On nonlinear Schrödinger equations with random initial data

1.   Introduction

Many physical phenomenon can be described by integral equations. Non-linear integral equations play a vital role to solve many mathematical problems arising in engineering and applied science. There are various techniques available in the literature for the existence of solutions of these equations. Many researchers used fixed points techniques to the existence of a unique solution to non-linear integral equations. For instance, refer to ^{[1,2,3,4,5,6,7]}. Especially, Shoaib et al. ^[2] studied fixed point results and its applications to the systems of non-linear integral and differential equations of arbitrary order, while Rashwan and Saleh ^[8] established fixed point results to find the existence of a unique common solution to a system of Urysohn integral equations. As opposed to that, Pathak et al. ^[9] and Rashwan and saleh ^[8] ensured the existence of a unique common solution to a system of Volterra-Hammerstein non-linear integral equations. Additionally, Baklouti and his co-authors have made significant contributions to related areas, including optimal preventive maintenance policies for solar photovoltaic systems (c.f, ^[15]) and quadratic Hom-Lie triple systems (c.f, ^[16,17]).

In the current article, existence of solutions for the following systems of Volterra-Hammerstein integral and the Urysohn integral equations are investigated, respectively:

where $x \in (a, b) \subseteq R$; $D, \xi_{i} \in C((a, b), R^{n})$ and $W_{i} : (a, b) \times (a, b) \times R^{n} \rightarrow R^{n}$, for $i = 1, 2;$

and

where $x \in (0, \infty), \lambda, \mu \in R, D, T_{i}, m(x, y), n(x, y), g_{i}(y, D(y))$ and $h_{i}(y, D(y))$ for $i = 1, 2$ are measurable functions with real values both in $x$ and $y$ on $(0, \infty).$ The most elaborated result in this area, known as the Banach fixed point theorem, ensures that a solution exists. The extensive usage of the fixed point theory, particularly in metric spaces ^[11], has then had an impact on the study of its evolution over the past few decades.

Due to important applications of the Banach contraction principle, many researchers generalized this principle by elaborating the underlying spaces or changing the contractive conditions. See ^[12,13,14] for details. In recent decades, scholars concentrated on applying such an aforementioned theorem to various generalized metric spaces, see ^[18,19]. Among these generalized spaces, there is the $b$-metric space, where the coefficient of the triangle inequality is $s\geq 1$. It was introduced by Bakhtin ^[13]. Moreover, in ^[26] Czerwik provided the Banach contraction principle on this space. Recently, Salmi and Noorani ^[21] presented several properties in these spaces and established some common fixed point theorems in ordered cone $b$-metric spaces. Khojasteh et al. ^[34] introduced the concept of a simulation function. This concept has been refined in ^[22] in order to guarantee the presence of a unique coincidence point for two non-linear mappings.

Later, the concept of a $b$-simulation function was introduced to ensure the existence and uniqueness of a fixed point. In ^[23], Olgun et al. presented the concept of a generalized $\Im$-contraction. In ^[24], Jawaher et al. utilized the idea of a $b$-simulation function and investigated some common fixed points for two contractive mappings. In the same direction, using the concept of a generalized $\Im_ b$-contraction with a $b$-simulation function, Rodjanadid et al. ^[23] proved some fixed point results in complete $b$-metric spaces.

Motivated by the above contributions, using $b$-simulation functions and $\Im_b$-contractions, some fixed point theorems are constructed. As applications of these findings, some examples and existence results for systems of integral equations are also discussed. We note that by using the presented work, some well known results can be deduced from the existence literature.

2.   Preliminaries

This section includes all those concepts (definitions, theorems, lemmas, etc.) which will help us to prove the main results of this manuscript. These concepts are taken from different papers, like ^[25,26,27] etc. Throughout the manuscript, the following notions and symbols will be utilized. $\digamma$, $\Omega$, $\beth$ and $\Im$ represent a non-empty set, a metric, a simulation function and a family of simulation functions, respectively. Also, the initials $\mathbb{R}$ and $\mathbb{N}$ in the sequel stand for the sets of all real and natural numbers, respectively.

