Skip to main content

Theory and Modern Applications

Best approximation of a nonlinear fractional Volterra integro-differential equation in matrix MB-space


In this article, we introduce a class of stochastic matrix control functions to stabilize a nonlinear fractional Volterra integro-differential equation with Ψ-Hilfer fractional derivative. Next, using the fixed-point method, we study the Ulam–Hyers and Ulam–Hyers–Rassias stability of the nonlinear fractional Volterra integro-differential equation in matrix MB-space.

1 Introduction

Fractional calculus is considered as a branch of mathematical analysis which deals with the investigation and applications of integrals and derivatives of arbitrary order. Therefore, fractional calculus is an extension of the integer-order calculus that considers integrals and derivatives of any real or complex order [1, 2], i.e., unifies and generalizes the notions of integer-order differentiation and n-fold integration.

Different forms of fractional operators have been introduced, like the Riemann–Liouville, Grinwald–Letnikov, Weyl, Caputo, Marchaud, and Hadamard fractional derivatives. The first approach is that Riemann-Liouville, which is based on iterating the classical integral operator n times and then considering the Cauchy’s formula where n! is replaced by the Gamma function, and hence the fractional integral of noninteger order is defined.

Fractional calculus has attracted the attention of many mathematicians, but also of some researchers in other areas like physics, chemistry, and engineering. As it is well known, several physical phenomena are often better described by fractional derivatives. This is mainly due to the fact that fractional operators take into consideration the evolution of the system, by taking the global correlation, and not only local characteristics. Moreover, integer-order calculus sometimes contradicts the experimental results, and therefore, derivatives of fractional order may be more suitable [35].

Very useful physical applications have given birth to the variable-order fractional calculus, for example, in modeling mechanical behaviors [6]. Nowadays, variable-order fractional calculus is particularly recognized as a useful and promising approach in the modeling of diffusion processes, in order to characterize time- or concentration-dependent anomalous diffusion, or diffusion processes in inhomogeneous porous media [7].

Results on existence and stability of solutions of implicit fractional differential equations can be found in [811].

By proposing the study of solution stability via fractional integrals and fractional derivatives, we can generalize the results and obtain the usual ones as particular cases. In this article, we study distribution functions with the ranges in a class of matrix algebras with the generalized triangular norms, to define MB-space and introduce a new class of matrix control functions. Also, we will use two recent fractional operators, that is, of general differentiation and integration [12].

These concepts help us study the Hyers–Ulam (in short HU) and Hyers–Ulam–Rassias (in short HUR) stability of fractional nonlinear Volterra integro-differential equation (in short VIDE),

$$ \textstyle\begin{cases} {}^{H}\mathbb{D}_{0+}^{\iota ,\kappa ;\varPsi }\mu (\varsigma )= \boldsymbol{F}(\varsigma ,\mu (\varsigma ))+\int _{0}^{\varsigma } \boldsymbol{H}(\varsigma ,\vartheta ,\mu (\varsigma ))\,d\vartheta , \\ \mathcal{I}_{0+}^{1-\gamma }\mu (0)=\sigma , \end{cases} $$

with \(\varsigma \in [0,T]\) and a continuous function (in short CF) \(\boldsymbol{F}(\varsigma ,\mu )\), also \(\boldsymbol{H}(\varsigma ,\vartheta ,\mu )\) is a CF with respect to ς, ϑ and μ on \([0,T]\times \mathbb{R}\times \mathbb{R}\), σ is a fixed number, \({}^{H}\mathbb{D}_{0+}^{\iota ,\kappa ;\varPsi }\mu (\cdot )\) is defined in (2.1) in which \(0<\iota <1\), \(0\leq \kappa \leq 1\), and \(\mathcal{I}_{0+}^{1-\gamma }(\cdot )\) is the Ψ-Riemann–Liouville fractional integral in which \(0\leq \gamma <1\) [12].

2 Preliminaries

Here, we let \(\Xi _{1}=[0,T]\), with \(T>0\), \(\Xi _{2}=(0,\infty )\), \(\Xi _{3}=(0,1]\), \(\Xi _{4}=[0,\infty ]\), and \(\Xi _{5}=[0,1]\) (note that \(\Xi _{5}^{\circ }=(0,1)\) denotes the interior of \(\Xi _{5}\)).


$$ \operatorname{diag}M_{n}(\Xi _{5})= \left\{ \begin{bmatrix} a_{1} & & \\ & \ddots & \\ & & a_{n} \end{bmatrix} =\operatorname{diag}[a_{1},\dots ,a_{n}], a_{1},\ldots ,a_{n}\in \Xi _{5} \right\} , $$

where \(\operatorname{diag}M_{n}(\Xi _{5})\) is equipped with the partial order relation:

$$\begin{aligned}& \boldsymbol{a}:=\operatorname{diag}[a_{1},\dots , a_{n}],\qquad \boldsymbol{b}:= \operatorname{diag}[b_{1},\dots , b_{n}] \in \operatorname{diag}M_{n}(\Xi _{5}), \\& \boldsymbol{a}\preceq \boldsymbol{b}\quad \Longleftrightarrow \quad a_{j} \leq b_{j} \quad \text{for every } j=1, \ldots , n. \end{aligned}$$

Also, \(\boldsymbol{a} \prec \boldsymbol{b}\) denotes that \(\boldsymbol{a} \preceq \boldsymbol{b}\) and \(\boldsymbol{a} \neq \boldsymbol{b}\); \(\boldsymbol{a} \ll \boldsymbol{b}\) and \(a_{j}< b_{j}\) for every \(j=1, \ldots ,n\). We define \(\boldsymbol{e}:=\operatorname{diag}[e,\ldots ,e] \) in \(\operatorname{diag}M_{n}(\Xi _{5})\) where \(e\in \Xi _{5}\). For example, \(\boldsymbol{1}=\operatorname{diag}[1,\ldots ,1]\) and \(\boldsymbol{0}=\operatorname{diag}[0,\ldots ,0]\).

Now, we extend the concept of triangular norms [13, 14] on \(\operatorname{diag}M_{n}(\Xi _{5})\).

Definition 2.1

A generalized triangular norm (in short GTN) on \(\operatorname{diag}M_{n}(\Xi _{5})\) is an operation \(\circledast : \operatorname{diag}M_{n}(\Xi _{5})\times \operatorname{diag}M_{n}(\Xi _{5}) \to \operatorname{diag}M_{n}(\Xi _{5})\) satisfying the following conditions:

  1. (a)

    \((\forall \boldsymbol{a} \in \operatorname{diag}M_{n}(\Xi _{5})) ( \boldsymbol{a}\circledast \textbf{1})=\boldsymbol{a})\) (boundary condition);

  2. (b)

    \((\forall (\boldsymbol{a},\boldsymbol{b})\in (\operatorname{diag}M_{n}(\Xi _{5}))^{2}) ( \boldsymbol{a}\circledast \boldsymbol{b} = \boldsymbol{b} \circledast \boldsymbol{a} )\) (commutativity);

  3. (c)

    \((\forall (\boldsymbol{a},\boldsymbol{b},\boldsymbol{c})\in (\operatorname{diag}M_{n}( \Xi _{5})^{3}) ( \boldsymbol{a}\circledast (\boldsymbol{b} \circledast \boldsymbol{c}) = (\boldsymbol{a}\circledast \boldsymbol{b})\circledast \boldsymbol{c} )\) (associativity);

  4. (d)

    \((\forall (\boldsymbol{a},\boldsymbol{a}^{\prime },\boldsymbol{b}, \boldsymbol{b}^{\prime })\in (\operatorname{diag}M_{n}(\Xi _{5}^{4}) ( \boldsymbol{a}\preceq \boldsymbol{a}^{\prime } \text{ and } \boldsymbol{b}\preceq \boldsymbol{b}^{\prime } \Longrightarrow \boldsymbol{a} \circledast \boldsymbol{b} \preceq \boldsymbol{a}^{ \prime }\circledast \boldsymbol{b}^{\prime } \) (monotonicity).

