Four Different Ulam-Type Stability for Implicit Second-Order Fractional Integro-Differential Equation with M-Point Boundary Conditions

Abstract

In this paper, we discuss the existence and uniqueness of a solution for the implicit two-order fractional integro-differential equation with m-point boundary conditions by applying the Banach fixed point theorem. Moreover, in the paper we establish the four different varieties of Ulam stability (Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam-Rassias stability, and generalized Hyers–Ulam–Rassias stability) for the given problem.

1. Introduction

The derivative and integral of a function are generalized to non-integer order in fractional calculus. Despite this calculus’s ancient origins, it has only recently been relevant in the fields of science and engineering (mathematical physics, hydrology, biology, and mechanics) [1,2,3,4,5]. The existence and singularity of solutions to differential equations with fractional derivatives have been discussed by some academics ([6,7,8] and references therein).

In 1940, Ulam delivered a lecture at Wisconsin University, introducing the concept of functional equations’ stability. Within this discussion, Ulam posed a question [9,10] that was initially addressed by Hyers in 1941 [11]. Consequently, this type of stability became known as Ulam–Hyers stability. Rassias significantly developed the concept of Ulam–Hyers stability in 1978 [12]. Since then, a functional equation’s stability has been extensively studied in a multitude of articles and through presentations delivered at various conferences [13,14,15].

The primary goal of this paper is to study the Ulam-type stability for the following fractional integro-differential problem with tow-orders (FIDP)

$${}^{c}D_{0+}^{\beta }\left(p{\left(t\right)}^c{D}_{0^{+}}^{\alpha }u\left(t\right)\right)=f\left(t,u\left(t\right),{\mathsf{\Xi }}_{k}u\left(t\right),{\mathsf{\Xi }}_{h}u\left(t\right)\right),$$

(1)

$${\mathsf{\Xi }}_{k}u\left(t\right)={\int }_0^{t}k{(t,s)}^c{D}_{0^{+}}^{\beta }\left(p{\left(s\right)}^c{D}_{0^{+}}^{\alpha }u\left(s\right)\right)ds,{\mathsf{\Xi }}_{h}u\left(t\right)={\int }_0^{t}h{(t,s)}^c{D}_{0^{+}}^{\beta }\left(p{\left(s\right)}^c{D}_{0^{+}}^{\alpha }u\left(s\right)\right)ds,$$

with the conditions

$$u\left(0\right)=u^{\prime }\left(0\right)=u^{\left(3\right)}\left(0\right)=0,u^{″}\left(T\right)=\sum _{i=1}^{m-2}{\eta }_{i}u\left({\xi }_i\right)and{}^{c}D^{\alpha }u\left(t\right){\mid }_{t=0}=u_{\alpha },$$

(2)

for all $t\in [0,T]$ and ${}^{c}D_{0+}^{\alpha }$ is the Caputo fractional derivative, $\alpha \in (3,4)$, $\beta \in \left(0,1\right)$ such that $3<\alpha +\beta <4$, $p\left(t\right)$ on $\left[0,T\right]$ and $k(t,s)$, $h(t,s)$ on $\left[0,T\right]\times \left[0,T\right]$ are real given functions, $f:\left[0,T\right]\times {\mathbb{R}}^3\to \mathbb{R}$ is a given function.

2. Preliminaries

In this section, we introduce some definitions and proprieties, which are used in this paper.

Definition 1 

([1,2]). The Riemann–Liouville fractional (arbitrary) integral of order $\alpha >0$ of the function $f\in L^1\left[0,T\right]$ is formally defined by

$$I_{0^{+}}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{\left(t-s\right)}^{\alpha -1}f\left(s\right)ds,$$

(3)

where Γ is the classical Gamma function.

Definition 2 

([1,2]). The Caputo fractional derivative of order $\alpha >0$ for a given function $f\left(t\right)$ on $\left[0,T\right]$ is defined by

$${}^{c}D_{0+}^{\alpha }f\left(t\right)=D_{0^{+}}^{\alpha }\left[f\left(t\right)-\sum _{k=0}^{n-1}{\displaystyle \frac{f^{\left(k\right)}\left(0\right)}{k!}}{t}^k\right],$$

(4)

where $n=\left[\alpha \right]+1,$$\left[\alpha \right]$ means the integer part of α and $D_{0^{+}}^{\alpha }$ is the Riemann- Liouville fractional derivative operator of order α defined by

$$D_{0^{+}}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(n-\alpha \right)}}{\left({\displaystyle \frac{d}{dt}}\right)}^n{\int }_0^t{\left(t-s\right)}^{n-1-\alpha }f\left(s\right)ds=D^n{I}_{0^{+}}^{n-\alpha }f\left(t\right),fort>0.$$

(5)

Lemma 1 

([16]). Let $\alpha >0,$$u\in C^n\left(\mathbb{R}\right)$, then

$$I^{\alpha }\left({}^{c}D^{\alpha }u\left(t\right)\right)=u\left(t\right)+c_0+c_{1}t+c_2{t}^2+...+c_{n-1}{t}^{n-1},$$

for $c_i\in \mathbb{R},$$i=0,1,...,n-1$, $n=\left[\alpha \right]+1$.

To define Ulam’s stability for fractional integro-differential problem (1) and (2).

Definition 3. 

If there exists a real number $c_f>0$ such that for each $ϵ>0$ and for each solution $w\in AC\left(\left[0,T\right],\mathbb{R}\right)$ of the inequality

$$\begin{array}{c}\hfill \left|{}^{c}D_{0+}^{\beta }\left(p{\left(t\right)}^c{D}_{0^{+}}^{\alpha }w\left(t\right)\right)-f\left(t,w\left(t\right),{\mathsf{\Xi }}_{k}w\left(t\right),{\mathsf{\Xi }}_{h}w\left(t\right)\right)\right|\le ϵ,\end{array}$$

(6)

there exists a solution $u\in AC\left(\left[0,T\right],\mathbb{R}\right)$ of the  FIDP  (1) and (2) with

$$\left|w\left(t\right)-u\left(t\right)\right|\le c_fϵ,t\in \left[0,T\right],$$

then the  FIDP  (1) and (2) is Hyers–Ulam stable.

Definition 4. 

