Fractional differential equation, as an excellent tool for describing memory and hereditary properties of various materials and processes in natural sciences and engineering, received a great deal of attention in the literature [1-4] and there were some works on the investigation of the solution of fractional differential equation [5, 6].

On the other hand, Riemann-Liouville fractional derivatives or integrals are strong tools for resolving some fractional differential problems in the real world. It is possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives or integrals which were verified by Heymans and Podlubny[7], and such initial conditions are more appropriate than physically interpretable initial conditions. For another, they considered the impulse response with Riemann-Liouville fractional derivatives as widely used in the fields of physics, such as viscoelasticity.

In recent years, many authors investigated the existence and stability of solutions to fractional differential equations with Caputo fractional derivative, and there were a lot of interesting and excellent results on this fields. However, there is still little literature on the existence and stability of solutions to Riemann-Liouville fractional differential equations. Three years ago, Weera Yukunthorn et al.[8] studied the existence and uniqueness of solutions to impulsive multiorders Riemann-Liouville fractional differential equations

$ \begin{equation*} \left\{\begin{array}{lcl} D_{t_{k}}^{\alpha_{k}}x(t) = f(t, x(t)), \quad t\in J, \quad t\neq t_{k}, \\ \tilde{\Delta}x(t_{k}) = \varphi_{k}(x(t_{k})), \quad \Delta^{\ast}x(t_{k}) = \varphi_{k}^{\ast}(x(t_{k})), \quad k = 1, 2, \cdots, m, \\ x(0) = 0, \quad D^{\alpha_{0}-1}x(0) = \beta, \end{array}\right. \end{equation*} $

where $ \beta\in R, 0 = t_{0}<t_{1}<\cdots<t_{k}<\cdots<t_{m}<t_{m+1} = T $, $ f: J\times R\rightarrow R $ is a continuous function, $ \varphi_{k}, \varphi_{k}^{\ast}\in C(R, R) $ for $ k = 1, 2, \cdots, m $, and $ D_{t_{k}}^{\alpha_{k}} $ is the Riemann-Liouville fractional derivative of order $ 1<\alpha_{k}< 2 $ on intervals $ J_{k} $ for $ k = 0, 1, 2, \cdots, m $. The notation $ \tilde{\Delta}x(t_{k}) $ is defined by

$ \tilde{\Delta}x(t_{k}) = I_{t_{k}}^{1-\alpha_{k}}x(t_{k}^{+})-I_{t_{k-1}}^{1-\alpha_{k-1}}x(t_{k}), \quad k = 1, 2, \cdots, m, $

and $ \Delta^{\ast}x(t_{k}) $ is defined by

$ \Delta^{\ast}x(t_{k}) = I_{t_{k}}^{2-\alpha_{k}}x(t_{k}^{+})-I_{t_{k-1}}^{2-\alpha_{k-1}}x(t_{k}), \quad k = 1, 2, \cdots, m, $

where $ I_{t_{k}}^{2-\alpha_{k}} $ is the Riemann-Liouville fractional integral of order $ 2-\alpha_{k}>0 $ on $ J_{k} $. By using Banach's fixed point theorem, the authors developed the existence theorem for such equations.

Motivated by this work, we use Mönch's fixed point theorem via measure of noncompactness as well as the basic theory of Ulam stability to investigate the existence and stability of solution to the following impulsive Riemann-Liouville fractional neutral function differential equation with infinite delay in a Banach space $ X $.

$ \left\{\begin{array}{lcl} D_{0^{+}}^{\beta}[x(t)-g(t, x_{t})] = f(t, x_{t}), \quad t\in [0, T], \quad t\neq t_{k}, \\ \Delta I_{0^{+}}^{2-\beta}x(t_{k}) = I_{k}(x(t_{k}^{-})), \quad \Delta I_{0^{+}}^{1-\beta}x(t_{k}) = J_{k}(x(t_{k}^{-})), \\ I_{0^{+}}^{2-\beta}[x(0)-g(0, x_{0})] = \varphi_{1}\in B_{v}, I_{0^{+}}^{1-\beta}[x(0)-g(0, x_{0})] = \varphi_{2}\in B_{v}, \end{array}\right. $

(1.1)

where $ k = 1, 2, \cdots, m $ and $ D_{0^{+}}^{\beta} $ is the Riemann-Liouville fractional derivative of order $ 1<\beta<2 $. $ 0 = t_{0}<t_{1}<\cdots<t_{k}<\cdots<t_{m}<t_{m+1} = T $, Let $ T_{k} = (t_{k}, t_{k+1}], k = 1, 2, \cdots, m, T_{0} = [0, t_{1}] $. $ f: J\times B_{v} $ and $ g: J\times B_{v} $ are given functions, where $ B_{v} $ is the phase space defined in Section 2. The impulsive functions $ I_{k}, J_{k}: X\rightarrow X \; (k = 1, 2, \cdots, m) $ is an appropriate functions. The notation $ \Delta I_{0^{+}}^{2-\beta}x(t_{k}), \Delta I_{0^{+}}^{1-\beta}x(t_{k}) $ is defined by

$ \Delta I_{0^{+}}^{2-\beta}x(t_{k}) = I_{0^{+}}^{2-\beta}x(t_{k}^{+})-I_{0^{+}}^{2-\beta}x(t_{k}^{-}), $

$ \Delta I_{0^{+}}^{1-\beta}x(t_{k}) = I_{0^{+}}^{1-\beta}x(t_{k}^{+})-I_{0^{+}}^{1-\beta}x(t_{k}^{-}), \quad k = 1, 2, \cdots m, $

where $ I_{0^{+}}^{2-\beta}, I_{0^{+}}^{1-\beta} $ is the Riemann-Liouville fractional integral of order $ 2-\beta, $ $ 1-\beta $. The histories $ x_{t}: (-\propto, 0]\rightarrow X $, defined by $ x_{t}(s) = x(t+s) $, $ s\leq0 $, belong to some abstract phase space $ B_{v} $.

The rest of the paper is organized as follows: in section 2, some basic definitions, notations and preliminary facts that are used throughout the paper are presented. In Section 3, we prove the solution of the equation and present the main results for problem (1.1).

In this section, we mention some notations, definitions, lemmas and preliminary facts needed to establish our main results.

Let $ X $ be a complex Banach space, whose norm is denoted by $ \|\cdot\| $. Let $ J = [0, T], J_{0} = [0, t_{1}], J_{k} = (t_{k}, t_{k+1}] $ for $ k = 1, 2, \cdots, m $. Let

$ \begin{align*} PC(J, X): = \{&x:J\rightarrow X, \; \mbox{is continuous everywhere except for some}\;t_{k} \mbox{ at which }\\& x(t_{k}^{+}) \;\mbox{and} \;x(t_{k}^{-})\;\mbox{exist, and}\; x(t_{k}^{-}) = x(t_{k}), k = 1, 2, \cdots, m\}. \end{align*} $

We introduce the space $ C_{2-\beta, k}(J_{k}, X): = \{x:J_{k}\rightarrow X: t^{2-\beta}x(t)\in C(J_{k}, X)\} $ with the norm $ \|x\|_{C_{2-\beta, k}} = \sup\limits_{t\in J_{k}}|t^{2-\beta}\|x(t)\|| $ and $ PC_{2-\beta} = \{x: J\rightarrow X: \;\mbox{for each}\;t\in J_{k} \;\mbox{and}\;t^{2-\beta}x(t)\in C(J_{k}, X), k = 0, 1, 2, \cdots, m\} $ with the norm

$ \|x\|_{PC_{2-\beta}} = \max{\sup\limits_{t\in J_{k}}|t^{2-\beta}\| x(t)\||: k = 0, 1, 2, \cdots, m}. $

Clearly $ PC_{2-\beta} $ is a Banach space. We use $ B_r(x, X) $ to denote the closed ball in $ X $ with center at $ x $ and radius $ r $.

