Rational interpolative contractions with applications in extended $b$-metric spaces

1.   Introduction and preliminaries

The concept of distance was axiomatically formulated in the beginning of the 19th century with the introduction of metric spaces, by Frechet and Haussdorff. Since then, many authors have developed this concept, with several results available in the literature. For the generalization of this concept, the axioms of the metric space have been relaxed in several ways (see^[1]), among which the notion of a $b$-metric space takes great importance. Bakhtin ^[2] (and, independently, Czerwik ^[3]) presented the idea of $b$-metric spaces and showed different results based on the existence of fixed points. For the sake of understanding, we present here the definition of a $b$-metric, also called a quasi-metric (see ^[4]).

Definition 1.1. (Czerwik ^[3]) Consider Q to be a non-empty set and $b$: $Q\times Q\rightarrow [0, +\infty)$ to be a self-map fulfilling the below prerequisites:

(1) $b(s, p) = 0$ $\Leftrightarrow$ $s = p$;

(2) $b(s, p) = b(p, s)$ for all $s, p\in Q$;

(3) $b(s, u)\leq q[b(s, p)+b(p, u)]$ for all $s, p, u\in Q$, where $q\geq1.$

The function $b$: $Q\times Q\rightarrow [0, +\infty)$ is called a $b$-metric, while the pair $(Q, b)$ is known as a $b$-metric space.

Example 1.1. ^[4] The space $M^p$[0, 1] (where $p\in(0, 1)$) of all real functions k(s), $s\in[0, 1]$ such that

together with the functional

is a $b$-metric space. Here, $q = 2^{\frac{1}{p}-1}.$

Example 1.2. ^[5] Let

for some $J\in N$ and $a\geq2$. Define a function $b$: $Q\times Q\rightarrow [0, +\infty)$ by

Accordingly, we obtain

for all $i, j, k\in\{1, 2, \ldots, J\}.$ The pair $(Q, b)$ forms a $b$-metric space for $a > 2.$ We can observe that the standard triangular inequality does not hold in this case.

The $b$-metric space shares many topological properties with traditional metric spaces but does not require continuity. Recently, Kamran et al. ^[6] presented a new generalization of metric spaces and proved some important fixed-point results in the newly defined space. Further more, Alqahtani et al. ^[7] studied common fixed point results on extended $b$-metric space.

Definition 1.2. ^[6] Consider Q to be a non-empty set and $\vartheta$: $Q\times Q\rightarrow [1, +\infty).$ A function $b_{\vartheta}$: $Q\times Q\rightarrow [0, +\infty)$ is said to be an extended $b$-metric if for all $s, t, u\in Q$ the given axioms are satisfied

(1) $b_{\vartheta}(s, t) = 0$ implies $s = t;$

(2) $b_{\vartheta}(s, t) = b_{\vartheta}(t, s);$

(3) $b_{\vartheta}(s, u)\leq \vartheta(s, u)[b_{\vartheta}(s, t)+b_{\vartheta}(t, u)].$

The pair $(Q, b_{\vartheta})$ is known as extended $b$-metric space.

Remark 1.1. ^[5] Suppose $\vartheta(s, t) = a, \ \hbox{for} \ a\geq 1,$ then it is obvious that the b-metric and extended $b$-metric spaces ($bMS$) will coincide. Note that either the $b$-metric or the extended $b$-metric need to be continuous like metric spaces.

Example 1.3. ^[5] Suppose $p\in(0, 1), q > 1$ and $Q = l_p(R)\cup l_q(R)$ equipped with the metric

Where

for $r = p, q,$ and

for $r = p, q,$ we can observe that $(Q, b_\vartheta)$ forms an extended bMS with

Example 1.4. ^[6] Let $G = \{1, 2, 3\}, \ \vartheta$: $G\times G \rightarrow [1, +\infty)$ and $b_{\vartheta}$: $G\times G\rightarrow [0, +\infty)$ as $\vartheta(s, t) = 1+s+t$ and

Example 1.5. ^[7] Let $G = [0, 1], \vartheta$: $G\times G\rightarrow [1, +\infty)$ and $b_\vartheta$: $G\times G \rightarrow [0, +\infty)$ be defined by

Now we are going to discuss some basic notions like convergence, completeness, and Cauchy sequence in extended $bMS$ that are defined as:

Definition 1.3. ^[6] Suppose (Q, $b_\vartheta$) be an extended $b$-metric space.