If there exists ${\theta }_f\in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ with ${\theta }_f\left(0\right)=0$ such that for each solution $w\in AC\left(\left[0,T\right],\mathbb{R}\right)$ of the inequality (6), for each $ϵ>0$ there exists a solution $u\in AC\left(\left[0,T\right],\mathbb{R}\right)$ of (1) and (2) with

$$\left|w\left(t\right)-u\left(t\right)\right|\le {\theta }_f\left(ϵ\right),t\in \left[0,T\right],$$

thus, the  FIDP  (1) and (2) is generalized Hyers–Ulam stable.

Definition 5. 

Let $\psi :[0,T]\to {\mathbb{R}}_{+}$ be continues function, for $t\in \left[0,T\right].$ The  FIDP  (1) and (2) is Hyers–Ulam–Rassias stable with respect to ψ if there exists $c_{f,\psi }>0$ where for every $ϵ>0$ and for each solution $w\in AC\left(\left[0,T\right],\mathbb{R}\right)$ of the inequality

$$\left|{}^{c}D_{0+}^{\beta }\left(p{\left(t\right)}^c{D}_{0^{+}}^{\alpha }w\left(t\right)\right)-f\left(t,w\left(t\right),{\mathsf{\Xi }}_{k}w\left(t\right),{\mathsf{\Xi }}_{h}w\left(t\right)\right)\right|\le ϵ\psi \left(t\right),$$

(7)

there exists a solution $u\in AC\left(\left[0,T\right],\mathbb{R}\right)$ of (1) and (2) with

$$\left|w\left(t\right)-u\left(t\right)\right|\le c_{f,\psi }ϵ\psi \left(t\right),t\in \left[0,T\right].$$

Definition 6. 

Let $\psi :[0,T]\to {\mathbb{R}}_{+}$ be continues function, for $t\in \left[0,T\right].$ The  FIDP  (1) and (2) is generalized stability according to the Hyers–Ulam–Rassias with respect to ψ if there exists $c_{f,\psi }>0$ such that for each solution $w\in AC\left(\left[0,T\right],\mathbb{R}\right)$ of the inequality

$$\left|{}^{c}D_{0+}^{\beta }\left(p{\left(t\right)}^c{D}_{0^{+}}^{\alpha }w\left(t\right)\right)-f\left(t,w\left(t\right),{\mathsf{\Xi }}_{k}w\left(t\right),{\mathsf{\Xi }}_{h}w\left(t\right)\right)\right|\le \psi \left(t\right),$$

(8)

there exists a solution $u\in AC\left(\left[0,T\right],\mathbb{R}\right)$ of (1) and (2) with

$$\left|w\left(t\right)-u\left(t\right)\right|\le c_{f,\psi }\psi \left(t\right),t\in \left[0,T\right].$$

Remark 1. 

It is clear that: $\left(i\right)$ Definition 3⟹ Definition 4, $\left(ii\right)$ Definition 5⟹ Definition 6 $\left(iii\right)$ Definition 5 for $\psi \left(t\right)=1$⟹ Definition 3.

Remark 2. 

A function $w\in AC\left(\left[0,T\right],\mathbb{R}\right)$ meets the inequality $\left(6\right)$ if and only if a function $g\in C\left(\left[0,T\right],\mathbb{R}\right)$ exists such that

(i) 

$\left|g\left(t\right)\right|\le ϵ$, $t\in \left[0,T\right]$.

(ii) 

${}^{c}D_{0+}^{\beta }\left(p{\left(t\right)}^c{D}_{0^{+}}^{\alpha }w\left(t\right)\right)=f\left(t,w\left(t\right),{\mathsf{\Xi }}_{k}w\left(t\right),{\mathsf{\Xi }}_{h}w\left(t\right)\right)+g\left(t\right),$$t\in \left[0,T\right].$

Remark 3. 

A function $w\in AC\left(\left[0,T\right],\mathbb{R}\right)$ meets the inequality $\left(8\right)$ if and only if a function $g\in C\left(\left[0,T\right],\mathbb{R}\right)$ exists such that

(i) 

$\left|g\left(t\right)\right|\le \psi \left(t\right)$, $t\in \left[0,T\right]$.

(ii) 

${}^{c}D_{0+}^{\beta }\left(p{\left(t\right)}^c{D}_{0^{+}}^{\alpha }w\left(t\right)\right)=f\left(t,w\left(t\right),{\mathsf{\Xi }}_{k}w\left(t\right),{\mathsf{\Xi }}_{h}w\left(t\right)\right)+g\left(t\right),$$t\in \left[0,T\right].$

Remark 4. 

A function $w\in AC\left(\left[0,T\right],\mathbb{R}\right)$ meets the inequality $\left(7\right)$ if and only if a function $g\in C\left(\left[0,T\right],\mathbb{R}\right)$ exists such that

(i) 

$\left|g\left(t\right)\right|\le ϵ\psi \left(t\right)$, $t\in \left[0,T\right]$.

(ii) 

${}^{c}D_{0+}^{\beta }\left(p{\left(t\right)}^c{D}_{0^{+}}^{\alpha }w\left(t\right)\right)=f\left(t,w\left(t\right),{\mathsf{\Xi }}_{k}w\left(t\right),{\mathsf{\Xi }}_{h}w\left(t\right)\right)+g\left(t\right),$$t\in \left[0,T\right].$

3. Existence and Uniqueness

In this section, by mentioning basic lemmas, we introduce conditions adequate for the existence and the uniqueness of solution to the FIDP (1) and (2) using Banach’s fixed point theorem.

Lemma 2. 

A function $u\in AC\left(\left[0,T\right],\mathbb{R}\right)$ is a solution of the  FIDP  (1) and (2) if and only if u is a solution of the following fractional integral equation for $t\in \left[0,T\right]$

$$\begin{array}{ccc}\hfill u\left(t\right)& =& {\int }_0^{T}G\left(t,s\right)\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}x\left(\tau \right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds\hfill \\ & & +{\displaystyle \frac{t^2}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}{\eta }_i{\int }_0^{T}G\left({\xi }_i,s\right)\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}x\left(\tau \right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds,\hfill \end{array}$$

(9)

where x is the solution of fractional integral equation

$$x\left(t\right)=f\left(t,u\left(t\right),{\int }_0^{t}k(t,s)x\left(s\right)ds,{\int }_0^{t}h(t,s)x\left(s\right)ds\right),$$

and $G(t,s)$ is the Green’s function defined by

$$G(t,s)=\left\{\begin{array}{c}\begin{array}{c}{\displaystyle \frac{{(t-s)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}}-{\displaystyle \frac{t^2{(T-s)}^{\alpha -3}}{2\mathsf{\Gamma }(\alpha -2)}},0<s<t<T,\hfill \\ -{\displaystyle \frac{t^2{(T-s)}^{\alpha -3}}{2\mathsf{\Gamma }(\alpha -2)}},0<t<s<T.\hfill \end{array}\hfill \end{array}\right.$$

(10)

Proof. 