Before introducing the fractional-order functional differential equation with infinite delay, we define the abstract phase space $ B_v $. Let $ v:(\infty, 0]\rightarrow(0, \infty) $ be a continuous function that satisfies $ l = \int_{-\infty}^0 v(t)dt<+\infty $. The Banach space $ (B_v, \|\cdot\|_{B_v}) $ induced by $ v $ is then given by

$ \begin{align*} B_v:& = \{\varphi:(-\infty, 0)\rightarrow X: \mbox{for any } c>0, \; \varphi(\theta)\; \mbox{ is a bounded and measurable}\\&\qquad \mbox{function on [-c, 0], and} \int_{-\infty}^0 v(s)\sup\limits_{s\leq\theta\leq0}\|\varphi(\theta)\|ds<+\infty\} \end{align*} $

endowed with the norm $ \|\varphi\|_{B_v}: = \int_{-\infty}^0 v(s)\sup\limits_{s\leq\theta\leq0}\|\varphi(\theta)\|ds. $

Define the following space

$ \begin{align*} B'_v:& = \{\varphi:(-\infty, T]\rightarrow X: \varphi_k\in C^1(J_k, X), k = 0, 1, 2, \cdots, m, \; \mbox{and there exist }\\& \qquad \varphi(t_k^-) \mbox{ and } \varphi(t_k^+) \mbox{ with } \varphi(t_k) = \varphi(t_k^-), \varphi_{0} = \phi\in B_{v}\}, \end{align*} $

where $ \varphi_k $ is the restriction of $ \varphi $ to $ J_k $, $ J_0 = [0, t_1] $, $ J_k = (t_k, t_{k+1}] $, $ k = 1, 2, \cdots, m. $

We use $ \|\cdot\|_{B'_v} $ to denote a seminorm in the space $ B'_v $ defined by

$ \|\varphi\|_{B'_v}: = \|\phi\|_{B_v}+\max\{\|\varphi_k\|_{J_k(2-\beta)}, \;\;k = 0, 1, \cdots, m\}, $

where

$ \phi = \varphi_{0}, \|\varphi_k\|_{J_k(2-\beta)} = \sup\limits_{s\in J_k}\{s^{2-\beta}\|\varphi(s)\|\}. $

Now we consider some definitions about fractional differential equations.

Definition 2.1    The Riemann-Liouville fractional derivative of order $ \alpha>0 $ of a continuous function $ f; (a, b)\rightarrow X $ is defined by

$ D_{a^{+}}^{\alpha}f(t) = \frac{1}{\Gamma(n-\alpha)}\left(\frac{d}{dt}\right)^{n}\int_{a}^{t}(t-s)^{n-\alpha-1}f(s)ds, \quad n-1<\alpha<n, \quad t\in (a, b), $

where $ n = [\alpha]+1 $, $ [\alpha] $ denotes the integer part of number $ \alpha $, provided the right-hand side is pointwise defined on $ (a, b) $, $ \Gamma $ is the gamma function.

Definition 2.2   The Riemann-Liouville fractional integral of order $ \alpha>0 $ of a continuous function $ f: (a, b)\rightarrow X $ is defined by

$ I_{a^{+}}^{\alpha}f(t) = \frac{1}{\Gamma (\alpha)}\int_{a}^{t}(t-s)^{\alpha-1}f(s)ds, \quad t\in (a, b), $

provided the right-hand side is pointwise defined on $ (a, b) $.

Lemma 2.1(see [9])   Let $ \alpha>0 $. Then for $ x\in C(a, b)\cap L(a, b), $ it holds

$ D_{a^{+}}^{\alpha}I_{a^{+}}^{\alpha}x(t) = x(t), $

$ I_{a}^{\alpha}D_{a}^{\alpha}x(t) = x(t)-\sum\limits_{j = 1}^{n}\frac{(I_{a}^{n-\alpha})^{(n-j)}x(a)} {\Gamma(\alpha-j+1)}(t-a)^{\alpha-j}, $

where $ n-1<\alpha<n $.

Lemma 2.2(see [9])  If $ \alpha\geq0 $ and $ \beta>0 $, then

$ I_{a^{+}}^{\alpha}(t-s)^{\beta-1} = \frac{\Gamma(\beta)}{\Gamma(\beta+\alpha)}(t-a)^{\beta+\alpha-1}, $

$ D_{a^{+}}^{\alpha}(t-s)^{\beta-1} = \frac{\Gamma(\beta)}{\Gamma(\beta-\alpha)}(t-a)^{\beta-\alpha-1}. $

Before investigating the solutions to equation (1.1), we consider a simplified version of (1.1), given by

$ \left\{\begin{array}{lcl} D_{0^{+}}^{\beta}x(t) = f(t), \quad t\in [0, T], \quad t\neq t_{k}, \\\\ \Delta I_{0^{+}}^{2-\beta}x(t_{k}) = y_k, \quad \Delta I_{0^{+}}^{1-\beta}x(t_{k}) = \overline{y_k}, \\\\ I_{0^{+}}^{2-\beta}x(0^{+}) = x_{0}, \quad I_{0^{+}}^{1-\beta}x(0^{+}) = x_{1}, \end{array}\right. $

(2.1)

where $ k = 1, 2, \cdots, m, x_0, x_1, y_k, \overline{y_k}\in X $ and $ D_{0^{+}}^{\beta} $ is the Riemann-Liouville fractional derivative of order $ 1<\beta<2 $.

Theorem 2.1   Let $ 1<\beta<2 $ and $ f: J\rightarrow X $ be continuous. If $ x\in PC_{2-\beta}(J, X) $ is a solution of (2.1) if and only if $ x $ is a solution of the following the fractional integral equation

$ \begin{equation*} x(t) = \left\{\begin{array}{lcl} \frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}f(s)ds+x_{1}\frac{t^{\beta-1}}{\Gamma(\beta)}+x_{0}\frac{t^{\beta-2}}{\Gamma(\beta-1)}, \quad t\in[0, t_{1}], \\ \frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}f(s)ds+x_{1}\frac{t^{\beta-1}}{\Gamma(\beta)}+x_{0}\frac{t^{\beta-2}}{\Gamma(\beta-1)}\\ +\sum\limits_{i = 1}^{k}\overline{y_i} \frac{t^{\beta-2}}{\Gamma(\beta-1)}(\frac{t}{\beta-1}-t_{i}) +\sum\limits_{i = 1}^{k}y_i\frac{t^{\beta-2}}{\Gamma(\beta-1)}, \quad t\in(t_{k}, t_{k+1}], \end{array} \right. \end{equation*} $

(2.2)

where $ k = 1, 2, \cdots, m $.

Proof  For all $ t\in(t_{k}, t_{k+1}] $ where $ k = 0, 1, \cdots, m $ by Lemma 2.1 and 2.2, we obtain

$ \begin{align*} D_{0^{+}}^{\beta}x(t) = &D_{0^{+}}^{\beta}\left[ \frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}f(s)ds+x_{1}\frac{t^{\beta-1}}{\Gamma(\beta)} +x_{0}\frac{t^{\beta-2}}{\Gamma(\beta-1)}\right.\\ &\left. +\sum\limits_{i = 1}^{k}\overline{y_i}\frac{t^{\beta-2}}{\Gamma(\beta-1)}(\frac{t}{\beta-1}-t_{i}) +\sum\limits_{i = 1}^{k}y_i\frac{t^{\beta-2}}{\Gamma(\beta-1)}\right]\\ = &D_{0^{+}}^{\beta}I_{0^{+}}^{\beta}f(t)+D_{0^{+}}^{\beta}\left[x_{1}\frac{t^{\beta-1}}{\Gamma(\beta)}+x_{0}\frac{t^{\beta-2}}{\Gamma(\beta-1)}\right.\\ &\left. +\sum\limits_{i = 1}^{k}\overline{y_i}\frac{t^{\beta-2}}{\Gamma(\beta-1)}(\frac{t}{\beta-1}-t_{i}) +\sum\limits_{i = 1}^{k}y_i\frac{t^{\beta-2}}{\Gamma(\beta-1)}\right]\\ = &f(t). \end{align*} $