For every \(\boldsymbol{a}, \boldsymbol{b} \in \operatorname{diag}M_{n}(\Xi _{5})\) and all sequences \(\{\boldsymbol{a}_{k}\}\) and \(\{\boldsymbol{b}_{k}\}\) converging to a and b, respectively, suppose we have

$$ \lim_{k}(\boldsymbol{a}_{k}\circledast \boldsymbol{b}_{k})= \boldsymbol{a}\circledast \boldsymbol{b}, $$

then, on \(\operatorname{diag}M_{n}(\Xi _{5})\) is continuous GTN (in short CGTN). Now we present some examples of CGTN:

(1) If \(\circledast _{M} : \operatorname{diag}M_{n}(\Xi _{5})\times \operatorname{diag}M_{n}( \Xi _{5}) \to \operatorname{diag}M_{n}(\Xi _{5})\) is defined by

$$ \boldsymbol{a}\circledast _{M} \boldsymbol{b}= \operatorname{diag}[t_{1}, \dots ,t_{n}]\circledast _{M} \operatorname{diag}[s_{1}, \dots ,s_{n}] =\operatorname{diag}\bigl[ \min \{t_{1},s_{1} \},\dots ,\min \{t_{n},s_{n}\}\bigr], $$

then \(\circledast _{M}\) is CGTN (minimum CGTN);

(2) If \(\circledast _{P} : \operatorname{diag}M_{n}(\Xi _{5})\times \operatorname{diag}M_{n}( \Xi _{5}) \to \operatorname{diag}M_{n}(\Xi _{5})\) is such that

$$ \boldsymbol{a}\circledast _{P} \boldsymbol{b}= \operatorname{diag}[t_{1}, \dots ,t_{n}]\circledast _{P} \operatorname{diag}[s_{1}, \dots ,s_{n}] =\operatorname{diag}[t_{1} \cdot s_{1},\dots ,t_{n}\cdot s_{n}], $$

then \(\circledast _{P}\) is CGTN (product CGTN);

(3) If \(\circledast _{L} : \operatorname{diag}M_{n}(\Xi _{5})\times \operatorname{diag}M_{n}( \Xi _{5}) \to \operatorname{diag}M_{n}(\Xi _{5})\) is defined by

$$ \begin{aligned} \boldsymbol{a}\circledast _{L} \boldsymbol{b}&= \operatorname{diag}[t_{1}, \dots ,t_{n}]\circledast _{L} \operatorname{diag}[s_{1}, \dots ,s_{n}] \\ &=\operatorname{diag}\bigl[ \max \{t_{1}+s_{1}-1,0 \},\dots ,\max \{t_{n}+s_{n}-1,0\}\bigr], \end{aligned} $$

then \(\circledast _{P}\) is CGTN (Lukasiewicz CGTN).

Now, we present some numerical examples:

$$\begin{aligned}& \begin{aligned} \operatorname{diag} \biggl[\frac{3}{7},1,\frac{4}{5} \biggr] \circledast _{M} \operatorname{diag} \biggl[0,\frac{1}{5}, \frac{2}{3} \biggr]&= \begin{bmatrix} \frac{3}{7} & & \\ & 1 & \\ & & \frac{4}{5} \end{bmatrix} \circledast _{M} \begin{bmatrix} 0 & & \\ & \frac{1}{5} & \\ & & \frac{2}{3} \end{bmatrix} = \begin{bmatrix} 0 & & \\ & \frac{1}{5} & \\ & & \frac{2}{3} \end{bmatrix} \\ &=\operatorname{diag} \biggl[0,\frac{1}{5}, \frac{2}{3} \biggr], \end{aligned} \\& \begin{aligned} \operatorname{diag} \biggl[\frac{3}{7},1,\frac{4}{5} \biggr] \circledast _{P} \operatorname{diag} \biggl[0,\frac{1}{5}, \frac{2}{3} \biggr]&= \begin{bmatrix} \frac{3}{7} & & \\ & 1 & \\ & & \frac{4}{5} \end{bmatrix} \circledast _{P} \begin{bmatrix} 0 & & \\ & \frac{1}{5} & \\ & & \frac{2}{3} \end{bmatrix} = \begin{bmatrix} 0 & & \\ & \frac{1}{5} & \\ & & \frac{8}{15} \end{bmatrix} \\ &=\operatorname{diag} \biggl[0,\frac{1}{5}, \frac{8}{15} \biggr], \end{aligned} \\& \begin{aligned} \operatorname{diag} \biggl[\frac{3}{7},1,\frac{4}{5} \biggr] \circledast _{L} \operatorname{diag} \biggl[0,\frac{1}{5}, \frac{2}{3} \biggr]&= \begin{bmatrix} \frac{3}{7} & & \\ & 1 & \\ & & \frac{4}{5} \end{bmatrix} \circledast _{L} \begin{bmatrix} 0 & & \\ & \frac{1}{5} & \\ & & \frac{2}{3} \end{bmatrix} = \begin{bmatrix} 0 & & \\ & \frac{1}{5} & \\ & & \frac{7}{15} \end{bmatrix} \\ &=\operatorname{diag} \biggl[0,\frac{1}{5}, \frac{7}{15} \biggr]. \end{aligned} \end{aligned}$$

Then we get

$$\begin{aligned}& \operatorname{diag} \biggl[\frac{3}{7},1,\frac{4}{5} \biggr] \circledast _{M} \operatorname{diag} \biggl[0,\frac{1}{5}, \frac{2}{3} \biggr] \\& \quad \succeq \operatorname{diag} \biggl[\frac{3}{7},1,\frac{4}{5} \biggr] \circledast _{P}\operatorname{diag} \biggl[0, \frac{1}{5},\frac{2}{3} \biggr] \\& \quad \succeq \operatorname{diag} \biggl[\frac{3}{7},1,\frac{4}{5} \biggr] \circledast _{L}\operatorname{diag} \biggl[0, \frac{1}{5},\frac{2}{3} \biggr]. \end{aligned}$$

We consider the set \(D^{+}\) of matrix-distribution-function-valued (MDF-valued), left continuous and increasing functions \(\varphi :{\mathbb{R}} \cup \{-\infty ,\infty \} \to \operatorname{diag}M_{n}(\Xi _{5})\) such that \(\varphi _{0}=\boldsymbol{0}\) and \(\varphi _{+\infty }=\boldsymbol{1}\). Now \(O^{+}\subseteq D^{+}\) are all (proper) functions \(\varphi \in D^{+}\) for which \(\ell ^{-}\varphi _{+\infty }=\boldsymbol{1}\) (\(\ell ^{-}\varphi _{ \tau }=\lim_{\varsigma \to \tau ^{-}}\varphi _{\varsigma }\)). Note that proper MDF-valued functions are the MDF-valued functions of real random variables (i.e., of those random variables g that a.s. take real values (\(P(|g|=\infty )=0\))).