Definition 2.1. ^[25] (Metric space) Let $\digamma\neq\emptyset$. A function $\Omega:\mathrm{\digamma}\times\mathrm{\digamma} \rightarrow \mathrm{[0, \infty)}$ is known as a metric on $\digamma,$ if for all $a, b, c \in\digamma$ the following conditions hold:

m1) $\Omega(a, b) = 0$ if and only if $a = b;$

m2) $\Omega(a, b) = \Omega(b, a);$

m3) $\Omega(a, b)\leq{\Omega(a, c)+\Omega(c, b)}.$

The pair $(\digamma, \Omega)$ is a metric space.

Definition 2.2. ^[26] ($b$-metric space) Let $\digamma\neq\emptyset$ and assume $b\geq{1}$. A function $\Omega_{b}:\mathrm{\digamma}\times\mathrm{\digamma}\rightarrow \mathrm{[0, \infty)}$ is called a $b$-metric on $\digamma$ if for all $a, e, c\in{\digamma}$, the following requirements are satisfied:

d1) $\Omega_{b}(a, e) = 0$ if and only if $a = e;$

d2) $\Omega_{b}(a, e) = \Omega_{b}(e, a);$

d3) $\Omega_{b}(a, e)\leq{b[\Omega_{b}(a, c)+\Omega_{b}(c, e)]}.$

The pair $(\digamma, \Omega_{b})$ is known as a $b$-metric space, in short $(bMS).$

Definition 2.3. ^[27] (Convergence, Cauchyness and Completeness) Let $\{ {f}_{n} \}$ be a sequence in a $b$-metric space $(\digamma, \Omega_{b}, b)$.

a) $\{{f}_{n}\}$ is called $b$-convergent if and only if there is $f\in{\digamma}$ such that $\Omega_{b}({f}_{n}, f) \rightarrow {0}$ as $n\rightarrow {\infty}.$

b) $\{{f}_{n}\}$ is a $b$-Cauchy sequence if and only if $\Omega_{b}({f}_{n}, {f}_{m})\rightarrow {0}$ as $n, m \rightarrow {\infty}.$

c) The $b$-metric space is complete if every $b$-Cauchy sequence is $b$-convergent.

Proposition 2.4. ^[27] The following assertions hold in a $b$-metric space $(\digamma, \Omega_{b}, b)$:

i) The limit of a $b$-convergent sequence is unique;

ii) Each $b$-convergent sequence is $b$-Cauchy;

iii) A $b$-metric is not continuous generally.

Definition 2.5. ^[28] (Simulation function) Let $\beth: {[0, \infty)}\times{[0, \infty)}\rightarrow \mathbb{R}$ be a function. If $\beth$ satisfies the criteria below:

($\beth1$) $\beth(0, 0)$ = $0;$

($\beth2$) $\beth(t, s) < s-t$ for all $t, s > 0;$

($\beth3$) if $\{ {t}_{n} \}$, $\{ {s}_{n} \}$ are sequences in $(0, \infty)$ such that

then

then it is referred as a simulation function. The set of all simulation functions is denoted by the symbol $\Im$.

Example 2.6. ^[28] Let $\beth: {[0, \infty)}\times{[0, \infty)}\rightarrow \mathrm{\digamma}$ be defined by $\beth(t, s) = {\lambda s}-{t}$ for all $t, s \in [0, \infty)$, where $\lambda \in [0, 1)$. Then $\beth$ is a simulation function.

Example 2.7. ^[28] Let $\beth: {[0, \infty)\times[0, \infty)\rightarrow \digamma}$ be defined by

$\beth(t, s) = \Psi(s)-\Phi(t)$ for all $t, s \in [0, \infty),$ where $\Psi, \Phi :[0, \infty)\rightarrow [0, \infty)$

are two continuous functions such that $\Psi(t) = \Phi(t) = 0$ iff $t = 0$

and $\Psi(t) < t\leq\Phi(t)$ for all $t > 0$. Here, $\beth$ is a simulation function.

Definition 2.8. ^[28] ($\Im$-contraction) Let $(\digamma, \Omega)$ be a metric space, $T:\digamma\rightarrow \digamma$ be a mapping and $\beth\in\Im$. $T$ is called a $\Im$-contraction with regard to $\beth$ if the following condition holds

If $T$ is a $\Im$-contraction with respect to $\beth\in\Im$, then $\Omega({T}{x}, {T}{y}) < \Omega(x, y)$ for all distinct $x, y \in\digamma$.

