)} \biggr) \Vert u -\tilde{u} \Vert _{C_{1-\alpha}^{\alpha}[0,\delta]} \\ &\quad\leq L \Vert u -\tilde{u} \Vert _{C_{1-\alpha}^{\alpha}[0,\delta]}. \end{aligned}$$

Since the space \(C_{1-\alpha}^{\alpha}[0,\delta]\) 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). □

Theorem 2.2

Let\(f : [0,T]\times\mathbb{R}^{n} \rightarrow\mathbb{R}\)be a function such that, for every\(u \in C_{1-\alpha}^{n\alpha} [0,T]\),

$$f\bigl(x,u(x),\mathcal{D}^{\alpha}u(x),\mathcal{D}^{2\alpha}u(x), \ldots ,\mathcal{D}^{(n-1)\alpha}u(x)\bigr) \in C_{\gamma}[0,T] $$

for some\(0 \leq\gamma< 1\), and also

$$\bigl\vert f(x,u_{1},\ldots,u_{n}) - f(x, \tilde{u}_{1},\ldots,\tilde {u}_{n}) \bigr\vert \leq L \sum_{i=1}^{n} \vert u_{i} - \tilde{u}_{i} \vert u_{i} \geq\tilde{u}_{i},\quad i=1,2,\ldots,n $$

for some\(L > 0\). Then there exists\(0 < \delta\leq T\)such that the problem (4) possesses a unique solution in\(C_{1-\alpha}^{n\alpha} [0,\delta]\).

Proof

The proof is the same as Theorem 2.1. □

Remark 2.1

Let \(f : [0,T]\times\mathbb{R}^{2} \rightarrow\mathbb{R}\) be a non-decreasing function in all its arguments except for the first argument such that

$$f(x,u,v) - f(x,\tilde{u},\tilde{v}) \leq L_{1} (u-\tilde{u}) + L_{2}(v - \tilde{v}) $$

for some \(L_{1},L_{2} > 0\) for all \(u \geq\tilde{u}\), \(v \geq \tilde{v}\) and \(x \in(0,T]\). Then

$$\bigl\vert f(x,u,v) - f(x,\tilde{u},\tilde{v}) \bigr\vert \leq L_{1} \vert u-\tilde{u} \vert + L_{2} \vert v - \tilde{v} \vert $$

for all \((u,v),(\tilde{u},\tilde{v}) \in\mathbb{R}^{2}\) and \(x \in(0,T]\).

Remark 2.2

Let \(f : [0,T]\times\mathbb{R}^{n} \rightarrow\mathbb{R}\) be a non-decreasing function in all its arguments except for the first argument such that

$$f(x,u_{1},\ldots,u_{n}) - f(x,\tilde{u}_{1}, \ldots,\tilde{u}_{n}) \leq L \sum_{i=1}^{n}(u_{i} - \tilde{u}_{i}) u_{i} \geq\tilde{u}_{i},\quad i=1,2,\ldots,n $$

for \(L > 0\) and \(x \in(0,T]\). Then

$$\bigl\vert f(x,u_{1},\ldots,u_{n}) - f(x, \tilde{u}_{1},\ldots,\tilde{u}_{n}) \bigr\vert \leq L \sum_{i=1}^{n} \vert u_{i} - \tilde{u}_{i} \vert $$

for all \((u_{1},\ldots,u_{n}),(\tilde{u}_{1},\ldots,\tilde{u}_{n}) \in \mathbb{R}^{n}\) and \(x \in(0,T]\).

Remark 2.3

By using Remark 2.1 and Remark 2.2, we can conclude that our theorems are really generalizations of Theorem 1.1 and Theorem 1.2.

Remark 2.4

In Theorem 1.1 and Theorem 1.2, we need to find a lower solution of (1) and (4), respectively, while we do not need this assumption in Theorem 2.1 and Theorem 2.2. In general, finding a lower solution of (1) and (4) is difficult.

Remark 2.5

The mapping f in Theorem 1.1 and Theorem 1.2 is non-decreasing in all its arguments except for the first argument, while this assumption is not required in our theorems.

Now, with the following examples, we show that our main theorems are generalizations of the main theorems of [1] that are Theorem1.1 and Theorem1.2.

Example 2.1

The linear initial value problem is given as follows:

$$ \left \{ \textstyle\begin{array}{l} \mathcal{D}u(x) = x^{2} - \frac{\nu_{1}\sin(2k)\mathcal{D}^{\alpha }u(x)}{8} - \frac{\nu_{2}\tanh(x)u(x)}{4}, \quad x \in(0,T], \\ \lim_{x \rightarrow0}x^{\frac{1}{2}}u(x)=a, \qquad\lim_{x \rightarrow 0}x^{\frac{1}{2}}\mathcal{D}^{\frac{1}{2}}u(x)= b. \end{array}\displaystyle \right . $$
(6)

This problem is a special case of (1) with \(\alpha= \frac {1}{2}\), \(T,a,b,\nu_{1},\nu_{2} > 0\), \(\max\{ \nu_{1},\nu_{2} \} < \frac {4}{(2 + \sqrt{\pi})}\) and \(f(x,u(x),\mathcal{D}^{\alpha}u(x)) = x^{2} - \frac{\nu_{1}\sin (2k)\mathcal{D}^{\alpha}u(x)}{8} - \frac{\nu_{2}\tanh(x)u(x)}{4}\). It is easy to see that

$$\bigl\vert f\bigl(x,u(x),\mathcal{D}^{\alpha}u(x)\bigr) - f \bigl(x,v(x),\mathcal {D}^{\alpha}v(x)\bigr) \bigr\vert \leq \frac{\nu_{2}}{4} \bigl\vert v(x) - u(x) \bigr\vert + \frac{\nu_{1}}{8} \bigl\vert \mathcal{D}^{\alpha}v(x) - \mathcal {D}^{\alpha}u(x) \bigr\vert . $$

Applying Theorem 2.1 the linear initial value problem (6) possesses a unique solution in \(C_{1-\alpha}^{\alpha}[0,\gamma]\). 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.

Example 2.2

Let \(\alpha= \frac{2}{3}\), \(n = 4\), \(a_{i} \geq0\), \(i = 1,2,3,4\) and

$$\begin{aligned} f\bigl(x,u(x),\mathcal{D}^{\alpha}u(x),\mathcal {D}^{2\alpha}u(x), \mathcal{D}^{3\alpha}u(x)\bigr) = & e^{x} - \frac{\zeta _{1}}{e^{(x+3\alpha)}}u(x) - \frac{\zeta_{2}}{e^{(x+3\alpha)}}\mathcal{D}^{\alpha}u(x) \\ &{}+\frac{\zeta_{3}}{e^{(x+3\alpha)}}\mathcal{D}^{2\alpha}u(x) - \frac{\zeta_{4}}{e^{(x+3\alpha)}}\mathcal{D}^{3\alpha}u(x). \end{aligned}$$

If \(\max_{1 \leq i \leq4} \lbrace\zeta_{i} \rbrace< \frac {e^{4}}{5.86}\), then, by applying Theorem (4), the problem

$$ \left \{ \textstyle\begin{array}{ll} \mathcal{D}^{\frac{8}{3}} u(x) = e^{x} - \frac{\zeta _{1}}{e^{(x+2)}}u(x) - \frac{\zeta_{2}}{e^{(x+2)}}\mathcal{D}^{\alpha }u(x) + \frac{\zeta_{3}}{e^{(x+2)}}\mathcal{D}^{2\alpha}u(x) - \frac{\zeta _{4}}{e^{(x+2)}}\mathcal{D}^{3\alpha}u(x), \\ \lim_{x \rightarrow0}x^{\frac{1}{3}}u(x)=a_{1},\qquad \lim_{x \rightarrow 0}x^{\frac{1}{3}}\mathcal{D}^{\frac{2}{3}}u(x)= a_{2},\\ \lim_{x \rightarrow0}x^{\frac{1}{3}}\mathcal{D}^{\frac{4}{3}}u(x)= a_{3},\qquad \lim_{x \rightarrow0}x^{\frac{1}{3}}\mathcal{D}^{2}u(x)= a_{4}, \end{array}\displaystyle \right . $$
(7)

where \(x \in(0,T]\), possesses a unique solution. On the other hand, since f is not increasing, Theorem 2.2 is not applicable here.

3 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 \(\ell_{p}\) 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:

$$C_{1-\alpha}(I , X) = \bigl\{ u\in C((0,T] , X):x^{1-\alpha}u \in C(I ,X)\bigr\} , $$

with the norm \(\Vert u \Vert_{C_{1-\alpha}(I , X)} = \max_{0 \leq x \leq T} \Vert x^{1 - \alpha}u(x) \Vert_{X}\).

Definition 3.2

We denote the spaces of continuous functions

$$C^{\alpha}_{1-\alpha}(I, X) = \bigl\{ u \in C_{1-\alpha}(I , X):\mathcal {D}^{\alpha}u \in C_{1-\alpha}(I,X) \bigr\} , $$

with the norm \(\Vert u \Vert_{C^{\alpha}_{1-\alpha}(I, X)} = \Vert u \Vert_{C_{1-\alpha}(I, X)} + \Vert\mathcal{D}^{\alpha}u \Vert _{C_{1-\alpha}(I, X)}\).