(1) A sequence $\{s_j\}_{j\in \mathbb{K}}$ in Q will converge to $t\in Q$, if for every $\zeta > 0$ there exists $K = K(\zeta)\in \mathbb{K}$ such that $b_\vartheta(s_j, s) < \zeta$ for all $j\geq K.$ In this case, we write

(2) A sequence $\{s_j\}_{j\in \mathbb{K}}$ in Q is known as Cauchy sequence if for every $\zeta > 0$ exists $K = K(\zeta)\in \mathbb{K}$ such that $b_\vartheta(s_j, s_m) < \zeta,$ for all $m, j\geq K.$

Definition 1.4. ^[5] Suppose every Cauchy sequence in Q is convergent, then the extended-bMS $(Q, b_\vartheta)$ is said to be complete.

Definition 1.5. ^[5] Let $(Q, b_\vartheta)$ be an extended-bMS and $\daleth$: $Q\rightarrow Q$ be a self-map. For $t_o\in Q$, the $orbit$ of $\daleth$ at $t_o$ is the set

The function $\daleth$ is known as orbitally continuous at a given point $e\in Q$ if

Besides that, suppose every Cauchy sequence $\{\daleth^jt_o\}$ in Q is convergent, then the extended-bMS $(Q, b_\vartheta)$ is called $\daleth$-orbitally complete.

Definition 1.6. ^[8] Suppose $(R, b_\vartheta)$ be an extended $b$-metric. The mapping $\daleth$: $R\rightarrow R$ is known as m-continuous, where $m = 1, 2, \ldots$, if

whenever ${t_n}$ is a sequence in R such that

Remark 1.2. ^[1] It is notable that every continuous function is orbitally continuous in Q and every complete extended-bMS is $\daleth$-orbitally complete for any $\daleth$: $Q\rightarrow Q$, but the converse is not necessarily true.

Besides that, it is obvious that 1-continuity results in 2-continuity, which in turn will result in 3-continuity, and so on; but the converse of this is not true. This might be clearer from this example: consider the self-mapping $\daleth$: $Q\rightarrow Q,$ where $Q = [0, +\infty),$ defined by

we can clearly see that $\daleth$ is discontinuous (at $s = 5$), while it is 2-continuous because $T^2 s = 5.$

Definition 1.7. ^[4] A self-mapping $\yen$: $[0, +\infty)\rightarrow [0, +\infty)$ is said to be a comparison function if it is increasing and $\yen^n(s)\rightarrow 0$ as $n\rightarrow +\infty$ for every $s\in [0, +\infty),$ where $\yen^n$ is the $n^{th}$ iterate of $\yen$.

Lemma 1.1. ^[4] Suppose $\yen$: $[0, +\infty)\rightarrow [0, +\infty)$ is a comparison function, then

(1) $\yen$ is continuous at 0;

(2) every iterate $\yen^k$ of $\yen, k\geq1$ is also a comparison function;

(3) $\yen(s) < s$ for all $s > 0.$

Definition 1.8. ^[4] Suppose $t\geq1$ be a real number. A self-mapping $\yen$: $[0, +\infty)\rightarrow [0, +\infty)$ is said to be a ($b$)-comparison function if it is increasing and if there exist $k_o\in N, a\in [0, 1)$ and a convergent non-negative series $\sum_{k = 1}^{+\infty}w_k$ such that

for $k\geq k_o$ and any $s\geq0.$

The set of all ($b$)-comparison functions is denoted by $\Theta.$ The ($b$)-comparison function is said to be a ($c$)-comparison function if we take $t = 1.$ It is easy to show that every ($c$)-comparison function is a ($b$)-comparison function, but the converse is not true. Another important property of ($b$)-comparison functions is presented by Berinde ^[4].

Lemma 1.2. ^[4] Suppose $\Xi$: $[0, +\infty)\rightarrow [0, +\infty)$ be a (b)-comparison function. Then:

(1) The series $\sum_{t = 0}^{\infty} h^t\Xi^t(s)$ converges for any $s\in[0, +\infty)$;

(2) The function $b_s$: $[0, +\infty)\rightarrow [0, +\infty)$ defined as

is increasing and is continuous at $s = 0$.