Theorem 2.9. ^[28] Suppose $(\digamma, \Omega)$ is a complete metric space and $T:\digamma\rightarrow \digamma$ is a $\Im$- contraction with respect to $\beth\in\Im$. Then $T$ has a unique fixed point u in $\digamma$ and for every ${x}_{0}\in\digamma$, the Picard sequence $\{ {x}_{n} \}$ (where ${x}_{n} = T{x}_{n-1}$ for all $n\in N$) converges to the fixed point of $T$.

Definition 2.10. ^[29] (Generalized $\Im$-contraction) Suppose $(\digamma, \Omega)$ is a metric space, $T:\digamma\rightarrow \digamma$ is a mapping and $\beth\in\Im$. Then $T$ is referred to as a generalized $\Im$-contraction with regard to $\beth$ If the following condition is satisfied:

where

Theorem 2.11. ^[29] Assume $(\digamma, \Omega)$ is a complete metric space and $T:\digamma\rightarrow \digamma$ is a generalized $\Im$-contraction with respect to $\beth\in\Im$, then $T$ has a fixed point in $\digamma$. Moreover, for every ${x}_{0}\in \digamma$, the Picard sequence $\{ {T^{n} {x}_{0}} \}$ converges to this fixed point.

Definition 2.12. ^[33] (b-simulation function) Let $(\digamma, \Omega_{b})$ be a $b$-metric space with a constant $b\geq 1$. A $b$-simulation function is a function $\beth:{[0, \infty)\times[0, \infty)\rightarrow R}$ satisfying the following conditions: $(\beth1)$ $\beth(t, s) < s-t$ for all $t, s > 0$;

$(\beth2)$ If $\{ {t}_{n} \}$, $\{ {s}_{n} \}$ are two sequences in $(0, \infty)$ such that

then

We represent the set of all $b$-simulation functions by the symbol $\Im _{b}$.

Some examples of $b$-simulation functions are as follows.

Example 2.13. ^[33] Let $t, s \in [0, \infty)$.

(1) $\beth(t, s) = \Psi(s)-\Phi(t)$, where \ $\Phi, \Psi : [0, \infty)\rightarrow [0, \infty)$ are two continuous functions such that $\Psi(t) = \Phi(t) = 0 \iff t = 0$ and $\Psi(t) < t\leq\Phi(t)$ for all $t > 0$;

(2) $\beth(t, s) = s\frac{y(t, s)}{z(t, s)}t$, where $y, z : [0, \infty)\times[0, \infty)\rightarrow [0, \infty)$ are two functions which are continuous respect to each variable, i.e., $y(t, s) > z(t, s)$ for all $t, s > 0$;

(3) $\beth(t, s)$ = $s-\Phi(s)-t$, where \ $\Phi: [0, \infty)\rightarrow [0, \infty)$ is a lower semi-continuous function such that $\Phi(t) = 0\Leftrightarrow t = 0;$

(4) $\beth(t, s) = s \Phi(s)-t$, where $\Phi : [0, \infty) \rightarrow [0, \infty)$ is such that ${\lim}_{t\rightarrow p^{+}}\Phi(t) < 1 \ \forall \ p > 0$;

(5) $\beth(t, s) = {\lambda s}- t$, where $\lambda\in[0, \infty)$.

Definition 2.14. ^[31] Let $(\digamma, \Omega_{b}, b)$ be a $b$-metric and $y, z$ be two self mappings on $\digamma$. Then the pair $\{ {y}, {z} \}$ is said to be compatible if

whenever $\{ {x}_{n} \}$ is a sequence in $\digamma$ such that

Lemma 2.15. ^[24] Let $(\digamma, \Omega_{b}, b)$ be a $b$-metric space. If there exist two sequences $\{ {f}_{n} \}$ and $\{ {r}_{n} \}$ such that

whenever $\{ {f}_{n} \}$ is a sequence in $\digamma$ such that

then

Theorem 2.16. ^[30] Let $T: \digamma\rightarrow \digamma$ be a mapping and $(\digamma, \Omega_{b})$ be a complete $b$-metric space with a constant $b \geq 1$. Assume there is a $b$-simulation function \ $\beth$ \ such that $\beth(b \Omega_{b}(T{f}, T{r}), \Omega_{b}(f, r))\geq 0$ for all $f, r \in \digamma$, then $T$ has a unique fixed point.