Thus, expression (2.2) satisfies the first equation of problem (2.1). For $ k = 1, 2, \cdots, m $, it follows from (2.1) that

$ \begin{align*} I_{0^{+}}^{2-\beta}x(t) = &I_{0^{+}}^{2-\beta}\left[I_{0^{+}}^{\beta}f(t)) +x_{1}\frac{t^{\beta-1}}{\Gamma(\beta)}+x_{0}\frac{t^{\beta-2}}{\Gamma(\beta-1)}\right.\\ &\left. +\sum\limits_{i = 1}^{k}\overline{y_i}\frac{t^{\beta-2}}{\Gamma(\beta-1)}(\frac{t}{\beta-1}-t_{i}) +\sum\limits_{i = 1}^{k}y_i\frac{t^{\beta-2}}{\Gamma(\beta-1)}\right]\\ = &I_{0^{+}}^{2}f(t)+x_{1}I_{0^{+}}^{2-\beta}(\frac{t^{\beta-1}}{\Gamma(\beta)}) +x_{0}I_{0^{+}}^{2-\beta}(\frac{t^{\beta-2}}{\Gamma(\beta-1)})\\ &+ \sum\limits_{i = 1}^{k}\overline{y_i}\left[I_{0^{+}}^{2-\beta}(\frac{t^{\beta-1}}{\Gamma(\beta)})-t_{i} I_{0^{+}}^{2-\beta}(\frac{t^{\beta-2}}{\Gamma(\beta-1)})\right] +\sum\limits_{i = 1}^{k}y_i I_{0^{+}}^{2-\beta}(\frac{t^{\beta-2}}{\Gamma(\beta-1)})\\ = &I_{0^{+}}^{2}f(t)+x_{1}t+x_{0}+ \sum\limits_{i = 1}^{k}\overline{y_i}(t-t_{i})+\sum\limits_{i = 1}^{k}y_i, \\ I_{0^{+}}^{1-\beta}x(t) = &I_{0^{+}}^{1-\beta}\left[I_{0^{+}}^{\beta}f(t) +x_{1}\frac{t^{\beta-1}}{\Gamma(\beta)}+x_{0}\frac{t^{\beta-2}}{\Gamma(\beta-1)}\right.\\ &\left. +\sum\limits_{i = 1}^{k}\overline{y_i}\frac{t^{\beta-2}}{\Gamma(\beta-1)}(\frac{t}{\beta-1}-t_{i}) +\sum\limits_{i = 1}^{k}y_i\frac{t^{\beta-2}}{\Gamma(\beta-1)}\right]\\ = &I_{0^{+}}f(t)+x_{1} +\sum\limits_{i = 1}^{k}\overline{y_i}. \end{align*} $

Therefore, we have

$ \begin{align*} \Delta I_{0^{+}}^{2-\beta}x(t_{k})& = I_{0^{+}}^{2-\beta}x(t_{k}^{+})-I_{0^{+}}^{2-\beta}x(t_{k}^{-})\\ & = \sum\limits_{i = 1}^{k}\overline{y_i}(t_{k}-t_{i})+\sum\limits_{i = 1}^{k}y_i- \sum\limits_{i = 1}^{k-1}\overline{y_i}(t_{k}-t_{i})-\sum\limits_{i = 1}^{k-1}y_i\\ & = y_{k}, \end{align*} $

$ \begin{align*} \Delta I_{0^{+}}^{1-\beta}x(t_{k})& = I_{0^{+}}^{1-\beta}x(t_{k}^{+})-I_{0^{+}}^{1-\beta}x(t_{k}^{-}) = \sum\limits_{i = 1}^{k}\overline{y_i}- \sum\limits_{i = 1}^{k-1}\overline{y_i} = \overline{y_k}. \end{align*} $

Consequently, all the conditions of problem (2.1) are satisfied. Hence, (2.2) is a solution of problem (2.1)

Next, based on Theorem, we consider the solutions of the Cauchy problem(1.1)

Definition 2.3   Suppose function $ x: (-\propto, T]\rightarrow X $. The solution of the fractional differential equation, given by

$ \begin{equation*} x(t) = \left\{\begin{array}{lcl} x_{0} = \phi\in B_{v}, \qquad t\in(-\propto, 0], \\ \frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}f(s, x_{s})ds+g(t, x_{t})\\ +\varphi_{2}\frac{t^{\beta-1}}{\Gamma(\beta)}+\varphi_{1}\frac{t^{\beta-2}}{\Gamma(\beta-1)}, t\in[0, t_{1}], \\ \frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}f(s, x_{s})ds+g(t, x_{t})+\varphi_{2}\frac{t^{\beta-1}}{\Gamma(\beta)}+\varphi_{1}\frac{t^{\beta-2}}{\Gamma(\beta-1)}\\ +\sum\limits_{i = 1}^{k}J_{i}(x(t_{i}^{-}))\frac{t^{\beta-2}}{\Gamma(\beta-1)}(\frac{t}{\beta-1}-t_{i})\\ +\sum\limits_{i = 1}^{k}I_{i}(x(t_{i}^{-}))\frac{t^{\beta-2}}{\Gamma(\beta-1)}, \quad t\in(t_{k}, t_{k+1}], \end{array} \right. \end{equation*} $

will be called a fundamental solution of problem (1.1).

Lemma 2.3(see [10])   Assume $ x\in B_{v}^{'} $, then for $ t\in J, \; x_{t}\in B_{v} $. Moreover

$ l\|x(t)\|\leq\|x_{t}\|_{B_{v}}\leq\|\phi\|_{B_{v}}+l\sup\limits_{s\in[0, t]} \|s^{2-\beta}x(s)\|, $

where $ l = \int_{-\propto}^{0}v(t)dt<+\propto, \phi = x_{0} $.

Next, we consider some definitions and properties of the measures of noncompactness.

The Hausdorff measure of noncompactness $ \beta(\cdot) $ defined on each bounded subset $ \mathscr{B} $ of Banach space $ X $ is given by

$ \beta(\mathscr{B}) = \inf\{\varepsilon >0; \mathscr{B} \;\;\mbox{ has a finite}\;\; \varepsilon-\;\mbox{net in}\;\; X \}. $

Some basic properties of $ \beta(\cdot) $ are given in the following lemma.

Lemma 2.4(see [11-13])   If $ X $ is a real Banach space and $ \mathscr{B}, \mathscr{D}\subset X $ are bounded, then the following properties are satisfied

(1) monotone: if for all bounded subsets $ \mathscr{B}, \mathscr{D} $ of $ X $, $ \mathscr{B}\subseteq\mathscr{D} $ implies $ \beta(\mathscr{B})\leq \beta(\mathscr{D}) $;

(2) nonsingular: $ \beta(\{x\}\cup\mathscr{B}) = \beta(\mathscr{B}) $ for every $ x \in X $ and every nonempty subset $ \mathscr{B}\subset X $;

(3) regular: $ \mathscr{B} $ is precompact if and only if $ \beta(\mathscr{B}) = 0 $;

(4) $ \beta(\mathscr{B}+\mathscr{D})\leq \beta(\mathscr{B})+\beta(\mathscr{D}) $, where $ \mathscr{B}+\mathscr{D} = \{x +y; x \in \mathscr{B}, y\in \mathscr{D}\} $;

(5) $ \beta(\mathscr{B}\cup\mathscr{D})\leq \max\{\beta(\mathscr{B}), \beta(\mathscr{D})\} $;

(6) $ \beta(\lambda\mathscr{B})\leq|\lambda|\beta(\mathscr{B}) $;