Throughout this section, we define \(I = (0,T]\), \(u_{0} = \lbrace u_{n}^{0}\rbrace_{n=1}^{\infty}\), \(u_{1} = \lbrace u_{n}^{1}\rbrace _{n=1}^{\infty}\), \(u(x) = \lbrace u_{n}(x)\rbrace_{n=1}^{\infty} \) and \(f(x,u(x),\mathcal {D}^{\alpha}u(x)) = \lbrace f_{n}(x,u(x),\mathcal{D}^{\alpha}u(x)) \rbrace_{n=1}^{\infty}\), which belongs to some Banach space \({(X,\Vert\cdot \Vert)}\). Therefore, one has system (5) as follows:

$$ \left \{ \textstyle\begin{array}{ll} \mathcal{D}^{2\alpha}u(x) = f(x,u(x),\mathcal{D}^{\alpha}u(x)),\quad x \in(0,T], \\ \lim_{x \rightarrow0}x^{1-\alpha}u(x)=u^{0}, \qquad\lim_{x \rightarrow 0}x^{1-\alpha}\mathcal{D}^{\alpha}u(x)= u^{1}, \quad n=1,2,\ldots, \end{array}\displaystyle \right . $$
(8)

where \(u_{n}(x)\), \(n =1, 2, 3, \dots\), are continuous on I, f is defined on \(I \times X \times X \longrightarrow X\) and \(f_{i}\) is a real valued function.

3.1 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 \(\Vert u \Vert= \sup\lbrace\vert u_{i}\vert: i = 1,2,3, \ldots\rbrace\).

Suppose that the following conditions are satisfied:

\((C_{1})\):

\(\lbrace u_{0}^{n}\rbrace_{n=1}^{\infty}\) and \(\lbrace u_{1}^{n}\rbrace_{n=1}^{\infty}\) belong to \(c_{0}\);

\((C_{2})\):

for any fixed u, \(f(x,u(x),\mathcal{D}^{\alpha}u(x))\) is measurable;

\((C_{3})\):

for each \(x \in I\), \(u(x) \in c_{0}\) and \(i = 1,2, \dots \), we have

$$\bigl\vert f_{i}\bigl(x,u(x),\mathcal{D}^{\alpha}u(x) \bigr) \bigr\vert \leq j_{i}(x) + k_{i}(x) \sup\bigl\lbrace \vert u_{n} \vert : n \geq i\bigr\rbrace , $$

where \(j_{i}(x)\) and \(k_{i}(x)\) are continuous real valued functions on I such that the sequence \(\lbrace k_{i}(x) \rbrace_{i =1}^{\infty}\) is equibounded on I and the sequence \((j_{i}(x))\) converges uniformly on I to the zero function identically;

\((C_{4})\):

the family of \(\lbrace f_{x}(u) \rbrace_{x \in I}\) where \(f_{x}(u) = f(x,u(x),\mathcal{D}^{\alpha}u(x))\) is equicontinuous at any point of the space \(c_{0}\).

Theorem 3.1

Under the conditions\((C_{1})\)–\((C_{4})\), with\((\frac{T^{\alpha}}{\varGamma(\alpha+1)}+\frac{T^{2 \alpha}}{\varGamma(2\alpha +1)} ) K < 1\), where\(\max_{i} \sup_{x \in I}\vert k_{i}(x) \vert\leq K\), the infinite system (5) possesses at least one solution\(\lbrace u_{n}(x)\rbrace_{n=1}^{\infty} =u(x) \in c_{0}\)for any\(x \in I\).

Proof

Suppose that \(u(x) = \lbrace u_{n}(x)\rbrace_{n=1}^{\infty}\) satisfies the boundary conditions of the infinite system (8). We define the operator \(\mathcal{A} : C^{\alpha}_{1-\alpha}(I, c_{0}) \longrightarrow C^{\alpha}_{1-\alpha}(I, c_{0})\) by

$$\mathcal{A}u(x) = u_{0} x^{\alpha-1} + u_{1} \frac{\varGamma(\alpha )}{\varGamma(2\alpha)} x^{2\alpha-1} + I^{2\alpha} f\bigl(x,u(x), \mathcal{D}^{\alpha}u(x)\bigr). $$

Applying \((C_{2})\), \(\mathcal{A}\) is well defined. We show that \(\mathcal{A}\) is bounded on \(C^{\alpha}_{1-\alpha}(I, c_{0})\),

$$\begin{aligned} & \bigl\Vert x^{1-\alpha}\mathcal{A}u(x) \bigr\Vert _{c_{0}} + \bigl\Vert x^{1-\alpha }\mathcal{D}^{\alpha}\mathcal{A}u(x) \bigr\Vert _{c_{0}} \\ &\quad= \biggl\Vert u_{0} + u_{1} \frac{\varGamma(\alpha)}{\varGamma(2\alpha)} x^{\alpha} + x^{1 - \alpha} I^{2\alpha} f\bigl(x,u(x), \mathcal{D}^{\alpha}u(x)\bigr) \biggr\Vert _{c_{0}} \\ &\quad\quad{}+ \bigl\Vert u_{1} +x^{1-\alpha} I^{\alpha} f \bigl(x,u(x),\mathcal{D}^{\alpha}u(x)\bigr) \bigr\Vert _{c_{0}} \\ & \quad\leq\sup_{i \geq1} \biggl\vert u_{0}^{i} + u_{1}^{i} \frac{\varGamma(\alpha )}{\varGamma(2\alpha)} x^{\alpha} +x^{1-\alpha} I^{2\alpha} f_{i}\bigl(x,u(x), \mathcal{D}^{\alpha}u(x)\bigr) \biggr\vert \\ &\qquad{}+ \sup_{i \geq1} \bigl\vert u_{1}^{i} + x^{1-\alpha} I^{\alpha} f_{i}\bigl(x,u(x), \mathcal{D}^{\alpha}u(x)\bigr) \bigr\vert \\ &\quad \leq \sup_{i \geq1} \bigl\vert u_{0}^{i} \bigr\vert + \sup_{i \geq1} \bigl\vert u_{1}^{i} \bigr\vert \biggl(\frac{\varGamma(\alpha)}{\varGamma(2\alpha)} x^{\alpha}+1 \biggr) \\ &\qquad{}+ \frac{x^{1-\alpha}}{\varGamma(2\alpha)} \sup_{i \geq1} \biggl\vert \int_{0}^{x} (x - \eta)^{2\alpha-1} f_{i}\bigl(\eta,u(\eta),\mathcal{D}^{\alpha}u(\eta)\bigr)\,d \eta \biggr\vert \\ &\qquad{}+ \frac{x^{1-\alpha}}{\varGamma(\alpha)} \sup_{i \geq1} \biggl\vert \int_{0}^{x} (x - \eta)^{\alpha-1} f_{i}\bigl(\eta,u(\eta), \mathcal{D}^{\alpha}u(\eta)\bigr)\,d \eta \biggr\vert \\ & \quad\leq \Vert u_{0} \Vert _{c_{0}} + \Vert u_{1} \Vert _{c_{0}} \biggl(\frac {\varGamma(\alpha)}{\varGamma(2\alpha)}T^{\alpha}+1 \biggr) \\ &\qquad{}+ \frac{T^{1-\alpha}}{ \varGamma(2\alpha)}\sup_{i \geq1}\biggl\vert \int_{0}^{x} (x - \eta )^{2\alpha-1} \bigl(j_{i}(x) + k_{i}(x) \sup\bigl\lbrace \vert u_{n} \vert : n \geq i\bigr\rbrace \bigr)\,d\eta\biggr\vert \\ &\quad\quad{}+ \frac{T^{1-\alpha}}{\varGamma(\alpha)}\sup_{i\geq1}\biggl\vert \int_{0}^{x} (x - \eta)^{\alpha-1} \bigl(j_{i}(x) + k_{i}(x) \sup\bigl\lbrace \vert u_{n} \vert : n \geq i\bigr\rbrace \bigr)\,d\eta\biggr\vert . \end{aligned}$$

Using \((C_{3})\), there exists \(J = \max_{i} \sup_{x \in I} \vert j_{i}(x) \vert\), therefore