Let ${}^{c}D_{0+}^{\beta }\left(p{\left(t\right)}^c{D}_{0^{+}}^{\alpha }u\left(t\right)\right)=x\left(t\right)$ in Equation (1), then

$$x\left(t\right)=f\left(t,u\left(t\right),{\int }_0^{t}k(t,s)x\left(s\right)ds,{\int }_0^{t}h(t,s)x\left(s\right)ds\right),$$

assume that u a solution of the FIDP (1) and (2) then we have $I_{0^{+}}^{\beta }\left({}^{c}D_{0+}^{\beta }\left(p{\left(t\right)}^c{D}_{0^{+}}^{\alpha }u\left(t\right)\right)\right)=I_{0^{+}}^{\beta }x\left(t\right)$, $\beta \in (0,1)$, then

$$\begin{array}{ccc}\hfill {}^{c}D_{0+}^{\alpha }u\left(t\right)& =& {\displaystyle \frac{1}{p\left(t\right)}}{I}_{0^{+}}^{\beta }x\left(t\right)+{\displaystyle \frac{c_0}{p\left(t\right)}},\hfill \end{array}$$

where $c_0$ is a real constant, for $t=0$, ${}^{c}D_{0+}^{\alpha }u\left(0\right)={\displaystyle \frac{c_0}{p_0}}=u_{\alpha }$ implies that $c_0=p_0{u}_{\alpha }$.

Applying $I_{0^{+}}^{\alpha }$, we get

$$\begin{array}{ccc}\hfill I_{0^{+}}^{\alpha }\left({}^{c}D_{0+}^{\alpha }u\left(t\right)\right)& =& I_{0^{+}}^{\alpha }\left({\displaystyle \frac{1}{p\left(t\right)}}{I}_{0^{+}}^{\beta }x\left(t\right)+{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(t\right)}}\right),\alpha \in (3,4),\hfill \end{array}$$

then

$$\begin{array}{ccc}\hfill u\left(t\right)& =& I_{0^{+}}^{\alpha }\left({\displaystyle \frac{1}{p\left(t\right)}}{I}^{\beta }x\left(t\right)+{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(t\right)}}\right)+a_0+a_{1}t+a_2{t}^2+a_3{t}^3,\hfill \end{array}$$

(11)

where $a_0$, $a_1$, $a_2$ and $a_3$ are some real constants. Using the boundary conditions $u\left(0\right)=u^{\prime }\left(0\right)=u^{\left(3\right)}\left(0\right)=0$, we find

$$a_0=a_1=a_3=0.$$

The boundary condition $u^{″}\left(T\right)={\sum }_{i=1}^{m-2}{\eta }_{i}u\left({\xi }_i\right)$ we get

$$\begin{array}{ccc}\hfill a_2& =& {\displaystyle \frac{1}{2-{\sum }_1^{m-2}{\eta }_i{\xi }_i^2}}\{{\displaystyle \frac{-1}{\mathsf{\Gamma }(\alpha -2)}}{\int }_0^T{\displaystyle \frac{{(T-s)}^{\alpha -3}}{p\left(s\right)}}\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}}x\left(\tau \right)d\tau +p_0{u}_{\alpha }\right)ds\hfill \\ & & +\sum _1^{m-2}{\eta }_i{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^{{\xi }_i}{\displaystyle \frac{{({\xi }_i-s)}^{\alpha -1}}{p\left(s\right)}}\left({\int }_0^s{\displaystyle \frac{{(s-\tau )}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}}x\left(\tau \right)d\tau +p_0{u}_{\alpha }\right)ds\}.\hfill \end{array}$$

Substituting $a_0,$$a_1$, $a_2$ and $a_3$ into $\left(11\right)$ we have $\left(9\right)$. □

Now, we present our existence result based on the Banach contraction fixed point theorem.

Theorem 1. 

Assume that

$\left(H_1\right)$$f:[0,T]\times {\mathbb{R}}^3\to \mathbb{R}$ is continuous, $k(t,s)$, $h(t,s)$ on $[0,T]\times [0,T]$ are continuous and there exist constants $L,l_k,l_h>0$ such that

$$\begin{array}{ccc}\hfill \left|f\left(t,u_1,u_2,u_3\right)-f\left(t,v_1,v_2,v_3\right)\right|& \le & L\left(\left|u_1-v_1\right|+\left|u_2-v_2\right|+\left|u_3-v_3\right|\right),\hfill \end{array}$$

for any $u_i,$$v_i\in \mathbb{R},$$(i=1,2,3)$ and $t\in \left[0,T\right],$

$$\underset{t,s\in [0,T]}{max}k(t,s)=l_k,\underset{t,s\in [0,T]}{max}h(t,s)=l_h,$$

$\left(H_2\right)$$p\in AC\left(\left[0,T\right],\mathbb{R}\right)$ such that $p\left(t\right)\ne 0$$t\in \left[0,T\right]$ with $p\left(0\right)=p_0.$

$\left(H_3\right)$ For ${inf}_{t\in \left[0,T\right]}\left|p\left(t\right)\right|=q$ we have

$$\zeta =L\mathsf{\Delta }<1,$$

(12)

where

$$\mathsf{\Delta }=\left({\displaystyle \frac{G_0{T}^{\beta +1}}{\mathsf{\Gamma }\left(\beta +1\right)q}}+{\displaystyle \frac{T^{\beta +3}{G}_0}{\left(2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2\right)\mathsf{\Gamma }\left(\beta +1\right)q}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|\right){\displaystyle \frac{1}{1-LT\left(l_k+l_h\right)}},$$

(13)

and ${max}_{t,s}G(t,s)=G_0.$ If $\left(H_1\right)-\left(H_3\right)$ are satisfied, then the  FIDP  (1) and (2) has a unique solution.

