*Equation* (1) *is Mittag-Leffler-Ulam-Hyers stable of first type, with respect to E*_{q}, if there exists a real number c > 0 *such that for each ε* > 0 *and for each solution y of the inequality*
$$|y(x)-\frac{1}{\text{\Gamma}(q)}{\displaystyle \underset{c}{\overset{x}{\int}}{(x-\tau )}^{q-1}}f(x,\tau ,y(\tau ),y(\alpha (\tau )))d\tau |\le \epsilon {E}_{q}({\tau}^{q}),$$(2)

*there exists a unique solution y*_{0} *of equation (1) satisfying the following inequality*:
$$|y(x)-{y}_{0}(x)|\le c\epsilon Eq({x}^{q}).$$

*Suppose that α*: [*a, b*] → [*a, b*] *is a continuous function such that α*(*x*) ≤ *x*, *for all x* ∈ [*a, b*] *and f*: [*a, b*] × [*a, b*] × ℝ × ℝ → ℝ *is a continuous function which satisfies the following Lipschitz condition*
$$|f(x,\tau ,{y}_{1}(\tau ),{y}_{1}(\alpha (\tau ))-f(x,\tau ,{y}_{2}(\tau ),{y}_{2}(\alpha (\tau ))|\le L|{y}_{1}-{y}_{2}|,$$(3)

*for any x, τ* ∈ [*a, b*] *and y*_{1}, *y*_{2} ∈ ℝ *and equation (2). Then, the equation (1) is Mittag-Leffler-Hyers-Ulam stable of the first type*.

Let us consider the space of continuous functions
$$X=\{g:[a,b]\to \mathbb{R}|\hspace{0.17em}g\hspace{0.17em}\text{is\hspace{0.17em}\hspace{0.17em}continuous}}\text{.}$$

Similar to the well-known Theorem 3.1 of [25], endowed with the generalized metric defined by
$$\begin{array}{cc}d(g,h)=\text{inf}\{K\in [0,\infty ]|& |g(x)-h(x)|\le K\epsilon {E}_{q}({x}^{q})\forall x\in [a,b]\},\end{array}$$(4)

it is known that (*X*, *d*) is a complete generalized metric space. Define an operator ∧ : *X* → *X* by
$$(\Lambda g)(x)\frac{1}{\text{\Gamma}(q)}{\displaystyle \underset{0}{\overset{x}{\int}}{(x-\tau )}^{q-1}g(x,\tau ,g(\tau ),g(\alpha (\tau )))d\tau ,}$$(5)

for all *g* ∈ *X* and *x* ∈ [*a, b*]. Since *g* is a continuous function, it follows that ∧*g* is also continuous and this ensures that ∧ is a well-defined operator. For any *g*, *h* ∈ *X*, let *K*_{gh} ∈ [0, ∞] such that
$$|g(x)-h(x)|\le {K}_{gh}\epsilon {E}_{q}({x}^{q})$$(6)

for any *x* ∈ [*a, b*]. From the definition of ∧, (3) and (6) we have
$$\begin{array}{rl}|(\mathrm{\Lambda}g)(x)-(\mathrm{\Lambda}h)(x)|& =\frac{1}{\mathrm{\Gamma}(q)}|\underset{0}{\overset{x}{\int}}{(x-\tau )}^{q-1}(f(x,\tau ,g(\tau ),g(\alpha (\tau ))-f(x,\tau ,h(\tau ),h(\alpha (\tau )))d\tau |\\ & \le \frac{1}{\mathrm{\Gamma}(q)}\underset{0}{\overset{x}{\int}}{(x-\tau )}^{q-1}|g(\tau )-h(\tau )|d\tau \le \frac{L{K}_{gh}\epsilon}{\mathrm{\Gamma}(q)}\underset{0}{\overset{x}{\int}}{(x-\tau )}^{q-1}{E}_{q}({\tau}^{q})d\tau \\ & \le \frac{L{K}_{gh}\epsilon}{\mathrm{\Gamma}(q)}\underset{0}{\overset{x}{\int}}{(x-\tau )}^{q-1}\sum _{k=0}^{\mathrm{\infty}}\frac{{\tau}^{kq}}{\mathrm{\Gamma}(kq+1)}d\tau \\ & =\frac{L{K}_{gh}\epsilon}{\mathrm{\Gamma}(q)}\sum _{k=0}^{\mathrm{\infty}}\frac{1}{\mathrm{\Gamma}(kq+1)}\underset{0}{\overset{x}{\int}}{(x-\tau )}^{q-1}{\tau}^{kq}d\tau \\ & =\frac{L{K}_{gh}\epsilon}{\mathrm{\Gamma}(q)}\sum _{k=0}^{\mathrm{\infty}}\frac{1}{\mathrm{\Gamma}(kq+1)}\underset{0}{\overset{x}{\int}}{(x-xt)}^{q-1}(xt{)}^{kq}xdt\\ & =\frac{L{K}_{gh}\epsilon}{\mathrm{\Gamma}(q)}\sum _{k=0}^{\mathrm{\infty}}\frac{{x}^{(k+1)q}}{\mathrm{\Gamma}(kq+1)}\underset{0}{\overset{x}{\int}}{(1-t)}^{q-1}{t}^{kq}dt\\ & =\frac{L{K}_{gh}\epsilon}{\mathrm{\Gamma}(q)}\sum _{k=0}^{\mathrm{\infty}}\frac{{x}^{(k+1)q}}{\mathrm{\Gamma}(kq+1)}\frac{\mathrm{\Gamma}(q)\mathrm{\Gamma}(kq+1)}{\mathrm{\Gamma}(q+kq+1))}\\ & =L{K}_{gh}\epsilon \sum _{k=0}^{\mathrm{\infty}}\frac{{x}^{(k+1)q}}{\mathrm{\Gamma}((k+1)q+1)}\le L{K}_{gh}\epsilon \sum _{k=0}^{\mathrm{\infty}}\frac{{x}^{nq}}{\mathrm{\Gamma}(nq+1)}=L{K}_{gh}\epsilon {E}_{q}({x}^{q})\end{array}$$