Remark 1.3. ^[5] Each (b)-comparison function $\Xi$ satisfies $\Xi(s) < s$ and

for each $s > 0.$

Definition 1.9. ^[9] Let $\alpha$: $Q\times Q \rightarrow [0, +\infty)$ be a mapping and $Q\neq \emptyset.$ A self-mapping $\daleth$: $Q\rightarrow Q$ is called $\alpha$-orbital admissible if for all $a\in Q,$ we have

Besides, the $\alpha$-orbital admissible function $\daleth$ is said to be triangular $\alpha$-orbital admissible if

Besides that, we say that the extended-$bMS$ $(Q, b_\vartheta)$ is $\alpha$-regular if for any sequence ${t_n}$ in Q such that

we have $\alpha(t_n, t)\geq1$ (for more details and examples, see ^[9]). Popescu ^[9] redefined the concept of $\alpha$-admissible mapping and triangular $\alpha$-admissible mapping. Qawagneha et al. ^[10] investigated common fixed points for pairs of triangular $\alpha$ -admissible mappings. The idea of interpolative contractions was very recently introduced by ^[11], and the well-known Kannan-type contractions were revisited in the context of interpolation. Subsequently, most famous contractions (Rus ^[4], Ćirić ^[12], Reich ^[13], Hardy and Rogers ^[14], Kannan ^[15], Bianchini and Grandolf ^[16]) have been revisited in this newly introduced context-(see ^[11,17,18]). Following this trend and using the idea of fractional interpolative contraction, Fulga ^[1] established some fixed-point results in the framework of $bMS.$ Additionally Debnath et al. studied interpolative Hardy-Rogers and Reich-Rus-Ćirić-type contractions in $b$-metric and rectangular $bMS$ ^[19].

Non-linear integral equations have emerged in various fields of science and engineering, offering powerful tools for modeling physical phenomena and solving problems in diverse areas such as physics, engineering, and economics. Various researchers have studied these equations using different approaches, some of which can be found in ^[20,21,22].

Motivated by the above contributions using fractional interpolative contractions some fixed-point results are studied in the setting of extended-$bMS$. The work here presented generalizes some well-known results from the existing literature. For the authenticity of the present work a key theorems is used to establish the existence of solutions for the Fredholm integral equations. The results obtained can be extended to investigate the existence of solutions for other integral equations (see^[20,21,23]).

2.   Main results

We initiate with the following definition of contractive mapping to prove the main results.

Definition 2.1. Let $(Q, b_\vartheta)$ be an extended-$bMS$. A mapping $\daleth$: $Q\rightarrow Q$ is known as $A_\daleth^l$-admissible interpolative contraction $(l = 1, 2)$ if $\exists$ $\psi\in \Theta$ and $\Omega$: $Q\times Q\rightarrow [0, +\infty)$ such that

where $p_j\geq0, \ j = 1, 2, 3, 4, 5,$ are such that $\sum_{j = 1}^{5}p_j = 1$ and

and

for any $s, a\in Q\backslash Fix_\daleth(Q), (Fix_\daleth(Q) = \{s\in Q | \daleth s = s\}).$

Theorem 2.1. Let $(Q, b_\vartheta)$ be an extended-$bMS$ and $\daleth$ be an $A_\daleth^1$-admissible interpolative contraction, assume that $\exists$ a sequence $\{q_j\}_{j\in \mathbb{N}}, q_j > 1,$ for all $j\in \mathbb{N}$, such that $\vartheta(a_j, a_m) < q_j$ for all $m > j$, and $\daleth$ also satisfies:

i) There exists $a_o\in Q$ such that $\alpha(a_o, \daleth a_o)\geq1;$

$ii) \daleth$ is $\alpha$-orbital admissible;

$iii_1) \daleth$ is orbitally continuous; or

$iii_2) \daleth$ is m-continuous for $m\geq1.$

Then, $\daleth$ possesses a fixed point $\varpi \in Q$ and the sequence $\{\daleth^m a_o\}$ converges to $\varpi.$

Proof. Suppose $a_o \in Q$ and the sequence $\{a_j\}$ be defined as $a_j = \daleth^ja_o$, $\forall$ $j\in \mathbb{N}.$ Suppose there exists $k\in \mathbb{N}$ such that

then, we have that $a_k$ is a fixed point of $\daleth$ and the proof is complete. Therefore, we suppose that $a_j\neq a_{j+1}$ for any $j\in \mathbb{N}.$ Using assumption $(ii)$, we obtain that $\daleth$ is $\alpha$-orbital admissible, so consider that we have