$$\begin{aligned} &\max_{x \in I} \bigl\Vert x^{1-\alpha}\mathcal{A}u(x) \bigr\Vert _{c_{0}} + \max_{x \in I} \bigl\Vert x^{1-\alpha}\mathcal{D}^{\alpha}\mathcal{A}u(x) \bigr\Vert _{c_{0}} \\ &\quad\leq\max_{x \in I} \biggl( \Vert u_{0} \Vert _{c_{0}} + \Vert u_{1} \Vert _{c_{0}} \biggl( \frac{\varGamma(\alpha)}{\varGamma(2\alpha)} x^{\alpha}+1 \biggr) \\ &\qquad{}+ \frac{x^{1-\alpha}}{\varGamma(2\alpha)}\sup_{i \geq1} \int _{0}^{x} (x - \eta)^{2\alpha-1} \bigl( \bigl\vert j_{i}(x) \bigr\vert + \bigl\vert k_{i}(x) \bigr\vert \sup\bigl\lbrace \vert u_{n} \vert : n \geq i\bigr\rbrace \bigr)\,d\eta \\ &\quad\quad{}+ \frac{x^{1-\alpha}}{\varGamma(\alpha)}\sup_{i \geq1} \int _{0}^{x} (x - \eta)^{\alpha-1} \bigl(j_{i}(x) + k_{i}(x) \sup\bigl\lbrace \vert u_{n} \vert : n \geq i\bigr\rbrace \bigr)\,d\eta. \biggr) \\ &\quad\leq \Vert u_{0} \Vert _{c_{0}} + \Vert u_{1} \Vert _{c_{0}} \biggl(\frac{\varGamma(\alpha)}{\varGamma(2\alpha)}t^{\alpha }+1 \biggr) \\ &\qquad{}+ \frac{1}{\varGamma(2\alpha)}\max_{x \in I}\sup _{i \geq 1}x^{1-\alpha} \int_{0}^{x} (x - \eta)^{2\alpha-1} \bigl( \bigl\vert j_{i}(x) \bigr\vert + \bigl\vert k_{i}(x) \bigr\vert \sup\bigl\lbrace \vert u_{n} \vert : n \geq i\bigr\rbrace \bigr)\,d\eta \\ &\qquad{}+ \frac{1}{\varGamma(\alpha)} \max_{x \in I}\sup _{i \geq 1}x^{1-\alpha} \int_{0}^{x} (x - \eta)^{\alpha-1} \bigl( \bigl\vert j_{i}(x) \bigr\vert + \bigl\vert k_{i}(x) \bigr\vert \sup\bigl\lbrace \vert u_{n} \vert : n \geq i\bigr\rbrace \bigr)\,d\eta \\ & \quad\leq \Vert u_{0} \Vert _{c_{0}} + \Vert u_{1} \Vert _{c_{0}} \biggl(\frac {\varGamma(\alpha)}{\varGamma(2\alpha)}T^{\alpha}+1 \biggr) + \biggl(\frac{T^{\alpha}}{\varGamma(\alpha+1)}+\frac{T^{2 \alpha }}{\varGamma(\alpha+1)} \biggr) J \\ &\qquad{}+ \biggl(\frac{T^{\alpha}}{\varGamma(2\alpha+1)}+\frac{T^{2\alpha }}{\varGamma(2\alpha+1)} \biggr) K \Vert u \Vert _{c_{0}}, \end{aligned}$$

and so

$$\begin{aligned} \Vert \mathcal{A}u \Vert \leq& \Vert u_{0} \Vert _{c_{0}} + \Vert u_{1} \Vert _{c_{0}} \biggl( \frac{\varGamma(\alpha)}{\varGamma(2\alpha )}T^{\alpha}+1 \biggr) + \biggl(\frac{T^{\alpha}}{\varGamma(\alpha+1)}+ \frac{T^{2\alpha }}{\varGamma(2\alpha+1)} \biggr) J \\ &{}+ \biggl(\frac{T^{\alpha}}{\varGamma(\alpha+1)}+\frac{T^{2 \alpha }}{\varGamma(2\alpha+1)} \biggr) K \Vert u \Vert . \end{aligned}$$

Then we conclude that

$$r = \frac{ \Vert u_{0} \Vert _{c_{0}} + \Vert u_{1} \Vert _{c_{0}} (\frac {\varGamma(\alpha)}{\varGamma(2\alpha)}T^{\alpha}+1 ) + (\frac{T^{\alpha}}{\varGamma(\alpha+1)}+\frac{T^{2\alpha }}{\varGamma(2\alpha+1)} ) J}{1- (\frac{T^{\alpha}}{\varGamma(\alpha+1)} +\frac{T^{2 \alpha}}{\varGamma(2\alpha+1)} ) K} $$

is the optimal solution of the inequality

$$\begin{aligned} &\Vert u_{0} \Vert _{c_{0}} + \Vert u_{1} \Vert _{c_{0}} \biggl(\frac{\varGamma (\alpha)}{\varGamma(2\alpha)}T^{\alpha}+1 \biggr) + \biggl(\frac{T^{\alpha}}{\varGamma(\alpha+1)}+\frac{T^{2 \alpha }}{\varGamma(\alpha+1)} \biggr) J \\ &\quad{}+ \biggl( \frac{T^{\alpha}}{\varGamma (2\alpha+1)} +\frac{T^{2\alpha}}{\varGamma(2\alpha+1)} \biggr) K r^{' } \leq r^{'}. \end{aligned}$$

Define the closed, bounded and convex set

$$B_{r} = \Bigl\lbrace u \in C^{\alpha}_{1-\alpha}(I, c_{0}): \Vert u \Vert \leq r, \lim_{x \rightarrow0} x^{1-\alpha}u_{n}(x)=u_{0}^{n}, \lim _{x \rightarrow0}x^{1-\alpha}\mathcal {D}^{\alpha}u_{n}(x)= u_{1}^{n}, \Bigr\rbrace , $$

where \(n=1,2,\dots\). Clearly, \(\mathcal{A}\) is bounded on \(B_{r}\). In the following, we show that \(\mathcal{A}\) is continuous on \(B_{r}\). We can write

$$\begin{aligned} &\bigl\Vert \mathcal{A}u(x) - \mathcal{A}v(x) \bigr\Vert _{c_{0}} \\ &\quad\leq \sup_{i \geq1} \frac{1}{\varGamma(2\alpha)} \int_{0}^{x} (x - \eta)^{2\alpha-1} \bigl\vert f_{i}\bigl(\eta,u(\eta ),\mathcal{D}^{\alpha}u(\eta) \bigr) - f_{i}\bigl(\eta,v(\eta),\mathcal {D}^{\alpha}v(\eta) \bigr) \bigr\vert \,d\eta \\ &\quad\leq \frac{1}{\varGamma(2\alpha)} \int_{0}^{x} (x - \eta )^{2\alpha-1} \bigl\Vert f\bigl(\eta,u(\eta),\mathcal{D}^{\alpha}u(\eta)\bigr) - f\bigl( \eta,v(\eta),\mathcal{D}^{\alpha}v(\eta)\bigr) \bigr\Vert _{c_{0}} \,d\eta. \end{aligned}$$

The family of \(\lbrace f_{x}(u) \rbrace_{x \in I}\) where \(f_{x}(u) = f(x,u(x),\mathcal{D}^{\alpha}u(x))\) is equicontinuous on \(c_{0}\). Bearing \((C_{4})\) in mind, we have

$$\forall v,u \in B_{r} \text{ and } \forall\epsilon>0, \quad\exists \delta> 0{:}\quad \Vert u - v \Vert \leq\delta \quad\Longrightarrow \quad\bigl\Vert f_{x}(u) -f_{x}(v) \bigr\Vert _{c_{0}} \leq \frac{\epsilon }{z}, \quad\forall x \in I, $$

where \(z = \frac{T^{2}}{\varGamma(2\alpha+1)}\). Therefore, we conclude that

$$\bigl\Vert \mathcal{A} u(x) - \mathcal{A} v(x) \bigr\Vert _{c_{0}} \leq \frac {1}{\varGamma(2\alpha)} \int_{0}^{x} (x - \eta)^{2\alpha-1} \bigl\Vert f_{\eta}(u) - f_{\eta}(v) \bigr\Vert _{c_{0}} \,d\eta< \epsilon, $$

which means that \(\mathcal{A}\) is continuous. Without loss of generality, we can suppose \(x_{1} > x_{2}\). There exist \(m_{1}\), \(m_{2}\) and \(m_{3}\) in \(\mathbb{R}^{+}\) such that

$$\begin{aligned} & \bigl\vert x_{1}^{\alpha-1} - x_{2}^{\alpha-1} \bigr\vert \leq m_{1} \vert x_{1} - x_{2} \vert , \quad\text{for all } x_{1} \leq\eta\leq x_{2}, \\ & \bigl\vert x_{1}^{\alpha-1} - x_{2}^{\alpha-1} \bigr\vert \leq m_{2} \vert x_{1} - x_{2} \vert , \quad\text{for all } x_{1} \leq\eta\leq x_{2},\\ & \bigl\vert (x_{2} - \eta)^{2\alpha- 1} - (x_{1} - \eta)^{2\alpha-1} \bigr\vert < m_{3} \vert x_{1} ? x_{2} \vert ,\quad\text{for all } x_{1} \leq\eta\leq x_{2}. \end{aligned}$$
(9)