In this work, we introduce generalized $\Im_b$-contraction pairs of self-mappings on a $b$-metric space. We will show that such mappings have a common fixed point. Some examples and applications are presented making effective the new concepts and obtained results. Well known results in literature are investigated and compared.

3.   Results via generalized $\Im_b$-contractions

This section includes the main work of this article. We initiate this section with the definition of generalized $\Im_b$-contractions and a related example. Before the proof of the main theorem, we prove some basic lemmas. For the support of the main theorem, some examples are presented. In the last of this section, some remarks are presented.

Definition 3.1. Consider a $b$-metric space $(\digamma, \Omega_{b}, b)$ with $b \geq 1$, $f_{1}, f_{2}: \digamma \rightarrow \digamma$ are two mappings and $\beth$ $\in$ $\Im_{b}$.

Then $f_{1}, f_{2}$ are called generalized $\Im_b$-contractions with respect to $\beth$ if the circumstance listed below is true

$\beth (b\Omega_{b}(f_{1}s, f_{1}t), M_{b}(s, t)) \geq 0$ for all $s, t \in \digamma$,

where

Example 3.2. Consider the be metric space $(\digamma, \Omega_b, b)$ with $ \digamma = ^[1,2] $and$ \Omega_{b} = (s-t)^2 $for all$ s, t\in \digamma $(Here, b = 2). Then the self-mappings$ f_1, f_2:\digamma \rightarrow \digamma $ defined by

are generalized $\Im_b$-contractions with respect to $\beth(s, t) = \frac{1}{2}s-t$. Indeed,

Lemma 3.3. Suppose $(\digamma, \Omega_{b}, b)$ is a $b$-metric space, and $f_{1}, f_{2}:\digamma\rightarrow \digamma$ are two generalized $\Im_{b}$-contractions. Suppose $f_{1}(\digamma) \subseteq f_{2}(\digamma)$ and there is a $b$-simulation function $\beth$ such that

then there is a sequence $\{ {a}_{n} \}$ in $\digamma$ such that

Proof. Assume ${t_{0}}$ is an arbitrary point. Since $f_{1}(\digamma)\subseteq f_{2}(\digamma)$, we can construct two sequences $\{ {t}_{n} \}$ and $\{ {s}_{n} \}$ such that ${s_{n}} = f_{1}({t_{n}}) = f_{2}({t}_{n+1})$ for every $n\in\mathbb{N}$. If there is ${{n}_{0}} \; \in N$ such that ${t_{n_{0}}} = {t_{n_{0}+1}}$, then it follows from the given inequality (3.1) and from $(\beth1)$ that for all $n\in\mathbb{N}$,

That is,

Therefore, by triangle inequality,

Since ${s}_{n_{0}} = {s}_{n_{0}+1}$ implies that $\Omega_{b}({s}_{n_{0}+1}, {s}_{n_{0}}) = 0$, consider

which implies that

Now, suppose that ${s}_{n} \neq {s}_{n+1}$ for every $n\in\mathbb{N}$. Then, it follows from (3.1) and $(\beth1)$ that for every $n\in\mathbb{N},$ we have

If $\Omega_{b}({s}_{n}, {s}_{n+1}) \geq \Omega_{b} ({s}_{n-1}, {s}_{n}),$ then $0 < \Omega_{b}({s}_{n}, {s}_{n+1}) - b \Omega_{b}({s}_{n}, {s}_{n+1})$. That is,

which is a contradiction. So we have

This implies that $\{\Omega_{b} ({s}_{n-1}, {s}_{n})\}$ is a decreasing sequence of positive real numbers. Thus, there is some $\Gamma \geq 0$, so that

Assume $\Gamma > 0$, so from the condition $(\beth2)$, with ${a}_{n} = \Omega_{b}({s}_{n}, {s}_{n+1})$ and ${b}_{n} = \Omega_{b} ({s}_{n-1}, {s}_{n})$, one writes

which is a contradiction. Hence, we get that $\Gamma = 0$. It ends the proof. □

Remark 3.4. Let $(\digamma, \Omega_{b}, b)$ be a $b$-metric space and assume $f_{1}, f_{2}: \digamma\rightarrow \digamma$ are two generalized $\Im_{b}$-contractions. Assume that $f_{1}(\digamma)\subseteq f_{2}(\digamma)$ and there is a $b$-simulation function $\beth$ such that