(7) if $ W \subset C(J;X) $ is bounded and equicontinuous, then $ t \rightarrow\beta(W(t)) $ is continuous on $ J $, and

$ \begin{align} \beta(W)\leq \max\limits_{t\in J} \beta(W(t)), \end{align} $

(2.1)

$ \begin{align} \beta\left(\int_0^tW(s)ds \right )\leq \int_0^t\beta (W(s))ds \;\; \mbox{for all} \;\; t\in J, \end{align} $

(2.2)

where

$ \int_0^tW(s)ds = \left\{ \int_0^t u(s)ds \;\; \mbox{for all}\;\; u\in W, t\in J \right\}; $

(8) if $ \{u_n\}_1^\infty $ is a sequence of Bochner integrable functions from $ J $ into $ X $ with $ \|u_n(t)\| \leq \widehat{m}(t) $ for almost all $ t \in J $ and every $ n \geq 1 $, where $ \widehat{m}(t)\in L(J;R^+) $, then the function $ \psi(t) = \beta(\{u_n\}_{n = 1}^\infty)) $ belongs to $ L(J;R^+) $ and satisfies

$ \begin{align} \beta \left(\left\{\int_0^t u_n(s)ds: n\geq 1 \right\}\right )\leq 2\int_0^t \psi(s)ds; \end{align} $

(2.3)

(9) if $ W $ is bounded, then for each $ \varepsilon>0 $, there is a sequence $ \{u_n\}_{n = 1}^\infty\subset W $ such that

$ \begin{align} \beta(W)\leq 2\beta(\{u_n\}_{n = 1}^\infty)+\varepsilon. \end{align} $

(2.4)

The following lemmas about the Hausdorff measure of noncompactness will be used in proving our main results.

Lemma 2.5(see [14])  Let $ D $ be a closed convex subset of a Banach Space $ X $ and $ 0\in D $. Assume that $ F: D\rightarrow X $ is a continuous map which satisfies the Mönch's condition, that is, $ M\subseteq D $ is countable, $ M\subseteq \overline{\text {co}}\left({0}\cup F(M)\right)\Rightarrow\overline{M} $ is compact. Then $ F $ has a fixed point in $ D $.

Next, we consider the Ulam stability for the equation.

Consider the following inequality

$ \|D_{0^{+}}^{\beta}[x(t)-g(t, x_{t})]-f(t, x_{t})\|<\varepsilon. $

Definition 2.4   Equation (1.1) is Hyers-Ulam stable if, for any $ \varepsilon > 0 $, there exists a solution $ y(t) $ which satisfies the above inequality and has the same initial value as $ x(t) $, where $ x(t) $ is a solution to (1.1). Then $ y(t) $ satisfies $ \|y(t)-x(t)\|<K\varepsilon $ in which $ K $ is a constant.

To prove our main results, we list the following basic assumptions of this paper.

(H₁) The function $ f: J\times B_{v}\rightarrow X $ satisfies the following conditions.

(ⅰ) $ f(\cdot, \phi) $ is measurable for all $ \phi\in B_{v} $ and $ f(t, \cdot) $ is continuous for a.e. $ t\in J $.

(ⅱ) There exist a constant $ \alpha_{1}\in (0, \alpha) $, $ m\in L^{\frac{1}{\alpha_{1}}}(J, R^{+}) $ and a positive integrable function $ \Omega: R^{+}\rightarrow R^{+} $ such that $ \|f(t, \phi)\|\leq m(t)\Omega (\|\phi\|_{B_{v}}) $ for all $ (t, \phi)\in J\times B_{v} $, where $ \Omega $ satisfies $ \liminf\limits_{n\rightarrow\propto}\frac{\Omega(n)}{n} = 0. $

(ⅲ) There exist a constant $ \alpha_{2}\in (0, \alpha) $ and a function $ \eta\in L^{\frac{1}{\alpha_{2}}}(J, R^{+}) $ such that, for any bounded subset $ F_{1}\subset B_{v} $,

$ \beta(f(t, F_{1}))\leq \eta(t)[\sup\limits_{\theta\in(-\propto, 0]}\beta(F_{1}(\theta))] $

for a.e. $ t\in J $, where $ F_{1}(\theta) = \{v(\theta): v\in F_{1}\} $ and $ \beta $ is the Hausdorff MNC.

(H₂) The function $ g: J\times B_{v}\rightarrow X $ satisfies the following conditions.

(ⅰ) $ g $ is continuous and there exist a constant $ H_{1} >0 $ and

$ \|t^{2-\beta}g(t, x)\|\leq H_{1}(1+\|x\|_{B_{v}}). $

(ⅱ) There exist a constant $ \alpha_{3}\in(0, \alpha) $ and $ g^{\ast}\in L^{\frac{1}{\alpha_{3}}}(J, R^{+}) $ such that, for any bounded subset $ F_{2}\subset B_{v} $,

$ \beta(g(t, F_{2}))\leq g^{\ast}(t)\sup\limits_{\theta\in(-\propto, 0]}\beta(F_{2}(\theta)), \quad G = \sup\limits_{t\in J}g^{\ast}(t). $

(H₃) $ I_{k}, J_{k}: X\rightarrow X, k = 1, 2, \cdots, m $ are continuous functions and satisfy

$ \begin{align*} &\|I_{k}(x)\|_{X}\leq c_{k}\|x\|_{B_{v}^{'}}, \quad \|J_{k}(x)\|_{X}\leq f_{k}\|x\|_{B_{v}^{'}}, \\ &\beta (t^{\beta-2}I_{k}(F_{3}))\leq K_{k}\sup\limits_{\theta\in(-\propto, T]}\beta(F_{3}(\theta)), \\ &\beta(t^{\beta-2}J_{k}(F_{4}))\leq M_{k}\sup\limits_{\theta\in(-\propto, T]}\beta(F_{4}(\theta)), \end{align*} $

where $ c_{k}, f_{k}, K_{k}, M_{k}>0 $. $ F_{3}, F_{4}\subset B_{v}^{'} $.

(H₄): $ H_{1}l+\frac{T^{\ast}}{\Gamma^{\ast}}\sum\limits_{i = 1}^{m}(f_{i}+c_{i})<1, $

$ M^{\ast} = \frac{2T^{\beta}}{\Gamma(\beta+1)}\|\eta\|_{L^{\frac{1}{\alpha_{2}}}(J, R^{+})}+G+ \frac{T^{\ast}}{\Gamma^{\ast}}\sum\limits_{i = 1}^{m}(M_{i}+K_{i})<1, $

where $ T^{\ast} = \max{\{1, T, T^{2}}\} $, $ \Gamma^{\ast} = \min{\{\Gamma(\beta+1), \Gamma(\beta), \Gamma(\beta-1)}\} $.

Theorem 3.1   Suppose conditions (H$ _{1})- $(H$ _{4}) $ are satisfied. Then system(1.1) has at least one solution on $ J $.