Proof. 

First, we denote by $X=AC\left(\left[0,T\right],\mathbb{R}\right)$ the Banach space of all continuous functions from $\left[0,T\right]$ into $\mathbb{R}$ with the sup norm ${∥u∥}_{\infty }={sup}_{t\in \left[0,T\right]}\left|u\left(t\right)\right|.$ Define the operator $A:C\left(\left[0,T\right],\mathbb{R}\right)\to C\left(\left[0,T\right],\mathbb{R}\right)$ by

$$\begin{array}{ccc}\hfill Au\left(t\right)& =& {\int }_0^{T}G\left(t,s\right)\times \left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}x\left(\tau \right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds\hfill \\ & & +{\displaystyle \frac{t^2}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}{\eta }_i{\int }_0^{T}G\left({\xi }_i,s\right)\times \left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}x\left(\tau \right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds,\hfill \end{array}$$

(14)

it is clear that the fixed points of the operator A are solution of FIDP (1) and (2).

From hypothesis $\left(H_1\right)$, we have

$$\left|f\left(t,u_1,u_2,u_3\right)-f\left(t,0,0,0\right)\right|\le L\left(\left|u_1\right|+\left|u_2\right|+\left|u_3\right|\right),$$

then

$$\left|f\left(t,u_1,u_2,u_3\right)\right|\le \left|f\left(t,u_1,u_2,u_3\right)-f\left(t,0,0,0\right)\right|+\left|f\left(t,0,0,0\right)\right|\le L\left(\left|u_1\right|+\left|u_2\right|+\left|u_3\right|\right)+M,$$

where $M={sup}_{t\in \left[0,T\right]}\left|f\left(t,0,0,0\right)\right|.$ Then, we define the nonempty, bounded and convex closed set by

$$B_r=\left\{u\in C\left(\left[0,T\right],R\right):∥u∥\le r\right\},$$

where

$$r\ge {\displaystyle \frac{\mathsf{\Delta }M+\frac{G_{0}T{p}_0{u}_{\alpha }}{q}+\frac{T^3{G}_0}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}{\sum }_{i=1}^{m-2}\left|{\eta }_i\right|\left(\frac{p_0{u}_{\alpha }}{q}\right)}{1-\mathsf{\Delta }L}},$$

(15)

we demonstrate that the operator A defined by $\left(14\right)$ meets the hypothesis of the fixed point theorem of Banach. The proof will be two steps.

Step 1. Maps bounded sets into bounded sets in $B_r$, i.e., $A\left(B_r\right)\subset B_r$

For each $t\in \left[0,T\right],$ by condition $\left(H_1\right)$, we have

$$\begin{array}{ccc}\hfill \left|Au\left(t\right)\right|& =& \left|{\int }_0^{T}G\left(t,s\right)\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}x\left(\tau \right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds\right.\hfill \\ & +& \left.{\displaystyle \frac{t^2}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}{\eta }_i{\int }_0^{T}G\left({\xi }_i,s\right)\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}x\left(\tau \right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds\right|\hfill \\ & \le & {\int }_0^T\left|G\left(t,s\right)\right|\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\left|x\left(\tau \right)\right|d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{\left|p\left(s\right)\right|}}\right)ds\hfill \\ & +& {\displaystyle \frac{\left|t^2\right|}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|{\int }_0^T\left|G\left({\xi }_i,s\right)\right|\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\left|x\left(\tau \right)\right|d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{\left|p\left(s\right)\right|}}\right)ds.\hfill \end{array}$$

(16)

We take $u\in AC\left(\left[0,T\right],\mathbb{R}\right),$ then for $t\in \left[0,T\right]$, we have

$$\begin{array}{ccc}\hfill \left|x\left(t\right)\right|& =& \left|f\left(t,u\left(t\right),{\int }_0^{t}k\left(\tau ,s\right)x\left(s\right)ds,{\int }_0^{t}h\left(\tau ,s\right)x\left(s\right)ds\right)\right|\hfill \\ & ⩽& L\left(\left|u\left(t\right)\right|+T\left(l_k+l_h\right)\left|x\left(t\right)\right|\right)+M,\hfill \end{array}$$

then

$$∥x∥⩽{\displaystyle \frac{L∥u∥+M}{1-LT\left(l_k+l_h\right)}}.$$

Equation $\left(16\right)$ then reduces to

$$\begin{array}{ccc}\hfill \left|Au\left(t\right)\right|& \le & G_0{\int }_0^T\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\left({\displaystyle \frac{L∥u∥+M}{1-LT\left(l_k+l_h\right)}}\right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{\left|p\left(s\right)\right|}}\right)ds\hfill \\ & +& {\displaystyle \frac{T^2}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|G_0{\int }_0^T\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\left({\displaystyle \frac{L∥u∥+M}{1-LT\left(l_k+l_h\right)}}\right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{\left|p\left(s\right)\right|}}\right)ds\hfill \\ & \le & G_{0}T\left({\displaystyle \frac{T^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)q}}\left({\displaystyle \frac{L∥u∥+M}{1-LT\left(l_k+l_h\right)}}\right)\right)+{\displaystyle \frac{G_{0}T{p}_0{u}_{\alpha }}{q}}\hfill \\ & +& {\displaystyle \frac{T^3{G}_0}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|\times \left({\displaystyle \frac{T^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)q}}\left({\displaystyle \frac{L∥u∥+M}{1-LT\left(l_k+l_h\right)}}\right)+{\displaystyle \frac{p_0{u}_{\alpha }}{q}}\right)\hfill \\ & \le & \left({\displaystyle \frac{G_0{T}^{\beta +1}}{\mathsf{\Gamma }\left(\beta +1\right)q}}+{\displaystyle \frac{T^{\beta +3}{G}_0}{\left(2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2\right)\mathsf{\Gamma }\left(\beta +1\right)q}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|\right){\displaystyle \frac{L}{1-LT\left(l_k+l_h\right)}}r\hfill \\ & +& \left({\displaystyle \frac{G_0{T}^{\beta +1}}{\mathsf{\Gamma }\left(\beta +1\right)q}}+{\displaystyle \frac{T^{\beta +3}{G}_0}{\left(2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2\right)\mathsf{\Gamma }\left(\beta +1\right)q}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|\right){\displaystyle \frac{M}{1-LT\left(l_k+l_h\right)}}\hfill \\ & +& {\displaystyle \frac{G_{0}T{p}_0{u}_{\alpha }}{q}}+{\displaystyle \frac{T^3{G}_0}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|\left({\displaystyle \frac{p_0{u}_{\alpha }}{q}}\right)\hfill \\ & \le & \mathsf{\Delta }Lr+\mathsf{\Delta }M+{\displaystyle \frac{G_{0}T{p}_0{u}_{\alpha }}{q}}+{\displaystyle \frac{T^3{G}_0}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|\left({\displaystyle \frac{p_0{u}_{\alpha }}{q}}\right)\le r.\hfill \end{array}$$