Then there is a sequence $\{ {s}_{n} \}$ in $\digamma$ such that

Proof. By a similar argument of Lemma 3.3 for every $n\in\mathbb{N}$, we have ${s}_{n} = f_{1} ({t}_{n}) = f_{2}({t}_{n+1})$. Hence, from (3.1) and $(\beth1)$, we have for all $m, n\in\mathbb{N}$,

□

Lemma 3.5. Let $(\digamma, \Omega_{b}, b)$ be a $b$-metric space and assume $f_{1}, f_{2} : \digamma\rightarrow \digamma$ be two generalized $\Im_{b}$- contractions. Assume that $f_{1}(\digamma)\subseteq f_{2}(\digamma)$ and there is a $b$-simulation function $\beth$ such that the inequality (3.1) holds. Then there exists a sequence $\{ {s}_{n} \}$ in $\digamma$, such that $\{ {s}_{n} \}$ is a bounded sequence.

Proof. By a similar argument of Lemma 3.3, when for some $n_0 \; {s}_{n_0} = {s}_{n_0+1}$ we have $\Omega_{b}({s}_{i}, {s}_{j}) \leq M$ for all $i, j = 0, 1, 2, \cdots,$ where

Let us assume that $s_{n} \neq {s}_{{n+1}}$ for each $n\in\mathbb{N}$ and suppose $\{{s}_{n}\}$ is a sequence which is not bounded. Then there is a subsequence $\{s\}_{n_k}$ of $\{ {s}_{n} \}$ such that for ${n}_{1} = 1$ and for each $k \in N$, ${n}_{k+1}$ is the minimum integer such that $\Omega_{b} ({s}_{{n}_{k+1}}, {s}_{n_k}) > 1$ and $\Omega_{b} ({s}_{m}, {s}_{n_k}) \leq 1$ for ${n}_{k} \leq m \leq {{{n}_{k+1}}-1}.$ By triangular inequality, we obtain

Letting $k \rightarrow \infty$ in the above inequality and using Lemma 3.3, we get

Again from Remark 3.4 we have

Now, as

then taking $k\rightarrow \infty$, one gets

i.e.,

That is,

Using the inequality (3.1) and $(\beth2)$ with ${a}_{k} = \Omega_{b}({s}_{{n}_{k+1}}, {s}_{n_k})$ and ${c}_{k} = M_{b} ({s}_{{{n}_{k+1}}-1}, {s}_{n_k-1}),$ we have

which is a contradiction. Hence, $\{{s}_{n}\}$ is a bounded sequence. □

Lemma 3.6. Suppose $(\digamma, \Omega_{b}, b)$ is a $b$-metric space and assume $f_{1}, f_{2} : \digamma\rightarrow \digamma$ are two generalized $\Im_{b}$-contractions. Assume that $f_{1}(\digamma)\subseteq f_{2}(\digamma)$ and there is a $b$-simulation function $\beth$ such that the inequality (3.1) holds. Then there is a sequence $\{{s}_{n}\}$ in $\digamma$, such that $\{ {s}_{n} \}$ is a Cauchy sequence.

Proof. Using a similar argument as in Lemma 3.3, we have for every $n\in\mathbb{N}$, ${s}_{n} = f_{1} ({t}_{n}) = f_{2} ({t}_{n+1}).$ If there is ${n}_{o} \in N$ such that ${s}_{n_{0}} = {s}_{n_{0}+1}$, then we have $\{ {s}_{n} \}$ is a cauchy sequence. Let us ${s}_{n} \neq {s}_{n+1}$ for every $n\in\mathbb{N}$ and let

Now, from Lemma 3.3, ${C}_{n} < \infty$ for every $n\in\mathbb{N}$. Since ${C}_{n}$ is a positive decreasing sequence, there is some $c \geq 0$ such that

Let us consider that $c > 0$. Then by the definition of ${C}_{n}$, for every $k \in N$ there are ${n}_{k}, {m}_{k} \in N$ such that ${m}_{k} > {n}_{k} \geq k$ and

Letting $k \rightarrow \infty$ in the inequality above, we have

and

By the inequality 3.1 and property $(\beth 1)$, we have

Letting $\lim_{k\rightarrow \infty}$ in the above inequality using Lemma 3.3,

and

We have

Then

From the above inequality and since $c > 0$ that is $b = 1$, then by the property $(\beth 2)$ with

and

we get

which is a contradiction, thus $c = 0$, i.e.,

This proves that $\{ {s}_{n} \}$ is a Cauchy sequence. □

We are now going to present our main result.