for all *x* ∈ [*a, b*]; that is, *d*(∧*g*, ∧*h*) ≤ *LK*_{gh}εE_{q}(*x*^{q}). Hence, we can conclude that *d*(∧*g*, ∧*h*) ≤ *Ld*(*g*, *h*) for any *g*, *h* ∈ *X*, and since 0 < *L* < 1, the strictly continuous property is verified.

Let us take *g*_{0} ∈ *X*. From the continuous property of *g*_{0} and ∧*g*_{0}, it follows that there exists a constant 0 < *K*_{1} < ∞ such that
$$|(\Lambda {g}_{0})(x)-{g}_{0}(x)|=|\frac{1}{\text{\Gamma}(q)}{\displaystyle \underset{0}{\overset{x}{\int}}{(x-\tau )}^{q-1}}f(x,\tau ,{g}_{0}(\tau ),{g}_{0}(\alpha (\tau )))d\tau -{g}_{0}(x)|\le {K}_{1}{E}_{q}({x}^{q}),$$

for all *x* ∈ [*a, b*], since *f* and *g*_{0} are bounded on [*a, b*] and min_{x∈[a, b]} *E*_{q}(*x*^{q}) > 0. Thus, (4) implies that *d*(∧*g*_{0}, *g*_{0}) < ∞.

Therefore, according to Theorem 2.5 (a), there exists a continuous function *y*_{0} : [*a, b*] → ℝ such that ∧^{n}*g*_{0} → *y*_{0} in (*X*, *d*) as *n* → ∞ and ∧*y*_{0} = *y*_{0}; that is, *y*_{0} satisfies the equation (1) for every *x* ∈ [*a, b*].

We will now prove that {*g* ∈ *X*|*d*(*g*_{0}, *g*) < ∞} = *X*. For any *g* ∈ *X*, since *g* and *g*_{0} are bounded on [*a, b*] and min_{x∈[a, b]} *E*_{q}(*x*^{q}) > 0, there exists a constant 0 < *C*_{g} < ∞ such that
$$|{g}_{0}(x)-g(x)|\le {C}_{g}{E}_{q}({x}^{q}),$$

for any *x* ∈ [*a, b*]. Hence, we have *d*(*g*_{0}, *g*) < ∞ for all *g* ∈ *X*; that is,
$$\{g\in X|d({g}_{0},g)<\infty \}=X.$$

Hence, in view of Theorem 2.5 (b), we conclude that y_{0} is the unique continuous function which satisfies the equation (1). Now we have *d*(*y*, ∧ *y*) ≤ *εE*_{q}(*x*^{q}). Finally, Theorem 2.5 (c) together with the above inequality imply that
$$d(y,{y}_{0})\le \frac{1}{1-L}d(\Lambda y,y)\le \frac{1}{1-L}\epsilon {E}_{q}({x}^{q}).$$