Taking for $t\in \left[0,T\right],$ we have

$$\left|Au\left(t\right)\right|\le r,$$

then

$$A\left(B_r\right)\subset B_r.$$

Step 2. A is contraction.

Now for $u,v\in AC\left(\left[0,T\right],\mathbb{R}\right)$, then for $t\in \left[0,T\right]$ we have

$$\begin{array}{c}\hfill Au\left(t\right)-Av\left(t\right)={\int }_0^{T}G\left(t,s\right)\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\left(x\left(\tau \right)-y\left(\tau \right)\right)d\tau \right)ds\\ \hfill +{\displaystyle \frac{T^2}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}{\eta }_i{\int }_0^{T}G\left({\xi }_i,s\right)\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\left(x\left(\tau \right)-y\left(\tau \right)\right)d\tau \right)ds,\end{array}$$

where $x,y\in AC\left(\left[0,T\right],\mathbb{R}\right)$ be such that

$$\begin{array}{ccc}\hfill x\left(t\right)& =& f\left(t,u\left(t\right),{\int }_0^{t}k\left(\tau ,s\right)x\left(s\right)ds,{\int }_0^{t}h\left(\tau ,s\right)x\left(s\right)ds\right),\hfill \\ \hfill y\left(t\right)& =& f\left(t,v\left(t\right),{\int }_0^{t}k\left(\tau ,s\right)y\left(s\right)ds,{\int }_0^{t}h\left(\tau ,s\right)y\left(s\right)ds\right),\hfill \end{array}$$

then

$$\begin{array}{c}\hfill \left|Au\left(t\right)-Av\left(t\right)\right|\le {\int }_0^T\left|G\left(t,s\right)\right|\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\left|x\left(\tau \right)-y\left(\tau \right)\right|d\tau \right)ds\\ \hfill +{\displaystyle \frac{T^2}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|{\int }_0^T\left|G\left({\xi }_i,s\right)\right|\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\left|x\left(\tau \right)-y\left(\tau \right)\right|d\tau \right)ds.\end{array}$$

(17)

From hypothesis $\left(H_1\right)$ we have

$$\begin{array}{ccc}\hfill \left|x\left(t\right)-y\left(t\right)\right|& =& \left|f\left(t,u\left(t\right),{\int }_0^{t}k\left(\tau ,s\right)x\left(s\right)ds,{\int }_0^{t}h\left(\tau ,s\right)x\left(s\right)ds\right)\right.\hfill \\ & & -\left.f\left(t,v\left(t\right),{\int }_0^{t}k\left(\tau ,s\right)y\left(s\right)ds,{\int }_0^{t}h\left(\tau ,s\right)y\left(s\right)ds\right)\right|\hfill \\ & ⩽& L\left(\left|u\left(t\right)-v\left(t\right)\right|+{\int }_0^{t}k\left(t,s\right)\left|x\left(s\right)-y\left(s\right)\right|ds+{\int }_0^{t}h\left(t,s\right)\left|x\left(s\right)-y\left(s\right)\right|ds\right)\hfill \\ & ⩽& L\left(∥u-v∥+T\left(l_k+l_h\right)∥x-y∥\right),\hfill \end{array}$$

thus

$$∥x-y∥\le {\displaystyle \frac{L}{1-LT\left(l_k+l_h\right)}}∥u-v∥.$$

We substitute in $\left(17\right)$, we have

$$\begin{array}{ccc}\hfill \left|Au\left(t\right)-Av\left(t\right)\right|& \le & \left({\displaystyle \frac{G_0{T}^{\beta +1}}{\mathsf{\Gamma }\left(\beta +1\right)q}}+{\displaystyle \frac{T^{\beta +3}{G}_0}{\left(2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2\right)\mathsf{\Gamma }\left(\beta +1\right)q}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|\right){\displaystyle \frac{L}{1-LT\left(l_k+l_h\right)}}∥u-v∥\hfill \\ & ⩽& \zeta ∥u-v∥,\hfill \end{array}$$

by $\zeta <1$, the operator A is a contraction. Hence, by Banach’s contraction principle, A has a unique fixed point which is a solution of the FIDP (1) and (2) on $[0,T]$. □

4. Ulam Types Stability

In this section, we first establish our Hyres–Ulam stability result and secondly our Hyres–Ulam–Rassias stability of the two-orders fractional integro differential Equation $\left(1\right)$.

4.1. Hyres–Ulam Stability

We need the following lemma.

Lemma 3. 

If $w\in AC\left(\left[0,T\right],\mathbb{R}\right)$ is a solution of the fractional differential inequality $\left(6\right)$ for each $ϵ>0$ and the initial condition $\left(2\right)$, then w is a solution of the following integral inequality

$$\left|w\left(t\right)-{\int }_0^{T}G\left(t,s\right)\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}\left(z\left(\tau \right)+g\left(\tau \right)\right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds\right.$$

$$\left.-{\displaystyle \frac{t^2}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}{\eta }_i{\int }_0^{T}G\left({\xi }_i,s\right)\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}\left(z\left(\tau \right)+g\left(\tau \right)\right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds\right|$$

$$\le \left({\displaystyle \frac{G_0{T}^{\beta +1}}{\mathsf{\Gamma }\left(\beta +1\right)q}}+{\displaystyle \frac{T^{\beta +3}{G}_0}{\left(2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2\right)\mathsf{\Gamma }\left(\beta +1\right)q}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|\right)ϵ,$$

where

$$z\left(t\right)=f\left(t,w\left(t\right),{\int }_0^{t}k\left(\tau ,s\right)z\left(s\right)ds,{\int }_0^{t}h\left(\tau ,s\right)z\left(s\right)ds\right).$$

Proof. 