Theorem 3.7. Consider $(\digamma, \Omega_{b}, b)$ be a complete $b$-metric space, and $f_{1}, f_{2}:\digamma\rightarrow \digamma$ be two generalized $\Im_{b}$-contractions with $f_{1}(\digamma)\subseteq f_{2}(\digamma)$ and the pair $(f_{1}, f_{2})$ is compatible. Assume that $\exists$ a $b$-simulation function $\beth$ such that 3.1 holds, that is,

If $f_{2}$ is continuous, then there is a coincidence point of $f_{1}$ and $f_{2}$, that is, there exists $t \in \digamma$ such that $f_{1}(t) = f_{2}(t)$. Moreover, if $f_{2}$ is one to one, then $f_{1}$ and $f_{2}$ have a unique common fixed point.

Proof. Consider ${x}_{0} \in \digamma$. Since $f_{1}(\digamma)\subseteq f_{2}(\digamma)$, we have for every $n\in\mathbb{N}$, ${s}_{n} = f_{1} ({t}_{n}) = f_{2}({t}_{n+1})$. Now, from Lemma 3.6, the sequence $\{ {s}_{n} \}$ is Cauchy and since $(\digamma, \Omega_{b}, b)$ is a complete $b$-metric space, there is some $s \in \digamma$ such that

that is,

We claim that s is a coincidence point of $f_{1}$ and $f_{2}$. Since $f_{2}$ is continuous, we have

Also, since $\{f_{1}, f_{2}\}$ is compatible, we have

Hence, by Lemma 2.15 we deduce

Using (3.1), we have

That is,

Letting $n \rightarrow \infty$,

But

which implies that

Thus,

that is

therefore $f_{1}(s) = f_{2} (s)$.

Now, assume there is $p \in \digamma$ such that $f_{1}(p) = f_{2}(p)$ then the inequality (3.1) and $(\beth2)$ imply that

where

Hence,

If $b > 1$ then $f_{1}(s) = f_{1}(p)$. If $b = 1$, by the condition $(\beth 2)$ with

and

we get

which is a contradiction. Therefore,

Now, suppose that $f_{2}$ is one to one. If $s, p$ are two coincidence points of $f_{1}$ and $f_{2}$, In this case, by the above argument we have

Since $f_{2}$ is one to one, it follows that $p = s$. Also, since $f_{2}(s) = f_{1}(s)$ and the pair $\{f_{1}, f_{2}\}$ is compatible we have

Therefore,

That is, $f_{1}(s)$ is a coincidence point of $f_{1}$ and $f_{2}$. Therefore, $f_{1}(s) = s$ and hence

That is, $f_{1}$ and $f_{2}$ have a unique common fixed point $s \in \digamma$. □

Corollary 3.8. Let $(\digamma, \Omega_{b}, b)$ be a complete $b$-metric space and $f_{1}, f_{2} : \digamma\rightarrow \digamma$ be two generalized $\Im_{b}$-contractions with $f_{1}(\digamma)\subseteq f_{2}(\digamma)$ and the pair $(f_{1}, f_{2})$ is compatible. Suppose that there is $\lambda \in (0, 1)$ such that

If $f_{2}$ is continuous, then there is a coincidence point of $f_{1}$ and $f_{2}$, that is, there is $t \in \digamma$ such that $f_{1}(t) = f_{2}(t)$. Moreover, if $f_{2}$ is one to one, then $f_{1}$ and $f_{2}$ have a unique common fixed point.

Proof. By taking the $b$-simulation function

The result follows from Theorem 3.7.

□

Example 3.9. Let $X = [0, \infty)$ and $d:X\times X \rightarrow X$ be defined by

Then, clearly $d$ is a $b$-metric on $X$ with $b = \frac{5}{4}$.