In \(D^{+}\), we define “” as follows:

$$ \varphi \precsim \phi \quad \Longleftrightarrow \quad \varphi _{\tau } \preceq \phi _{\tau },\quad \forall \tau \in \mathbb{R}. $$

Also for each \(\varsigma \in \mathbb{R}\),

$$ \nabla ^{\varsigma }_{\tau }= \textstyle\begin{cases} {\mathbf{0}}, & \text{if } \tau \leq \varsigma , \\ {\mathbf{1}}, & \text{if } \tau >\varsigma , \end{cases} $$

belongs to \(D^{+}\) and for every MDF-valued φ we have \(\varphi \precsim \nabla ^{0}\) [13, 15]. For example,

$$ \varphi _{\tau }= \textstyle\begin{cases} {\mathbf{0}}, & \text{if } \tau \leq 0, \\ \operatorname{diag} [1-e^{-\tau },\frac{\tau }{1+\tau },e^{-\frac{1}{\tau }} ], & \text{if } \tau >0, \end{cases} $$

is an MDF-valued function in \(\operatorname{diag}M_{3}(\Xi _{5})\). Note that \(\varphi _{\tau }=\operatorname{diag}[\varphi _{1,\tau },\dots ,\varphi _{n,\tau }]\), with \(\varphi _{i,\tau }\) being distribution functions, is MDF-valued.

Definition 2.2

Consider the CGTN , a linear space W, and MDF-valued \(\Omega :W\to O^{+}\). In this case, we define a matrix Menger normed space (MMN-space) \((W,\Omega ,\circledast )\) as follows:

(MMN1) \(\Omega ^{w}_{\tau }=\nabla ^{0}_{\tau }\) for all \(\tau >0\) if and only if \(w=0\);

(MMN2) \(\Omega ^{\alpha w}_{\tau }=\Omega ^{w}_{\frac{\tau }{|\alpha |}}\) for all \(w\in W\) and \(\alpha \in \mathbb{C}\) with \(\alpha \neq 0\);

(MMN3) \(\Omega ^{w+w'}_{\tau +\varsigma }\succeq \Omega ^{w}_{\tau }\circledast \Omega ^{w'}_{\varsigma }\) for all \(w,w'\in W\) and \(\tau ,\varsigma \geq 0\).

A complete MMN-space is called MMB-space.

For example, the MDF-valued Ω given by

$$ \Omega ^{w}_{\tau }= \textstyle\begin{cases} {\mathbf{0}}, & \text{if } \tau \leq 0, \\ \operatorname{diag} [\exp (-\frac{ \Vert w \Vert }{\tau }),\frac{\tau }{\tau + \Vert w \Vert }, \exp (-\frac{ \Vert w \Vert }{\tau }) ], & \text{if } \tau >0, \end{cases} $$

is a matrix Menger norm and \((W,\Omega ,\circledast _{M})\) is an MMN-space; here \((W,\|\cdot \|)\) is a normed linear space.

Approximation of functional equations was studied in MN-spaces, fuzzy metric spaces, and random multi-normed space [16, 17]. Also stability results for stochastic fractional differential and integral equations were considered in [1827].

Theorem 2.3

([28, 29])

Let \((U, \rho )\) be a complete \(\Xi _{4}\)-valued metric space and let \(\Lambda : U \rightarrow U\) be a strictly contractive function with Lipschitz constant \(\iota <1\). Then, for a given element \(\xi \in U\), either

$$\begin{aligned} \rho \bigl(\Lambda ^{n}\xi ,\Lambda ^{n+1}\xi \bigr) = \infty , \end{aligned}$$

for each \(n\in \mathbb{N}\) or there is \(n_{0}\in \mathbb{N}\) such that

  1. (i)

    \(\rho (\Lambda ^{n}\xi ,\Lambda ^{n+1}\xi )<\infty \), for every \(n\ge n_{0}\);

  2. (ii)

    the fixed point \(\zeta ^{*}\) of Λ is the limit point of the sequence \(\{ \Lambda ^{n} \xi \} \);

  3. (iii)

    in the set \(V= \{ \zeta \in U \mid \rho (\Lambda ^{n_{0}}\xi ,\zeta )< \infty \} \), \(\zeta ^{*}\) is the unique fixed point of Λ;

  4. (iv)

    \((1-\iota )\rho (\zeta ,\zeta ^{\ast } ) \le \rho (\zeta , \Lambda \zeta )\) for every \(\zeta \in V\).

Definition 2.4


The Gamma function Γ is defined by

$$\begin{aligned} \Gamma (z)= \int _{0}^{\infty }e^{-\varsigma }\varsigma ^{z-1}\,d \varsigma , \quad z\in \mathbb{C}, Re(z)>0. \end{aligned}$$

Let \(\iota \in \mathring{\Xi _{5}}\), let Δ be an integrable function on \(\Xi _{1}\) and \(\varPsi \in C^{1}(\Xi _{1})\) an increasing function with \(\varPsi '(\varsigma )\neq 0\), for each \(\varsigma \in \Xi _{1}\). The Ψ-Hilfer fractional derivative is defined by [15]

$$\begin{aligned} {}^{H}\mathbb{D}_{0+}^{\iota ,\kappa ;\varPsi }\Delta ( \varsigma )= \mathcal{I}_{0+}^{\kappa (1-\iota );\varPsi } \biggl( \frac{1}{\varPsi '(\varsigma )} \frac{d}{d\varsigma } \biggr) \mathcal{I}_{0+}^{(1-\kappa )(1-\iota );\varPsi }\Delta ( \varsigma ). \end{aligned}$$

Definition 2.5

If for every continuously differentiable function \(\Delta (\varsigma )\) and MDF-valued φ satisfying

$$\begin{aligned} \Omega ^{ ({}^{H}\mathbb{D}_{0+}^{\iota ,\kappa ;\varPsi }\Delta ( \varsigma )-\boldsymbol{F}(\varsigma ,\Delta (\varsigma ))-\int _{0}^{ \varsigma }\boldsymbol{H}(\varsigma ,\vartheta ,\Delta (\varsigma ))\,d \vartheta )}_{\tau }\succeq \varphi ^{\varsigma }_{\tau }, \end{aligned}$$

for every \(\varsigma \in \Xi _{1}\) and \(\tau \in \Xi _{2}\), there exist a solution \(\Delta _{0}(\varsigma )\) of the VIDE Eq. (1.1) and a fixed number \(\lambda >0\) with

$$\begin{aligned} \Omega ^{ (\Delta (\varsigma )-\Delta _{0}(\varsigma ) )}_{ \tau }\succeq \varphi ^{\varsigma }_{\frac{\tau }{\lambda }}, \end{aligned}$$

for every \(\varsigma \in \Xi _{1}\) and \(\tau \in \Xi _{2}\), where λ is independent of \(\Delta (\varsigma )\) and \(\Delta _{0}(\varsigma )\), then (1.1) has the HUR stability.

3 Main results

Consider the following hypotheses:

(H0) Assume that M, \(L_{\boldsymbol{F}}\), \(L_{\boldsymbol{H}}\) are positive real numbers with \(2M (\max \{ L_{\boldsymbol{F}},L_{\boldsymbol{H}}\} )\in \mathring{\Xi _{5}}\) and let \(\boldsymbol{F}:\Xi _{1}\times \mathbb{R}\rightarrow \mathbb{R}\) and \(\boldsymbol{H}:\Xi _{1}\times \Xi _{1}\times \mathbb{R}\rightarrow \mathbb{R}\) be CFs satisfying

$$\begin{aligned} \Omega ^{ (\boldsymbol{F}(\varsigma ,\Delta _{1})-\boldsymbol{F}( \varsigma ,\Delta _{2}) )}_{\tau }\succeq \Omega ^{\Delta _{1}- \Delta _{2}}_{\frac{\tau }{L_{\boldsymbol{F}}}}, \end{aligned}$$

for all \(\varsigma \in \Xi _{1}\), \(\Delta _{1},\Delta _{2}\in \mathbb{R}\) and \(\tau \in \Xi _{2}\), and

$$\begin{aligned} \Omega ^{ (\boldsymbol{H}(\varsigma ,\vartheta ,\Delta _{1})- \boldsymbol{H}(\varsigma ,\vartheta ,\Delta _{2}) )}_{\tau } \succeq \Omega ^{\Delta _{1}-\Delta _{2}}_{ \frac{\tau }{L_{\boldsymbol{H}}}}, \end{aligned}$$

for all \(\varsigma ,\vartheta \in \Xi _{1}\), \(\Delta _{1},\Delta _{2}\in \mathbb{R}\) and \(\tau \in \Xi _{2}\).