Let $w\in AC\left(\left[0,T\right],\mathbb{R}\right)$ be a solution of the inequality (6) for each $ϵ$ then, from Remark 2 and for some continuous function $g\left(t\right)$ such that $\left|g\left(t\right)\right|⩽ϵ,t\in \left[0,T\right]$ we have

$$\begin{array}{ccc}\hfill \left|w\left(t\right)-Aw\left(t\right)\right|& ⩽& \left|w\left(t\right)-{\int }_0^{T}G\left(t,s\right)\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}\left(z\left(\tau \right)+g\left(\tau \right)\right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds\right.\hfill \\ & & \left.-{\displaystyle \frac{t^2}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}{\eta }_i{\int }_0^{T}G\left({\xi }_i,s\right)\times \left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}\left(z\left(\tau \right)+g\left(\tau \right)\right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds\right|\hfill \\ & \le & {\int }_0^T\left|G\left(t,s\right)\right|{\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\left|g\left(\tau \right)\right|d\tau ds\hfill \\ & & +{\displaystyle \frac{T^2}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|{\int }_0^T\left|G\left({\xi }_i,s\right)\right|{\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\left|g\left(\tau \right)\right|d\tau ds\hfill \\ & ⩽& \left({\displaystyle \frac{G_0{T}^{\beta +1}}{\mathsf{\Gamma }\left(\beta +1\right)q}}+{\displaystyle \frac{T^{\beta +3}{G}_0}{\left(2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2\right)\mathsf{\Gamma }\left(\beta +1\right)q}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|\right)ϵ.\hfill \end{array}$$

□

The main result of this subsection.

Theorem 2. 

Assume that $\left(H_1\right)-\left(H_3\right)$ hold, then the FIDP  (1) and (2) is Hyers–Ulam stable.

Proof. 

Under $\left(H_1\right)-\left(H_3\right)$, the FIDP (1) and (2) has unique solution in $AC\left(\left[0,T\right],\mathbb{R}\right).$ Let $w\in$$AC\left(\left[0,T\right],\mathbb{R}\right)$ be a solution of the inequality (6), then by Lemma 3 we have for each $t\in \left[0,T\right]$

$$\begin{array}{cc}\hfill \left|w\left(t\right)-u\left(t\right)\right|& ⩽\left|w\left(t\right)-{\int }_0^{T}G\left(t,s\right)\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}x\left(\tau \right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{t^2}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}{\eta }_i{\int }_0^{T}G\left({\xi }_i,s\right)\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}x\left(\tau \right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds\right|\hfill \\ \hfill & ⩽\left({\displaystyle \frac{G_0{T}^{\beta +1}}{\mathsf{\Gamma }\left(\beta +1\right)q}}+{\displaystyle \frac{T^{\beta +3}{G}_0}{\left(2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2\right)\mathsf{\Gamma }\left(\beta +1\right)q}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|\right)ϵ\hfill \\ \hfill & +{\int }_0^T\left|G\left(t,s\right)\right|{\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\left|z\left(\tau \right)-x\left(\tau \right)\right|d\tau ds\hfill \\ \hfill & +{\displaystyle \frac{T^2}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|{\int }_0^T\left|G\left({\xi }_i,s\right)\right|{\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\left|z\left(\tau \right)-x\left(\tau \right)\right|d\tau ds\hfill \\ \hfill & ⩽\left({\displaystyle \frac{G_0{T}^{\beta +1}}{\mathsf{\Gamma }\left(\beta +1\right)q}}+{\displaystyle \frac{T^{\beta +3}{G}_0}{\left(2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2\right)\mathsf{\Gamma }\left(\beta +1\right)q}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|\right)ϵ+\mathsf{\Delta }L\left|w\left(t\right)-u\left(t\right)\right|,\hfill \end{array}$$

then, we have

$$\left|w\left(t\right)-u\left(t\right)\right|⩽{\displaystyle \frac{\frac{G_0{T}^{\beta +1}}{\mathsf{\Gamma }\left(\beta +1\right)q}+\frac{T^{\beta +3}{G}_0}{\left(2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2\right)\mathsf{\Gamma }\left(\beta +1\right)q}{\sum }_{i=1}^{m-2}\left|{\eta }_i\right|}{1-\mathsf{\Delta }L}}ϵ,$$

where $\mathsf{\Delta }$ is defined by (13).

Then, there exists a real number $c_f={\displaystyle \frac{\frac{G_0{T}^{\beta +1}}{\mathsf{\Gamma }\left(\beta +1\right)q}+\frac{T^{\beta +3}{G}_0}{\left(2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2\right)\mathsf{\Gamma }\left(\beta +1\right)q}{\sum }_{i=1}^{m-2}\left|{\eta }_i\right|}{1-\mathsf{\Delta }L}}>0$ such that

$$\left|w\left(t\right)-u\left(t\right)\right|⩽c_fϵ.$$

Thus, FIDP (1) and (2) has the Hyers–Ulam stability. □

4.2. Hyres–Ulam–Rassias Stability

The following lemma will be used in the subsequent proof of the Hyers–Ulam–Rassias stability result.

Lemma 4. 

If $w\in AC\left(\left[0,T\right],\mathbb{R}\right)$ is a solution of the fractional differential inequality (7), it follows that w a solution of the following integral inequality

$$\left|w\left(t\right)-{\int }_0^{T}G\left(t,s\right)\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}z\left(\tau \right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds\right.$$

$$\left.-{\displaystyle \frac{t^2}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}{\eta }_i{\int }_0^{T}G\left({\xi }_i,s\right)\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}z\left(\tau \right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds\right|$$

$$\le {\displaystyle \frac{G_0ϵ}{\mathsf{\Gamma }\left(\beta \right)q}}{\int }_0^t{\int }_0^s{(s-\tau )}^{\beta -1}\psi \left(\tau \right)d\tau ds$$

$$+\left({\displaystyle \frac{ϵG_0}{\mathsf{\Gamma }\left(\beta \right)q}}+{\displaystyle \frac{ϵT^2{G}_0}{\left(2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2\right)\mathsf{\Gamma }\left(\beta \right)q}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|\right){\int }_0^T{\int }_0^s{(s-\tau )}^{\beta -1}\psi \left(\tau \right)d\tau ds,$$

where

$$z\left(t\right)=f\left(t,w\left(t\right),{\int }_0^{t}k\left(\tau ,s\right)z\left(s\right)ds,{\int }_0^{t}h\left(\tau ,s\right)z\left(s\right)ds\right).$$

Proof. 