Here, we observe that when $s = \frac{3}{2}$ and $u = 2$ (both belong to $[1, \infty)$ and $t\in [0, \infty)$), we have

and

Hence, $d(s, u)\neq d(s, t)+d(t, u)$. Hence, $d$ is a $b$-metric with $b = \frac{5}{4}(>1)$, but it is not a metric. We now define $f, g:X\rightarrow X$ by

and

Clearly, $f$ and $g$ are $b$-continuous functions. Now, we define $\beth:[0, \infty)\times[0, \infty)\rightarrow [0, \infty)$ by $\beth(x, y) = \frac{4}{5}y-x.$ We have the following possible cases:

Case 1: $s, t\in [0, 1).$

In this case, $d(fs, ft) = 3+\frac{1}{s+t}$ and $M_{b}(s, t) = 8$

Case 2: $s, t\in [1, \infty)$.

Here, $d(fs, ft) = 3+\frac{1}{s+t}$ and $M_{b} = 8$. We have

Case 3: $s\in [0, 1)$ and $t\in [1, \infty)$.

Here, $d(fs, ft) = 3+\frac{1}{s+t}$ and $M_{b} = 8$. Also,

Case 4: $s\in [1, \infty)$ and $t\in [0, 1)$.

In this case $d(fs, ft) = 3+\frac{1}{s+t}$ and $M_{b} = 8$. Also,

So the pair $\{f, g\}$ is a generalized $\Im_b$-contraction. It satisfies all the conditions of Theorem 3.7. Hence, $f$ and $g$ have a common unique fixed point.

Example 3.10. Let $\digamma = [0, 1]$ be endowed with the $b$-metric $\Omega_{b} (s, t) = (s-t)^{2}$, where $b = 2$. Define $f_{1}$ and $f_{2}$ on $\digamma$ by

and

Obviously $f_{1}(\digamma)\subseteq f_{2}(\digamma)$ and furthermore the pair $\{f_{1}, f_{2}\}$ is compatible. Consider the $b$-simulation function given as

For all $s, t \in \digamma$ we have

Now,

As all the requirements of Theorem 3.7 are satisfied, so $f_{1}$ and $f_{2}$ have a unique common fixed point, which is 0.

Example 3.11. Take $\digamma = [0, 1]$. Define $\Omega_{b}: \digamma\times\digamma\rightarrow R$ by $\Omega_{b}(s, t) = (s-t)^{2}$. Clearly, $(\digamma, \Omega_{b})$ is a complete $b$-metric with $b = 2$.

Now, we define the functions $f_{1}, f_{2} : [0, 1]\rightarrow [0, 1]$ by

and

Clearly,

and furthermore the pair is $[f_{1}, f_{2}]$ is compatible. Now, consider the $b$-simulation function $\beth : [0, \infty)\times[0, \infty)\rightarrow R$ defined by

We have

Thus, all the assumption are satisfied of Theorem 3.7, and hence $f_{1}$ and $f_{2}$ have a unique common fixed point, which is 0.

Remark 3.12. If in Lemma 3.3, $M_{b} = \Omega_{b}(f_{2}t, f_{2}s)$ in inequality (3.1), then we will get Lemma 3.1 of ^[24].

Remark 3.13. If in Remark 3.4, $M_{b} = \Omega_{b}(f_{2}t, f_{2}s)$ in inequality (3.1), then we will get Remark 3.2 of ^[24].

Remark 3.14. If in Lemma 3.5, $M_{b} = \Omega_{b}(f_{2}t, f_{2}s)$ in inequality (3.1), then we will get Lemma 3.3 of ^[24].

Remark 3.15. If in Lemma 3.6, $M_{b} = \Omega_{b}(f_{2}t, f_{2}s)$ in inequality (3.1), then we will get Lemma 3.4 of ^[24].

Remark 3.16. If in Theorem 3.7, $M_{b} = \Omega_{b}(f_{2}t, f_{2}s)$ in inequality (3.1), then we will get Theorem 3.5 of ^[24].

4.   A system of non linear Urysohn integral equations

In this section, we present an application of our result to integral equations. Namely, we study the existence of the unique common solution of a system of non linear Urysohn integral equations.