Theorem 3.1

Suppose \((H0)\) holds and consider a nondecreasing function \(\varPsi \in C(\Xi _{1})\) with \(\varPsi '(\varsigma )\neq 0\) and a CDF \(\Delta :\Xi _{1}\rightarrow \mathbb{R}\) satisfying

$$\begin{aligned} \Omega ^{ ({}^{H}\mathbb{D}_{0+}^{\iota ,\kappa ;\varPsi }\Delta ( \varsigma )-\boldsymbol{F}(\varsigma ,\Delta (\varsigma ))-\int _{0}^{ \varsigma }\boldsymbol{H}(\varsigma ,\vartheta ,\Delta (\vartheta ))\,d \vartheta )}_{\tau }\succeq \varphi ^{\varsigma }_{\tau }, \end{aligned}$$

for all \(\varsigma ,\vartheta \in \Xi _{1}\), \(\Delta \in \mathbb{R,}\) and \(\tau \in \Xi _{2}\), where φ is MDF-valued with

$$\begin{aligned} \mathcal{I}_{0+}^{\iota ;\varPsi }\Delta (\varsigma ):= \frac{1}{\Gamma (\iota )} \int _{0}^{\varsigma }\varPsi '(\xi ) \bigl( \varPsi (\varsigma )-\varPsi (\xi )\bigr)^{\iota -1}\Delta (\xi )\,d\xi , \end{aligned}$$


$$\begin{aligned} \Omega ^{\Delta (\varsigma )}_{\tau } \succeq \varphi ^{\varsigma }_{ \tau } \quad \Longrightarrow \quad \Omega ^{\mathcal{I}_{0+}^{\iota ;\varPsi } \Delta (\varsigma )}_{\tau } \succeq \varphi ^{\varsigma }_{ \frac{\tau }{M}} , \qquad \inf_{\xi \in \Xi _{1}} \varphi ^{\xi }_{ \frac{\tau }{T}}\succeq \varphi ^{\varsigma }_{\tau }, \end{aligned}$$

for each \(\varsigma \in \Xi _{1}\) and \(\tau \in \Xi _{2}\). Then, we can find a unique CF \(\Delta _{0}:\Xi _{1}\rightarrow \mathbb{R}\) such that

$$\begin{aligned} \Delta _{0}(\varsigma ) &= \frac{(\varPsi (\varsigma )-\varPsi (0))^{\gamma -1}}{\Gamma (\gamma )} \sigma \\ &\quad {}+\mathcal{I}_{0+}^{\iota ;\varPsi }\boldsymbol{F}\bigl( \varsigma ,\Delta _{0}( \varsigma )\bigr) \\ &\quad {}+\mathcal{I}_{0+}^{\iota ;\varPsi } \biggl[ \int _{0}^{\xi } \boldsymbol{H}\bigl(\varsigma , \vartheta ,\Delta _{0}(\vartheta )\bigr)\,d \vartheta \biggr], \end{aligned}$$

with \(\mathcal{I}_{0+}^{1-\gamma ;\varPsi }\Delta (0)=\sigma \), \(\iota \in \mathring{\Xi _{5}}\), \(\kappa \in \Xi _{5}\), and

$$\begin{aligned} \Omega ^{ (\Delta (\varsigma )-\Delta _{0}(\varsigma ) )}_{ \tau }\succeq \varphi ^{\varsigma }_{ \frac{M\tau }{1-2M (\max \{ L_{\boldsymbol{F}},L_{\boldsymbol{H}}\} )}}, \end{aligned}$$

for each \(\varsigma \in \Xi _{1}\) and \(\tau \in \Xi _{2}\).


For \(\alpha ,\beta \in U\), we set

$$\begin{aligned} \rho (\alpha ,\beta )=\inf \bigl\{ \lambda \in \Xi _{4}:\Omega ^{ (\alpha (\varsigma )-\beta (\varsigma ) )}_{\tau }\succeq \varphi ^{\varsigma }_{\frac{\tau }{\lambda }} \bigr\} , \end{aligned}$$

for each \(\varsigma \in \Xi _{1}\) and \(\tau \in \Xi _{2}\), where

$$\begin{aligned} U= \{ \alpha :\Xi _{1}\rightarrow \mathbb{R} \text{ is a CF} \} . \end{aligned}$$

Let \(\Lambda :U\rightarrow U\) be given by

$$\begin{aligned} \Lambda \alpha (\varsigma ) &= \frac{(\varPsi (\varsigma )-\varPsi (0))^{\gamma -1}}{\Gamma (\gamma )} \sigma \\ &\quad {}+\mathcal{I}_{0+}^{\iota ;\varPsi }\boldsymbol{F}\bigl( \varsigma ,\alpha ( \varsigma )\bigr) \\ &\quad {}+\mathcal{I}_{0+}^{\iota ;\varPsi } \biggl[ \int _{0}^{\xi } \boldsymbol{H}\bigl(\varsigma , \vartheta ,\alpha (\vartheta )\bigr)\,d\vartheta \biggr], \end{aligned}$$

for all \(\alpha \in \Xi _{1}\) and \(\varsigma \in \Xi _{1}\).

First we show that Λ is strictly contractive on U. Let \(\lambda _{\alpha \beta }\in \Xi _{4}\) be a fixed number with \(\rho (\alpha ,\beta )\leq \lambda _{\alpha \beta }\) for any \(\alpha ,\beta \in U\), so from Eq. (3.8) we have

$$\begin{aligned} \Omega ^{ (\alpha (\varsigma )-\beta (\varsigma ) )}_{\tau } \succeq \varphi ^{\varsigma }_{\frac{\tau }{\lambda _{\alpha \beta }}}, \end{aligned}$$

Let \(0=\varpi _{1}<\varpi _{2}<\cdots <\varpi _{k}=T\), \(\Delta \xi _{i}=\varpi _{i}-\varpi _{i-1}=\frac{\vert T-0\vert }{k}\), \(i=1,2,\dots ,k\) and \(\Vert \Delta \xi \Vert =\max_{1\leq i\leq k} (\Delta \xi _{i} )\), for each \(\varsigma ,\xi \in \Xi _{1}\) and \(\tau \in \Xi _{2}\). From Eqs. (3.2), (3.5), and (3.10), we have