Proof   We define the operator $ \Gamma: B_{v}^{'} \rightarrow B_{v}^{'} $ by

$ \begin{equation*} \Gamma x(t) = \left\{\begin{array}{lcl} x_{0} = \phi\in B_{v}, \qquad t\in(-\propto, 0]\\ \frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}f(s, x_{s})ds+g(t, x_{t})+ \varphi_{2}\frac{t^{\beta-1}}{\Gamma(\beta)}+\varphi_{1}\frac{t^{\beta-2}}{\Gamma(\beta-1)}, t\in[0, t_{1}], \\ \frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}f(s, x_{s})ds+g(t, x_{t})+\varphi_{2}\frac{t^{\beta-1}}{\Gamma(\beta)}+\varphi_{1}\frac{t^{\beta-2}}{\Gamma(\beta-1)}\\ +\sum\limits_{i = 1}^{k}J_{i}(x(t_{i}^{-}))\frac{t^{\beta-2}}{\Gamma(\beta-1)}(\frac{t}{\beta-1}-t_{i})\\ +\sum\limits_{i = 1}^{k}I_{i}(x(t_{i}^{-}))\frac{t^{\beta-2}}{\Gamma(\beta-1)}, \quad t\in(t_{k}, t_{k+1}]. \end{array} \right. \end{equation*} $

The operator $ \Gamma $ has a fixed point if and only if system (1.1) has a solution. For $ \phi\in B_{v} $, denote

$ \begin{equation*} \hat{\phi}(t) = \left\{\begin{array}{lcl}\phi(t), \qquad t\in(-\propto, 0], \\ 0, \qquad t\in J. \end{array} \right. \end{equation*} $

Then $ \hat{\phi}(t)\in B_{v}^{'} $. Let

$ x(t) = y(t)+\hat{\phi}(t), \quad -\propto<t\leq T. $

It is easy to see that $ y $ satisfies $ y_{0} = 0, t\in(-\propto, 0] $ and

$ \begin{equation*} y(t) = \left\{\begin{array}{lcl} \frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}f(s, y_{s}+\hat{\phi_{s}})ds+g(t, y_{t}+\hat{\phi_{t}})\\ +\varphi_{2}\frac{t^{\beta-1}}{\Gamma(\beta)}+\varphi_{1}\frac{t^{\beta-2}}{\Gamma(\beta-1)}, t\in[0, t_{1}], \\ \frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}f(s, y_{s}+\hat{\phi_{s}})ds+g(t, y_{t}+\hat{\phi_{t}})+\varphi_{2}\frac{t^{\beta-1}}{\Gamma(\beta)}+\varphi_{1}\frac{t^{\beta-2}}{\Gamma(\beta-1)}\\ +\sum\limits_{i = 1}^{k}J_{i}(y(t_{i}^{-})+\hat{\phi{(t_{i}^{-})}})\frac{t^{\beta-2}}{\Gamma(\beta-1)}(\frac{t}{\beta-1}-t_{i})\\ +\sum\limits_{i = 1}^{k}I_{i}(y(t_{i}^{-})+\hat{\phi{(t_{i}^{-})}})\frac{t^{\beta-2}}{\Gamma(\beta-1)}, \quad t\in(t_{k}, t_{k+1}] \end{array} \right. \end{equation*} $

if and only if $ x(t) $ satisfies $ x(t) = \phi(t), t\in(-\propto, 0] $ and

$ \begin{equation*} x(t) = \left\{\begin{array}{lcl} \frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}f(s, x_{s})ds+g(t, x_{t})\\ +\varphi_{2}\frac{t^{\beta-1}}{\Gamma(\beta)}+\varphi_{1}\frac{t^{\beta-2}}{\Gamma(\beta-1)}, t\in[0, t_{1}], \\ \frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}f(s, x_{s})ds+g(t, x_{t})+\varphi_{2}\frac{t^{\beta-1}}{\Gamma(\beta)}+\varphi_{1}\frac{t^{\beta-2}}{\Gamma(\beta-1)}\\ +\sum\limits_{i = 1}^{k}J_{i}(x(t_{i}^{-}))\frac{t^{\beta-2}}{\Gamma(\beta-1)}(\frac{t}{\beta-1}-t_{i})\\ +\sum\limits_{i = 1}^{k}I_{i}(x(t_{i}^{-}))\frac{t^{\beta-2}}{\Gamma(\beta-1)}, \quad t\in(t_{k}, t_{k+1}]. \end{array} \right. \end{equation*} $

Define the Banach space $ (B_{v}^{''}, \|\cdot\|_{B_{v}^{''}}), $ induced by $ B_{v}^{'} $

$ B_{v}^{''} = \{y: y\in B_{v}^{'}, y_{0} = 0\} $

with the norm

$ \|y(t)\|_{B_{v}^{''}} = \sup\{s^{2-\beta}\|y(s)\|_{X}, s\in[0, T]\}. $

Let $ B_{r} = \{y\in B_{v}^{''} : \|y\|_{B_{v}^{'}}\leq r\} $. Then for each $ r $, $ B_{r} $ is a bounded, close and convex subset. For any $ y\in B_{r} $, it follows from Lemma 2.3 that

$ \begin{align*} \|y_{t}+\hat{\phi}_{t}\|_{B_{v}}&\leq\|y_{t}\|_{B_{v}}+\|\hat{\phi}_{t}\|_{B_{v}}\\ &\leq l\sup\limits_{s\in[0, t]}s^{2-\beta}\|x(s)\|+\|\phi\|_{B_{v}}\\ &\leq lr+\|\phi\|_{B_{v}} = r'. \end{align*} $

We define the operator $ N: B_{v}^{''}\rightarrow B_{v}^{''} $ by

$ \begin{equation*} Ny(t) = \left\{\begin{array}{lcl} \frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}f(s, y_{s}+\hat{\phi_{s}})ds+g(t, y_{t}+\hat{\phi_{t}})\\ +\varphi_{2}\frac{t^{\beta-1}}{\Gamma(\beta)}+\varphi_{1}\frac{t^{\beta-2}}{\Gamma(\beta-1)}, t\in[0, t_{1}], \\ \frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}f(s, y_{s}+\hat{\phi_{s}})ds\\+g(t, y_{t}+\hat{\phi_{t}})+\varphi_{2}\frac{t^{\beta-1}}{\Gamma(\beta)}+\varphi_{1}\frac{t^{\beta-2}}{\Gamma(\beta-1)}\\ +\sum\limits_{i = 1}^{k}J_{i}(y(t_{i}^{-})+\hat{\phi{(t_{i}^{-})}})\frac{t^{\beta-2}}{\Gamma(\beta-1)}(\frac{t}{\beta-1}-t_{i})\\ +\sum\limits_{i = 1}^{k}I_{i}(y(t_{i}^{-})+\hat{\phi{(t_{i}^{-})}})\frac{t^{\beta-2}}{\Gamma(\beta-1)}, \quad t\in(t_{k}, t_{k+1}]. \end{array} \right. \end{equation*} $

Step 1   We prove that there exists some $ r>0 $ such that $ N(B_{r})\subset B_{r} $. If this is not true, then, for each positive integer $ r $, there exist $ y_{r}\in B_{r} $ and $ t_{r}\in(-\propto, T] $ such that $ \|(Ny_{r})(t_{r})\|>r $. On the other hand, it follows from the assumption that

$ \begin{align*} t_{r}^{2-\beta}\|N(y_{r}(t_{r}))\|&\leq\left\| \frac{1}{\Gamma(\beta)}\int_{0}^{t_{r}}(t_{r}-s)^{\beta-1}t_{r}^{2-\beta}f(s, (y_{r})_{s}+\hat{\phi}_{s})ds\right\|\\ &\quad+\left\|t_{r}^{2-\beta}g(t_{r}, (y_{r})_{t_{r}}+\hat{\phi}_{t_{r}})\right\|\\ &\quad+\left\|\varphi_{2}\frac{t_{r}^{2-\beta+\beta-1}}{\Gamma(\beta)}\right\| +\left\|\varphi_{1}\frac{t_{r}^{2-\beta+\beta-2}}{\Gamma(\beta-1)}\right\| \\ &\quad+\left\|\sum\limits_{i = 1}^{k}J_{i}(y_{r}(t_{i}^{-})+\hat{\phi{(t_{i}^{-})}})\frac{t_{r}^{2-\beta+\beta-2}}{\Gamma(\beta-1)}(\frac{t_{r}}{\beta-1}-t_{i})\right\| \\&\quad+\left\|\sum\limits_{i = 1}^{k}I_{i}(y_{r}(t_{i}^{-})+\hat{\phi{(t_{i}^{-})}})\frac{t_{r}^{2-\beta+\beta-2}}{\Gamma(\beta-1)}\right\| \\ &\leq \left\|\frac{T^{2-\beta}}{\Gamma(\beta)}\int_{0}^{t_{r}}(t_{r}-s)^{\beta-1}f(s, (y_{r})_{s}+\hat{\phi}_{s})ds\right\| +H_{1}(1+\left\|(y_{r})_{t_{r}}+\hat{\phi_{t}}\right\|_{B_{v}})\\ &\quad+\frac{T}{\Gamma(\beta)}\left\|\varphi_{2}\right\| +\frac{1}{\Gamma(\beta-1)}\|\varphi_{1}\|\\&\quad+\frac{T}{\Gamma(\beta)}\left\|\sum\limits_{i = 1}^{k}J_{i}(y_{r}(t_{i}^{-})+\hat{\phi{(t_{i}^{-})}})\right\|\\ &\quad+\frac{1}{\Gamma(\beta-1)}\left\|\sum\limits_{i = 1}^{k}I_{i}(y_{r}(t_{i}^{-})+\hat{\phi{(t_{i}^{-})}})\right\|\\ &\leq\frac{T^{2}}{\Gamma(\beta+1)}m(t)\Omega(r')+H_{1}+H_{1}r'\\ &\quad+\frac{T^{\ast}}{\Gamma^{\ast}}(\|\varphi_{2}\|+\|\varphi_{1}\|)+\left(\frac{T^{\ast}}{\Gamma^{\ast}} \sum\limits_{i = 1}^{m}f_{i}+c_{i}\right)r. \end{align*} $