This means that the equation (1) is Mittag-Leffler-Hyers-Ulam stable. □

*Consider the following fractional order system*
$${}^{C}{D}_{t}^{\frac{1}{2}}x(t)=\frac{1}{5}\frac{{x}^{2}(t-1)}{1+{x}^{2}(t-1)}+\frac{1}{5}sin(2x(t)),t\in [0,1]$$

*and set x*(0) = 0. *The following inequality holds*:
$$|{}^{C}{D}_{t}^{\frac{1}{2}}y(t)-f(t,y(t),y(t-1))|\text{\hspace{0.17em}}\le \epsilon {E}_{\frac{1}{2}}({t}^{\frac{1}{2}}).$$

*By Remark 15, x*(0) = 0, $L=\frac{2}{5}$ *and above inequality, all assumptions in Theorem 3.2 are satisfied. So our fractional integral is Mittag-Leffler-Hyers-Ulam stable of the first type and*
$$|y(t)-x(t)|\le C\epsilon {E}_{\frac{1}{2}}({t}^{\frac{1}{2}}).$$

Next, we use the Chebyshev norm ||.||_{c} to derive the above similar result for the equation (1).

*Suppose that α*: [*a, b*] → [*a, b*] *is a continuous function such that α*(*x*) ≤ *x*, *for all x* ∈ [*a, b*] *and f* : [*a, b*] × [*a, b*] × ℝ × ℝ → ℝ *is a continuous function which additionally satisfies the following Lipschitz condition*
$$|f(x,\tau ,{y}_{1}(\tau ),{y}_{1}(\alpha (\tau ))-f(x,\tau ,{y}_{2}(\tau ),{y}_{2}(\alpha (\tau ))|\le L|{y}_{1}-{y}_{2}|$$

*for any x, τ* ∈ [*a, b*] *and y*_{1}, *y*_{2} ∈ ℝ *and equation (2). Also suppose that* 0 < 2*LE*_{q}(*b*) < 1. *Then, the initial integral equation (1) is Mittag-Leffler-Hyers-Ulam stable of the first type via the Chebyshev norm*.

Just like the discussion in Theorem 3.2, we only prove that ∧ defined in (5) is a contraction mapping on *X* with respect to the Chebyshev norm:
$$\begin{array}{rl}|(\mathrm{\Lambda}g)(x)-(\mathrm{\Lambda}h)(x)|& =\frac{1}{\mathrm{\Gamma}(q)}|\underset{0}{\overset{x}{\int}}{(x-\tau )}^{q-1}f(x,\tau ,g(\tau ),g(\alpha (\tau ))-f(x,\tau ,h(\tau ),h(\alpha (\tau )))d\tau |\\ & \le \frac{1}{\mathrm{\Gamma}(q)}\underset{0}{\overset{x}{\int}}{(x-\tau )}^{q-1}(\underset{x\in [a,b]}{max}|g(\tau )-h(\tau )|+\underset{x\in [a,b]}{max}|g(\alpha (\tau ))-h(\alpha (\tau ))|)d\tau \\ & \le \frac{2L}{\mathrm{\Gamma}(q)}{\u2225g-h\u2225}_{c}\underset{0}{\overset{x}{\int}}{(x-\tau )}^{q-1}d\tau \le \frac{2L{b}^{q}}{\mathrm{\Gamma}(q+1)}{\u2225g-h\u2225}_{c}\le 2L{\u2225g-h\u2225}_{c}{E}_{q}(b)\end{array}$$

for all *x* ∈ [*a, b*]; that is, *d*(∧*g*, ∧*h*) ≤ 2*L*||*g* − *h*||_{c} *E*_{q}(*b*). Hence, we can conclude that *d*(∧*g*, ∧*h*) ≤ 2*LE*_{q}(*b*)*d*(*g*, *h*) for any *g*, *h* ∈ *X*. By letting 0 < 2*L**E*_{q}(*b*) < 1, the strictly continuous property is verified. Now by proceeding a proof similar to the proof of Theorem 3.2, we have
$$d(y,{y}_{0})\le \frac{1}{1-2L{E}_{q}(b)}d(\Lambda y,y)\le \frac{1}{1-2L{E}_{q}(b)}\epsilon {E}_{q}({x}^{q})\le C\epsilon {E}_{q}({x}^{q}),$$

which means that the equation (1) is Mittag-Leffler-Hyers-Ulam stable of the first type via the Chebyshev norm. □

In the following Theorem we have used the Bielecki norm
$${\Vert x\Vert}_{B}:=\underset{t\in J}{\text{max}}|x(t)|{e}^{-\theta t},\theta >0,J\subset {\mathbb{R}}_{+}$$

to derive the similar Theorem 3.2 for the fundamental equation (1) via the Bielecki norm.