Applying (9), for any \(u \in B_{r}\), we have

$$\begin{aligned} &\bigl\Vert \mathcal{A}u(x_{1}) - \mathcal{A}u(x_{2}) \bigr\Vert _{c_{0}} \\ &\quad=\sup_{i \geq1} \biggl\vert u_{0}^{i} \bigl(x_{1}^{\alpha-1} - x_{2}^{\alpha- 1}\bigr) + u_{1}^{i} \frac{\varGamma(\alpha)}{\varGamma(2\alpha)} \bigl(x_{1}^{2\alpha -1} - x_{2}^{2\alpha- 1}\bigr) \\ &\qquad{}+ \frac{1}{\varGamma(2\alpha)} \biggl( \int_{0}^{x_{1}} (x_{1} - \eta )^{2\alpha-1} f_{i}\bigl(\eta,u(\eta),\mathcal{D}^{\alpha}u(\eta)\bigr)\,d\eta \\ &\qquad{}- \int_{0}^{x_{2}} (x_{2} - \eta)^{2\alpha-1} f_{i}\bigl(\eta,u(\eta ),\mathcal{D}^{\alpha}u(\eta)\bigr)\,d\eta \biggr) \biggr\vert \\ &\quad\leq\sup_{i\geq1} \bigl\vert u_{0}^{i} \bigl(x_{1}^{\alpha-1} - x_{2}^{\alpha- 1} \bigr) \bigr\vert + \sup_{i\geq1} \biggl\vert u_{1}^{i} \frac{\varGamma(\alpha)}{ \varGamma(2\alpha)} \bigl(x_{1}^{2\alpha-1} - x_{2}^{2\alpha- 1}\bigr) \biggr\vert \\ &\qquad{}+\frac{1}{\varGamma(2\alpha)} \sup_{i\geq1} \biggl\vert \int _{0}^{x_{1}} \bigl((x_{2} - \eta)^{2\alpha-1}-(x_{1} - \eta)^{2\alpha -1} \bigr) f_{i}\bigl(\eta,u(\eta),\mathcal{D}^{\alpha}u(\eta)\bigr)\,d \eta \\ &\qquad{}+ \int_{x_{1}}^{x_{2}} (x_{2} - \eta)^{2\alpha-1} f_{i}\bigl(\eta,u(\eta ),\mathcal{D}^{\alpha}u(\eta)\bigr)\,d\eta \biggr\vert \\ &\quad\leq \Vert u_{0} \Vert \vert x_{2} -x_{1} \vert m_{1} + \Vert u_{1} \Vert \frac {\varGamma(\alpha)}{\varGamma(2\alpha)} m_{2} \vert x_{2} -x_{1} \vert \\ &\qquad{}+ \frac{1}{\varGamma(2\alpha)} ( \sup_{i\geq1} \int _{0}^{x_{1}} \vert x_{2} -x_{1} \vert m_{3} (j_{i}(\eta) + k_{i}(\eta) \sup \bigl\lbrace \bigl\vert u_{n}(\eta) \bigr\vert : n \geq i \bigr\rbrace \,d\eta \\ &\qquad{}+ \int_{x_{1}}^{x_{2}} (x_{2} - \eta)^{2\alpha-1} \bigl(j_{i}(\eta) + k_{i}(\eta ) \sup\bigl\lbrace \bigl\vert u_{n}(\eta) \bigr\vert : n \geq i \bigr\rbrace \,d\eta \bigr) \\ &\quad \leq \Vert u_{0} \Vert \vert x_{2} -x_{1} \vert m_{1} + \Vert u_{1} \Vert \frac {\varGamma(\alpha)}{\varGamma(2\alpha)} m_{2} \vert x_{2} -x_{1} \vert \\ &\qquad{}+\frac{1}{\varGamma(2\alpha)} \bigl( m_{3} J + K \Vert u \Vert \bigr) \vert x_{1} - x_{2} \vert + \frac{1}{\varGamma(2\alpha+ 1)} \bigl( J + K \Vert u \Vert \bigr) (x_{1} - x_{2})^{2\alpha}, \end{aligned}$$

which tends to zero when \(x_{1} \longrightarrow x_{2}\). Thus, we deduce that \(\mathcal{A} \) is equicontinuous on \(B_{r}\).

Setting \(\bar{B} = \operatorname{conv}(\mathcal{A}(B_{r}))\), clearly \(\bar{B} \subset B_{r}\). Let \(Y \subset\bar{B}\), then \(\mathcal{A}\) 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 \(C^{\alpha}_{1-\alpha}(I, c_{0})\), Proposition 1.7 and Theorem 1.8, we have

$$\chi_{C^{\alpha}_{1-\alpha}(I, c_{0})}(Y) = \sup_{x \in I} \chi _{c_{0}} \bigl(Y(x)\bigr). $$

Recalling Theorem 1.8, for any \(u \in Y\), we observe

$$\begin{aligned} \chi_{c_{0}}\bigl(\mathcal{A} u(x)\bigr) =& \lim _{i \rightarrow\infty} \Bigl\lbrace \sup_{u \in B} \Bigl( \max_{n \geq i} \mathcal{A} u_{n}(x) \Bigr) \Bigr\rbrace \\ \leq& \lim_{i \rightarrow\infty} \biggl\lbrace \sup_{u \in B} \biggl( \max_{n \geq i} \biggl\vert u_{0}^{n} x^{\alpha-1} u_{1}^{n} \frac{\varGamma(\alpha)}{\varGamma(2\alpha)} x^{2\alpha-1} \\ &{}+\frac{1}{\varGamma(2\alpha)} \int_{0}^{x} (x-\eta)^{2\alpha -1} f_{n}\bigl(x,u(x),\mathcal{D}^{\alpha}u(x)\bigr) \,d\eta \biggr\vert \biggr) \biggr\rbrace \\ \leq& \lim_{i \rightarrow\infty} \biggl\lbrace \sup_{u \in B} \biggl( \max_{n \geq i} \frac{1}{\varGamma(2\alpha)} \biggl\vert \int_{0}^{x} (x-\eta)^{2\alpha-1} \bigl(j_{n}(\eta) \\ &{}+ k_{n}(\eta) \sup\bigl\lbrace \bigl\vert u_{k}( \eta) \bigr\vert : k \geq n \bigr\rbrace \bigr) \,d\eta \biggr\vert \biggr) \biggr\rbrace \\ \leq& K \lim_{i \rightarrow\infty} \biggl\lbrace \sup_{u \in B} \biggl( \max_{n \geq i} \frac{1}{\varGamma(2\alpha)} \biggl\vert \int_{0}^{x} (x-\eta)^{2\alpha-1} \bigl( \sup\bigl\lbrace \bigl\vert u_{k}(\eta) \bigr\vert : k \geq n \bigr\rbrace \bigr) \,d\eta \biggr\vert \biggr) \biggr\rbrace \\ \leq&\frac{K T^{2\alpha}}{\varGamma(2\alpha+1)} \lim_{i \rightarrow\infty} \Bigl\lbrace \sup _{u \in B} \Bigl( \max_{n \geq i} \bigl\vert u_{n}(x) \bigr\vert \Bigr) \Bigr\rbrace . \end{aligned}$$

Therefore

$$\sup_{x \in I} \chi_{c_{0}}\bigl(\mathcal{A} u(x) \bigr) \leq\frac{KT^{2\alpha }}{\varGamma(2\alpha+1)}\sup_{x \in I} \lim _{i \rightarrow\infty} \Bigl\lbrace \sup_{u \in B} \Bigl( \max_{n \geq i} \bigl\vert u_{n}(x) \bigr\vert \Bigr) \Bigr\rbrace $$

and

$$\chi_{C^{\alpha}_{1-\alpha}(I, c_{0})}\bigl(\mathcal{A} u(x)\bigr) \leq\frac {KT^{2\alpha}}{\varGamma(2\alpha+1)} \chi_{C^{\alpha}_{1-\alpha}(I, c_{0})}(Y). $$

As \(\frac{KT^{2\alpha}}{\varGamma(2\alpha+1)} < 1\), applying Lemma 1.6, \(\mathcal{A}\) possesses at least one fixed point in \(\mathcal{A}\), which is a solution for (5) in the space \(C^{\alpha}_{1-\alpha}(I, c_{0})\). □

Now, with the following example, we clarify the main result of this subsection.

Example 3.1

The system of fractional differential equation is given as follows:

$$ \left \{ \textstyle\begin{array}{ll} \mathcal{D}^{2}u_{n}(x) = \frac{x\sin(\frac{1}{n}+1) + \tan(\arctan (x)) u_{n}(x) }{n+1},\quad x \in(0,1], \\ \lim_{x \rightarrow0}x^{1-\alpha}u_{n}(x)=\frac{a}{n}, \qquad\lim_{x \rightarrow0}x^{1-\alpha}\mathcal{D}^{\alpha}u_{n}(x)= \frac{b}{n}, n=1,2,\ldots, \end{array}\displaystyle \right . $$
(10)

where \(u_{n}(x)\), \(n =1, 2, 3, \dots\) are continuous on I. This system is a special case of (5) with \(\alpha= T= 1\),

$$f_{n}\bigl(x,u(x),\mathcal{D}^{\alpha}u(x)\bigr) = \frac{x\sin(\frac{1}{n}+1) + \tan(\arctan(x)) u_{n}(x) }{n+1}, $$

and \(a,b \geq0\). Obviously, the conditions \((C_{1})\) and \((C_{2})\) are satisfied. Hereafter, we show that \(f(x,u(x),\mathcal{D}^{\alpha}u(x)) \in c_{0}\). For any \(x\in(0,1]\) and \(u \in c_{0}\), we conclude that

$$\begin{aligned} \lim_{n \rightarrow\infty}f_{n}\bigl(x,u(x), \mathcal{D}^{\alpha}u(x)\bigr) = & \lim_{n \rightarrow\infty} \frac{x\sin(\frac{1}{n}+1) + \tan (\arctan(x)) u_{n}(x) }{n+1} \\ \leq& \lim_{n \rightarrow\infty}\frac{\sin(\frac{1}{n}+1) + \sup_{n \leq1} \vert u_{n}(x) \vert }{n+1} = 0. \end{aligned}$$

Also, clearly \(\vert f_{n}(x,u(x),\mathcal{D}^{\alpha}u(x))\vert\leq j_{n}(x) + k_{n}(x) \sup\lbrace\vert u_{k}\vert: k \geq n\rbrace\), where

$$j_{n}(x) = \frac{x\sin(\frac{1}{n}+1) }{n+1}, \qquad k_{n}(x) = \frac{ \tan (\arctan(x))}{n+1}. $$