So we have

$ \begin{align*} r&<\|Ny_{r}(t_{r})\|_{B_{v}^{''}}\\ &\leq\frac{T^{2}}{\Gamma(\beta+1)}m(t)\Omega(r')+H_{1}+H_{1}r'\\ &\quad+\frac{T^{\ast}}{\Gamma^{\ast}}(\|\varphi_{2}\|+\|\varphi_{1}\|)+\left(\frac{T^{\ast}}{\Gamma^{\ast}} \sum\limits_{i = 1}^{m}f_{i}+c_{i}\right)r. \end{align*} $

Dividing both sides by $ r $ and taking $ r\rightarrow+\propto $ from

$ \lim\limits_{r\rightarrow\propto}\frac{r^{'}}{r} = \lim\limits_{r\rightarrow\propto}\frac{lr+\|\phi\|_{B_{v}}}{r} = l $

and

$ \lim\limits_{n\rightarrow\propto}\inf\frac{\Omega(n)}{n} = 0 $

yields

$ H_{1}l+\frac{T^{\ast}}{\Gamma^{\ast}}\sum\limits_{i = 1}^{m}(f_{i}+c_{i})<1. $

This contradicts (H₄). Thus, for some number $ r, N(B_{r})\subset B_{r} $.

Step 2   $ N $ is continuous on $ B_{r} $. Let $ \{y^{n}\}_{n = 1}^{+\propto}\subset B_{r} $, with $ y^{n}\rightarrow y $ in $ B_{r} $ as $ n\rightarrow +\propto $. Then, by using hypotheses (H$ _{1}) $, (H$ _{2}) $ and (H$ _{3}) $, we have

(ⅰ)

$ f(s, y_{s}^{n}+\hat{\phi}_{s})\rightarrow f(s, y_{s}+\hat{\phi}_{s}), \quad n\rightarrow \propto. $

(ⅱ)

$ g\left(t, y_{t}^{n}+\hat{\phi}_{t}\right)\rightarrow g\left(t, y_{t}+\hat{\phi}_{t}\right), \quad n\rightarrow\propto. $

(ⅲ)

$ \|I_{i}(y^{n}(t_{i}^{-})+\hat{\phi}(t_{i}^{-}))-I_{i}(y(t_{i}^{-})+\hat{\phi}(t_{i}^{-}))\|\rightarrow 0, $

$ \|J_{i}(y^{n}(t_{i}^{-})+\hat{\phi}(t_{i}^{-}))-J_{i}(y(t_{i}^{-})+\hat{\phi}(t_{i}^{-}))\|\rightarrow 0, \quad n\rightarrow \propto, i = 1, 2, \cdots, m. $

Now, for every $ t\in[0, t_{1}] $, we have

$ \begin{align*} t^{2-\beta}\|Ny^{n}(t)-Ny(t)\|\leq&\left\|\frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}t^{2-\beta}[f(s, y^{n}_{s}+\hat{\phi}_{s}) -f(s, y_{s}+\hat{\phi}_{s})]ds\right\|\\ &+\left\|t^{2-\beta}[g(t, y_{t}^{n}+\hat{\phi}_{t})-g(t, y_{t}+\hat{\phi}_{t})\right\|\\ \leq&\frac{T^{2}}{\Gamma(\beta+1)}\left\|f(s, y^{n}_{s}+\hat{\phi}_{s})-f(s, y_{s}+\hat{\phi}_{s})\right\|\\ &+T^{2-\beta}\left\|g(t, y_{t}^{n}+\hat{\phi}_{t})-g(t, y_{t}+\hat{\phi}_{t})\right\|\\ \rightarrow& 0\quad (n\rightarrow \propto). \end{align*} $

Moreover, for all $ t\in(t_{k}, t_{k+1}], k = 1, 2, \cdots, m $, we have

$ \begin{align*} t^{2-\beta}\|Ny^{n}(t)-Ny(t)\|\leq&\frac{T^{2}}{\Gamma(\beta+1)}\left\|f(s, y^{n}_{s}+\hat{\phi}_{s})-f(s, y_{s}+\hat{\phi}_{s})\right\|\\ &+T^{2-\beta}\left\|g(t, y_{t}^{n}+\hat{\phi}_{t})-g(t, y_{t}+\hat{\phi}_{t})\right\|\\ &+\frac{T}{\Gamma(\beta)}\sum\limits_{i = 1}^{k}\left\|J_{i}(y^{n}(t_{i}^{-})+\hat{\phi}(t_{i}^{-}))-J_{i}(y(t_{i}^{-})+\hat{\phi}(t_{i}^{-}))\right\|\\ &+\frac{1}{\Gamma(\beta-1)}\sum\limits_{i = 1}^{k}\left\|I_{i}(y^{n}(t_{i}^{-})+\hat{\phi}(t_{i}^{-}))-I_{i}(y(t_{i}^{-})+\hat{\phi}(t_{i}^{-}))\right\|\\ \rightarrow& 0\quad (n\rightarrow \propto). \end{align*} $

We thus obtain

$ \|Ny^{n}-Ny\|_{B_{v}^{''}}\rightarrow 0 \;\;\mbox{as} \;\;n\rightarrow \propto $

implying that $ N $ is continuous on $ B_{r} $.

Step 3   The map $ N(B_{r}) $ is equicontinuous on $ J $. The functions $ \{Ny: y\in B_{r}\} $ are equicontinuous at $ t = 0 $. For $ t_{1}, t_{2}\in J_{k}, t_{1}<t_{2}, k = 0, 1, 2, \cdots, m $ and $ y\in B_{r} $ we have

$ \begin{align*} t^{2-\beta}\|Ny(t_{1})-Ny(t_{2})\|\leq& C_{1}(t_{1})t_{2}^{2-\beta}\|Ny(t_{1})-Ny(t_{2})\|\\ \leq& C_{1}(t_{1})\|t_{1}^{2-\beta}Ny(t_{1})-t_{2}^{2-\beta}Ny(t_{2})\|\\ &+C_{1}(t_{1})\|t_{2}^{2-\beta}Ny(t_{2})-t_{1}^{2-\beta}Ny(t_{2})\|\\ \leq& C_{1}(t_{1})\|t_{1}^{2-\beta}Ny(t_{1})-t_{2}^{2-\beta}Ny(t_{2})\|\\ &+C_{1}(t_{1})\|Ny(t_{2})\|\|t_{2}^{2-\beta}-t_{1}^{2-\beta}\|, \end{align*} $

where there exist $ C_{1}(t_{1})>0 $. The right side is independent of $ y\in B_{r} $ and tend to zero as $ t_{1}\rightarrow t_{2} $ since $ t^{2-\beta}Ny(t)\in C(J_{k, X)} $ and $ \|t_{2}^{2-\beta}-t_{1}^{2-\beta}\|\rightarrow 0 $ as $ t_{1}\rightarrow t_{2} $. So $ \|Ny(t_{1})-Ny(t_{2})\|_{B_{v}^{''}}\rightarrow0 $ as $ t_{1}\rightarrow t_{2} $. Hence, $ N(B_{r}) $ is equicontinuous on $ J $.