$$\begin{aligned} &\Omega ^{ (\int _{0}^{\xi }\boldsymbol{H}(\varsigma ,\vartheta , \alpha (\vartheta ))-\boldsymbol{H}(\varsigma ,\vartheta ,\beta ( \vartheta ))\,d\vartheta )}_{\tau } \\ &\quad = \Omega ^{ (\lim _{\Vert \Delta \xi \Vert \to 0}\sum _{i=1}^{k} \boldsymbol{H}(\varsigma ,\varpi _{i},\alpha (\varpi _{i}))- \boldsymbol{H}(\varsigma ,\varpi _{i},\beta (\varpi _{i}))\Delta \xi _{i} )}_{\tau } \\ &\quad = \lim_{\Vert \Delta \xi \Vert \to 0}\Omega ^{ (\sum _{i=1}^{k} (\boldsymbol{H}(\varsigma ,\varpi _{i},\alpha (\varpi _{i}))- \boldsymbol{H}(\varsigma ,\varpi _{i},\beta (\varpi _{i}))\Delta \xi _{i} ) )}_{\tau } \\ &\quad \succeq \lim_{\Vert \Delta \xi \Vert \to 0}\circledast _{M} \Omega ^{ (\boldsymbol{H}(\varsigma ,\varpi _{i},\alpha (\varpi _{i}))- \boldsymbol{H}(\varsigma ,\varpi _{i},\beta (\varpi _{i}))\Delta \xi _{i} )}_{\frac{\tau }{k}} \\ &\quad \succeq \inf_{\xi \in \Xi _{1}}\Omega ^{ ( \boldsymbol{H}(\varsigma ,\xi ,\alpha (\xi ))-\boldsymbol{H}( \varsigma ,\xi ,\beta (\xi )) )}_{\frac{\tau }{k\Delta \xi _{i}}} \\ &\quad \succeq \inf_{\xi \in \Xi _{1}}\Omega ^{ ( \boldsymbol{H}(\varsigma ,\xi ,\alpha (\xi ))-\boldsymbol{H}( \varsigma ,\xi ,\beta (\xi )) )}_{\frac{\tau }{T}} \\ &\quad \succeq \inf_{\xi \in \Xi _{1}}\varphi ^{\xi }_{ \frac{\tau }{T\lambda _{\alpha \beta }L_{\boldsymbol{H}}}} \\ &\quad \succeq \varphi ^{\varsigma }_{ \frac{\tau }{\lambda _{\alpha \beta }L_{\boldsymbol{H}}}}, \end{aligned}$$

Then, by Eqs. (3.1), (3.4), (3.5), (3.9), (3.10), and (3.11), we have

$$\begin{aligned} &\Omega ^{ (\Lambda \alpha (\varsigma )-\Lambda \beta ( \varsigma ) )}_{\tau } \\ &\quad =\Omega ^{ (\frac{1}{\Gamma (\iota )}\int _{0}^{\varsigma } \varPsi '(\xi ) (\varPsi (\varsigma )-\varPsi (\xi ))^{\iota -1} (\boldsymbol{F}(\xi ,\alpha (\xi ))-\boldsymbol{F}(\xi ,\beta ( \xi )) +\int _{0}^{\xi }\boldsymbol{H}(\varsigma ,\vartheta ,\alpha ( \vartheta ))-\boldsymbol{H}(\varsigma ,\vartheta ,\beta (\vartheta ))\,d \vartheta ) \,d\xi )}_{\tau } \\ &\quad \succeq \Omega ^{ (\mathcal{I}_{0+}^{\iota ;\varPsi } ( \boldsymbol{F}(\xi ,\alpha (\xi ))-\boldsymbol{F}(\xi ,\beta (\xi )) ) \,d\xi )}_{\frac{\tau }{2}} \circledast _{M} \Omega ^{ (\mathcal{I}_{0+}^{\iota ;\varPsi } (\int _{0}^{\xi } \boldsymbol{H}(\varsigma ,\vartheta ,\alpha (\vartheta ))- \boldsymbol{H}(\varsigma ,\vartheta ,\beta (\vartheta ))\,d\vartheta ) )}_{\frac{\tau }{2}} \\ &\quad \succeq \varphi ^{\varsigma }_{ \frac{\tau }{2M\lambda _{\alpha \beta }L_{\boldsymbol{F}}}} \circledast _{M} \varphi ^{\varsigma }_{ \frac{\tau }{2M\lambda _{\alpha \beta }L_{\boldsymbol{H}}}} \\ &\quad \succeq \varphi ^{\varsigma }_{ \frac{\tau }{2M\lambda _{\alpha \beta } (\max \{ L_{\boldsymbol{F}},L_{\boldsymbol{H}}\} )}}, \end{aligned}$$

and we conclude that

$$\begin{aligned} \rho (\Lambda \alpha ,\Lambda \beta )\leq 2M\lambda _{\alpha \beta } \bigl(\max \{ L_{\boldsymbol{F}},L_{\boldsymbol{H}}\} \bigr), \end{aligned}$$

for all \(\varsigma \in \Xi _{1}\) and \(\tau \in \Xi _{2}\). Hence, we deduce that \(\rho (\Lambda \alpha ,\Lambda \beta )\leq [2M (\max \{ L_{ \boldsymbol{F}},L_{\boldsymbol{H}}\} )]\rho (\alpha ,\beta )\) for any \(\alpha ,\beta \in U\), where \(2M (\max \{ L_{\boldsymbol{F}},L_{\boldsymbol{H}}\} )\in \mathring{\Xi _{5}}\).

From Eq. (3.9), we can find a fixed number \(\lambda \in \Xi _{2}\) such that

$$\begin{aligned} &\Omega ^{ (\Lambda \beta (\varsigma )-\beta _{0}(\varsigma ) )}_{\tau } \\ &\quad =\Omega ^{ ( \frac{(\varPsi (\varsigma )-\varPsi (0))^{\gamma -1}}{\Gamma (\gamma )} \sigma +\mathcal{I}_{0+}^{\iota ;\varPsi }\boldsymbol{F}(\varsigma , \beta _{0}(\varsigma ))+\mathcal{I}_{0+}^{\iota ;\varPsi } [\int _{0} ^{\xi }\boldsymbol{H}(\varsigma ,\vartheta ,\beta _{0}(\vartheta ))\,d \vartheta ]-\beta _{0}(\varsigma ) )}_{\tau } \\ &\quad \succeq \varphi ^{\varsigma }_{\frac{\tau }{\lambda }}, \end{aligned}$$

for arbitrary \(\beta _{0}\in U\), for all \(\varsigma \in \Xi _{1}\) and \(\tau \in \Xi _{2}\). The boundedness property of

$$ \boldsymbol{F}\bigl(\varsigma ,\beta _{0}(\varsigma )\bigr),\qquad \boldsymbol{H}\bigl( \varsigma ,\vartheta ,\beta _{0}(\vartheta ) \bigr),\qquad \beta _{0}( \varsigma ), $$

\(\min_{\varsigma \in \Xi _{1}}\varphi ^{\varsigma }_{\tau }>0\), and Eq. (3.8) imply that \(\rho (\Lambda \beta ,\beta _{0})<\infty \). From Theorem 2.3, there exists a CF \(\Delta _{0}:\Xi _{1}\rightarrow \mathbb{R}\) such that \(\Lambda ^{n}\Delta _{0}\rightarrow \Delta _{0}\) in \((U,\rho )\) and \(\Lambda \Delta _{0}=\Delta _{0}\).

Since β and \(\Delta _{0}\) are bounded on \(\Xi _{1}\) for each \(\beta \in U\) and \(\min_{\varsigma \in \Xi _{1}}\varphi ^{\varsigma }_{\tau }>0\), we have a fixed number \(\lambda _{\beta }\in \Xi _{4}\) with

$$\begin{aligned} \Omega ^{ (\beta _{0}(\varsigma )-\beta (\varsigma ) )}_{ \tau } \succeq \varphi ^{\varsigma }_{\frac{\tau }{\lambda _{\beta }}}, \end{aligned}$$

for any \(\varsigma \in \Xi _{1}\) and \(\tau \in \Xi _{2}\). Thus \(\rho (\beta _{0},\beta )<\infty \) for any \(\beta \in U\).

Therefore, \(U=\{\beta \in U:\rho (\beta _{0},\beta )<\infty \}\). Also Theorem 2.3 and Eq. (3.6) imply the uniqueness of \(\Delta _{0}\).