Let $w\in AC\left(\left[0,T\right],\mathbb{R}\right)$ be a solution of the inequality (7) for each $ϵ$ then, from Remark 4 and for some continuous function $g\left(t\right)$ such that $\left|g\left(t\right)\right|⩽ϵ,t\in \left[0,T\right]$, in each $t\in \left[0T\right]$, we have

$$\begin{array}{ccc}\hfill \left|w\left(t\right)-Aw\left(t\right)\right|& ⩽& \left|w\left(t\right)-{\int }_0^{T}G\left(t,s\right)\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}\left(z\left(\tau \right)+g\left(\tau \right)\right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds\right.\hfill \\ & & \left.-{\displaystyle \frac{t^2}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}{\eta }_i{\int }_0^{T}G\left({\xi }_i,s\right)\times \left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}\left(z\left(\tau \right)+g\left(\tau \right)\right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds\right|\hfill \\ & \le & {\int }_0^t{\displaystyle \frac{{(t-s)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\psi \left(\tau \right)ϵd\tau ds+{\int }_0^T{G}_0{\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\psi \left(\tau \right)ϵd\tau ds\hfill \\ & & +{\displaystyle \frac{T^2}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|{\int }_0^T\left|G\left({\xi }_i,s\right)\right|{\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\psi \left(\tau \right)ϵd\tau ds\hfill \\ & \le & {\displaystyle \frac{G_0ϵ}{\mathsf{\Gamma }\left(\beta \right)q}}{\int }_0^t{\int }_0^s{(s-\tau )}^{\beta -1}\psi \left(\tau \right)d\tau ds\hfill \\ & & +\left({\displaystyle \frac{ϵG_0}{\mathsf{\Gamma }\left(\beta \right)q}}+{\displaystyle \frac{ϵT^2{G}_0}{\left(2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2\right)\mathsf{\Gamma }\left(\beta \right)q}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|\right){\int }_0^T{\int }_0^s{(s-\tau )}^{\beta -1}\psi \left(\tau \right)d\tau ds.\hfill \end{array}$$

□

Now we are ready to announce our Hyres–Ulam–Rassias stability result.

Theorem 3. 

Assume that $\left(H_1\right)-\left(H_3\right)$ hold and $\left(H_4\right)$ The function $\psi \in AC\left(\left[0,T\right],\mathbb{R}\right)$ is increasing and there exists ${\gamma }_{\psi }>0$ such that, for each $t\in \left[0,T\right]$, we have

$${\int }_0^t{\int }_0^s{\left(s-\tau \right)}^{\beta -1}\psi \left(\tau \right)d\tau ds⩽{\gamma }_{\psi }\psi \left(t\right),$$

then the  FIDP  (1) and (2) is Hyers–Ulam–Rassias stable with respect to ${\mathsf{\Pi }}_{\psi }$, where ${\mathsf{\Pi }}_{\psi }$ is defined by:

$${\mathsf{\Pi }}_{\psi }\left(t\right)=\psi \left(t\right)+{\displaystyle \frac{\left(2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2+T^2{\sum }_{i=1}^{m-2}\left|{\eta }_i\right|\right)}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\psi \left(T\right),$$

Proof. 

Under $\left(H_1\right)-\left(H_3\right)$, the FIDP (1) and (2) has unique solution in $AC\left(\left[0,T\right],\mathbb{R}\right).$ Let $w\in AC\left(\left[0,T\right],\mathbb{R}\right)$ be a solution of the inequality (7), then for each $t\in \left[0,T\right]$

$$\begin{array}{cc}\hfill \left|w\left(t\right)-u\left(t\right)\right|& ⩽\left|w\left(t\right)-{\int }_0^{T}G\left(t,s\right)\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}x\left(\tau \right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{t^2}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}{\eta }_i{\int }_0^{T}G\left({\xi }_i,s\right)\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)p\left(s\right)}}x\left(\tau \right)d\tau +{\displaystyle \frac{p_0{u}_{\alpha }}{p\left(s\right)}}\right)ds\right|\hfill \\ \hfill \le & {\displaystyle \frac{G_0ϵ}{\mathsf{\Gamma }\left(\beta \right)q}}{\gamma }_{\psi }\psi \left(t\right)+\left({\displaystyle \frac{ϵG_0}{\mathsf{\Gamma }\left(\beta \right)q}}+{\displaystyle \frac{ϵT^2{G}_0}{\left(2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2\right)\mathsf{\Gamma }\left(\beta \right)q}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|\right){\gamma }_{\psi }\psi \left(T\right)\hfill \\ \hfill & +{\int }_0^T\left|G\left(t,s\right)\right|\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\left|z\left(\tau \right)-x\left(\tau \right)\right|d\tau \right)ds\hfill \\ \hfill & +{\displaystyle \frac{T^2}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|{\int }_0^T\left|G\left({\xi }_i,s\right)\right|\left({\int }_0^s{\displaystyle \frac{{\left(s-\tau \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)\left|p\left(s\right)\right|}}\left|z\left(\tau \right)-x\left(\tau \right)\right|d\tau \right)ds\hfill \\ \hfill & ⩽{\displaystyle \frac{G_0ϵ}{\mathsf{\Gamma }\left(\beta \right)q}}{\gamma }_{\psi }\psi \left(t\right)+\left({\displaystyle \frac{G_0}{\mathsf{\Gamma }\left(\beta \right)q}}+{\displaystyle \frac{T^2{G}_0}{\left(2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2\right)\mathsf{\Gamma }\left(\beta \right)q}}\sum _{i=1}^{m-2}\left|{\eta }_i\right|\right){\gamma }_{\psi }ϵ\psi \left(T\right)\hfill \\ \hfill & +\mathsf{\Delta }L\left|w\left(t\right)-u\left(t\right)\right|\hfill \\ \hfill & ⩽{\displaystyle \frac{G_0ϵ}{\mathsf{\Gamma }\left(\beta \right)q}}{\gamma }_{\psi }{\mathsf{\Pi }}_{\psi }\left(t\right)+\mathsf{\Delta }L\left|w\left(t\right)-u\left(t\right)\right|,\hfill \end{array}$$