Let us consider the integral equations

and

where (i) $f, r_{1}$, $r_{2}$ and $g$ are unknown functions for each $x\in[a, b].$

(ii) $k_{1}$ and $k_{2}$ are kernels defined for $x, t\in[a, b]$.

Let us denote

and

Assume that

● ($A_{1}$) $\vartheta_{1}f(x) + r_{1}(x) + r_{2}(x) - \vartheta_{2}(\vartheta_{1}f(x)+r_{1}(x)) + r_{2}(x) = 0$

● ($A_{2}$) $r_{1}(x) - r_{2}(x) + \vartheta_{1}f(x)-\vartheta_{1}g(x) = 0.$

We will ensure the existence of a unique common solution of (4.1) and (4.2) that belong to $G = (C[a, b], R^n)$ (the set of continuous mappings defined on $[a, b]$). For this, define the continuous mappings $T_{1}, T_{2}:G \rightarrow G$ by

and

where $f, g, r_{1}, r_{2}\in G.$ We claim $T_{1}\subseteq T_{2}.$

Proof. If we show that $T_{2}(T_{1}f(x)+r_{2}(x)) = T_{1}f(x)$, then it is conformed that $T_{1}\subseteq T_{2}$. Hence,

Using $(A_{1})$,

It shows that

We endow on $G$ the $b$-metric (with $b = 2$) given as $\Omega_{b}(x, y) = |x-y|^2.$ Here, $(G, \Omega_{b})$ is complete. Further, let us suppose that $k_{1}, k_{2}:[a, b]\times[a, b]\times R^n \rightarrow R^n$ are continuous functions satisfying

where

□

Theorem 4.1. Under the conditions $(A_{1})$, $(A_{2})$ and (4.3), the Eqs (4.1) and (4.2) have a unique common solution.

Proof. For $f, g\in (G, R^n)$ and $x\in[a, b]$, we define the continuous mappings $T_{1}, T_{2}:G \rightarrow G$ by

and

Then we have

This shows that all the requirements of our main theorem are satisfied, i.e., $2\Omega_{b}(T_{1}(f), T_{1}(g)) \leq M_{b}(f, g)$. Therefore, the integral equations (4.1) and (4.2) have a unique common solution.

□

5.   A system of non linear Volterra-Hammerstein integral equations

In this section, we give a second application and we ensure the existence of the unique common solution of a system of non linear Volterra-Hammerstein integral equations.

Let us take $\digamma = (L(0, \infty), R)$ the space of real-valued measurable functions on $(0, \infty)$. Consider

and

for all $x, y \in (0, \infty)$, where $\lambda, \mu \in\mathbb{R}$, and $D, \tau_{1}, \tau_{2}, m(x, y), n(x, y), g_{1}, g_{2}, h_{1}, and \ h_{2}$ are measurable functions with real values in $S$ and $r$ on $(0, \infty)$,

for $i = 1, 2$

with

Assume that

We consider the $b$-metric space $\Omega_{b} (x, y) = | x - y |^{2}.$

Theorem 5.1. Under the assumption $(c_{1})$ and the condition (5.3), the system of non linear Volterra-Hammerstein integral equations has a unique common solution.

Proof. Note that the system of non linear Volterra-Hammerstein integral equations (5.1) and (5.2) has a unique common solution if and only if the system of operator $f_{1}$ and $f_{2}$ has a unique common fixed point.

Now,

This shows that all the requirements of our main theorem are satisfied, and therefore the integral equations (5.1) and (5.2) have a unique common solution. □

6.   Conclusions

Rodjanadid et al.^[23] used the idea of generalized $\Im_b$-contractions with $b$-simulation functions and proved some fixed point results in complete $b$-metric spaces. Jawaher et al.^[24] utilized the idea of $b$-simulation functions and investigated some common fixed points for two contractive mappings. In this manuscript, we combined these two ideas and proved some common fixed points for two contractive mappings using the idea of generalized $\Im_b$-contractions with $b$-simulation functions in $b$-metric spaces. Different examples and applications are given to demonstrate the validity of the concept and the degree of applicability of our findings. Many applied problems can be described by systems of Fredholm and Volterra integral equations. The presented results can be utilized to study the existence of unique common solutions of these systems.

Acknowledgments

The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work grant code: 23UQU4331214DSR002.

Conflict of interest

The authors declare that they have no conflict of interest.