Using Eqs. (3.3), (3.5), and (3.9), we have

$$\begin{aligned} &\Omega ^{ (\Delta (\varsigma )- \frac{(\varPsi (\varsigma )-\varPsi (0))^{\gamma -1}}{\Gamma (\gamma )} \sigma -\mathcal{I}_{0+}^{\iota ;\varPsi }\boldsymbol{F}(\varsigma , \Delta (\varsigma ))-\mathcal{I}_{0+}^{\iota ;\varPsi } [\int _{0}^{ \xi }\boldsymbol{H}(\varsigma ,\vartheta ,\Delta (\vartheta ))\,d \vartheta ] )}_{\tau }\succeq \varphi ^{\varsigma }_{ \frac{\tau }{M}}. \end{aligned}$$

Then, we obtain

$$\begin{aligned} \Omega ^{ (\Delta (\varsigma )-\Lambda \Delta (\varsigma ) )}_{\tau }\succeq \varphi ^{\varsigma }_{\frac{\tau }{M}}, \end{aligned}$$

for any \(\varsigma \in \Xi _{1}\) and \(\tau \in \Xi _{2}\), which implies

$$\begin{aligned} \rho (\Delta ,\Lambda \Delta )\leq M. \end{aligned}$$

From Theorem 2.3 and Eq. (3.13), we deduce that

$$\begin{aligned} \rho (\Delta ,\Delta _{0})\leq \frac{1}{1-2M(\max \{ L_{\boldsymbol{F}},L_{\boldsymbol{H}}\})}\rho ( \Lambda \Delta ,\Delta )\leq \frac{M}{1-2M(\max \{ L_{\boldsymbol{F}},L_{\boldsymbol{H}}\})}, \end{aligned}$$

which implies Eq. (3.7). □

Theorem 3.2

Assume that \(\iota ,\kappa \in \mathring{\Xi _{5}}\) and consider a nondecreasing function \(\varPsi \in C^{1}(\Xi _{1})\) with \(\varPsi '(\varsigma )\neq 0\) for all \(\varsigma \in \Xi _{1}\). Also let \(L_{\boldsymbol{F}},L_{\boldsymbol{H}}\in \Xi _{2}\) be fixed numbers such that \(2M (\max \{ L_{\boldsymbol{F}}, TL_{\boldsymbol{H}}\} ) \in \mathring{\Xi _{5}}\). Consider CFs \(\boldsymbol{F}:\Xi _{1}\times \mathbb{R}\rightarrow \mathbb{R}\) and \(\boldsymbol{H}:\Xi _{1}\times \Xi _{1}\times \mathbb{R}\rightarrow \mathbb{R}\) satisfying Eqs. (3.1) and (3.2), respectively. If for \(\varepsilon \geq 0\), \(\tau \in \Xi _{2}\), \(\boldsymbol{\epsilon }_{\tau }:=\operatorname{diag} [e^{- \frac{\varepsilon }{\tau }},\ldots ,e^{-\frac{\varepsilon }{\tau }} ] \), a CDF \(\Delta :\Xi _{1}\rightarrow \mathbb{R}\) satisfies

$$\begin{aligned} \Omega ^{ ({}^{H}\mathbb{D}_{0+}^{\iota ,\kappa ;\varPsi }\Delta ( \varsigma )-\boldsymbol{F}(\varsigma ,\Delta (\varsigma ))-\int _{0}^{ \varsigma }\boldsymbol{H}(\varsigma ,\vartheta ,\Delta (\vartheta ))\,d \vartheta )}_{\tau } \succeq \boldsymbol{\epsilon }_{\tau }, \end{aligned}$$

for all \(\varsigma ,\vartheta \in \Xi _{1}\), \(\Delta \in \mathbb{R,}\) and \(\tau \in \Xi _{2}\), then we can find a unique CF \(\Delta _{0}:\Xi _{1}\rightarrow \mathbb{R}\) satisfying Eq. (3.6) and

$$\begin{aligned} \Omega ^{ (\Delta (\varsigma )-\Delta _{0}(\varsigma ) )}_{ \tau } \succeq \boldsymbol{ \epsilon }_{ \frac{M\tau }{1-2M (\max \{ L_{\boldsymbol{F}},TL_{\boldsymbol{H}}\} )}}, \end{aligned}$$

for all \(\varsigma \in \Xi _{1}\) and \(\Delta \in \mathbb{R}\).


Let \(U= \{ \alpha :\Xi _{1}\rightarrow \mathbb{R} \text{ is a CF} \} \). Consider the \(\Xi _{4}\)-valued metric on U defined by

$$\begin{aligned} \rho (\alpha ,\beta )=\inf \bigl\{ \lambda \in \Xi _{4}:\Omega ^{ (\alpha (\varsigma )-\beta (\varsigma ) )}_{\tau } \succeq \boldsymbol{ \epsilon }_{\frac{\tau }{\lambda }} \bigr\} , \end{aligned}$$

for each \(\varsigma \in \Xi _{1}\) and \(\tau \in \Xi _{2}\). In [15] the authors proved the completeness of \((U,\rho )\).

Let \(\Lambda :U\rightarrow U\) be given by

$$\begin{aligned} \Lambda \alpha (\varsigma ) &= \frac{(\varPsi (\varsigma )-\varPsi (0))^{\gamma -1}}{\Gamma (\gamma )} \sigma \\ &\quad {}+\mathcal{I}_{0+}^{\iota ;\varPsi }\boldsymbol{F}\bigl( \varsigma ,\alpha ( \varsigma )\bigr) \\ &\quad {}+\mathcal{I}_{0+}^{\iota ;\varPsi } \biggl[ \int _{0}^{\xi } \boldsymbol{H}\bigl(\varsigma , \vartheta ,\alpha (\vartheta )\bigr)\,d\vartheta \biggr], \end{aligned}$$

for all \(\varsigma \in \Xi _{1}\).

Let \(\alpha ,\beta \in U\) and consider a fixed number \(\lambda _{\alpha \beta }\in \Xi _{4}\) such that \(\rho (\alpha ,\beta )\leq \lambda _{\alpha \beta }\) and

$$\begin{aligned} \Omega ^{ (\alpha (\varsigma )-\beta (\varsigma ) )}_{\tau } \succeq \boldsymbol{ \epsilon }_{\frac{\tau }{\lambda _{\alpha \beta }}}, \end{aligned}$$

Let \(0=\varpi _{1}<\varpi _{2}<\cdots <\varpi _{k}=T\), \(\Delta \xi _{i}=\varpi _{i}-\varpi _{i-1}=\frac{\vert T-0\vert }{k}\), \(i=1,2,\dots ,k\), and \(\Vert \Delta \xi \Vert =\max_{1\leq i\leq k} (\Delta \xi _{i} )\), for each \(\varsigma \in \Xi _{1}\) and \(\tau \in \Xi _{2}\).

From Eqs. (3.2) and (3.17), we have

$$\begin{aligned} &\Omega ^{ (\int _{0}^{\xi }\boldsymbol{H}(\varsigma ,\vartheta , \alpha (\vartheta ))-\boldsymbol{H}(\varsigma ,\vartheta ,\beta ( \vartheta ))\,d\vartheta )}_{\tau } \\ &\quad = \Omega ^{ (\lim _{\Vert \Delta \xi \Vert \to 0}\sum _{i=1}^{k} \boldsymbol{H}(\varsigma ,\varpi _{i},\alpha (\varpi _{i}))- \boldsymbol{H}(\varsigma ,\varpi _{i},\beta (\varpi _{i}))\Delta \xi _{i} )}_{\tau } \\ &\quad = \lim_{\Vert \Delta \xi \Vert \to 0}\Omega ^{ (\sum _{i=1}^{k} (\boldsymbol{H}(\varsigma ,\varpi _{i},\alpha (\varpi _{i}))- \boldsymbol{H}(\varsigma ,\varpi _{i},\beta (\varpi _{i}))\Delta \xi _{i} ) )}_{\tau } \\ &\quad \succeq \lim_{\Vert \Delta \xi \Vert \to 0}\circledast _{M} \Omega ^{ (\boldsymbol{H}(\varsigma ,\varpi _{i},\alpha (\varpi _{i}))- \boldsymbol{H}(\varsigma ,\varpi _{i},\beta (\varpi _{i}))\Delta \xi _{i} )}_{\frac{\tau }{k}} \\ &\quad \succeq \inf_{\xi \in \Xi _{1}}\Omega ^{ ( \boldsymbol{H}(\varsigma ,\xi ,\alpha (\xi ))-\boldsymbol{H}( \varsigma ,\xi ,\beta (\xi )) )}_{\frac{\tau }{k\Delta \xi _{i}}} \\ &\quad \succeq \inf_{\xi \in \Xi _{1}}\Omega ^{ ( \boldsymbol{H}(\varsigma ,\xi ,\alpha (\xi ))-\boldsymbol{H}( \varsigma ,\xi ,\beta (\xi )) )}_{\frac{\tau }{T}} \\ &\quad \succeq \inf_{\xi \in \Xi _{1}}\boldsymbol{\epsilon }_{ \frac{\tau }{T\lambda _{\alpha \beta }L_{\boldsymbol{H}}}} \\ &\quad = \boldsymbol{\epsilon }_{ \frac{\tau }{T\lambda _{\alpha \beta }L_{\boldsymbol{H}}}}, \end{aligned}$$