with

$${\mathsf{\Pi }}_{\psi }\left(t\right)=\psi \left(t\right)+{\displaystyle \frac{\left(2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2+T^2{\sum }_{i=1}^{m-2}\left|{\eta }_i\right|\right)}{2-{\sum }_{i=1}^{m-2}{\eta }_i{\xi }_i^2}}\psi \left(T\right),$$

then, we have

$$\left|w\left(t\right)-u\left(t\right)\right|\le {\displaystyle \frac{G_0ϵ}{\mathsf{\Gamma }\left(\beta \right)q\left(1-\mathsf{\Delta }L\right)}}{\gamma }_{\psi }{\mathsf{\Pi }}_{\psi }\left(t\right),$$

where $\mathsf{\Delta }$ is defined by (13).

Then, there exists a real number

$$c_{f,\psi }={\displaystyle \frac{G_0{\gamma }_{\psi }}{\mathsf{\Gamma }\left(\beta \right)q\left(1-\mathsf{\Delta }L\right)}}>0,$$

such that

$$\left|w\left(t\right)-u\left(t\right)\right|⩽c_{f,\psi }ϵ{\mathsf{\Pi }}_{\psi }\left(t\right).$$

Thus, FIDP (1) and (2) has the Hyers–Ulam–Rassias stability with respect to ${\mathsf{\Pi }}_{\psi }$. □

5. Example

To clarify to the reader the importance of our results, we present the following examble.

Example 1. 

Consider the following problem

$$\left\{\begin{array}{c}{}^{c}D_{0+}^{{\displaystyle \frac{1}{2}}}\left({{\displaystyle \frac{1}{1+t}}}^c{D}_{0^{+}}^{{\displaystyle \frac{7}{2}}}u\left(t\right)\right)=f\left(t,u\left(t\right),{\int }_0^{t}k\left(t,s\right)x\left(s\right)ds,{\int }_0^{t}h\left(t,s\right)x\left(s\right)ds\right),t\in \left[0,1\right],\\ u\left(0\right)=u^{\prime }\left(0\right)=u^{\left(3\right)}\left(0\right)=0,u^{″}\left(1\right)=\eta \xi \mathit{and}{}^{c}D^{{\displaystyle \frac{7}{2}}}{u\left(t\right)|}_{t=0}=u_{{\displaystyle \frac{7}{2}}},\end{array}\right.$$

(18)

where $x\left(t\right){=}^c{D}_{0^{+}}^{{\displaystyle \frac{1}{2}}}\left({{\displaystyle \frac{1}{1+t}}}^c{D}_{0^{+}}^{{\displaystyle \frac{7}{2}}}u\left(t\right)\right)$, $k\left(t,s\right)=e^{-\left(s-t\right)}$ and $h\left(t,s\right)=e^{-{\displaystyle \frac{s-t}{2}}},$ 

with

$$\begin{array}{cc}\hfill & f\left(t,u\left(t\right),{\int }_0^{t}k\left(t,s\right)x\left(s\right)ds,{\int }_0^{t}h\left(t,s\right)x\left(s\right)ds\right)\hfill \\ \hfill & ={\displaystyle \frac{2+u\left(t\right)+{\int }_0^t{e}^{-\left(s-t\right)}x\left(s\right)ds+{\int }_0^t{e}^{-\frac{s-t}{2}}x\left(s\right)ds}{\pi e^{t+1}\left(1+u\left(t\right)+{\int }_0^t{e}^{-\left(s-t\right)}x\left(s\right)ds+{\int }_0^t{e}^{-\frac{s-t}{2}}x\left(s\right)ds\right)}},\hfill \end{array}$$

Clearly, the function f is jointly continuous.

For any $u_i,v_i\in \mathbb{R},\left(i=1,2,3\right)$ and $t\in \left[0,1\right]$

$$\left|f\left(t,u_1,u_2,u_3\right)-f\left(t,v_1,v_2,v_3\right)\right|\le {\displaystyle \frac{1}{\pi e}}\left(\left|u_1-v_1\right|+\left|u_2-v_2\right|+\left|u_3-v_3\right|\right).$$

Then $L={\displaystyle \frac{1}{\pi e}},$$G_0={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{7}{2}\right)}},$${max}_{t,s\in \left[0,1\right]}k\left(t,s\right)=e=l_k,$${max}_{t,s\in \left[0,1\right]}h\left(t,s\right)=\sqrt{e}=l_h.$

Substituting L, $l_k$, $l_h$, $G_0$ into Δ yields

$$\mathsf{\Delta }=\left({\displaystyle \frac{1}{{\left(\mathsf{\Gamma }\left(\frac{7}{2}\right)\right)}^2}}+{\displaystyle \frac{1}{3\left(2-\frac{1}{3}{\left(\frac{3}{4}\right)}^2\right){\left(\mathsf{\Gamma }\left(\frac{7}{2}\right)\right)}^2}}\right)\times {\displaystyle \frac{1}{1-\frac{1}{\pi e}\left(e+\sqrt{e}\right)}}=3.0850$$

then

$$\varsigma =\mathsf{\Delta }L\simeq {\displaystyle \frac{3.0850}{\pi e}}=0.36125<1.$$

By Theorem 2,  FIDP  (18) is Hyers–Ulam stable.

6. Conclusions

This research study has a number of important contributions. First, we prove that the integration Equation (9) and FIDP (1) and (2) are equivalent. The focus of our investigation is this significant link. The existence and uniqueness of solutions for boundary value issues involving implicit two-order fractional integro-differential equations are then established. The use of the Banach fixed-point theorem forms the foundation of this accomplishment. Additionally, we learn about and investigate a number of stability characteristics, such as Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam–Rassias stability, and generalized Hyers–Ulam–Rassias stability. If boundary conditions of the fractional integral type are added, implicit two-order fractional integro-differential equations of fractional order are also subject to these stability features. Last but not least, we wrap up the article with a concrete example that effectively illustrates the usefulness and applicability of the obtained results.