Moreover, \(j_{n}(x)\) converges uniformly to zero and \(k_{n}(x)\) is equibounded by \(K = \frac{1}{2}\). Now, we are going to check the conditions \((C_{4})\). For any \(x \in(0,1]\) and \(u,v\in c_{0}\) with \(\Vert u(x) - v(x) \Vert< \delta\), we have

$$\begin{aligned} \bigl\Vert f_{x}(u) - f_{x}(v) \bigr\Vert _{c_{0}} = & \bigl\Vert f\bigl(x,u(x),\mathcal {D}^{\alpha}u(x)\bigr) - f\bigl(x,v(x),\mathcal{D}^{\alpha}v(x)\bigr) \bigr\Vert \\ = &\sup_{n \geq1} \bigl\vert f_{n}\bigl(x,u(x), \mathcal{D}^{\alpha}u(x)\bigr) - f_{n}\bigl(x,v(x), \mathcal{D}^{\alpha}v(x)\bigr) \bigr\vert \\ = &\sup_{n \geq1} \biggl\vert \frac{x\sin(\frac{1}{n}+1) + \tan (\arctan(x)) u_{n}(x) }{n+1} \\ &{}-\frac{x\sin(\frac{1}{n}+1) + \tan(\arctan(x)) v_{n}(x) }{n+1} \biggr\vert \\ \leq&\frac{1}{2} \sup_{n \geq1} \bigl\vert u_{n}(x) - v_{n}(x) \bigr\vert \\ \leq& \frac{1}{2} \bigl\Vert u(x) - v(x) \bigr\Vert _{c_{0}} < \epsilon. \end{aligned}$$

Applying Theorem 3.1, hence the system of fractional differential equation (5) possesses at least one solution in \(C^{\alpha}_{1-\alpha}(I, c_{0})\).

3.2 Solution in space \(l_{p}\)

In this subsection, let \(X = \ell_{p}\). For a real number \(p \geq1\), the space denoted by \(\ell_{p}\) is the Banach sequence space, when equipped with the following norm:

$$\Vert u \Vert _{P} = \Biggl( \sum_{i=1}^{\infty} \vert u_{i} \vert ^{P} \Biggr)^{\frac{1}{p}}. $$

In the following, we show that the infinite system (5) has at least on solution in the space \(\ell_{p}\), when the following conditions are satisfied:

\((C'_{1})\):

\(u_{0}\) and \(u_{1}\) belong to \(\ell_{p}\);

\((C'_{2})\):

\(f : I \times\ell_{p} \longrightarrow\ell_{p}\) is continuous;

\((C'_{3})\):

for each \(x \in[0,t]\), \(u(x) \in\ell_{p}\) and \(i = 1,2, \dots\), we have

$$\bigl\vert f_{i}\bigl(x,u(x),\mathcal{D}^{\alpha}u(x) \bigr) \bigr\vert ^{p} \leq j_{i}(x) + k_{i}(x) \vert u_{n} \vert ^{p}, $$

where \(j_{i}(x)\) and \(k_{i}(x)\) are continuous nonnegative functions on I such that the sequence \(\lbrace k_{i}(x) \rbrace_{i=1}^{\infty}\) is equibounded on I, \(\lim_{i \longrightarrow\infty}\sup k_{i}(x)\) is integrable over I and the series \(\sum_{i=1}^{\infty}j_{i}(x)\) converges uniformly on I;

\((C'_{4})\):

the family of \(\lbrace f_{x}(u) \rbrace_{x \in I}\), where \(f_{x}(u) = f(x,u(x),\mathcal{D}^{\alpha}u(x))\) is equicontinuous at any point of the space \(\ell_{p}\).

Theorem 3.2

Under the conditions\((C'_{1})\)–\((C'_{4})\), if

$$\biggl(\frac{T^{2\alpha}}{\varGamma(2\alpha)((2\alpha-1)p+1)^{\frac {1}{p}}} + \frac{T^{\alpha}}{\varGamma(\alpha)((\alpha-1)p+1)^{\frac {1}{p}}} \biggr) K^{\frac{1}{p}} < 1, $$

the infinite system (5) possesses at least one solution\(\lbrace u_{n}(x)\rbrace_{n=1}^{\infty} =u(x) \in\ell_{P}\)for any\(x \in I\), where\(J = \sup_{x \in I} \vert j(x)\vert\), \(j(x) = \sum_{i=1}^{\infty}j_{i}(x)\)and\(k_{i}(x)\)is equibounded byK.

Proof

Suppose that \(u(x) = \lbrace u_{n}(x)\rbrace_{n=1}^{\infty}\) satisfies the boundary conditions of the infinite system (5). We define the operator \(\mathcal{A} : C^{\alpha}_{1-\alpha}(I,\ell _{p}) \longrightarrow C^{\alpha}_{1-\alpha}(I, \ell_{p})\) by

$$\mathcal{A}u(x) = u_{0} x^{\alpha-1} + u_{1} \frac{\varGamma(\alpha )}{\varGamma(2\alpha)} x^{2\alpha-1} + I^{2\alpha} f\bigl(x,u(x), \mathcal{D}^{\alpha}u(x)\bigr). $$

Applying \((C'_{2})\), \(\mathcal{A}\) is well defined. We show that \(\mathcal{A}\) is bounded on \(C^{\alpha}_{1-\alpha}(I, \ell_{p})\).

$$\begin{aligned} \bigl\Vert x^{1-\alpha}\mathcal{A}u(x) \bigr\Vert _{p} =& \biggl\Vert u_{0} + u_{1} \frac{\varGamma(\alpha)}{\varGamma(2\alpha)} x^{\alpha} + x^{1 - \alpha} I^{2\alpha} f\bigl(x,u(x), \mathcal{D}^{\alpha}u(x)\bigr) \biggr\Vert _{p} \\ \leq& \Vert u_{0} \Vert _{p} + \Vert u_{1} \Vert _{p} \biggl( \frac {\varGamma(\alpha)}{\varGamma(2\alpha)} T^{\alpha} + 1 \biggr) \\ &{}+ \bigl\Vert x^{1 - \alpha} I^{2\alpha} f\bigl(x,u(x), \mathcal{D}^{\alpha}u(x)\bigr) \bigr\Vert _{p} + \bigl\Vert x^{1-\alpha} I^{\alpha} f\bigl(x,u(x),\mathcal{D}^{\alpha}u(x)\bigr) \bigr\Vert _{p} \\ \leq& \Vert u_{0} \Vert _{p} + \Vert u_{1} \Vert _{p} \biggl( \frac {\varGamma(\alpha)}{\varGamma(2\alpha)} T^{\alpha} \biggr) \\ &{}+T^{\frac{p-1}{p}} \frac{x^{1-\alpha}}{\varGamma(2\alpha)} \biggl( \sum _{i \geq1} \int_{0}^{x} \bigl\vert (x - \eta)^{2\alpha-1} \bigr\vert ^{p} \bigl\vert f_{i}\bigl(\eta,u(\eta ),\mathcal{D}^{\alpha}u(\eta)\bigr) \bigr\vert ^{P} \,d\eta \biggr)^{\frac{1}{p}} \\ \leq& \Vert u_{0} \Vert _{p} + \Vert u_{1} \Vert _{p} \biggl( \frac {\varGamma(\alpha)}{\varGamma(2\alpha)} T^{\alpha} \biggr) \\ &{}+ T^{\frac{p-1}{p}} \frac{x^{1-\alpha}}{\varGamma(2\alpha)} \biggl( \sum _{i \geq1} \int_{0}^{x} \bigl\vert (x - \eta)^{2\alpha-1} \bigr\vert ^{p} \bigl(j_{i}(\eta) + k_{i} (\eta) \bigl\vert u_{i}(\eta) \bigr\vert ^{P} \bigr) \,d\eta \biggr)^{\frac{1}{p}} \\ \leq& \Vert u_{0} \Vert _{p} + \Vert u_{1} \Vert _{p} \biggl( \frac {\varGamma(\alpha)}{\varGamma(2\alpha)} T^{\alpha} \biggr) \\ &{}+ T^{\frac{p-1}{p}} \frac{x^{1-\alpha}}{\varGamma(2\alpha)} \biggl(J \frac{x^{(2\alpha -1)p+1}}{(2\alpha-1)p+1} + K \bigl\Vert u(x) \bigr\Vert _{p}^{p} \frac{x^{(2\alpha-1)p+1}}{(2\alpha-1)p+1} \biggr)^{\frac{1}{p}} \\ \leq& \Vert u_{0} \Vert _{p} + \Vert u_{1} \Vert _{p} \biggl( \frac {\varGamma(\alpha)}{\varGamma(2\alpha)} T^{\alpha} \biggr) \\ &{}+ \frac{T^{\frac{p-1}{p}} x^{1-\alpha}}{\varGamma(2\alpha)} \biggl( J^{\frac{1}{p}}\frac{x^{(2\alpha-1)+\frac{1}{p}}}{ ((2\alpha-1)p+1)^{\frac{1}{p}}} + K^{\frac{1}{p}} \bigl\Vert u(x) \bigr\Vert _{p} \frac{x^{(2\alpha-1)+\frac{1}{p}}}{((2\alpha-1)p+1)^{\frac {1}{p}}} \biggr) \\ \leq& \Vert u_{0} \Vert _{p} + \Vert u_{1} \Vert _{p} \biggl( \frac {\varGamma(\alpha)}{\varGamma(2\alpha)} T^{\alpha} \biggr) \\ &{}+ \frac{T^{(1 + \alpha)} J^{\frac{1}{p}}}{\varGamma(2\alpha)((2\alpha-1)p+1)^{\frac{1}{p}}} + \frac{T^{(2 \alpha)}K^{\frac{1}{p}}}{\varGamma(2\alpha)((2\alpha-1)p+1) ^{\frac{1}{p}}} \bigl\Vert x^{1 -\alpha}u(x) \bigr\Vert _{p} \end{aligned}$$