Then, by Eqs. (3.1), (3.16), and (3.17), we have

$$\begin{aligned} &\Omega ^{ (\Lambda \alpha (\varsigma )-\Lambda \beta ( \varsigma ) )}_{\tau } \\ &\quad =\Omega ^{ (\frac{1}{\Gamma (\iota )}\int _{0}^{\varsigma } \varPsi '(\xi ) (\varPsi (\varsigma )-\varPsi (\xi ))^{\iota -1} (\boldsymbol{F}(\xi ,\alpha (\xi ))-\boldsymbol{F}(\xi ,\beta ( \xi )) +\int _{0}^{\xi }\boldsymbol{H}(\varsigma ,\vartheta ,\alpha ( \vartheta ))-\boldsymbol{H}(\varsigma ,\vartheta ,\beta (\vartheta ))\,d \vartheta ) \,d\xi )}_{\tau } \\ &\quad \succeq \Omega ^{ (\mathcal{I}_{0+}^{\iota ;\varPsi } ( \boldsymbol{F}(\xi ,\alpha (\xi ))-\boldsymbol{F}(\xi ,\beta (\xi )) ) \,d\xi )}_{\frac{\tau }{2}} \circledast _{M} \Omega ^{ (\mathcal{I}_{0+}^{\iota ;\varPsi } (\int _{0}^{\xi } \boldsymbol{H}(\varsigma ,\vartheta ,\alpha (\vartheta ))- \boldsymbol{H}(\varsigma ,\vartheta ,\beta (\vartheta ))\,d\vartheta ) )}_{\frac{\tau }{2}} \\ &\quad \succeq \boldsymbol{\epsilon }_{ \frac{\tau }{2M\lambda _{\alpha \beta }L_{\boldsymbol{F}}}} \circledast _{M} \boldsymbol{\epsilon }_{ \frac{\tau }{2MT\lambda _{\alpha \beta }L_{\boldsymbol{H}}}} \\ &\quad \succeq \boldsymbol{\epsilon }_{ \frac{\tau }{2M\lambda _{\alpha \beta } (\max \{ L_{\boldsymbol{F}},TL_{\boldsymbol{H}}\} )}}, \end{aligned}$$

for each \(\varsigma \in \Xi _{1}\) and \(\tau \in \Xi _{2}\). Therefore \(\rho (\Lambda \alpha ,\Lambda \beta )\leq [2M (\max \{ L_{ \boldsymbol{F}},TL_{\boldsymbol{H}}\} )]\rho (\alpha ,\beta )\) for any \(\alpha ,\beta \in U\), where \(2M (\max \{ L_{\boldsymbol{F}},TL_{\boldsymbol{H}}\} ) \in \mathring{\Xi _{5}}\).

From Eq. (3.16), we can find a fixed number \(\lambda \in \Xi _{2}\) such that

$$\begin{aligned} &\Omega ^{ (\Lambda \beta (\varsigma )-\beta _{0}(\varsigma ) )}_{\tau } \\ &\quad =\Omega ^{ ( \frac{(\varPsi (\varsigma )-\varPsi (0))^{\gamma -1}}{\Gamma (\gamma )} \sigma +\mathcal{I}_{0+}^{\iota ;\varPsi }\boldsymbol{F}(\varsigma , \beta _{0}(\varsigma ))+\mathcal{I}_{0+}^{\iota ;\varPsi } [\int _{0} ^{\xi }\boldsymbol{H}(\varsigma ,\vartheta ,\beta _{0}(\vartheta ))\,d \vartheta ]-\beta _{0}(\varsigma ) )}_{\tau } \\ &\quad \succeq \boldsymbol{\epsilon }_{\frac{\tau }{\lambda }}, \end{aligned}$$

for arbitrary \(\beta _{0}\in U\), for all \(\varsigma \in \Xi _{1}\) and \(\tau \in \Xi _{2}\). The boundedness property of

$$ \boldsymbol{F}\bigl(\varsigma ,\beta _{0}(\varsigma )\bigr), \boldsymbol{H}\bigl( \varsigma ,\vartheta ,\beta _{0}(\vartheta ) \bigr), \beta _{0}( \varsigma ) $$

and Eq. (3.15) imply that \(\rho (\Lambda \beta ,\beta _{0})<\infty \). From Theorem 2.3, there exists a CF \(\Delta _{0}:\Xi _{1}\rightarrow \mathbb{R}\) such that \(\Lambda ^{n}\Delta _{0}\rightarrow \Delta _{0}\) in \((U,\rho )\) and \(\Lambda \Delta _{0}=\Delta _{0}\). Using a method similar to that in the proof of Theorem 3.1, we get \(\{\beta \in U:\rho (\beta _{0},\beta )<\infty \}=U\). Also Theorem 2.3 and Eq. (3.6) imply the uniqueness of \(\Delta _{0}\).

Now, using Eq. (3.3) and [12, Theorem 5], we have

$$\begin{aligned} &\Omega ^{ (\Delta (\varsigma )- \frac{(\varPsi (\varsigma )-\varPsi (0))^{\gamma -1}}{\Gamma (\gamma )} \sigma -\mathcal{I}_{0+}^{\iota ;\varPsi }\boldsymbol{F}(\varsigma , \Delta _{0}(\varsigma ))-\mathcal{I}_{0+}^{\iota ;\varPsi } [ \int _{0}^{\xi }\boldsymbol{H}(\varsigma ,\vartheta ,\Delta _{0}( \vartheta ))\,d\vartheta ] )}_{\tau }\succeq \boldsymbol{\epsilon }_{\frac{\tau }{M}}, \end{aligned}$$

for all \(\varsigma \in \Xi _{1}\), which implies

$$\begin{aligned} \rho (\Delta ,\Lambda \Delta )\leq M. \end{aligned}$$

From Theorem 2.3 and Eq. (3.8), we deduce that

$$\begin{aligned} \Omega ^{ (\Delta (\varsigma )-\Delta _{0}(\varsigma ) )}_{ \tau } \succeq \boldsymbol{\epsilon }_{ \frac{M\tau }{1-2M (\max \{ L_{\boldsymbol{F}},TL_{\boldsymbol{H}}\} )}}, \end{aligned}$$

which implies Eq. (3.14) for all \(\varsigma \in \Xi _{1}\). □

4 Conclusions

We introduced a new model of stochastic matrix control functions which helped us to stabilize a pseudo-nonlinear fractional Volterra integral equation and get better approximation for it. In fact, two kinds of novel stability concepts, i.e., Hyers–Ulam–Rassias and Hyers–Ulam stability, of a fractional Volterra integral equation with delay are proved by using an alternative fixed point theorem in generalized complete metric spaces and the concept of stochastic matrix control functions in a matrix MB-space.