Step 4   Mönch's condition holds.

Let $ N = N_{1}+N_{2}+N_{3} $, where

$ \begin{align*} N_{1}y(t) = &\frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}f(s, y_{s}+\hat{\phi_{s}})ds, \\ N_{2}y(t) = &g(t, y_{t}+\hat{\phi_{t}})+\varphi_{2}\frac{t^{\beta-1}}{\Gamma(\beta)}+\varphi_{1}\frac{t^{\beta-2}}{\Gamma(\beta-1)}, \\ N_{3}y(t) = &\sum\limits_{i = 1}^{k}J_{i}(y(t_{i}^{-})+\hat{\phi{(t_{i}^{-})}})\frac{t^{\beta-2}}{\Gamma(\beta-1)}(\frac{t}{\beta-1}-t_{i})\\ &+\sum\limits_{i = 1}^{k}I_{i}(y(t_{i}^{-})+\hat{\phi{(t_{i}^{-})}})\frac{t^{\beta-2}}{\Gamma(\beta-1)}. \end{align*} $

Assume $ W\subseteq B_{r} $ is countable and $ W\subseteq \overline{\text {co}}\left(\{0\}\cup N(W)\right) $. We show that $ \beta(W) = 0 $, where $ \beta $ is the Hausdorff MNC. Without loss of generality, we may suppose that $ W = \{y^{n}\}_{n = 1}^{\propto} $. Since $ N(W) $ is equicontinuous on $ J_{k} $, $ W\subseteq \overline{\text {co}}\left(\{0\}\cup N(W)\right) $ is equicontinuous on $ J_{k} $ as well.

Using Lemma 2.4, (H₁)(ⅲ), (H₂)(ⅱ),(H₃), we have

$ \begin{align*} \beta\left(\{N_{1}y^{n}(t)\}_{n = 1}^{\propto}\right)\leq&\frac{2}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}\eta(s) [\sup\limits_{-\propto<\theta\leq 0}\beta(\{y_{s}^{n}(\theta)\}_{n = 1}^{\propto})]ds\\ \leq&\frac{2T^{\beta}}{\Gamma(\beta+1)}\|\eta\|_{L^{\frac{1}{\alpha_{2}}}(J, R^{+})} \sup\limits_{-\propto<\theta\leq 0}\beta(\{y_{s}^{n}(\theta)\}_{n = 1}^{\propto}), \\ \beta\left(\{N_{2}y^{n}(t)\}_{n = 1}^{\propto}\right)\leq& \beta(g(t, y^{n}_{t}+\hat{\phi_{t}}))\\ \leq& G\sup\limits_{-\propto<\theta\leq 0}\beta(\{y_{t}^{n}(\theta)\}_{n = 1}^{\propto}), \\ \beta(\{N_{3}y^{n}(t)\}_{n = 1}^{\propto})\leq& \frac{T}{\Gamma(\beta)}\beta(\{\sum\limits_{i = 1}^{k}t^{\beta-2}J_{i}(y^{n}(t_{i}^{-})+\hat{\phi{(t_{i}^{-})}})\}_{n = 1}^{\propto})\\ &+\frac{1}{\Gamma(\beta-1)}\beta(\{\sum\limits_{i = 1}^{k}t^{\beta-2}I_{i}(y^{n}(t_{i}^{-})+\hat{\phi{(t_{i}^{-})}})\}_{n = 1}^{\propto})\\ \leq&\frac{T^{\ast}}{\Gamma^{\ast}}(\beta(\{\sum\limits_{i = 1}^{k}t^{\beta-2}J_{i}(y^{n}(t_{i}^{-})+\hat{\phi{(t_{i}^{-})}})\}_{n = 1}^{\propto})\\ &+\beta(\{\sum\limits_{i = 1}^{k}t^{\beta-2}I_{i}(y^{n}(t_{i}^{-})+\hat{\phi{(t_{i}^{-})}})\}_{n = 1}^{\propto}))\\ \leq&\frac{T^{\ast}}{\Gamma^{\ast}}\sum\limits_{i = 1}^{m}(M_{i}+K_{i})\sup\limits_{-\propto<\theta\leq 0}\beta(\{y^{n}(\theta)\}_{n = 1}^{\propto}). \end{align*} $

We thus obtain

$ \begin{align*} \beta\left(\{Ny^{n}(t)\}_{n = 1}^{\propto}\right)\leq&\beta\left(\{N_{1}y^{n}(t)\}_{n = 1}^{\propto}\right) +\beta\left(\{N_{2}y^{n}(t)\}_{n = 1}^{\propto}\right)+\beta\left(\{N_{3}y^{n}(t)\}_{n = 1}^{\propto}\right)\\ \leq&\frac{2T^{\beta}}{\Gamma(\beta+1)}\|\eta\|_{L^{\frac{1}{\alpha_{2}}}(J, R^{+})} \sup\limits_{-\propto<\theta\leq 0}\beta(\{y_{s}^{n}(\theta)\}_{n = 1}^{\propto})\\ &+G\sup\limits_{-\propto<\theta\leq 0}\beta(\{y_{t}^{n}(\theta)\}_{n = 1}^{\propto})\\ &+\frac{T^{\ast}}{\Gamma^{\ast}}\sum\limits_{i = 1}^{m}(M_{i}+K_{i})\sup\limits_{-\propto<\theta\leq 0}\beta(\{y^{n}(\theta)\}_{n = 1}^{\propto})\\ \leq&\left(\frac{2T^{\beta}}{\Gamma(\beta+1)}\|\eta\|_{L^{\frac{1}{\alpha_{2}}}(J, R^{+})}+G\right.\\ &\left.+\frac{T^{\ast}}{\Gamma^{\ast}}\sum\limits_{i = 1}^{m}(M_{i}+K_{i})\right)\beta(\{y^{n}(t)\}_{n = 1}^{\propto})\\ = &M^{\ast}\beta\left(\{y^{n}(t)\}_{n = 1}^{\propto}\right), \end{align*} $

where $ M^{\ast} $ is defined in assumption (H$ _{4}) $. Since $ W $ and $ N(W) $ are equicountinuous on every $ J_{k} $, it follows from Lemma 2.4 that the inequality implies $ \beta(NW)\leq M^{\ast}\beta(W) $. Thus, from Mönch's condition, we have

$ \begin{equation*} \beta(W)\leq\beta(\overline{\text {co}}\{0\}\cup N(W)) = \beta(NM)\leq M^{\ast}\beta(W). \end{equation*} $

Since $ M^{\ast}<1 $, we get $ \beta(W) = 0 $. It follows that $ W $ is relatively compact. Using Lemma 2.5, we know that $ N $ has a fixed point $ y $ in $ W $. So the theorem is proved.

(H₅) The function $ g(t, x) $ satisfies the condition that $ |g(t, x)-g(t, y)|\leqslant L|x-y| $, $ L $ is a constant and $ 0<Ll<1 $.

Theorem 3.2   Suppose conditions (H$ _{1} $)(H$ _{3} $)(H$ _{4} $)(H$ _{5}) $ are satisfied. Then system(1.1) has at least one solution on $ J $ and this solution is Ulam stable.