and

$$\begin{aligned} &\bigl\Vert x^{1-\alpha}\mathcal{D}^{\alpha}\mathcal{A}u(x) \bigr\Vert _{p} \\ &\quad \leq \Vert u_{1} \Vert _{p} + \bigl\Vert x^{1-\alpha} I^{\alpha} f \bigl(x,u(x),\mathcal{D}^{\alpha}u(x)\bigr) \bigr\Vert _{p} \\ &\quad\leq \Vert u_{1} \Vert _{p} + \frac{x^{1-\alpha}}{\varGamma(\alpha )} \biggl( \sum_{i \geq1} \biggl\vert \int_{0}^{x} (x - \eta)^{\alpha-1} f_{i}\bigl(\eta,u(\eta),\mathcal{D}^{\alpha}u(\eta)\bigr)\,d \eta \biggr\vert ^{p} \biggr)^{\frac{1}{p}} \\ &\quad\leq \Vert u_{1} \Vert _{p} + T^{\frac{p-1}{p}} \frac{x^{1-\alpha }}{\varGamma(\alpha)} \biggl( \sum_{i \geq1} \int_{0}^{x} \bigl\vert (x - \eta)^{\alpha-1} \bigr\vert ^{P} \bigl\vert f_{i}\bigl(\eta,u(\eta),\mathcal {D}^{\alpha}u(\eta)\bigr) \bigr\vert ^{P} \,d\eta \biggr)^{\frac{1}{p}} \\ &\quad\leq \Vert u_{1} \Vert _{p} + T^{\frac{p-1}{p}} \frac{x^{1-\alpha }}{\varGamma(\alpha)} \biggl( \sum_{i \geq1} \int_{0}^{x} \bigl\vert (x - \eta)^{\alpha-1} \bigr\vert ^{P} \bigl(j_{i}( \eta)+ k_{i} (\eta) \bigl\vert u_{i}(\eta ) \bigr\vert ^{P}\bigr) \,d\eta \biggr)^{\frac{1}{p}} \\ &\quad\leq \Vert u_{1} \Vert _{p} + T^{\frac{p-1}{p}} \frac{x^{1-\alpha }}{\varGamma(\alpha)} \biggl( J \frac{x^{(\alpha-1)p+1}}{(\alpha-1)p+1} + K \bigl\Vert u(x) \bigr\Vert _{p}^{P} \frac{x^{(\alpha-1)p+1}}{(\alpha-1)p+1} \biggr)^{\frac{1}{p}} \\ &\quad\leq \Vert u_{1} \Vert _{p} + T^{\frac{p-1}{p}} \frac{x^{1-\alpha }}{\varGamma(\alpha)} \biggl( J^{\frac{1}{p}} \frac{x^{(\alpha-1)+\frac{1}{p}}}{((\alpha-1)p+1)^{\frac{1}{p}}} + K^{\frac{1}{p}} \bigl\Vert u(x) \bigr\Vert _{p} \frac{x^{(\alpha-1)+\frac{1}{p}}}{((\alpha-1)p+1)^{\frac{1}{p}} } \biggr) \\ &\quad\leq \Vert u_{1} \Vert _{p} + \frac{TJ^{\frac{1}{p}}}{\varGamma (\alpha)((\alpha-1)p+1)^{\frac{1}{p}}} + \frac{T^{\alpha}K^{\frac{1}{p}}}{\varGamma(\alpha)((\alpha -1)p+1)^{\frac{1}{p}}} \bigl\Vert x^{1-\alpha}u(x) \bigr\Vert _{p}. \end{aligned}$$

Therefore, we have

$$\begin{aligned} &\max_{x \in I} \bigl\Vert x^{1-\alpha}\mathcal{A}u(x) \bigr\Vert _{p} + \max_{x \in I} \bigl\Vert x^{1-\alpha}\mathcal{D}^{\alpha}\mathcal{A}u(x) \bigr\Vert _{p} \\ &\quad\leq\max_{x \in I} \Vert u_{0} \Vert _{p} + \max_{x \in I} \Vert u_{1} \Vert _{p} \biggl( \frac{\varGamma(\alpha)}{\varGamma(2\alpha)} T^{\alpha} + 1 \biggr) + \frac{T^{(1 + \alpha)} J^{\frac{1}{p}}}{\varGamma (2\alpha)((2\alpha-1)p+1)^{\frac{1}{p}}} \\ &\qquad{} + \frac{T^{(2 \alpha)}K^{\frac{1}{p}}}{\varGamma(2\alpha )((2\alpha-1)p+1)^{\frac{1}{p}}} \max_{x \in I} \bigl\Vert x^{1 -\alpha }u(x) \bigr\Vert _{p} \\ &\qquad{}+ \frac{TJ^{\frac{1}{p}}}{\varGamma(\alpha)((\alpha -1)p+1)^{\frac{1}{p}}} + \frac{T^{\alpha}K^{\frac{1}{p}}}{ \varGamma(\alpha)((\alpha-1)p+1)^{\frac{1}{p}}} \max_{x \in I} \bigl\Vert x^{1-\alpha}u(x) \bigr\Vert _{p}, \end{aligned}$$

and so

$$\begin{aligned} \Vert \mathcal{A}u \Vert \leq& \Vert u_{0} \Vert _{p} + \Vert u_{1} \Vert _{p} \biggl( \frac{\varGamma(\alpha)}{\varGamma(2\alpha )}T^{\alpha}+1 \biggr) \\ &{}+ \biggl(\frac{T^{1 + \alpha}}{\varGamma(2\alpha)((2\alpha -1)p+1)^{\frac{1}{p}}} + \frac{T}{\varGamma(\alpha)((\alpha -1)p+1)^{\frac{1}{p}}} \biggr) J^{\frac{1}{p}} \\ &{}+ \biggl(\frac{T^{2\alpha}}{\varGamma(2\alpha)((2\alpha -1)p+1)^{\frac{1}{p}}} + \frac{T^{\alpha}}{\varGamma(\alpha)((\alpha -1)p+1)^{\frac{1}{p}}} \biggr) K^{\frac{1}{p}} \bigl\Vert u(x) \bigr\Vert . \end{aligned}$$

Then we see that

$$r = \frac{ \Vert u_{0} \Vert _{p} + \Vert u_{1} \Vert _{p} (\frac {\varGamma(\alpha)}{\varGamma(2\alpha)}T^{\alpha}+1 ) + (\frac{T^{1 + \alpha}}{\varGamma(2\alpha)((2\alpha -1)p+1)^{\frac{1}{p}}} + \frac{T}{\varGamma(\alpha)((\alpha -1)p+1)^{\frac{1}{p}}} ) J^{\frac{1}{p}}}{1- (\frac{T^{2\alpha}}{\varGamma(2\alpha )((2\alpha-1)p+1)^{\frac{1}{p}}} + \frac{T^{\alpha}}{\varGamma (\alpha)((\alpha-1)p+1) ^{\frac{1}{p}}} ) K^{\frac{1}{p}}} $$

is the optimal solution of the inequality

$$\begin{aligned} &\Vert u_{0} \Vert _{p} + \Vert u_{1} \Vert _{p} \biggl(\frac{\varGamma (\alpha)}{\varGamma(2\alpha)}T^{\alpha}+1 \biggr) \\ &\quad{}+ \biggl(\frac{T^{1 + \alpha}}{\varGamma(2\alpha)((2\alpha -1)p+1)^{\frac{1}{p}}} + \frac{T}{\varGamma(\alpha)((\alpha -1)p+1)^{\frac{1}{p}}} \biggr)J^{\frac{1}{p}} \\ &\quad{}+ \biggl(\frac {T^{2\alpha}}{\varGamma(2\alpha)((2\alpha-1)p+1)^{\frac{1}{p}}} + \frac{T^{\alpha}}{\varGamma(\alpha) ((\alpha-1)p+1)^{\frac{1}{p}}} \biggr) K^{\frac{1}{p}}r^{' }\leq r^{' }. \end{aligned}$$