Availability of data and materials

Not applicable.


  1. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) 204

    MATH  Google Scholar 

  2. Sousa, J.V.d.C., de Oliveira, E.C., Magna, L.A.: Fractional calculus and the ESR test. AIMS Math. 2(4), 692–705 (2017)

    Article  Google Scholar 

  3. Wang, J.R., Feckan, M., Zhou, Y.: A survey on impulsive fractional differential equations. Fract. Calc. Appl. Anal. 19(4), 806–831 (2016)

    Article  MathSciNet  Google Scholar 

  4. Liang, X., Gao, F., Zhou, C.-B., Wang, Z., Yang, X.-J.: An anomalous diffusion model based on a new general fractional operator with the Mittag-Leffler function of Wiman type. Adv. Differ. Equ. 2018, Paper No. 25, 11 pp. (2018)

    Article  MathSciNet  Google Scholar 

  5. Gumah, G., Al-Omari, S., Baleanu, D.: Soft computing technique for a system of fuzzy Volterra integro-differential equations in a Hilbert space. Appl. Numer. Math. 152, 310–322 (2020)

    Article  MathSciNet  Google Scholar 

  6. Sun, H.-G., Sheng, H., Chen, Y.-Q., Chen, W., Yu, Z.-B. A dynamic-order fractional dynamic system. Chin. Phys. Lett. 30(4), 046601 (2013)

    Article  Google Scholar 

  7. Sun, H., Chen, W., Li, C., Chen, Y.: Finite difference schemes for variable-order time fractional diffusion equation. Int. J. Bifurc. Chaos, 22(04), 1250085 (2012)

    Article  MathSciNet  Google Scholar 

  8. Wang, J.R., Zhou, Y.: Mittag-Leffler–Ulam stabilities of fractional evolution equations. Appl. Math. Lett. 25(4), 723–728 (2012)

    Article  MathSciNet  Google Scholar 

  9. Sousa, J.V.d.C., Kucche, K.D., de Oliveira, E.C.: On the Ulam–Hyers stabilities of the solutions of Ψ-Hilfer fractional differential equation with abstract Volterra operator. Math. Methods Appl. Sci. 42(9), 3021–3032 (2019)

    Article  MathSciNet  Google Scholar 

  10. Muniyappan, P., Rajan, S.: Stability of a class of fractional integro-differential equation with nonlocal initial conditions. Acta Math. Univ. Comen. 87(1), 85–95 (2018)

    MathSciNet  MATH  Google Scholar 

  11. Selvam, A.G.M., Baleanu, D., Alzabut, J., Vignesh, D., Abbas, S.: On Hyers–Ulam Mittag-Leffler stability of discrete fractional Duffing equation with application on inverted pendulum. Adv. Differ. Equ. 2020, 456 (2020)

    Article  MathSciNet  Google Scholar 

  12. Sousa, J.V.d.C., de Oliveira, E.C.: On the Ψ-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)

    Article  MathSciNet  Google Scholar 

  13. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland Series in Probability and Applied Mathematics. North-Holland, New York (1983)

    MATH  Google Scholar 

  14. Hadžić, O., Pap, E.: Fixed Point Theory in Probabilistic Metric Spaces. Mathematics and Its Applications, vol. 536. Kluwer Academic, Dordrecht (2001)

    Book  Google Scholar 

  15. Saadati, R.: Random Operator Theory. Elsevier, London (2016)

    MATH  Google Scholar 

  16. El-Moneam, M.A., Ibrahim, T.F., Elamody, S.: Stability of a fractional difference equation of high order. J. Nonlinear Sci. Appl. 12(2), 65–74 (2019)

    Article  MathSciNet  Google Scholar 

  17. Madadi, M., Saadati, R., De la Sen, M.: Stability of unbounded differential equations in Menger k-normed spaces: a fixed point technique. Mathematics 8(3), 400 (2020)

    Article  Google Scholar 

  18. Constantinescu, C.D., Ramirez, J.M., Zhu, W.R.: An application of fractional differential equations to risk theory. Finance Stoch. 23(4), 1001–1024 (2019)

    Article  MathSciNet  Google Scholar 

  19. El-Sayed, A.M.A., Gaafar, F.M.: Existence and uniqueness of solution for Sturm–Liouville fractional differential equation with multi-point boundary condition via Caputo derivative. Adv. Differ. Equ. 2019, Paper No. 46, 17 pp. (2019)

    Article  MathSciNet  Google Scholar 

  20. Jiang, J., O’Regan, D., Xu, J., Fu, Z.: Positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions. J. Inequal. Appl. 2019, Paper No. 204, 18 pp. (2019)

    Article  MathSciNet  Google Scholar 

  21. Sene, N.: Stability analysis of the generalized fractional differential equations with and without exogenous inputs. J. Nonlinear Sci. Appl. 12(9), 562–572 (2019)

    Article  MathSciNet  Google Scholar 

  22. Sene, N.: Global asymptotic stability of the fractional differential equations. J. Nonlinear Sci. Appl. 13(3), 171–175 (2020)

    MathSciNet  Google Scholar 

  23. Ali, A., Shah, K., Li, Y., Khan, R.A.: Numerical treatment of coupled system of fractional order partial differential equations. J. Math. Comput. Sci. 19, 74–85 (2019)

    Article  Google Scholar 

  24. Pap, E., Park, C., Saadati, R.: Additive σ-random operator inequality and rhom-derivations in fuzzy Banach algebras. Sci. Bull. “Politeh.” Univ. Buchar., Ser. A, Appl. Math. Phys. 82(2), 3–14 (2020)

    MathSciNet  Google Scholar 

  25. Asaduzzaman, M., Kilicman, A., Ali, M.Z.: Presence and diversity of positive solutions for a Caputo-type fractional order nonlinear differential equation with an advanced argument. J. Math. Comput. Sci. 23, 230–244 (2021)

    Article  Google Scholar 

  26. Chaharpashlou, R., Saadati, R., Atangana, A.: Ulam–Hyers–Rassias stability for nonlinear Ψ-Hilfer stochastic fractional differential equation with uncertainty. Adv. Differ. Equ. 2020, Paper No. 339, 10 pp. (2020)

    Article  MathSciNet  Google Scholar 

  27. Madadi, M., Saadati, R., Park, C., Rassias, J.M.: Stochastic Lie bracket (derivation, derivation) in MB-algebras. J. Inequal. Appl. 2020, Paper No. 141, 15 pp. (2020)

    Article  MathSciNet  Google Scholar 

  28. Cădariu, L., Radu, V.: Fixed points and the stability of Jensen’s functional equation. JIPAM. J. Inequal. Pure Appl. Math. 4(1), Article ID 4 (2003)

    MathSciNet  MATH  Google Scholar 

  29. Diaz, J.B., Margolis, B.: A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74, 305–309 (1968)

    Article  MathSciNet  Google Scholar 

  30. Sousa, J.V.d.C., de Oliveira, E.C.: On a new operator in fractional calculus and applications. J. Fixed Point Theory Appl. 20(3), Paper No. 96, 21 pp. (2018)

    Article  MathSciNet  Google Scholar 

Download references


The authors are thankful to the area editor and anonymous referees for giving valuable comments and suggestions.


No funding.

Author information

Authors and Affiliations



All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Corresponding authors

Correspondence to Reza Chaharpashlou or Reza Saadati.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chaharpashlou, R., Saadati, R. Best approximation of a nonlinear fractional Volterra integro-differential equation in matrix MB-space. Adv Differ Equ 2021, 118 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: