Some existence results for a nonlinear q-integral equations via M.N.C and fixed point theorem Petryshyn

Abstract

The paper focuses on establishing sufficient conditions for the existence of the solutions in some functional q-integral equations, particularly in Banach spaces. In this method, the technique of measures of noncompactness and Petryshyn’s fixed point theorem in Banach space is used. We provide some examples of equations, which confirm that our result is applicable to a wide class of integral equations.

1 Introduction

Nonlinear integral equations (N$\mathbb{IE}$) can describe many phenomena in nature. Many researchers consider this subject an important branch of mathematics. It helps solve many problems in physics, engineering, and other fields. In fact, the existence of a solution for N$\mathbb{IE}$ is one of the most important challenges for researchers. Authors have proposed many techniques. But, among them, Darbo’s fixed point theorem ($F.P.T$) has become more popular. The concept of measure of noncompactness (M.N.C) plays a basic role in it. The M.N.C method serves as the foundation for almost all methods used for this purpose [1–9].

In 1910, Jackson introduced the q-integral [10]. Afterward, he published many works on it. There are several applications of q-calculus in physical problems such as molecular problems [11], elementary particle physics, and chemical physics [12, 13]. One can find basic definitions and properties of quantum calculus in the book [14]. Several researchers have analyzed the existence of solutions of q-integral equations [15–23]. In [24], the authors discussed the q-integral equation (q-$\mathbb{IE}$s)

$$ \mathcal{C}(\zeta ) = \Delta \bigg( \zeta , \mathcal{C} \left ( a ( \zeta ) \right ), \tfrac{ \mathfrak{f}\left ( \zeta , \mathcal{C} \left ( b (\zeta ) \right ) \right ) }{ \Gamma _{q} (\varsigma )} \int _{0}^{\zeta} ( \zeta -qs)^{(\varsigma -1)} \mathfrak{u}(s, \mathcal{C}(s))\, \mathrm{d}_{q} s \bigg), $$

(1)

for $\zeta \in \Upsilon _{\circ}:=\lbrack 0,1 \rbrack$, where $\varsigma >1$, $0 < q < 1$, $a, b : \Upsilon _{\circ }\to \Upsilon _{\circ }$, $\mathfrak{f},\mathfrak{u} : \Upsilon _{\circ }\times \mathbb{R}\to \mathbb{R}$ and $\Delta : \Upsilon _{\circ}\times \mathbb{R}^{2} \to \mathbb{R}$. In fact, they provided sufficient conditions for the existence of a solutions of $\mathbb{IE}$,

$$ \mathcal{C}(\zeta ) = \Delta \big( \zeta , \mathcal{C} \left ( a ( \zeta ) \right ), \mathfrak{f} \left ( \zeta , \mathcal{C} \left ( b ( \zeta ) \right ) \right ) I_{q}^{\varsigma} \mathfrak{u}(\zeta , \mathcal{C}(\zeta )) \big), $$

(2)

by using the definition of q-fractional integral and a general version of Darbo’s $F.P.T$ [24]. In [25], the author examined quadratic $\mathbb{IE}$s as follows

$$ \mathcal{C}(\zeta ) = \mathfrak{f}(\zeta , \mathcal{C}(\zeta )) + \tfrac{ (\mathcal{A}\mathcal{C}) (\zeta )}{ \Gamma _{q} ( \varsigma ) } \int _{0}^{\zeta} k(\zeta ,s) ( \zeta -qs)^{(\varsigma -1)} ( \mathcal{B} \mathcal{C})(s)\, \mathrm{d}_{q}s, \qquad \zeta \in \Upsilon _{\circ}, $$

(3)

where $\varsigma >0$, $0 < q < 1$, $\mathcal{A}, \mathcal{B} : C(\Upsilon _{\circ}) \to C(\Upsilon _{ \circ})$ and $k: \Upsilon _{\circ}^{2} \to \mathbb{R}$. They demonstrated that Petryshyn’s theorem is much more useful than Darbo’s theorem. In fact, it dose not require to confirm the operator used, which maps a closed convex subset onto itself [26, 27].

In this paper, we employ the idea of Kazemi and Ezzati [28] and authors of [29–32], to study the existence of solution for a class of functional nonlinear q-integral equations (Nq-$\mathbb{IE}$s) as follows

$$\begin{aligned} \chi (\zeta ) & = \nabla \bigg( \zeta , \chi \left (a(\zeta ) \right ), \varrho \left (\zeta , \chi \left (c(\zeta ) \right ) \right ), \\ & \quad \int _{0}^{\eta (\zeta )} \tfrac{s^{\beta} (\eta (\zeta )-qs)^{ ( \varsigma -1)} }{ \Gamma _{q}(\varsigma )} p \left ( \zeta ,s,\chi \left (b (s) \right ) \right ) \, \mathrm{d}_{q}s \bigg), \end{aligned}$$

(4)

for $\zeta \in \Upsilon : = \lbrack 0, \alpha \rbrack$ and $0< q<1$, where $\beta >0$, $\varsigma >1$, $\chi \in C(\Upsilon )$, $\nabla \in C(\Upsilon \times \mathbb{R}^{3})$, $\varrho \in C(\Upsilon \times \mathbb{R})$, $p\in C(\Upsilon \times [0,\Omega ]\times \mathbb{R})$, and $a, b, c :\Upsilon \to \Upsilon$, $\eta : \Upsilon \to \mathbb{R}^{+}$ are continuous functions with $\eta (\zeta )\leq \Omega$, for each $\zeta \in \Upsilon$.

The paper is structured as follows. In Sect. 2, we collect some definitions, lemmas and theorems, which are essential to prove our main results. In Sect. 3.1, we establish and prove a new existence theorem by utilized Petryshyn’s $F.P.T$ for Nq-$\mathbb{IE}$ (4). In Sect. 3.2, we also give some examples to support our main theorem. Finally, in Sect. 4, we conclude the paper.

2 Auxiliary facts and notations

In this section, we review some definitions and theorems. We do this by stating some extra facts. Let X be a Banach space, $N_{r_{0}}$ be a ball of radius $r_{0}$, $\bar{N_{r_{0}}}$ sphere in X with radius $r_{0}$, $\mathcal{B}_{X}$ be a bounded subset of X and $\operatorname{Fix} ( \mathcal{T})$ be the set of fixed points of $\mathcal{T}$ in X.

In what follows, we recall some basic facts on quantum calculus, also present extra properties that we will use later. For $0 < q <1$, we define $[ \chi ]_{q} = \frac{1-q^{\chi}}{ 1 - q}$, $\chi \in \mathbb{R}$. The q-analogue of the power function $(1-\vartheta )^{k}$ with $k \in \{0,1,2,\ldots \}$ is

$$ ( 1 - \vartheta )^{(0)}=1, \qquad (1-\vartheta )^{(k)}=\prod _{i=0}^{k-1}(1- \vartheta q^{i}),\quad k\in N,\, \vartheta \in \mathbb{R}. $$

(5)

In fact, by using (5), the q-Gamma function is expressed by

$$ \Gamma _{q} (\varsigma ) = (1 - q)^{1-\varsigma} \prod _{k=0}^{ \infty }\frac{ 1 - q^{k+1} }{1 -q^{ \varsigma + k-1}}. $$

(6)

Note that, $\Gamma _{q} (\chi +1) = [\chi ]_{q} \Gamma _{q} (\chi )$ [33, Lemma 1]. For any $m,n>0$,

$$ B_{q}(m,n) = \int _{\Upsilon _{\circ}} \zeta ^{(m-1)}(1-q\zeta )^{( n-1)} \, \mathrm{d}_{q} \zeta , $$

(7)

is called the q-beta function. Also, the q-integral of a function χ defined on the interval ϒ is given by

$$ \int _{0}^{\nu} \chi (\zeta ) \, \mathrm{d}_{q} \zeta = \nu (1-q) \sum _{i=0}^{\infty} \chi (\nu q^{i} ) q^{i}, \qquad \nu \in \Upsilon . $$

(8)

Lemma 2.1

([34])

Let $m,n>0$, and $0< q<1$. Then, we have

$$ \int _{0}^{\nu} ( \nu -q\zeta )^{(m-1)}\zeta ^{(n)}\, \mathrm{d}_{q} \zeta =\nu ^{m+n}B_{q} (m,n+1). $$

(9)

Definition 2.2

([35])

Let $W \subset \mathcal{B}_{X}$, then $\phi (W) = \inf \big\{ \tau \, :\, \text{there exist a finite number of sets}\ \text{of diameter } \leq \tau \text{ that can cover } W \big\}$, is said to be the Kuratowski M.N.C.

Definition 2.3

([36])

Let $W \subset \mathcal{B}_{X}$, then $\pi (W) = \inf \big\{ \tau >0 \, : \, W \text{ has a finite }\tau \text{ net in } X \big\}$, is said to be the Hausdorff M.N.C.

Definition 2.4

([36])

For $W\subset \mathcal{B}_{X}$, the M.N.C.s ϕ and π fulfill $\pi (W) \leq \phi (W) \leq 2 \pi (W)$.

For $h\in C(\Upsilon )$ and $\tau \geq 0$ denoted by $\delta (h, \tau )$, the modulus of continuity of the function h, i.e., $\delta (h,\tau ) = \sup \, \big\{| h(\zeta ) - h(\hat{\zeta})| \, : \, |\zeta -\hat{\zeta}| \leq \tau \big\}$. The uniformly continuous h on ϒ implies that $\delta (h,\tau ) \to 0$ as $\tau \to 0$.

Theorem 2.5

([37])

For $W\subset \mathcal{B}_{X}$, the M.N.C in $C(\Upsilon )$ is denoted by

$$ \pi (W) = \limsup _{\tau \to 0}\delta ( \zeta ,\tau ), \qquad \zeta \in W. $$

(10)

Theorem 2.6

([38])

Let $W,Y \subset \mathcal{B}_{X}$ and $\varsigma \in \mathbb{R}$. Then

• 1. $\pi (W\cup Y) = \max \big\{ \pi (W), \pi (Y) \big\}$;

• 2. $\pi (W+Y) \leq \pi ( W) + \pi (Y)$ where $W+Y = \big\{m+u \, : \, m\in W, u \in Y\big\}$;

• 3. $\pi ( W )\leq \pi (Y)$, for $W \subset Y$;

• 4. $\pi (\varsigma W) = |\varsigma |\pi (W)$, where $\varsigma W = \big\{\varsigma m \, : \, m\in W\big\}$;

• 5. $\pi ( W )\leq \pi (Y)$, for $W \subset Y$;

• 6. $\pi \left ( \overline{\operatorname{co}}(W) \right ) = \pi ( W )$.

Recall that $\Lambda \in C(X)$ is called a κ-set contraction if for $\Lambda \subset \mathcal{B}_{X}$, Λ is bounded and $\phi (\Lambda K) \leq \kappa \phi (K)$ for all $0<\kappa <1$. Moreover, if $\phi (\Lambda K)\leq \phi (K)$, Λ is called condensing map [29, 30].

Theorem 2.7

(Petryshyn’s $F.P.T$ [38])

Let $\Lambda :N_{r} \to X$ be a condensing mapping, satisfying the following boundary condition, if $\Lambda (x) = \kappa x$, for some $x\in \bar{N_{r}}$, with $\kappa \leq 1$, then $\operatorname{Fix}_{ N_{r}}( \Lambda )\neq \varnothing$.

3 Main results

3.1 An existence theorem for q-$\mathbb{IE}$s

In this section, we will study the existence of q-$\mathbb{IE}$s (4) under the following assumptions:

• S1) There exist nonnegative constant $k_{1}$, $k_{2}$, $k_{3}$ and $k_{4}$ with $k_{1}+ k_{2}k_{4}\leq 1$ s.t.

  $$ |\nabla (\zeta , \zeta _{1},\zeta _{2}, \zeta _{3})-\nabla (\zeta , \hat{\zeta _{1}}, \hat{\zeta _{2}},\hat{\zeta _{3}})| \leq k_{1}| \zeta _{1}-\hat{\zeta _{1}}|+k_{2}|\zeta _{2}-\hat{\zeta _{2}}|+k_{3}| \zeta _{3}-\hat{\zeta _{3}}|, $$

  (11)

  and

  $$ |\varrho (x,y) - \varrho (x,\hat{y})|\leq k_{4}|y - \hat{y}|; $$

  (12)

• S2) For $r_{0}\geq 0$, the operator ∇ satisfies the following condition

  $$\begin{aligned} \sup \, & \Big\{ |\nabla (\zeta ,x,y,z)| \, : \, \zeta \in \Upsilon , \, x,y\in [ - r_{0}, r_{0}], \, \\ & \quad |z| \leq \tfrac{ \Omega ^{\varsigma + \beta}}{\Gamma _{q}(\varsigma )} L B_{q}( \varsigma , \beta +1) \Big\} {\leq r_{0},} \end{aligned}$$

  (13)

  where $L = \sup \Big\{ |p(\zeta ,s,z)| \, : \, \zeta \in \Upsilon , \, s \in [0,\Omega ], z\in [ - r_{0}, r_{0}] \Big\}$.

To prove the main results, we first establish the following key lemma.

Lemma 3.1

The following inequality holds:

$$\begin{aligned} \bigg|\int _{0}^{\eta (\zeta )} s^{\beta} & (\eta (\zeta )-qs)^{( \varsigma -1)} p(\zeta ,s,\chi (b(s))) \, \mathrm{d}_{q}s \\ & \quad - \int _{0}^{\eta (\hat{\zeta})} s^{\beta}(\eta (\hat{\zeta}) -qs )^{ (\varsigma -1)} p(\hat{\zeta},s,\chi (b(s))) \, \mathrm{d}_{q}s \bigg| \\ &\leq \hat{\delta}(p,\tau ) \Omega ^{ \varsigma + \beta} B_{q} ( \varsigma , \beta +1) +\, 2 L B_{q} ( \varsigma ,\beta +1) \hat{\delta}(\eta ,\tau )^{\varsigma +\beta}, \end{aligned}$$

(14)

where

$$\begin{aligned} \hat{\delta}(p,\tau ) & = \sup \Big\{ | p( \zeta ,x,y) - p( \hat{\zeta},x,y)| \, : \, |\zeta -\hat{\zeta}|\leq \tau , \, \\ & \quad x \in [0,\Omega ], \, y\in [ - r_{0}, r_{0}]\Big\} , \\ \hat{\delta}(\eta ,\tau ) & = \sup \Big\{ |\eta (\zeta )-\eta ( \hat{\zeta})| \, :\, \zeta , \hat{\zeta}\in \Upsilon , \, |\zeta - \hat{\zeta}|\leq \tau \Big\} . \end{aligned}$$

(15)

Proof

We have

$$\begin{aligned} \bigg|\int _{0}^{\eta (\zeta )} & s^{\beta}(\eta (\zeta )- qs)^{ ( \varsigma -1)} p(\zeta ,s,\chi (b(s)))\, \mathrm{d}_{q}s \\ & \quad - \int _{0}^{\eta (\hat{\zeta})}s^{\beta} ( \eta (\hat{\zeta})-qs)^{( \varsigma -1)} p(\hat{\zeta},s, \chi (b(s))) \, \mathrm{d}_{q}s\bigg| \\ &\leq \bigg| \int _{0}^{ \eta (\zeta )} s^{\beta}(\eta (\zeta )-qs)^{( \varsigma -1)} p(\zeta ,s, \chi (b(s))) \, \mathrm{d}_{q}s \\ & \quad - \int _{0}^{\eta (\zeta )} s^{ \beta}(\eta (\zeta )-qs)^{ ( \varsigma -1)} p ( \hat{\zeta},s, \chi (b(s)) )\, \mathrm{d}_{q}s \bigg| \\ &\quad + \bigg| \int _{0}^{ \eta (\zeta )} s^{\beta}(\eta (\zeta )-qs)^{ ( \varsigma -1)} p ( \hat{\zeta}, s, \chi (b(s)))\, \mathrm{d}_{q}s \\ & \quad - \int _{0}^{\eta (\hat{\zeta})} s^{ \beta}(\eta (\zeta ) - qs)^{( \varsigma -1)} p (\hat{\zeta},s, \chi (b(s))) \, \mathrm{d}_{q} s \bigg| \\ &\quad + \bigg| \int _{0}^{\eta ( \hat{\zeta})} s^{\beta}(\eta ( \zeta ) - qs)^{(\varsigma -1)} p(\hat{\zeta},s, \chi (b(s)))\, \mathrm{d}_{q}s \\ & \quad - \int _{0}^{\eta (\hat{\zeta})} s^{\beta}(\eta (\hat{\zeta})-qs)^{ ( \varsigma -1)} p ( \hat{\zeta},s, \chi (b(s)))\, \mathrm{d}_{q}s \bigg| \\ &\leq \int _{0}^{\eta (\zeta )} s^{\beta}(\eta (\zeta )-qs)^{( \varsigma -1)} {\Big| p(\zeta ,s,\chi (b(s)))- p( \hat{\zeta},s, \chi (b(s)))\Big|\, \mathrm{d}_{q}s} \\ &\quad + \int _{ \eta (\hat{ \zeta})}^{ \eta (\zeta )} s^{\beta} ( \eta (\zeta )-qs)^{(\varsigma -1)} \Big| p(\hat{\zeta},s, \chi (b(s))) \Big| \, \mathrm{d}_{q}s \\ & \quad + \int _{0}^{\eta (\hat{\zeta})} s^{\beta} \Big[ (\eta ( \hat{\zeta})-qs)^{(\varsigma -1)}-(\eta (\zeta )-qs)^{(\varsigma -1)} \Big] \, \left | p( \hat{\zeta},s, \chi (b(s))) \right |\, \mathrm{d}_{q}s \\ &\leq \hat{\delta}(p,\tau ) \int _{0}^{\eta (\zeta )} s^{\beta}(\eta ( \zeta ) - qs)^{(\varsigma -1)} \, \mathrm{d}_{q}s \\ & \quad + L \bigg[ \int _{0}^{\eta (\hat{\zeta})} s^{\beta}(\eta ( \hat{\zeta})-qs)^{(\varsigma -1)} \, \mathrm{d}_{q}s - \int _{0}^{ \eta (\hat{\zeta})} s^{\beta}(\eta (\zeta )-qs)^{(\varsigma -1)}\, \mathrm{d}_{q}s \\ & \quad + \int _{0}^{\eta (\zeta )} s^{\beta}( \eta (\zeta ) -qs)^{ ( \varsigma -1)}\, \mathrm{d}_{q}s - \int _{0}^{\eta (\hat{\zeta})} s^{ \beta}( \eta (\zeta ) -qs)^{ ( \varsigma -1)}\, \mathrm{d}_{q}s\bigg] \\ & = \hat{\delta}(p,\tau ) (\eta (\zeta ))^{\varsigma +\beta} B_{q}( \varsigma ,\beta +1) +L \Big[ B_{q}(\varsigma ,\beta +1) (\eta ( \hat{\zeta}))^{\varsigma +\beta} \\ & \quad + B_{q}(\varsigma , \beta +1) (\eta (\zeta ))^{ \varsigma + \beta} + 2 (\eta (\zeta )-\eta (\hat{\zeta}))^{\varsigma +\beta} \Big] \\ &\leq B_{q}(\varsigma ,\beta +1) \Big[ \hat{\delta}(p,\tau ) (\eta ( \zeta ))^{\varsigma +\beta} + 2L (\eta (\zeta ) - \eta (\hat{ \zeta}))^{ \varsigma +\beta} \Big] \\ &\leq B_{q}(\varsigma ,\beta +1) \Big[ \hat{\delta}(p,\tau ) \Omega ^{ \varsigma +\beta} + 2L (\hat{\delta}( \eta ,\tau ))^{\varsigma + \beta} \Big]. \end{aligned}$$

This complete the proof. □

Theorem 3.2

Under the tacit assumptions (S1)–(S2), the q-$\mathbb{IE}$s (4) has at least one solution in $C(\Upsilon )$.

Proof

Let

$$ \left \{ \textstyle\begin{array}{l} \Sigma : N_{\tau} \to X, \\ \displaystyle (\Sigma \chi )(\zeta )= \nabla \bigg( \zeta , a(\chi ( \zeta )), \varrho ( \zeta , c(\chi (\zeta ))), \\ \displaystyle \quad \quad \quad \int _{0}^{ \eta (\zeta )} \tfrac{ s^{\beta}(\eta ( \zeta )-qs)^{ ( \varsigma -1)}}{ \Gamma _{q}( \varsigma )} p(\zeta , s, \chi (b(s)))\, \mathrm{d}_{ q}s \bigg). \end{array}\displaystyle \right . $$

(16)

We divided the proof into several steps.

Step 1. The operator Σ is continuous on the ball $N_{r_{0}}$. Consider $\tau >0$ and any $\chi , \hat{\chi}\in N_{r_{0}}$ s.t. $\|\chi - \hat{\chi}\|<\tau$. We have

$$\begin{aligned} |(\Sigma \chi )(\zeta ) & - (\Sigma \hat{\chi})(\zeta )| \\ & = \bigg| \nabla \bigg( \zeta , \chi ( a ( \zeta )), \varrho (\zeta , \chi (c(\zeta ))), \\ & \quad \int _{0}^{\eta (\zeta )} \tfrac{s^{\beta}(\eta (\zeta )-qs)^{ (\varsigma -1)}}{ \Gamma _{q}(\varsigma )} p(\zeta ,s,\chi (b(s))) \, \mathrm{d}_{q}s \bigg) \\ & \quad -\nabla \bigg( \zeta , \hat{\chi}(a(\zeta )), \varrho (\zeta , \hat{\chi}(c(\zeta ))), \\ & \quad \int _{0}^{\eta (\zeta )} \tfrac{s^{\beta}(\eta (\zeta )-qs)^{ (\varsigma -1)}}{ \Gamma _{q}( \varsigma )} p(\zeta ,s, \hat{\chi}(b(s)))\, \mathrm{d}_{q}s \bigg)\bigg| \\ & \leq \bigg| \nabla \bigg(\zeta , \chi (a( \zeta )), \varrho (\zeta , \chi (c( \zeta ))), \\ & \quad \int _{0}^{\eta (\zeta )} \tfrac{s^{\beta} ( \eta (\zeta )-qs)^{ ( \varsigma -1)}}{\Gamma _{q}( \varsigma )} p(\zeta ,s, \chi (b(s))) \, \mathrm{d}_{q}s\bigg) \\ & \quad - \nabla \bigg( \zeta , \hat{\chi}(a(\zeta )), \varrho ( \zeta , \chi (c( \zeta ))), \\ & \quad \int _{0}^{\eta (\zeta )} \tfrac{s^{\beta}(\eta (\zeta )-qs)^{(\varsigma -1)}}{\Gamma _{q}( \varsigma )} p(\zeta ,s,\chi (b(s)) ) \, \mathrm{d}_{q}s\bigg)\bigg| \\ & \quad + \bigg| \nabla \bigg(\zeta , \hat{\chi}(a( \zeta )), \varrho (\zeta ,\chi (c(\zeta ))), \\ & \quad \int _{0}^{\eta (\zeta )} \tfrac{s^{\beta} (\eta (\zeta )-qs)^{(\varsigma -1)}}{\Gamma _{q}(\varsigma )} p(\zeta ,s, \chi (b(s))) \, \mathrm{d}_{q}s \bigg) \\ & \quad - \nabla \bigg(\zeta , \hat{\chi}( a( \zeta )), \varrho ( \zeta , \hat{\chi}(c( \zeta ))), \\ & \quad \int _{0}^{\eta (\zeta )} \tfrac{s^{\beta}(\eta (\zeta )-qs)^{ ( \varsigma -1)}}{\Gamma _{q}(\varsigma )} p (\zeta ,s, \chi (b(s))) \, \mathrm{d}_{q}s\bigg) \bigg| \\ &\quad + \bigg| \nabla \bigg(\zeta , \hat{\chi}(a(\zeta )), \varrho ( \zeta , \hat{\chi} ( c (\zeta ))), \\ & \quad \int _{0}^{\eta (\zeta )} \tfrac{s^{\beta}(\eta (\zeta )-qs)^{ ( \varsigma -1)}}{ \Gamma _{q} ( \varsigma )} p(\zeta ,s, \chi (b(s)) ) \, \mathrm{d}_{q}s\bigg) \\ & \quad - \nabla \bigg( \zeta , a( \hat{\chi}( \zeta )), \varrho ( \zeta , \hat{\chi}(c(\zeta ))), \\ & \quad \int _{0}^{\eta (\zeta )} \tfrac{s^{\beta}(\eta (\zeta )-qs)^{(\varsigma -1)}}{\Gamma _{q}(\varsigma )} p(\zeta ,s, \hat{\chi}(b(s))) \, \mathrm{d}_{q}s\bigg)\bigg| \\ & \leq k_{1} \left | \chi ( a ( \zeta ))- \hat{\chi}( a( \zeta )) \right | + k_{2} \left | \varrho (\zeta , \chi ( c(\zeta )))- \varrho ( \zeta , \hat{\chi}( c( \zeta )))\right | \\ & \quad + k_{3} \int _{0}^{\eta (\zeta )} \tfrac{s^{\beta}(\eta (\zeta )-qs)^{(\varsigma -1)} }{\Gamma _{q}( \varsigma )} \left | p ( \zeta ,s, \chi (b(s))) - p ( \zeta ,s, \hat{\chi}(b(s))) \right |\, \mathrm{d}_{q}s \\ & \leq k_{1} \left | \chi (a(\zeta )) - \hat{\chi}(a( \zeta ))\right | + k_{2}k_{4} \left | \chi ( c(\zeta )) - \hat{\chi}(c(\zeta )) \right | \\ & \quad + \tfrac{k_{3}}{ \Gamma _{q}( \varsigma )}\delta (p, \tau ) \eta (\zeta )^{(\varsigma +\beta )} B_{q} (\varsigma , \beta +1) \\ & \leq ( k_{1}+ k_{2}k_{4} ) \|\chi - \hat{\chi}\| + \tfrac{k_{3}}{\Gamma _{q} (\varsigma )} \delta (p,\tau ) \Omega ^{ \varsigma +\beta} B_{q}(\varsigma ,\beta +1), \end{aligned}$$

where

$$\begin{aligned} \delta (p,\tau ) & = \sup \Big\{ |p(\zeta ,x,y) - p(\zeta ,x,\hat{y})| \, : \, \zeta \in \Upsilon ,\, x\in [0,\Omega ], \, \\ & \quad y,\hat{y} \in [-r_{0},r_{0}], \, |y - \hat{y}|\leq \tau \Big\} . \end{aligned}$$

(17)

The uniformly continuity of $p(\cdot ,\cdot , \cdot )$ on the subset $\Upsilon \times [0,\Omega ]\times \mathbb{R}$ implies that $\lim _{\tau \to 0}\delta (p,\tau )=0$. This shows that the operator Σ is continuous on $N_{\tau}$.

Step 2. The condensation condition operator Σ. Let $E_{\tau}\in \mathcal{B}_{X}$ and $\zeta ,\hat{\zeta}\in \Upsilon$ s.t. $\zeta \leq \hat{\zeta}$ and $\zeta -\hat{\zeta}\leq \tau$ for $\tau >0$, the Lemma 3.1 implies that

$$\begin{aligned} |(\Sigma \chi ) (\zeta ) & -(\Sigma \chi )( \hat{\zeta})| \\ & = \bigg| \nabla \bigg( \zeta , a(\chi (\zeta )), \varrho (\zeta , \chi (c( \zeta ))), \\ & \quad \int _{0}^{\eta (\zeta )} \tfrac{s^{\beta}(\eta (\zeta )-qs)^{(\varsigma -1)}}{\Gamma _{q}(\varsigma )} p(\zeta ,s, \chi (b( s))) \, \mathrm{d}_{q}s\bigg) \\ & \quad - \nabla \bigg(\hat{\zeta}, f( \hat{\zeta}, \chi (a( \hat{\zeta})) ), \varrho (\zeta , \chi (c(\hat{\zeta}))), \\ & \quad \int _{0}^{\eta ( \hat{\zeta})} \tfrac{s^{\beta}(\eta ( \hat{\zeta})-qs)^{ (\varsigma -1)}}{ \Gamma _{q}( \varsigma )} p(\hat{\zeta},s, \chi (b(s))) \, \mathrm{d}_{q}s \bigg)\bigg| \\ & \leq \bigg| \nabla \bigg(\zeta , \chi (a( \zeta )),\varrho (\zeta , \chi (c( \zeta ))), \\ & \quad \int _{0}^{ \eta (\zeta )} \tfrac{ s^{\beta} ( \eta (\zeta )-qs)^{(\varsigma -1)} }{\Gamma _{q}(\varsigma )} p(\zeta ,s, \chi (b(s))) \, \mathrm{d}_{q}s \bigg) \\ & \quad - \nabla \bigg(\zeta , \chi (a( \zeta )),\varrho (\zeta , \chi (c( \zeta ))), \\ & \quad \int _{0}^{\eta (\hat{\zeta})} \tfrac{s^{\beta}(\eta (\hat{\zeta})-qs)^{( \varsigma -1)}}{ \Gamma _{q}( \varsigma )} p( \hat{\zeta},s, \chi (b(s))) \, \mathrm{d}_{q}s\bigg)\bigg| \\ & \quad + \bigg|\nabla \bigg( \zeta , \chi (a(\zeta )),\varrho ( \zeta , \chi (c( \zeta ))), \\ & \quad \int _{0}^{\eta (\hat{\zeta})} \tfrac{s^{\beta}( \eta (\hat{\zeta})-qs)^{ ( \varsigma -1)}}{\Gamma _{q}(\varsigma )} p( \hat{\zeta},s, \chi (b(s))) \, \mathrm{d}_{q}s\bigg) \\ & \quad - \nabla \bigg(\zeta , \chi (a( \zeta )),\varrho (\hat{\zeta}, \chi (c(\hat{\zeta}))), \\ & \quad \int _{0}^{\eta (\hat{\zeta})} \tfrac{ s^{\beta} (\eta (\hat{ \zeta})-qs)^{ (\varsigma -1)} }{ \Gamma _{q}(\varsigma )} p(\hat{\zeta},s , \chi (b(s)))\, \mathrm{d}_{q}s \bigg)\bigg| \\ & \quad + \bigg|\nabla \bigg(\zeta , \chi (a( \hat{\zeta}))), \varrho (\hat{\zeta}, \chi (c( \hat{\zeta}))), \\ & \quad \int _{0}^{ \eta ( \hat{\zeta})} \tfrac{ s^{\beta}(\eta (\hat{\zeta})-qs)^{ (\varsigma -1)} }{\Gamma _{q}(\varsigma )} p(\hat{\zeta},s, \chi (b(s)))\, \mathrm{d}_{q} s \bigg)\bigg| \\ & \quad - \nabla \bigg(\zeta , \chi (a( \hat{\zeta}))), \varrho ( \hat{\zeta}, \chi ( c( \hat{\zeta}))), \\ & \quad \int _{0}^{\eta (\hat{\zeta})} \tfrac{s^{\beta}( \eta (\hat{\zeta})-qs)^{ ( \varsigma -1)} }{ \Gamma _{q} ( \varsigma )} p (\hat{\zeta},s, \chi (b(s))) \, \mathrm{d}_{q}s \bigg) \bigg| \\ &\quad + \bigg|\nabla \bigg(\zeta , \chi ( a (\hat{\zeta})), \varrho ( \hat{\zeta}, \chi ( c( \hat{\zeta}))), \\ & \quad \int _{0}^{\eta (\hat{\zeta})} \tfrac{s^{\beta}( \eta (\hat{\zeta})-qs)^{ ( \varsigma -1)} }{ \Gamma _{q}( \varsigma )} p( \hat{\zeta},s, \chi (b( s))) \, \mathrm{d}_{q}s \bigg) \\ &\quad - \nabla \bigg(\hat{\zeta}, \chi ( a ( \hat{\zeta})), \varrho ( \hat{\zeta}, \chi ( c( \hat{\zeta}))), \\ & \quad \int _{0}^{\eta (\hat{\zeta})} \tfrac{s^{\beta}(\eta (\hat{\zeta})-qs)^{(\varsigma -1)} }{\Gamma _{q} ( \varsigma )} p(\hat{\zeta},s, \chi (b( s))) \, \mathrm{d}_{q}s\bigg)\bigg| \\ &\leq \tfrac{k_{3}}{\Gamma _{q}(\varsigma )} \bigg|\int _{0}^{\eta ( \zeta )} s^{\beta}(\eta (\zeta )-qs)^{(\varsigma -1)} p(\zeta ,s, \chi (b( s))) \, \mathrm{d}_{q}s \\ & \quad - \int _{0}^{ \eta (\hat{\zeta})} s^{\beta}(\eta ( \hat{ \zeta})-qs)^{ (\varsigma -1)} p(\hat{\zeta},s, \chi (b( s))) \, \mathrm{d}_{q}s \bigg| \\ &\quad + k_{2} \left |\varrho (\zeta , \chi (c( \zeta )) - \varrho ( \zeta , \chi (c( \hat{t}))\right | \\ & \quad + k_{2}\left |\varrho (\zeta , \chi (c( \zeta ))-\varrho ( \hat{\zeta}, \chi (c(\hat{\zeta})))\right | \\ & \quad + k_{1} \left | \chi (a( \zeta )) - \chi (a( \hat{\zeta})) \right | +\hat{\delta}(\nabla ,\tau ) \\ &\leq \tfrac{k_{3}\hat{\delta} (p,\tau )}{ \Gamma _{q}(\varsigma )} \Omega ^{\varsigma +\beta} B_{q}( \varsigma ,\beta +1) + \tfrac{2k_{3}L}{\Gamma _{q}(\varsigma )} B_{q}( \varsigma , \beta +1) \hat{\delta}(\eta ,\tau )^{\beta +\varsigma} \\ &\quad + k_{2} k_{4} \left | \chi (c(\zeta )) - \chi (c( \hat{\zeta}) \right | + k_{2}\hat{\delta}(\varrho ,\tau ) \\ & \quad +k_{1} \left | \chi (a( \zeta )) - \chi (a( \hat{\zeta})) \right | +\hat{\delta}(\nabla ,\tau ), \end{aligned}$$

where

$$\begin{aligned} \hat{\delta}(\nabla ,\tau )& =\sup \Big\{ |\nabla (\zeta ,x,y,z)- \nabla (\hat{\zeta},x,y,z)| \, :\, |\zeta -\hat{\zeta}| \leq \tau , \, \\ & \quad x,y\in [-r_{0},r_{0}], \, z\in [-\Omega L,\Omega L] \Big\} , \\ \hat{\delta}(\varrho ,\tau )& = \sup \Big\{ |\varrho (\zeta ,x)- \varrho (\hat{\zeta},x)| \, : \, |\zeta -\hat{\zeta}| \leq \tau , x \in [-r_{0},r_{0}] \Big\} . \end{aligned}$$

(18)

So,

$$\begin{aligned} \delta (\Sigma \chi ,\tau )& \leq \tfrac{k_{3}\hat{\delta}(p,\tau )}{\Gamma _{q}(\varsigma )}\Omega ^{ \varsigma +\beta}B_{q}(\varsigma ,\beta +1)+ \tfrac{2k_{3}L}{\Gamma _{q}(\varsigma ) } B_{q} (\varsigma ,\beta +1) \hat{\delta}(\eta ,\tau )^{\beta +\varsigma} \\ & \quad +k_{1}\delta (\chi ,\delta ( a (\zeta ),\tau ))+k_{2}k_{4} \delta (\chi ,\delta ( c( \zeta ),\tau ))+\hat{\delta}(\nabla ,\tau ). \end{aligned}$$

(19)

Now, taking limit of (19) equation as $\tau \to 0$ would give easily following inequality

$$ \delta (\Sigma \chi ,\tau )\leq (k_{1}+k_{2}k_{4})\delta (\chi ,\tau ), $$

(20)

this provide $\pi (\Sigma E_{\tau})\leq (k_{1}+k_{2}k_{4})\pi (E_{\tau})$. Therefore, ∇ is condensing map.

Step 3. The boundary condition. Suppose that $z\in \bar{N_{r_{0}}}$. If $\|\Sigma z\|=k\|z\|=kr_{0}$, the condition (S2) implies that

$$\begin{aligned} |(\Sigma z)(\zeta )| & = \bigg|\nabla \bigg( \zeta , a(\chi (\zeta )), \varrho (\zeta ,\chi (c(\zeta ))), \\ & \quad \int _{0}^{\eta (\zeta )} \tfrac{ s^{\beta} (\eta (\zeta )-qs)^{ ( \varsigma -1)} }{\Gamma _{q}(\varsigma )} p(\zeta ,s, \chi (b(s))) \, \mathrm{d}_{q}s \bigg)\bigg|\leq r_{0}, \qquad \forall \zeta \in \Upsilon , \end{aligned}$$

(21)

hence $\|\Sigma z\|\leq r_{0}$, i.e., $k\leq 1$. This complete the proof. □

Corollary 3.1

Consider the features of Nq-$\mathbb{IE}$ (4) with the following form

$$ \chi (\zeta ) = \nabla \bigg( \zeta ,\chi (a(\zeta )), \tfrac{ \varrho (\zeta , \chi (c(\zeta )))}{ \Gamma _{q}( \varsigma )} \int _{0}^{\zeta}s^{\beta}(\zeta -qs)^{ (\varsigma -1)} p(\zeta ,s, \chi (b(s)))\, \mathrm{d}_{q}s \bigg), $$

(22)

for $\zeta \in \Upsilon$, $0< q<1$ and $\eta (\zeta ) =\zeta$. It is obvious that $\chi \in C(\Upsilon )$, $\nabla \in C(\Upsilon \times \mathbb{R}^{)}$, $a, b, c :\Upsilon \to \Upsilon$, $\varrho \in C(\Upsilon \times \mathbb{R})$, $p(\zeta ,s,x)\in C(\Upsilon ^{2}\times \mathbb{R})$, and $\eta \in C(\Upsilon , \mathbb{R}^{+})$ with $\eta (\zeta )\leq \Omega$ for $\zeta \in \Upsilon$. Now, assume that the following conditions are met:

• (P1) there exist positive constant $l_{i}$, $i = 1,\ldots ,5$ s.t.

  $$ |\nabla (\zeta ,x,y)-\nabla (\zeta ,\bar{x},\bar{y})|\leq l_{1} | x - \bar{x}| + l_{2} |y - \bar{y}|,\qquad |\nabla (\zeta ,0,0)|\leq l_{3}, $$

  (23)

  and

  $$ |\varrho (\zeta ,x)-\varrho (\zeta ,\bar{x})|\leq l_{4} |x-\bar{x}|, \qquad |\varrho (\zeta ,0)|\leq l_{5}; $$

  (24)

• (P2) $\exists c_{1} , c_{2} >0$ s.t.

  $$ |p(\zeta ,s,x)|\leq c_{1} + c_{2} |x|,\qquad \forall \zeta ,s\in \Upsilon , \, x\in \mathbb{R}. $$

  (25)

Then Nq-$\mathbb{IE}$ (22) has at least one solution in $C(\Upsilon )$, whenever

$$ l_{1} + \tfrac{1}{ \Gamma _{q}(\varsigma )}l_{2} (l_{4} +l_{5}) \, \Omega ^{\varsigma +\beta} B_{q} ( \varsigma , \beta +1) (c_{1}+c_{2}){ < 1.} $$

(26)

Proof

One can easily check, (P1) implied that (S1) in theorem 3.2. Now, we prove that (S2) holds. Let $r_{0}>0$, $\|\chi \|\leq r_{0}$. We obtain

$$\begin{aligned} |\chi (\zeta )| &= \bigg|\nabla \bigg( \zeta , \chi (a(\zeta )), \tfrac{\varrho (\zeta , \chi (c(\zeta )))}{\Gamma _{q}(\varsigma )} \int _{0}^{\eta (\zeta )} s^{\beta}(\zeta -qs)^{(\varsigma -1)} p( \zeta ,s, \chi (b(s)))\, \mathrm{d}_{q}s\bigg) \bigg| \\ &\leq \bigg|\nabla \bigg(\zeta , \chi (a(\zeta )), \tfrac{\varrho (\zeta , \chi (c( \zeta )))}{\Gamma _{q}(\varsigma )} \int _{0}^{ \eta (\zeta )} s^{\beta} ( \zeta -qs)^{(\varsigma -1)} p( \zeta ,s, \chi (b(s)))\, \mathrm{d}_{q}s\bigg) \\ & \quad - \nabla (\zeta ,0,0)\bigg|+ \big|\nabla (\zeta ,0,0)\big| \\ & \leq l_{1} \|\chi (a(\zeta ))\| \\ & \quad +l_{2}\bigg| \tfrac{\varrho (\zeta ,\chi (c(\zeta )))}{\Gamma _{q}(\varsigma )} \int _{0}^{\eta (\zeta )} s^{\beta}(\zeta -qs)^{(\varsigma -1)} p( \zeta ,s, \chi (b(s)))\, \mathrm{d}_{q}s\bigg| +l_{3} \\ &\leq l_{1} \|\chi (a(\zeta ))\| + \tfrac{l_{2}}{\Gamma _{q}(\varsigma )} \left ( |\varrho (\zeta , \chi (c(\zeta ))) - \varrho (\zeta ,0)| + |\varrho (\zeta ,0)| \right ) \\ & \quad \cdot \bigg| \int _{0}^{\eta (\zeta )} s^{\beta}( \zeta -qs)^{( \varsigma -1)} p(\zeta ,s, \chi (b(s))) \, \mathrm{d}_{q}s\bigg| \\ &\leq l_{1} \|\chi (a(\zeta ))\|+ \tfrac{l_{2}}{ \Gamma _{q}(\varsigma )} \Big(l_{4} \|\chi (c(\zeta )) \|+l_{5} \Big) \\ & \quad \cdot \eta (\zeta )^{\varsigma +\beta} B_{q} ( \varsigma , \beta +1)\Big( c_{1}+c_{2}\|c(\chi (\zeta ))\| \Big). \end{aligned}$$

Hence, $r_{0}$ in (S2) is real number that satisfies

$$ l_{1} r_{0} + \tfrac{1}{\Gamma _{q} ( \varsigma )}l_{2} \left ( l_{4} r_{0}+l_{5} \right ) \Omega ^{ \varsigma +\beta}B_{q}(\varsigma , \beta +1) \left (c_{1}+c_{2} r_{0} \right )\leq r_{o}. $$

(27)

Now, we define the continuous function $\Xi : \Upsilon _{\circ }\to \mathbb{R}$ as follows

$$ \Xi (\tau )=l_{1} r_{0} + \tfrac{1}{\Gamma _{q}(\varsigma )}l_{2} \big(l_{4}r_{0}+l_{5}\big) \Omega ^{\varsigma +\beta}B_{q}(\varsigma , \beta +1)\big(c_{1}+c_{2} r_{0}\big)-r_{0}. $$

(28)

The property (26) implies that $\Xi (0)\cdot \Xi (1)<0$, then $\exists r_{0} \in (0,1)$ s.t. $\Xi (r_{0})=0$. □

Remark 3.1

Jleli et al. in [24], introduced a functional q-$\mathbb{IE}$s (1) that includes the assumptions (A1)–(A9). They proved the functional q-integral equation has at least one solution in $C(\Upsilon _{\circ})$. By utilization of suggestion technique in Theorem 3.2 and Corollary 3.1, one can achieve the desired result with the least conditions compared to Jleli’s article [24].

Corollary 3.2

Assume that

$$\begin{aligned} \chi (\zeta ) & = f (\zeta ,\chi (a(\zeta ))) + \varphi \bigg( \zeta , \varrho (\zeta , \chi (c(\zeta ))), \\ & \quad \int _{0}^{\eta (\zeta )} \tfrac{s^{\beta}(\eta (\zeta )-qs)^{(\varsigma -1)}}{\Gamma _{q}( \varsigma )} p(\zeta ,s, \chi (b(s))) \, \mathrm{d}_{q}s\bigg), \end{aligned}$$

(29)

for $\zeta \in \Upsilon$, where $\chi \in C(\Upsilon )$, $\varphi \in C(\Upsilon \times \mathbb{R}^{2})$, $f, \varrho \in C(\Upsilon \times \mathbb{R})$, $p\in C(\Upsilon \times [0,\Omega ]\times \mathbb{R})$, $a, b, c :\Upsilon \to \Upsilon$, $\eta : \Upsilon \to \mathbb{R}^{+}$ are continuous functions with $\eta (\zeta )\leq \Omega$ for $\zeta \in \Upsilon$ and let

• (I) There exist nonnegative constant $k_{1}$, $k_{2}$, $k_{3}$ and $k_{4}$ with $k_{1}k_{3}+k_{4}\leq 1$ s.t.

  $$\begin{aligned}& |\varphi (\zeta , \zeta _{1}, \zeta _{2})-\varphi (\zeta , \hat{\zeta _{1}},\hat{\zeta _{2}})|\leq k_{1}| \zeta _{1}- \hat{\zeta _{1}}|+k_{2}|\zeta _{2}-\hat{\zeta _{2}}|, \end{aligned}$$

  (30)
  $$\begin{aligned}& |\varrho (x,y)-\varrho (x,\hat{y})|\leq k_{3}|y-\hat{y}|,\qquad |f(s, y) - f(s, \bar{y})| \leq k_{4}|y - \bar{y}|; \end{aligned}$$

  (31)

• (II) For $r_{0}\geq 0$, the operator φ satisfies the following condition

  $$ \sup \Big\{ |\varphi (\zeta ,y,z)| \, : \, \zeta \in \Upsilon , \, y \in [-r_{0},r_{0}],\, |z| \leq \tfrac{\Omega ^{\varsigma +\beta}}{\Gamma _{q}(\varsigma )}\,L\, B_{q}( \varsigma ,\beta +1) \Big\} \leq r_{0}, $$

  (32)

  where L is defined by assumptions (S2).

Then Nq-$\mathbb{IE}$ (29) has at least one solution in $C( \Upsilon )$.

Proof

The proof is done similarly to the previous Corollary 3.1. □

3.2 Illustrative examples

In this section, based on the explained approach, we incline to present some examples by using Maple software to validate Theorem 3.2.

Example 3.1

Consider the following Nq-$\mathbb{IE}$

$$\begin{aligned} \chi (\zeta ) & = \tfrac{ \left | \zeta \right | e^{ - \zeta ^{2}} }{ 10 \left ( \sqrt{5} + \left | \zeta \right | \right ) } \tan ^{-1}\left ( \chi \left ( \tfrac{\sqrt{3}}{4} \zeta \right ) \right ) + \tfrac{ 1 + \zeta ^{3} }{ \sqrt{2}\left (3 + 4 \zeta ^{2} \right ) } \sin \left ( \chi \left (\tfrac{2\zeta}{9}\right ) \right ) \\ & \quad + \int _{0}^{ \sqrt{\zeta}} s^{ \ln (10)/3} \tfrac{\left ( \sqrt{\zeta} - qs \right )^{ \left (1/2 \right )} }{ \Gamma _{q} \left ( 3/2 \right ) } \tfrac{(\zeta +1)\chi \left ( s/\sqrt{5} \right )}{2 +s^{2}} \, \mathrm{d}_{q}s, \end{aligned}$$

(33)

for $\zeta \in \Upsilon$, $\alpha =1$, and $q\in \{ \frac{1}{3}, \frac{1}{2}, \frac{4}{5} \} \subseteq (0,1)$. Clearly, $\beta = \frac{1}{ 3} \ln (10)>0$ and $\varsigma =\frac{3}{2} >1$. According to the definition of the problem (4), we take $a(z) = \tfrac{\sqrt{3}}{4} z$,

$$\begin{aligned} & p(\zeta , z, b(z)) = \tfrac{(\zeta +1) \chi (b(z))}{2 + z^{2} }, \qquad b(z) = \tfrac{\sqrt{5}}{5}z, \\ &\varrho (\zeta , \chi (c(z)))= \tfrac{ 1 + \zeta ^{3} }{ \sqrt{2}\left (3 + 4 \zeta ^{2} \right ) } \sin \left ( \chi (c(z) \right ),\qquad c(z)= \tfrac{2}{9}z, \end{aligned}$$

(34)

and $\eta (z)=\sqrt{z}$. It is easy to examine that $\varrho \in C(\Upsilon \times \mathbb{R})$, $p\in C(\Upsilon \times [0,\Omega ]\times \mathbb{R})$, $\Omega =1$, $a, b, c \in C(\Upsilon , \Upsilon )$, and $\eta \in C(\Upsilon , \mathbb{R}^{+})$ with $\eta (\zeta )\leq \Omega$, for each $\zeta \in \Upsilon$. Thus, for $\mathfrak{u}_{i}$, $\hat{\mathfrak{u}}_{i}$, $i=1,2,3$ and $\zeta \in \Upsilon$, by using Lemma 3.1 and Eq. (13), we have

$$\begin{aligned} |\nabla (\zeta , \mathfrak{u}_{1}, \mathfrak{u}_{2}, \mathfrak{u}_{3}) & - \nabla (\zeta , \hat{\mathfrak{u}}_{1}, \hat{\mathfrak{u}}_{2}, \hat{\mathfrak{u}}_{3})| \\ & = \bigg| \tfrac{ \left | \zeta \right | e^{-\zeta ^{2}} }{ 10 \left ( \sqrt{5} + \left | \mathfrak{u}_{1} \right | \right ) } \tan ^{-1}\left ( \mathfrak{u}_{1} \right ) + \tfrac{ 1 + \zeta ^{3}}{ \sqrt{2}\left (3 + 4 \zeta ^{2} \right ) } \sin \left ( \mathfrak{u}_{2} \right ) \\ & \quad + \int _{0}^{ \sqrt{\zeta}} s^{ \ln (10)/3} \tfrac{\left ( \sqrt{\zeta} - qs \right )^{ \left (1/2\right )} }{ \Gamma _{q} \left ( 3/2 \right ) } \tfrac{(\zeta +1) \mathfrak{u}_{3}}{ 2 +s^{2}}\, \mathrm{d}_{q}s \\ & \quad - \bigg[\, \tfrac{ \left | \zeta \right | e^{ - \zeta ^{2}}}{ 10 \left ( \sqrt{5} + \left | \zeta \right | \right ) } \tan ^{-1}\left ( \hat{\mathfrak{u}}_{1} \right ) + \tfrac{ 1 + \zeta ^{3} }{ \sqrt{2}\left (3 + 4 \zeta ^{2} \right ) } \sin \left ( \hat{\mathfrak{u}}_{2} \right ) \\ & \quad + \int _{0}^{ \sqrt{\zeta}} s^{ \ln (10)/3} \tfrac{\left ( \sqrt{\zeta} - qs \right )^{ \left (1/2\right )} }{ \Gamma _{q} \left ( 3/2 \right ) } \tfrac{(\zeta +1) \hat{\mathfrak{u}}_{3}}{ 2 +s^{2}}\, \mathrm{d}_{q}s \bigg] \, \bigg| \\ & \leq \left | \tfrac{ \left | \zeta \right |e^{ - \zeta ^{2}} }{ 10 \left ( \sqrt{5} + \left | \zeta \right | \right ) } \right | \, \left | \tan ^{-1} \left ( \mathfrak{u}_{1} \right )- \tan ^{-1} \left ( \hat{\mathfrak{u}}_{1} \right ) \right | \\ & \quad + \left | \tfrac{ 1 + \zeta ^{3}}{ \sqrt{2} \left (3 + 4 \zeta ^{2} \right ) } \right | \, \left | \sin \left ( \mathfrak{u}_{2} \right ) - \sin \left ( \hat{\mathfrak{u}}_{2} \right ) \right | \\ & \quad + \bigg| \int _{0}^{ \sqrt{\zeta}} s^{ \ln (10)/3} \tfrac{\left ( \sqrt{\zeta} - qs \right )^{ \left (1/2\right )} }{ \Gamma _{q} \left ( 3/2 \right ) } \left | \tfrac{(\zeta +1) \mathfrak{u}_{3}}{ 2 +s^{2}} - \tfrac{(\zeta +1) \hat{\mathfrak{u}}_{3}}{ 2 +s^{2}} \right | \, \mathrm{d}_{q}s \bigg| \\ & \leq \tfrac{1}{10\sqrt{5}} \, \left | \mathfrak{u}_{1} - \hat{\mathfrak{u}}_{1} \right | + \tfrac{\sqrt{2}}{3} \left | \mathfrak{u}_{2} - \hat{\mathfrak{u}}_{2} \right | \\ & \quad + \tfrac{3}{2\Gamma _{q} \left ( 3/2 \right ) } \, B_{q}\left ( \tfrac{3}{2}, \tfrac{3+\ln (10)}{3} \right ) \left | \mathfrak{u}_{3} - \hat{\mathfrak{u}}_{3} \right | \\ & \leq k_{1}| \mathfrak{u}_{1} - \hat{\mathfrak{u}}_{1}| + k_{2} | \mathfrak{u}_{2} - \hat{\mathfrak{u}}_{2}| + k_{3} | \mathfrak{u}_{3} - \hat{\mathfrak{u}}_{3}|, \end{aligned}$$

(35)

where $k_{1}= \frac{1}{10\sqrt{5}}$, $k_{2}= \frac{\sqrt{2}}{3}$,

$$ k_{3} = \tfrac{3}{2\Gamma _{q} \left ( 3/2 \right ) } \, B_{q} \left ( \tfrac{3}{2},1+ \tfrac{1}{3}\ln (10) \right )\simeq \left \{ \textstyle\begin{array}{r@{\quad}l} 1.5833, & q= \frac{1}{3}, \\ 1.5842, & q= \frac{1}{2}, \\ 1.5802, & q= \frac{4}{5}, \end{array}\displaystyle \right . $$

(36)

and

$$\begin{aligned} |\varrho (\zeta ,x) - \varrho (\zeta ,\hat{x})|& = \left | \tfrac{ 1 + \zeta ^{3} }{ \sqrt{2}\left (3 + 4 \zeta ^{2} \right ) } \sin (x)- \tfrac{ 1 + \zeta ^{3} }{ \sqrt{2}\left ( 3 + 4 \zeta ^{2} \right ) } \sin (\hat{x}) \right | \\ & \leq \left | \tfrac{ 1 + \zeta ^{3} }{ \sqrt{2} \left (3 + 4 \zeta ^{2} \right ) } \right | \, \left | \sin (x) - \sin (\hat{x}) \right | \leq k_{4}|x- \hat{x} |, \end{aligned}$$

(37)

where $k_{4}=\frac{\sqrt{2}}{3}$. Hence, $k_{1}+k_{2}k_{4} \simeq 0.2669<1$. One can easily find that the assumption (S1) of Theorem 3.2 satisfies. Now, we check that (S2) also holds. Suppose that $\|\chi (\zeta )\|\leq r_{0}$, then

$$\begin{aligned} |\chi (\zeta ) | & =\bigg| \tfrac{ \left | \zeta \right | e^{ - \zeta ^{2}} }{ 10 \left ( \sqrt{5} + \left | \zeta \right | \right ) } \tan ^{-1}\left ( \chi \left ( \tfrac{\sqrt{3}}{4} \zeta \right ) \right ) + \tfrac{ 1 + \zeta ^{3} }{ \sqrt{2}\left (3 + 4 \zeta ^{2} \right ) } \sin \left ( \chi \left (\tfrac{2\zeta}{9} \right )\right ) \\ & \quad + \int _{0}^{ \sqrt{\zeta}} s^{ \ln (10)/3} \tfrac{\left ( \sqrt{\zeta} - qs \right )^{ \left (1/2 \right )} }{ \Gamma _{q} \left ( 3/2 \right ) } \tfrac{(\zeta +1)\chi \left ( s/\sqrt{5} \right )}{2 +s^{2}} \, \mathrm{d}_{q}s\bigg| \\ &< \tfrac{r_{0}}{10\sqrt{5}} + \tfrac{\sqrt{2} r_{0}}{3} + \tfrac{1}{\Gamma _{q} \left ( 3/2 \right ) } B_{q} \left ( \tfrac{3}{2}, 1+\tfrac{\ln (10)}{3} \right ) r_{0} \leq r_{0}. \end{aligned}$$

(38)

Hence, (S2) holds if $r_{0}\geq 1.0556,\, 1.0562,\, 1.0535$ for $q=\frac{1}{3}, \, \frac{1}{2},\, \frac{4}{5}$, respectively. These data are shown in Table 1. It can be seen that for $q=\frac{1}{2}$, the value of $r_{0}$ is at a higher level. In fact, for $0< q<0.5$, the value of $r_{0}$ is ascending, and for $0.5< q<1$, the value of $r_{0}$ is descending. Here, to calculate q-Gamma, we have considered the sum of the result in the definition, up to 120. But the values up to 4 decimal places have been observed the same, they have been displayed up to step 45. These changes can be seen in Figs. 1a and 1b. The data in Table 2 show that when the values of q are close to zero, with the number of repetitions of the continuous sum, the gamma product of q results.

Figure 1

Graphical representation of $k_{3}$ and suitable $r_{0}>0$ of Nq-$\mathbb{IE}$ (33) for three different values of $q \in \{ \frac{1}{3},\, \frac{1}{2}, \, \frac{4}{5}\}$ in Example 3.1

Table 1 Numerical results of $\Gamma _{q} \left ( \varsigma \right )$, $B_{q} \left ( \varsigma , \beta +1 \right )$, $k_{3}$ and suitable $r_{0}>0$ of Nq-$\mathbb{IE}$ (33) for three different values of $q\in \{ \frac{1}{3},\, \frac{1}{2}, \, \frac{4}{5}\}$ in Example 3.1

Table 2 Numerical results of $\Gamma _{q} \left ( \varsigma \right )$, $B_{q} \left ( \varsigma , \beta +1 \right )$, $k_{3}$ and suitable $r_{0}>0$ of Nq-$\mathbb{IE}$ (39) for three different values of $q\in \{ \frac{1}{8},\, \frac{1}{6}, \, \frac{1}{4}\}$ in Example 3.2

It can be seen that for $q=\frac{1}{2}$, the value of $r_{0}$ is at a higher level. In fact, for $0< q<0.5$, the value of $r_{0}$ is ascending, and for $0.5< q<1$, the value of $r_{0}$ is descending. Here, to calculate q-Gamma, we have considered the sum of the result in the definition, up to 120. But the values up to 4 decimal places have been observed the same, they have been displayed up to step 45. These changes can be seen in Figs. 1a and 1b. The data in Table 2 show that when the values of q are close to zero, with the number of repetitions of the continuous sum, the gamma product of q results. This implies that the equation has at least one solution in $C(\Upsilon _{\circ})$.

Example 3.2

Based on the proposed problem, we take Nq-$\mathbb{IE}$

$$\begin{aligned} \chi (\zeta ) & = \tfrac{\chi (\zeta ) \, e^{-\zeta ^{-3}}}{ \sqrt{2}( 4 + \zeta ^{4})}+ \tfrac{\sin \left (\chi \left ( \zeta/7 \right )\right ) }{ 4 ( 2 + \zeta ^{2})} \\ & \quad + \int _{0}^{\zeta ^{2}} \tfrac{ s^{2} \left ( \zeta ^{2} - qs \right )^{ \left (3/ 2 \right ) }}{\Gamma _{ q} \left ( 5/2 \right )} \, \left ( \tfrac{s^{2}}{2} + \tfrac{ s e^{-2\zeta} \sin \left ( \chi \left (\zeta/3\right ) \right )}{ 2 + \left | \zeta \right |} \right ) \, \mathrm{d}_{q}s, \end{aligned}$$

(39)

for $\zeta \in \Upsilon$ with $\alpha =1$ and $q\in \{ \frac{1}{8}, \frac{1}{6}, \frac{1}{4}\}$. It can be seen that $\beta =2$, $\varsigma =\frac{5}{2}$, $a(z) = z$,

$$\begin{aligned} & p(\zeta , z, b(z)) = \tfrac{z^{2}}{2} + \tfrac{z e^{ - 2 \zeta} \sin \left (b(z)\right )}{ 2 + \left | z \right |}, \qquad b(z) = \tfrac{\zeta}{3} , \\ &\varrho (\zeta , \chi (c(z)))= \tfrac{\sin \chi \left ( c(z) \right ) }{ 4 ( 2 + \zeta ^{2})} , \qquad c(z)= \tfrac{z}{7}, \end{aligned}$$

(40)

$\eta (z)=z^{2}$, and can be examine easily that $\varrho \in C(\Upsilon \times \mathbb{R})$, $p\in C(\Upsilon \times [0,\Omega ]\times \mathbb{R})$, $\Omega =1$, $a, b, c \in C(\Upsilon , \Upsilon )$, and $\eta \in C(\Upsilon , \mathbb{R}^{+})$ with $\eta (\zeta )\leq \Omega$, for each $\zeta \in \Upsilon$. Thus, for $\mathfrak{u}_{i}$, $\hat{\mathfrak{u}}_{i}$, $i=1,2,3$ and $\zeta \in \Upsilon$, by using Lemma 3.1 and Eq. (13), similar to the method of the previous example, we get

$$\begin{aligned} |\nabla ( \zeta , \mathfrak{u}_{1}, \mathfrak{u}_{2}, \mathfrak{u}_{3}) & -\nabla (\zeta , \hat{ \mathfrak{u}}_{1}, \hat{ \mathfrak{u}}_{2}, \hat{ \mathfrak{u}}_{3})| \\ & = \bigg| \tfrac{\mathfrak{u}_{1} \, e^{-\zeta ^{-3}}}{ \sqrt{2}( 4 + \zeta ^{4})} + \tfrac{\sin \left (\mathfrak{u}_{2}\right ) }{ 4 ( 2 + \zeta ^{2})} \\ & \quad + \int _{0}^{\zeta ^{2}} \tfrac{ s^{2} \left ( \zeta ^{2} - qs \right )^{ \left (3/2 \right ) }}{\Gamma _{ q} \left ( 5/2 \right )} \, \left ( \tfrac{s^{2}}{2} + \tfrac{ s e^{-2\zeta} \sin \left ( \mathfrak{u}_{3} \right )}{ 2 + \left | \zeta \right |} \right ) \, \mathrm{d}_{q}s \\ & \quad - \bigg[ \tfrac{\hat{\mathfrak{u}}_{1} \, e^{-\zeta ^{-3}}}{ \sqrt{2}( 4 + \zeta ^{4})} + \tfrac{\sin \left (\hat{\mathfrak{u}}_{2}\right ) }{ 4 ( 2 + \zeta ^{2})} \\ & \quad + \int _{0}^{\zeta ^{2}} \tfrac{ s^{2} \left ( \zeta ^{2} - qs \right )^{ \left ( 3/2 \right ) }}{ \Gamma _{ q} \left ( 5/2 \right )} \, \left ( \tfrac{s^{2}}{2} + \tfrac{ s e^{-2\zeta} \sin \left ( \hat{\mathfrak{u}}_{3} \right )}{ 2 + \left | \zeta \right |} \right ) \, \mathrm{d}_{q}s \bigg] \bigg| \\ & \leq \left | \tfrac{ e^{ - \zeta ^{-3}}}{ \sqrt{2}( 4 + \zeta ^{4} ) } \right | \, \left | \mathfrak{u}_{1} - \hat{ \mathfrak{u}}_{1} \right | + \left | \tfrac{1}{ 4 ( 2 + \zeta ^{2})} \right | \, \left | \sin \left ( \mathfrak{u}_{2}\right ) - \sin \left (\hat{\mathfrak{u}}_{2}\right ) \right | \\ & \quad + \int _{0}^{\zeta ^{2}} \tfrac{ s^{2} \left ( \zeta ^{2} - qs \right )^{ \left (3/2 \right ) }}{\Gamma _{ q} \left ( 5/2 \right )} \, \left | \tfrac{s}{e^{2 \zeta} \left (2 + \left |\zeta \right |\right )} \right | \, \left | \sin \left ( \mathfrak{u}_{3} \right ) - \sin \left ( \hat{\mathfrak{u}}_{3} \right ) \right | \, \mathrm{d}_{q}s \\ & \leq k_{1}| \mathfrak{u}_{1} - \hat{\mathfrak{u}}_{1}| + k_{2} | \mathfrak{u}_{2} - \hat{\mathfrak{u}}_{2}| + k_{3} | \mathfrak{u}_{3} - \hat{\mathfrak{u}}_{3}|, \end{aligned}$$

(41)

where $k_{1}= \tfrac{\sqrt{2}}{8}$, $k_{2}= \tfrac{1}{8}$,

$$ k_{3} = \frac{1}{2\Gamma _{q} \left ( 5/2 \right ) } \, B_{q} \left ( \tfrac{5}{2},3 \right ) \simeq \left \{ \textstyle\begin{array}{r@{\quad}l} 0.3645, & q= \frac{1}{8}, \\ 0.3277, & q= \frac{1}{6}, \\ 0.2646, & q= \frac{1}{4}, \end{array}\displaystyle \right . $$

(42)

and

$$\begin{aligned} |\varrho (\zeta ,x) - \varrho (\zeta ,\hat{x})|& = \left | \tfrac{\sin x }{ 4 ( 2 + \zeta ^{2})} - \tfrac{\sin \hat{x} }{ 4 ( 2 + \zeta ^{2})} \right | \\ & \leq \left | \tfrac{1}{ 4 ( 2 + \zeta ^{2})} \right | \, \left | \sin (x) - \sin (\hat{x}) \right | \leq k_{4}|x-\hat{x} |, \end{aligned}$$

(43)

where $k_{4}= \frac{1}{8}$. Figure 2a shows the curves of $k_{3}$ for different values of q. It implies that $k_{1} + k_{2} k_{4} \simeq 0.2951<1$. It follows that the assumption (S1) of Theorem 3.2 satisfies. Now, we check the assumption (S2). Assume that $\|\chi (\zeta )\|\leq r_{0}$, then

$$\begin{aligned} |\chi (\zeta ) | & = \bigg| \tfrac{ \chi (\zeta ) \, e^{-\zeta ^{-3}}}{ \sqrt{2}( 4 + \zeta ^{4})}+ \tfrac{ \sin \left (\chi \left ( \zeta/7 \right )\right ) }{ 4 ( 2 + \zeta ^{2})} \\ & \quad + \int _{0}^{\zeta ^{2}} \tfrac{ s^{2} \left ( \zeta ^{2} - qs \right )^{ \left (3/ 2 \right ) }}{\Gamma _{ q} \left ( 5/2 \right )} \, \left ( \tfrac{s^{2}}{2} + \tfrac{ s e^{-2\zeta} \sin \left ( \chi \left (\zeta/3\right ) \right )}{ 2 + \left | \zeta \right |} \right ) \, \mathrm{d}_{q}s \bigg| \\ &< \tfrac{r_{0}}{4\sqrt{2}} + \tfrac{r_{0}}{8} + \tfrac{r_{0}+1}{2\Gamma _{q} \left ( 5/2 \right ) } B_{q} \left ( \tfrac{5}{2}, 3\right )\leq r_{0}. \end{aligned}$$

(44)

Hence, (S2) holds if $r_{0}\geq 0.7291,\, 0.6554,\,0.5292$ for $q =\frac{1}{8}, \, \frac{1}{6},\, \frac{1}{4}$, respectively. Table 2 shows the results. As can be seen in the curves drawn in Fig. 2b, as the value of q increases, the chosen number for $r_{0}$ decreases. This implies that, from Theorem 3.2, the equation (39) has at least one solution in $C(\Upsilon _{\circ})$.

Figure 2

Graphical representation of $k_{3}$ and suitable $r_{0}>0$ of Nq-$\mathbb{IE}$ (33) for three different values of $q\in \{ \frac{1}{3},\, \frac{1}{2}, \, \frac{4}{5}\}$ in Example 3.2

In the next example, we consider an equation that is more complicated.

Example 3.3

Consider the following Nq-$\mathbb{IE}$

$$\begin{aligned} \chi (\zeta ) & = \tfrac{e^{ - 2 \zeta}}{3} + \tfrac{ \cos ^{2}( \chi ( \sqrt{\zeta}) ) }{ 4\ln (\sqrt{13} +\zeta ^{2}) } + \tfrac{| \chi ( \sqrt{\zeta})| }{25\left ( 1 +| \chi (\sqrt{\zeta})|\right )} \\ & \quad + \int _{0}^{\zeta ^{2}} \tfrac{ s^{2} \left ( \zeta ^{2} - q s \right )^{(3/2)} }{ \Gamma _{q} \left ( 5/2 \right )} \left [\tfrac{1 + \chi (\sqrt{\zeta})}{12 + s \zeta} \right ]^{ 1/2} \, \mathrm{d}_{q}s, \end{aligned}$$

(45)

for $\zeta \in [0, \alpha ]$, $\alpha =1$ and $q\in \{ \frac{7}{11}, \, \frac{8}{11}, \, \frac{9}{11}\}$. Here $\beta =2$, $\varsigma =\frac{5}{2}$, $a(z) = b(z) =c(z)=\sqrt{z}$, $\eta (z)=z^{2}$ and

$$p(\zeta ,z,b(z))={\left[\frac{1+\chi (b(z))}{12+z\zeta }\right]}^{1/2},\varrho (\zeta ,\chi (c(z)))=\frac{|\chi (c(z))|}{25(1+|\chi (c(z))|)}.$$

(46)

Clearly, $\varrho \in C(\Upsilon \times \mathbb{R})$, $p\in C(\Upsilon \times [0,\Omega ]\times \mathbb{R})$, $\Omega =1$, $a, b, c \in C(\Upsilon , \Upsilon )$, and $\eta \in C(\Upsilon , \mathbb{R}^{+})$ with $\eta (\zeta )\leq \Omega$, for each $\zeta \in \Upsilon$. Now, thanks to the Lemma 3.1 and Eq. (13), for $\mathfrak{u}_{i}$, $\hat{\mathfrak{u}}_{i}$, $i=1,2,3$ and $\zeta \in \Upsilon$, we obtain

$$\begin{aligned} |\nabla ( \zeta , \mathfrak{u}_{1}, \mathfrak{u}_{2}, \mathfrak{u}_{3}) & - \nabla ( \zeta , \hat{ \mathfrak{u} }_{1}, \hat{ \mathfrak{u}}_{2}, \hat{ \mathfrak{u}}_{3})| \\ & = \bigg| \tfrac{e^{-2\zeta}}{3} + \tfrac{ \cos ^{2}( \mathfrak{u}_{1}) }{ 4\ln (\sqrt{13} +\zeta ^{2}) } + \tfrac{ | \mathfrak{u}_{2} | }{ 25\left (1 +|\mathfrak{u}_{2}| \right )} \\ & \quad + \int _{0}^{\zeta ^{2}} \tfrac{ s^{2} \left ( \zeta ^{2} - qs \right )^{(3/2)} }{ \Gamma _{q} \left ( 5/2 \right ) } \left [ \tfrac{1+\mathfrak{u}_{3}}{12 + s \zeta} \right ]^{ 1/2} \, \mathrm{d}_{q}s \\ & \quad - \bigg[ \tfrac{e^{-2\zeta}}{3} + \tfrac{ \cos ^{2}( \hat{\mathfrak{u}}_{1}) }{ 4\ln (\sqrt{13} +\zeta ^{2}) } + \tfrac{| \hat{\mathfrak{u}}_{2} | }{25 \left (1 + |\hat{ \mathfrak{u}}_{2}| \right )} \\ & \quad + \int _{0}^{\zeta ^{2}} \tfrac{ s^{2}\left ( \zeta ^{2} - qs \right )^{(3/2)} }{ \Gamma _{q} \left ( 5/2 \right ) } \left [ \tfrac{1 + \hat{ \mathfrak{u}}_{3} }{12 + s \zeta} \right ]^{ 1/2 } \, \mathrm{d}_{q}s \bigg] \bigg| \\ & \leq \left | \tfrac{1}{ 4\ln (\sqrt{13} +\zeta ^{2}) } \right |\, \left | \cos ^{2}( \mathfrak{u}_{1}) - \cos ^{2}( \hat{\mathfrak{u}}_{1}) \right | + \tfrac{1}{25} \left | \tfrac{| \mathfrak{u}_{2} | }{ 1 +|\mathfrak{u}_{2}|} - \tfrac{| \hat{\mathfrak{u}}_{2} | }{ 1 + |\hat{\mathfrak{u}}_{2}|} \right | \\ & \quad + \int _{0}^{\zeta ^{2}} \tfrac{ s^{2} \left ( \zeta ^{2} - qs \right )^{ ( 3/2)} }{ \Gamma _{q} \left ( 5/2 \right ) } \left (\tfrac{1}{\sqrt{12 + s \zeta}} \right ) \left | \sqrt{1+ \mathfrak{u}_{3}} - \sqrt{1 + \hat{ \mathfrak{u}}_{3}} \right | \, \mathrm{d}_{q}s \\ & \leq \tfrac{1}{ 2\ln \sqrt{13} } \, \left | \cos ( \mathfrak{u}_{1}) - \cos ( \hat{\mathfrak{u}}_{1}) \right | + \tfrac{1}{25}\left | \mathfrak{u}_{2} - \hat{\mathfrak{u}}_{2} \right | \\ & \quad + \tfrac{1}{ \sqrt{12} \Gamma _{q} \left ( 5/2 \right ) } \int _{0}^{ \zeta ^{2}} s^{2}\left ( \zeta ^{2} - qs \right )^{ ( 3/2)} \left | \tfrac{\mathfrak{u}_{3} - \hat{ \mathfrak{u}}_{3}}{2} \right | \, \mathrm{d}_{q}s \\ & = \tfrac{1}{ 2\ln \sqrt{13} } \, \left | -2 \sin \left ( \tfrac{\mathfrak{u}_{1}+ \hat{\mathfrak{u}}_{1}}{2} \right ) \sin \left (\tfrac{\mathfrak{u}_{1} - \hat{\mathfrak{u}}_{1}}{2} \right ) \right | + \tfrac{1}{25} \left | \mathfrak{u}_{2} - \hat{\mathfrak{u}}_{2} \right | \\ & \quad + \tfrac{1}{4 \sqrt{3}\, \Gamma _{q} \left ( 5/2 \right ) } \int _{0}^{\zeta ^{2}} s^{2}\left ( \zeta ^{2} - qs \right )^{( 3/2)} \left | \mathfrak{u}_{3} - \hat{ \mathfrak{u}}_{3} \right | \, \mathrm{d}_{q}s \\ & \leq \tfrac{\left | \mathfrak{u}_{1} - \hat{\mathfrak{u}}_{1}\right | }{ \ln \sqrt{13} } +\tfrac{1}{25} \left | \mathfrak{u}_{2} - \hat{\mathfrak{u}}_{2} \right | \\ & \quad + \int _{0}^{\zeta ^{2}} \tfrac{s^{2}\left ( \zeta ^{2} - qs \right )^{(3/2)} }{4 \sqrt{3}\, \Gamma _{q} \left ( 5/2 \right ) } \left | \mathfrak{u}_{3} - \hat{ \mathfrak{u}}_{3} \right | \, \mathrm{d}_{q}s \\ &\leq k_{1}| \mathfrak{u}_{1} - \hat{\mathfrak{u}}_{1}| + k_{2} | \mathfrak{u}_{2} - \hat{\mathfrak{u}}_{2}| + k_{3} | \mathfrak{u}_{3} - \hat{\mathfrak{u}}_{3}|, \end{aligned}$$

(47)

where $k_{1}= \frac{1}{\ln \sqrt{13} }$, $k_{2}= \frac{1}{25}$,

$$ k_{3} = \tfrac{1}{4 \sqrt{3}\, \Gamma _{q} \left ( 5/2 \right ) } \, B_{q}\left ( \tfrac{5}{2},3 \right ) \simeq \left \{ \textstyle\begin{array}{r@{\quad}l} 0.4994, & q= \frac{7}{11}, \\ 0.5310, & q= \frac{8}{11}, \\ 0.6388, & q= \frac{9}{11}, \end{array}\displaystyle \right . $$

(48)

and

$$ |\varrho (\zeta ,x) - \varrho (\zeta ,\hat{x})| = \left | \tfrac{| x|}{ 25\left (1 + |x|\right )} - \tfrac{| \hat{x}|}{ 25\left (1 + |\hat{x}|\right )} \right |\leq k_{4}|x- \hat{x} |, $$

(49)

where $k_{4}= \frac{1}{25}$. We plot the curve of $k_{3}$ in Fig. 3a. It implies that $k_{1} + k_{2} k_{4} \simeq 0.3914 <1$. It follows that the assumption (S1) of Theorem 3.2 satisfies. Now, we check the assumption (S2). Assume that $\|\chi (\zeta )\|\leq r_{0}$, then

$$\begin{aligned} |\chi (\zeta ) | & = \bigg| \tfrac{e^{ - 2 \zeta}}{3} + \tfrac{ \cos ^{2}( \chi ( \sqrt{\zeta}) ) }{ 4\ln (\sqrt{13} +\zeta ^{2}) } + \tfrac{| \chi ( \sqrt{\zeta})| }{25\left ( 1 +| \chi (\sqrt{\zeta})|\right )} \\ & \quad + \int _{0}^{\zeta ^{2}} \tfrac{ s^{2} \left ( \zeta ^{2} - q s \right )^{(3/2)} }{ \Gamma _{q} \left ( 5/2 \right )} \left [\tfrac{ 1+ \chi (\sqrt{\zeta})}{12 + s \zeta} \right ]^{ 1/2} \, \mathrm{d}_{q}s \bigg| \\ & < \tfrac{1}{3} + \tfrac{1}{4\ln \sqrt{13}} + \tfrac{r_{0}}{25}+ \tfrac{ r_{0}}{4 \sqrt{3}\, \Gamma _{q} \left ( 5/2 \right ) } B_{q} \left ( \tfrac{5}{2}, 3\right )\leq r_{0}. \end{aligned}$$

(50)

Hence, (S2) holds if $r_{0}\geq 0.1996,\, 0.1604,\,0.1295$ for $q =\frac{7}{11}, \, \frac{8}{11},\, \frac{9}{11}$, respectively. Table 3 shows the results. As can be seen in the curves drawn in Fig. 3b, as the value of q increases, the chosen number for $r_{0}$ decreases. This implies that, from Theorem 3.2, the equation (45) has at least one solution in $C(\Upsilon _{\circ})$.

Figure 3

Graphical representation of $k_{3}$ and suitable $r_{0}>0$ of Nq-$\mathbb{IE}$ (45) for three different values of $q\in \{ \frac{1}{3},\, \frac{1}{2}, \, \frac{4}{5} \}$ in Example 3.3

Table 3 Numerical results of $\Gamma _{q} \left ( \varsigma \right )$, $B_{q} \left ( \varsigma , \beta +1 \right )$, $k_{3}$ and suitable $r_{0}>0$ of Nq-$\mathbb{IE}$ (45) for three different values of $q\in \{ \frac{7}{11},\, \frac{8}{11}, \, \frac{9}{11}\}$ in Example 3.3

Remark 3.2

Since there are no constants $c_{1}$ and $c_{2}$ satisfying the inequalities (Sublinear condition),

$$ |p(\zeta ,s,x)|\leq c_{1}+c_{2}|x|, \qquad \forall \, \zeta ,s\in \Upsilon , x\in \mathbb{R}, $$

(51)

the results in [24] is inapplicable to the $\mathbb{IE}$ (45).

4 Conclusion

Given the presence of insolvable issues in q-$\mathbb{IE}$s, it is imperative to ascertain the presence of a solution within these equations. Consequently, a multitude of researchers have published papers in this field, outlining their methodologies and the associated outcomes. In this specific paper, the authors present an innovative approach grounded in the use of M.N.C and $F.P.T$ of Petryshyn. This approach presents several advantages over similar methodologies, such as streamlined conditions and the elimination of the requirement to verify the operator’s mapping of a closed convex subset onto itself.