Define the closed, bounded and convex set

$$C_{r} = \Bigl\lbrace u \in C^{\alpha}_{1-\alpha}(I, \ell_{p}): \Vert u \Vert \leq r, \lim_{x \rightarrow0}x^{1-\alpha}u_{n}(x)=u_{0}^{n}, \lim_{x \rightarrow 0}x^{1-\alpha}\mathcal{D}^{\alpha}u_{n}(x)= u_{1}^{n}, \Bigr\rbrace , $$

where \(n=1,2,\dots\). Clearly, \(\mathcal{A}\) is bounded on \(C_{r}\). In the following, we show that \(\mathcal{A}\) is continuous on \(C_{r}\). For any \(u,v \in C_{r}\), applying Hölder’s inequality, we can write

$$\begin{aligned} &\bigl\Vert \mathcal{A}u(x) - \mathcal{A}v(x) \bigr\Vert _{p}^{p}\\ &\quad\leq\sum_{i \geq1} \biggl\vert \int_{0}^{x} \frac{(x - \eta)^{2\alpha-1}}{\varGamma(2\alpha)} \bigl(f_{i}\bigl(\eta ,u(\eta),\mathcal{D}^{\alpha}u(\eta) \bigr) - f_{i}\bigl(\eta,v(\eta),\mathcal{D}^{\alpha}v(\eta) \bigr) \bigr)\,d\eta \biggr\vert ^{p} \\ &\quad\leq T^{p-1} \sum_{i \geq1} \int_{0}^{x} \biggl\vert \frac{(x - \eta)^{2\alpha-1}}{\varGamma(2\alpha)} \biggr\vert ^{p} \bigl\vert f_{i}\bigl(\eta,u(\eta),\mathcal{D}^{\alpha}u(\eta )\bigr) - f_{i}\bigl(\eta,v(\eta),\mathcal{D}^{\alpha}v(\eta)\bigr) \bigr\vert ^{p} \,d\eta \\ &\quad\leq T^{p-1} \int_{0}^{x} \frac{(x - \eta)^{(2\alpha -1)p}}{\varGamma(2\alpha)^{p}}\,d\eta \bigl\Vert f\bigl(\eta,u(\eta),\mathcal {D}^{\alpha}u(\eta)\bigr) - f\bigl( \eta,v(\eta),\mathcal{D}^{\alpha}v(\eta)\bigr) \bigr\Vert _{p}^{p} \\ &\quad\leq\frac{T^{2\alpha}}{\varGamma(2\alpha)^{p}((2\alpha-1)p +1)} \bigl\Vert f\bigl(\eta,u(\eta), \mathcal{D}^{\alpha}u(\eta)\bigr) - f\bigl(\eta,v(\eta), \mathcal{D}^{\alpha}v(\eta)\bigr) \bigr\Vert _{p}^{p}. \end{aligned}$$

The family of \(\lbrace f_{x}(u) \rbrace_{x \in I}\) where \(f_{x}(u) = f(x,u(x),\mathcal{D}^{\alpha}u(x))\) is equicontinuous on \(\ell_{p}\). Bearing \((C'_{4})\) in mind, for all \(x \in I\), we have

$$\forall v,u \in B_{r} \text{ and } \forall\epsilon>0,\quad \exists \delta> 0 \text{ with } \Vert u - v \Vert \leq\delta\quad \text{such that} \quad\bigl\Vert f_{x}(u) -f_{x}(v) \bigr\Vert _{p} \leq\frac{\epsilon}{z}, $$

where \(z = \frac{T^{\frac{2\alpha}{p}}}{\varGamma(2\alpha)((2\alpha-1)p +1)^{\frac{1}{p}}}\). Therefore, we see that

$$\bigl\Vert \mathcal{A} u(x) - \mathcal{A} v(x) \bigr\Vert _{p} \leq \frac {t^{\frac{2\alpha}{p}}}{\varGamma(2\alpha) ((2\alpha-1)p +1)^{\frac{1}{p}}} \bigl\Vert f_{\eta}(u) - f_{\eta}(v) \bigr\Vert _{p} < \epsilon, $$

which means that \(\mathcal{A}\) is continuous.

Without loss of generality, we can suppose \(x_{1} > x_{2}\). Applying (9), for any \(u \in C_{r}\), we have

$$\begin{aligned} &\bigl\Vert \mathcal{A}u(x_{1}) - \mathcal{A}u(x_{2}) \bigr\Vert _{p}\\ &\quad \leq \bigl\Vert u_{0} \bigl(x_{1}^{\alpha-1} - x_{2}^{\alpha- 1} \bigr) \bigr\Vert _{p} + \biggl\Vert u_{1} \frac{\varGamma(\alpha)}{\varGamma (2\alpha)} \bigl(x_{1}^{2\alpha-1} - x_{2}^{2\alpha- 1}\bigr) \biggr\Vert _{p} \\ &\qquad{}+ \biggl\Vert \frac{1}{\varGamma(2\alpha)} \int_{0}^{x_{1}} \bigl((x_{2} - \eta)^{2\alpha-1}-(x_{1} - \eta)^{2\alpha-1} \bigr) f\bigl( \eta,u(\eta),\mathcal{D}^{\alpha}u(\eta)\bigr)\,d\eta \biggr\Vert _{p} \\ &\qquad{}+ \biggl\Vert \int_{x_{1}}^{x_{2}} (x_{2} - \eta)^{2\alpha-1} f\bigl(\eta ,u(\eta),\mathcal{D}^{\alpha}u(\eta) \bigr)\,d\eta \biggr\Vert _{p} \\ &\quad \leq \Vert u_{0} \Vert _{p} \vert x_{2} -x_{1} \vert m_{1} + \Vert u_{1} \Vert _{p} \frac{\varGamma(\alpha)}{\varGamma(2\alpha)} m_{2} \vert x_{2} -x_{1} \vert \\ &\quad\quad{}+ \frac{T^{\frac{p-1}{p}}}{\varGamma(2\alpha)} \biggl( \sum_{i\geq1} \int_{0}^{x_{1}} \vert x_{2} -x_{1} \vert ^{P} m_{3}^{P} \bigl(j_{i}(\eta) + k_{i}(\eta) \bigl\vert u_{i}(\eta) \bigr\vert ^{p}\bigr) \,d\eta \biggr)^{\frac{1}{p}} \\ &\quad\quad{}+ \frac{T^{\frac{p-1}{p}}}{\varGamma(2\alpha)} \biggl( \sum_{i\geq1} \int_{x_{1}}^{x_{2}} (x_{2} - \eta)^{(2\alpha-1)P} \bigl(j_{i}(\eta) + k_{i}(\eta) \bigl\vert u_{i}(\eta) \bigr\vert ^{P}\bigr) \,d\eta \biggr)^{\frac{1}{p}} \\ &\quad \leq \Vert u_{0} \Vert _{p} \vert x_{2} -x_{1} \vert m_{1} + \Vert u_{1} \Vert _{p} \frac{\varGamma(\alpha)}{\varGamma(2\alpha)} m_{2} \vert x_{2} -x_{1} \vert \\ &\qquad{}+\frac{T^{\frac{p-1}{p}}}{\varGamma(2\alpha)} \biggl( m_{3}^{P} \int _{0}^{x_{1}} \vert x_{2} -x_{1} \vert ^{P} \sum_{i\geq1} j_{i}(\eta)\,d\eta \\ &\qquad{}+ \int_{0}^{x_{1}} \vert x_{2} -x_{1} \vert ^{P} \lim_{n \rightarrow\infty }\sup k_{i}(\eta)\sum_{i\geq1} \bigl\vert u_{i}(\eta) \bigr\vert ^{P} \,d\eta \biggr)^{\frac{1}{p}} \\ &\qquad{}+\frac{T^{\frac{p-1}{p}}}{\varGamma(2\alpha)} \biggl( \int _{x_{1}}^{x_{2}} (x_{2} - \eta)^{(2\alpha-1)P}\sum_{i\geq1} j_{i}( \eta) \,d\eta \\ &\qquad{}+ \int_{x_{1}}^{x_{2}} (x_{2} - \eta)^{(2\alpha-1)p}\lim_{n \rightarrow \infty}\sup k_{i}(\eta) \sum_{i\geq1} \bigl\vert u_{i}(\eta) \bigr\vert ^{P} \,d\eta \biggr)^{\frac{1}{p}} \\ &\quad \leq \Vert u_{0} \Vert _{p} \vert x_{2} -x_{1} \vert m_{1} + \Vert u_{1} \Vert _{p} \frac{\varGamma(\alpha)}{\varGamma(2\alpha)} m_{2} \vert x_{2} -x_{1} \vert \\ &\qquad{}+\frac{T^{\frac{p-1}{p}}}{\varGamma(2\alpha)} \bigl(TJm_{3}^{P} \vert x_{2} -x_{1} \vert ^{P} + TK \Vert u \Vert ^{p} \vert x_{2} -x_{1} \vert ^{P} \bigr)^{\frac{1}{p}} \\ &\qquad{}+\frac{T^{\frac{p-1}{p}}}{\varGamma(2\alpha)} \biggl(\frac { \vert x_{2} -x_{1} \vert ^{(2\alpha-1)p+1} }{(2\alpha-1)p+1}J + \frac { \vert x_{2} -x_{1} \vert ^{(2\alpha-1)p+1} }{(2\alpha-1)p+1}K \Vert u \Vert ^{p} \biggr)^{\frac{1}{p}} \\ &\quad \leq \Vert u_{0} \Vert _{p} \vert x_{2} -x_{1} \vert m_{1} + \Vert u_{1} \Vert _{p} \frac{\varGamma(\alpha)}{\varGamma(2\alpha)} m_{2} \vert x_{2} -x_{1} \vert \\ &\qquad{}+\frac{TJ^{\frac{1}{p}}m_{3}}{\varGamma(2\alpha)} \vert x_{2} -x_{1} \vert + \frac{TJK^{\frac{1}{p}}}{\varGamma(2\alpha)} \vert x_{2} -x_{1} \vert \\ &\qquad{}+\frac{J^{\frac{1}{p}}T^{1-\frac{1}{p}}}{\varGamma(2\alpha )((2\alpha-1)p+1)^{\frac{1}{p}}} \vert x_{2} -x_{1} \vert ^{(2\alpha-1)p+1}, \end{aligned}$$

which tends to zero when \(x_{1} \longrightarrow x_{2}\). Thus, we deduce that \(\mathcal{A} \) is equicontinuous on \(C_{r}\).

Setting \(\bar{C} = \operatorname{conv}(\mathcal{A}(C_{r}))\), obviously \(\bar{C} \subset C_{r}\). Let \(Y \subset\bar{C}\), then \(\mathcal{A}\) 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 \(C^{\alpha}_{1-\alpha}(I, c_{0})\), Proposition 1.7 and Theorem 1.9, we have

$$\chi_{C^{\alpha}_{1-\alpha}(I,\ell_{p})}(Y) = \sup_{x \in I}\chi _{\ell_{p}} \bigl(Y(x)\bigr). $$

For any \(u \in Y\), we obtain

$$\begin{aligned} \chi_{\ell_{p}}\bigl(\mathcal{A} u(x)\bigr) = &\lim _{i \rightarrow\infty} \biggl\lbrace \sup_{u \in B} \biggl( \sum_{n \geq i} \bigl\vert \mathcal{A} u_{n}(x) \bigr\vert ^{p} \biggr)^{\frac{1}{p}} \biggr\rbrace \\ \leq& \lim_{i \rightarrow\infty} \biggl\lbrace \sup_{u \in B} \biggl( \sum_{n \geq i} \biggl\vert u_{n}^{0} x^{\alpha-1} + u_{n}^{1} \frac{\varGamma(\alpha)}{\varGamma(2\alpha)} x^{2\alpha -1} \\ &{}+\frac{1}{\varGamma(2\alpha)} \int_{0}^{x} (x-\eta )^{2\alpha-1} f_{n}\bigl(x,u(x),\mathcal{D}^{\alpha}u(x)\bigr) \,d\eta \biggr\vert ^{p} \biggr)^{\frac{1}{p}} \biggr\rbrace \\ \leq& \lim_{i \rightarrow\infty} \biggl\lbrace \sup_{u \in B} \frac{1}{\varGamma(2\alpha)} \biggl( \sum_{n \geq i} \biggl\vert \int_{0}^{x} (x-\eta)^{2\alpha -1} f_{n}\bigl(x,u(x),\mathcal{D}^{\alpha}u(x)\bigr) \,d\eta \biggr\vert ^{p} \biggr)^{\frac{1}{p}} \biggr\rbrace \\ \leq& \lim_{i \rightarrow\infty} \biggl\lbrace \sup_{u \in B} \frac{T^{\frac{1-p}{p}}}{\varGamma(2\alpha)} \biggl( \sum_{n \geq i} \int_{0}^{x} \bigl\vert (x- \eta)^{2\alpha-1} \bigr\vert ^{p} \bigl(j_{i}(\eta) + k_{i}(\eta) \bigl\vert u_{i}(\eta) \bigr\vert ^{p} \bigr) \,d\eta \biggr)^{\frac{1}{p}} \biggr\rbrace \\ \leq&\lim_{i \rightarrow\infty} \biggl\lbrace \sup_{u \in B} \frac{T^{\frac{1-p}{p}}}{\varGamma(2\alpha)} \biggl( \int_{0}^{x} \bigl\vert (x- \eta)^{2\alpha-1} \bigr\vert ^{p} \sum _{n \geq i} j_{i}(\eta) \,d\eta \\ &{}+ \int_{0}^{x} \bigl\vert (x- \eta)^{2\alpha -1} \bigr\vert ^{p} k_{i}(\eta) \sum _{n \geq i} \bigl\vert u_{i}(\eta) \bigr\vert ^{p} \biggr) \,d\eta )^{\frac{1}{p}} \biggr\rbrace . \end{aligned}$$

Then we get

$$\sup_{x \in I}\chi_{\ell_{p}}\bigl(\mathcal{A} u(x) \bigr) \leq\sup_{x \in I}\frac{T^{\frac{1-p}{p}}}{ \varGamma(2\alpha)}\lim _{i \rightarrow\infty} \biggl\lbrace \sup_{u \in B} \biggl( \frac{T^{(2\alpha-1)p+1)}}{ ((2\alpha-1)p+1)} K \sum_{n \geq i} \bigl\vert u_{i}(\eta) \bigr\vert ^{p} \biggr)^{\frac{1}{p}} \biggr\rbrace . $$

Therefore

$$\sup_{x \in I} \chi_{\ell_{p}}\bigl(\mathcal{A} u(x) \bigr) \leq\frac{K^{\frac {1}{p}}T^{2\alpha}}{ \varGamma(2\alpha)((2\alpha-1)p+1)^{\frac{1}{p}} }\sup_{x \in I} \lim _{i \rightarrow\infty} \Bigl\lbrace \sup_{u \in B} \Bigl( \max_{n \geq i} \bigl\vert u_{i}(\eta) \bigr\vert ^{p} \Bigr)^{\frac{1}{p}} \Bigr\rbrace $$

and

$$\chi_{C^{\alpha}_{1-\alpha}(I,\ell_{p})}\bigl(\mathcal{A} u(x)\bigr) \leq \frac{K^{\frac{1}{p}} T^{2\alpha}}{\varGamma(2\alpha) ((2\alpha-1)p+1)^{\frac{1}{p}}} \chi_{C^{\alpha}_{1-\alpha}(I, \ell_{p})}(Y). $$

As \(\frac{K^{\frac{1}{p}} T^{2\alpha}}{\varGamma(2\alpha)((2\alpha-1)p+1)^{\frac{1}{p}}} < 1\), hence, applying Lemma 1.6, \(\mathcal{A}\) admits at least one fixed point in \(\mathcal{A}\) which is a solution for (5) in the space \(C^{\alpha}_{1-\alpha}(I, \ell_{p})\). □

Example 3.2

The system of fractional differential equation in the space \(\ell_{2}\) is given as follows:

$$ \left \{ \textstyle\begin{array}{ll} \mathcal{D} u_{n}(x) = x^{n}\tanh(-x) + \sum_{m=n}^{\infty}\frac {u_{m}(x)e^{x}}{mn},\quad x \in(0,\frac{1}{2}], \\ \lim_{x \rightarrow0}x^{1-\alpha}u_{n}(x)=\frac{a}{n}, \qquad\lim_{x \rightarrow0}x^{1-\alpha}\mathcal{D}^{\alpha}u_{n}(x)= \frac{b}{n},\quad n=1,2,\ldots. \end{array}\displaystyle \right . $$
(11)

This system is a special case of (5) with \(a,b \geq0\), \(\alpha= 1\), \(T = \frac{1}{2}\) and

$$f_{n}\bigl(x,u(x),\mathcal{D}^{\alpha}u(x)\bigr) = x^{n}\tanh(-x) + \sum_{m=n}^{\infty} \frac{u_{m}(x)e^{x}}{mn}. $$

Obviously, the conditions \((C'_{1})\) and \((C'_{2})\) are satisfied. For every \(x\in(0,\frac{1}{2}]\) and \(u \in\ell_{2}\), we have

$$\begin{aligned} \bigl\vert f_{n}\bigl(x,u(x),\mathcal{D}^{\alpha}u(x) \bigr)^{2} \bigr\vert = & \Biggl\vert x^{n}\tanh(-x) + \sum_{m=n}^{\infty}\frac{u_{m}(x)e^{x}}{mn} \Biggr\vert \\ \leq& \bigl\vert x^{n}\tanh(-x) \bigr\vert ^{2} + \sum_{m=n}^{\infty} \biggl\vert \frac{u_{m}(x)e^{x}}{mn} \biggr\vert ^{2} \\ \leq& x^{2n} + \frac{\pi^{2} e^{2x}}{6n^{2}} \bigl\vert u_{n}(x) \bigr\vert ^{2}. \end{aligned}$$

Therefore, f satisfies condition \((C'_{3})\) with \(j_{i}(x) = x^{2n}\) and \(k_{i}(t) = \frac{\pi^{2} e^{2x}}{6n^{2}}\) in which the functions \(j_{i}(x)\) are continuous, \(\sum_{i\geq1}j_{i}(x)\) converges uniformly to \(\frac{1}{1 - x^{2}}\) and \(\lim_{i \rightarrow\infty} k_{i}(x) = 0\), that is, it is integrable over I. Now, we are going to check condition \((C'_{4})\). For any \(\epsilon >0\), \(x \in(0,1]\) and \(u,v\in\ell_{2}\), choose \(\delta= \frac{\epsilon\sqrt{6}}{e^{2}\pi}\) with \(\Vert u(x) - v(x) \Vert < \delta\), we have

$$\begin{aligned} \sum_{n \geq1} \bigl\vert f_{n} \bigl(x,u(x),\mathcal{D}^{\alpha}u(x)\bigr) - f_{n} \bigl(x,v(x),\mathcal{D}^{\alpha}v(x)\bigr) \bigr\vert ^{2} \leq&\sum_{n \geq1} \biggl\vert \sum _{m \geq n} \frac{(u_{m}(x) - v_{m}(x))e^{2x}}{mn} \biggr\vert ^{2} \\ \leq&\sum_{n \geq1}\frac{e^{2x}}{n^{2}} \sum _{m \geq n} \frac{ \vert u_{m}(x) - v_{m}(x) \vert ^{2}}{m^{2}} \\ \leq&\sum_{n \geq1}\frac{e^{2x}}{n^{2}} \bigl\Vert u(x) - v(x) \bigr\Vert ^{2}_{\ell_{2}} \\ \leq& \bigl\Vert u(x) - v(x) \bigr\Vert ^{2}_{\ell_{2}} \frac{e^{2}\pi^{2}}{6} < \epsilon. \end{aligned}$$

Applying Theorem 3.2, hence the system of fractional differential equation (11) possesses at least one solution in \(C^{\alpha}_{1-\alpha}(I, \ell_{2})\).