*Suppose that α*: [*a, b*] → [*a, b*] *is a continuous function such that α*(*x*) ≤ *x, for all x* ∈ [*a, b*] *and f*: [*a, b*] × [*a, b*] × ℝ × ℝ → ℝ *is a continuous function which additionally satisfies the Lipschitz condition*
$$|f(x,\tau ,{y}_{1}(\tau ),{y}_{1}(\alpha (\tau ))-f(x,\tau ,{y}_{2}(\tau ),{y}_{2}(\alpha (\tau ))|\le L|{y}_{1}-{y}_{2}|$$

*for any x, τ* ∈ [*a, b*] *and y*_{1}, *y*_{2} ∈ ℝ *and equation (2). Also suppose that* $0<\frac{2L}{\text{\Gamma}(q)}\frac{{b}^{q}{e}^{\theta b}}{\sqrt{2\theta (2q-1)}}<1$. *Then, equation (1) is Mittag-Leffler-Hyers-Ulam stable of the first type via the Bielecki norm*.

Just like the discussion in Theorem 3.4, we only prove that ∧ defined in (5) is a contraction mapping on X with respect to the Bielecki norm:
$$\begin{array}{rl}& |(\mathrm{\Lambda}g)(x)-(\mathrm{\Lambda}h)(x)|=\frac{1}{\mathrm{\Gamma}(q)}|\underset{0}{\overset{x}{\int}}{(x-\tau )}^{q-1}(f(x,\tau ,g(\tau ),g(\alpha (\tau ))-f(x,\tau ,h(\tau ),h(\alpha (\tau )))d\tau |\\ & \le \frac{1}{\mathrm{\Gamma}(q)}\underset{0}{\overset{x}{\int}}{(x-\tau )}^{q-1}{e}^{\theta \tau}(\underset{x\in [a,b]}{max}|g(\tau )-h(\tau )|{e}^{-\theta \tau}+\underset{x\in [a,b]}{max}|g(\alpha (\tau ))-h(\alpha (\tau ))|{e}^{-\theta \tau})d\tau \\ & \le \frac{1}{\mathrm{\Gamma}(q)}{\u2225g-h\u2225}_{B}\underset{0}{\overset{x}{\int}}{(x-\tau )}^{q-1}{e}^{\theta \tau}\le \frac{2L}{\mathrm{\Gamma}(q)}{\u2225g-h\u2225}_{B}[(\underset{0}{\overset{x}{\int}}{(x-\tau )}^{2(q-1)}d\tau {)}^{\frac{1}{2}}(\underset{0}{\overset{x}{\int}}{({e}^{2\theta \tau}d\tau )}^{\frac{1}{2}}]\\ & \le \frac{2L}{\mathrm{\Gamma}(q)}{\u2225g-h\u2225}_{B}\frac{{b}^{q}{e}^{\theta b}}{\sqrt{2\theta (2q-1)}}\end{array}$$

for all *x* ∈ [*a, b*]; that is, $d(\Lambda g,\Lambda h)\le \frac{2L}{\text{\Gamma}(q)}\frac{{b}^{q}{e}^{\theta b}}{\sqrt{2\theta (2q-1)}}{\Vert g-h\Vert}_{B}$. Hence, we can conclude that $d(\Lambda g,\Lambda h)\le \frac{2L}{\text{\Gamma}(q)}\frac{{b}^{q}{e}^{\theta b}}{\sqrt{2\theta (2q-1)}}d(g,h)$ for any *g*, *h* ∈ *X*. By letting $0<\frac{2L}{\text{\Gamma}(q)}\frac{{b}^{q}{e}^{\theta b}}{\sqrt{2\theta (2q-1)}}<1$, the strictly continuous property is verified. Now by a similar process with Theorem 3.4, we have
$$d(y,{y}_{0})\le \frac{1}{1-\frac{2L}{\text{\Gamma}(q)}\frac{{b}^{q}{e}^{\theta b}}{\sqrt{2\theta (2q-1)}}}d(\Lambda y,y)\le \frac{1}{1-\frac{2L}{\text{\Gamma}(q)}\frac{{b}^{q}{e}^{\theta b}}{\sqrt{2\theta (2q-1)}}}\epsilon {E}_{q}({x}^{q})\le C\epsilon {E}_{q}({x}^{q}),$$

which means that equation (1) is Mittag-Leffler-Hyers-Ulam stable of the first type via the Bielecki norm. □

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