Proof   It is easy to see that the solution satisfies condition (H$ _{2}) $ when the solution satisfies condition (H$ _{5}) $. By using Theorem 3.1, we can prove the existence of this solution. Then we consider the inequality

$ E\|D_{0^{+}}^{\beta}[x(t)-g(t, x_{t})]-f(t, x_{t})\|^2<\varepsilon. $

Suppose there exists a function $ f_1(t, y_t) $ satisfies $ \|f(t, x_t)-f_1(t, y_t)\|<\varepsilon, $ Then for the equation

$ \left\{\begin{array}{lcl} D_{0^{+}}^{\beta}[x(t)-g(t, x_{t})] = f(t, x_{t}), \quad t\in [0, T], \quad t\neq t_{k}, \\ \Delta I_{0^{+}}^{2-\beta}x(t_{k}) = I_{k}(x(t_{k}^{-})), \quad \Delta I_{0^{+}}^{1-\beta}x(t_{k}) = J_{k}(x(t_{k}^{-})), \\ I_{0^{+}}^{2-\beta}[x(0)-g(0, x_{0})] = \varphi_{1}\in B_{v}, \quad I_{0^{+}}^{1-\beta}[x(0)-g(0, x_{0})] = \varphi_{2}\in B_{v}. \end{array}\right. $

We have the fundamental solution of this equation as

$ \begin{equation*} y(t) = \left\{\begin{array}{lcl} y_{0} = \phi\in B_{v}, \qquad t\in(-\propto, 0], \\ \frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}f(s, y_{s})ds+g(t, y_{t})\\ +\varphi_{2}\frac{t^{\beta-1}}{\Gamma(\beta)}+\varphi_{1}\frac{t^{\beta-2}}{\Gamma(\beta-1)}, t\in[0, t_{1}], \\ \frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}f(s, y_{s})ds+g(t, y_{t})+\varphi_{2}\frac{t^{\beta-1}}{\Gamma(\beta)}+\varphi_{1}\frac{t^{\beta-2}}{\Gamma(\beta-1)}\\ +\sum\limits_{i = 1}^{k}J_{i}(y(t_{i}^{-}))\frac{t^{\beta-2}}{\Gamma(\beta-1)}(\frac{t}{\beta-1}-t_{i})\\ +\sum\limits_{i = 1}^{k}I_{i}(y(t_{i}^{-}))\frac{t^{\beta-2}}{\Gamma(\beta-1)}, \quad t\in(t_{k}, t_{k+1}]. \end{array} \right. \end{equation*} $

It is obvious to see that the solution is Ulam stable in the interval $ (-\propto, 0] $, so, first, let's have a look at the interval $ t\in(0, t_{1}] $,

$ \begin{align*} t^{2-\beta}\|x(t)-y(t)\|& = |\frac{t^{2-\beta}}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}(f(s, x_{s})-f(s, y_{s}))ds|+t^{2-\beta}|g(t, x_{t})-g(t, y_{t})| \\ &\leq\frac{T^{2-\beta}\varepsilon}{\Gamma(\beta)}\int_{0}^{t_{1}}(t_{1}-s)^{\beta-1}ds+Llt^{2-\beta}\|x(t)-y(t)\|. \end{align*} $

So, we have

$ \begin{align*} t^{2-\beta}\|x(t)-y(t)\|&\leq\frac{T^{2-\beta}\varepsilon}{\Gamma(\beta)(1-Ll)}\int_{0}^{t_{1}}(t_{1}-s)^{\beta-1}ds. \end{align*} $

Here $ K = \frac{T^{2-\beta}}{\Gamma(\beta)(1-Ll)}\int_{0}^{t_{1}}(t_{1}-s)^{\beta-1}ds $.

Second, consider the interval $ t\in(t_{1}, t_{2}] $,

$ \begin{align*} t^{2-\beta}\|x(t)-y(t)\| = &\frac{t^{2-\beta}}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}(f(s, x_{s})-f(s, y_{s}))ds+t^{2-\beta}|g(t, x_{t})-g(t, y_{t})| \\ & +\sum\limits_{i = 1}^{k}(J_{i}(x(t_{i}^{-}))-J_{i}(y(t_{i}^{-})))\frac{1}{\Gamma(\beta-1)}(\frac{t}{\beta-1}-t_{i})\\ & +\sum\limits_{i = 1}^{k}(I_{i}(x(t_{i}^{-}))-I_{i}(y(t_{i}^{-})))\frac{1}{\Gamma(\beta-1)}\\ \leq&\frac{T^{2-\beta}\varepsilon }{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}ds+Llt^{2-\beta}\|x(t)-y(t)\|\\ & +\sum\limits_{i = 1}^{k}(J_{i}(x(t_{i}^{-}))-J_{i}(y(t_{i}^{-})))\frac{1}{\Gamma(\beta-1)}(\frac{t}{\beta-1}-t_{i})\\ & +\sum\limits_{i = 1}^{k}(I_{i}(x(t_{i}^{-}))-I_{i}(y(t_{i}^{-})))\frac{1}{\Gamma(\beta-1)}. \end{align*} $

As we have had the conclusion that in the interval $ t\in(0, t_{1}] $ that $ |y(t)-x(t)|<K\varepsilon $, so we have

$ |I_{i}(x(t_{i}^{-}))-I_{i}(y(t_{i}^{-}))|<K_{1}\varepsilon, $

$ |J_{i}(x(t_{i}^{-}))-J_{i}(y(t_{i}^{-}))|<K_{2}\varepsilon, $

due to $ I_{k}, J_{k} $ are continuous functions.

So

$ \begin{align*} t^{2-\beta}\|x(t)-y(t)\|\leq&\frac{T^{2-\beta}\varepsilon }{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}ds+t^{2-\beta}Ll\|x(t)-y(t)\|\\ & +\sum\limits_{i = 1}^{k}(J_{i}(x(t_{i}^{-}))-J_{i}(y(t_{i}^{-})))\frac{1}{\Gamma(\beta-1)}(\frac{t}{\beta-1}-t_{i})\\ & +\sum\limits_{i = 1}^{k}(I_{i}(x(t_{i}^{-}))-I_{i}(y(t_{i}^{-})))\frac{1}{\Gamma(\beta-1)}\\ \leq&\frac{T^{2-\beta}\varepsilon }{\Gamma(\beta)}\int_{0}^{t_{2}}(t_{2}-s)^{\beta-1}ds+Llt^{2-\beta}\|x(t)-y(t)\|\\ & +\sum\limits_{i = 1}^{k}\frac{K_{1}\varepsilon }{\Gamma(\beta-1)}(\frac{t}{\beta-1}-t_{1})+\frac{K_{2}k\varepsilon }{\Gamma(\beta-1)}. \end{align*} $

So

$ \begin{align*} t^{2-\beta}\|x(t)-y(t)\| \leq&\frac{T^{2-\beta}\varepsilon }{\Gamma(\beta)(1-Ll)}\int_{0}^{t_{2}}(t_{2}-s)^{\beta-1}ds\\ &+\sum\limits_{i = 1}^{k}\frac{K_{1}\varepsilon }{\Gamma(\beta-1)(1-Ll)}(\frac{t_{2}}{\beta-1}-t_{1})\\ &+\frac{K_{2}k\varepsilon }{\Gamma(\beta-1)(1-Ll)}. \end{align*} $

So, in the interval $ t\in(t_1, t_2], $

$ \begin{align*} K = &\frac{T^{2-\beta}}{\Gamma(\beta)(1-Ll)}\int_{0}^{t_{2}}(t_{2}-s)^{\beta-1}ds\\ &+\sum\limits_{i = 1}^{k}\frac{K_{1}}{\Gamma(\beta-1)(1-Ll)}(\frac{t_{2}}{\beta-1}-t_{1})\\ &+\frac{K_{2}k}{\Gamma(\beta-1)(1-Ll)}. \end{align*} $

In this way, when $ t $ is in the interval $ t\in(t_{i-1}, t_{i}] $ can be proved.