On the other hand, we have that

We mention in the beginning that $\daleth$ is an $A_\daleth^1$-admissible interpolative contraction, so from (2.1) we get

So,

Therefore,

i.e.,

If exists $m_o\in \mathbb{N}$ such that

then the above inequality becomes

i.e.,

so,

but it is a contradiction, so for any $j\in \mathbb{N},$

Furthermore, returning to inequality (2.4), we have

Let $r\in \mathbb{N}$ and $j < m,$ then by (2.5) together with the condition $(iii)$ of extended-$bMS$, we obtain

Let

we deduce

Consider the series

Let

we have

From Lemma 1.2, we have that $\sum_{k = 0}^{\infty}\psi^k(b_\vartheta(a_o, a_1))q^k$ converges. For the convergence of series using comparison criteria, we get that

converges, and hence

As a result, we say that $\{a_j\}_{j\in N}$ is a Cauchy sequence in a $\daleth$-orbitally complete extended-$bMS$. Hence, there exists a point $\varpi\in Q$, such that

We can declare that $\varpi$ is a fixed point of the self-mapping $\daleth$ under of any hypothesis, $(iii_1)$ or $(iii_2).$ Indeed,

Moreover,

for every $m\geq1.$

If $\daleth$ is m-continuous, then

and by (2.6), it follows that $\daleth \varpi = \varpi.$ Suppose $\daleth$ is considered to be orbitally continuous on Q, then

Therefore, $\varpi\in Fix_\daleth(Q).$ □

Theorem 2.2. Let $(Q, b_\vartheta)$ be an extended $bMS.$ Suppose there exists a sequence $\{q_j\}, q_j > 1,$ for all $j \in \mathbb{N}$ such that $\vartheta(a_j, a_m) < q_j,$ for all $m > j,$ and $\daleth$ is $A_\daleth^2$-admissible interpolative contraction, and $\daleth$ also satisfies:

$i)$ There exists $a_o\in Q$ such that $\alpha(a_o, \daleth a_o)\geq1;$

$ii) \daleth$ is $\alpha$-orbital admissible;

$iii_1) \daleth$ is orbitally continuous; or

$iii_2) \daleth$ is m-continuous for $m\geq1$.

Then $\daleth$ has a fixed point $\varpi \in Q.$

Proof. From the proof of the above theorem, for $a_o \in Q,$ we construct the sequence $\{a_j\},$ where

for any $j\in N.$ Since $a_{j-1}\neq a_j$ for any $j\in N$, keeping in mind that $\daleth$ is assumed to be $A_\daleth^2$-admissible interpolative contraction, we have

where

Since, by assumption, it follows that $\alpha(a_{j-1}, a_j)\geq1$ for all $j\in \mathbb{N},$ we have

Therefore,

i.e.,

Furthermore, keeping in mind $\psi_2$, we obtain

and using the same method as in the proof of Theorem 2.1, we can see that the sequence $\{a_j\}$ is Cauchy. Furthermore, since $(Q, b_\vartheta)$ is considered to be $\daleth$-orbitally complete, we can find a point $\varpi \in Q$ such that

Consider that $\daleth$ is m-continuous, we have

and suppose that $\daleth$ is orbitally continuous, we obtain

it means that $\varpi$ is a fixed point of $\daleth$. □

The following corollaries are observed from the above results.

Corollary 2.1. Suppose $(Q, b_\vartheta)$ be a complete extended $b$-metric space. Suppose that there exists a sequence $\{p_j\}_{j\in N}, \ p_j > 1$ for all $j\in \mathbb{N}$ such that $\vartheta(s_j, s_m) < q_j$ for all $m > j$ and $\daleth$: $Q\rightarrow Q$ be a mapping such that

For any $s, v\in Q\backslash Fix(Q),$ where $A_\daleth^l, l = 1, 2$ is defined by (2.2) and (2.3), and $\psi\in \Theta.$ Then, $\daleth$ has a fixed point $\varpi\in Q$ provided that:

i) There exists $u_o\in Q$ such that $\alpha(u_o, \daleth u_o)\geq 1$;

$ii) \daleth$ is $\alpha$-orbital admissible;

$iii_1) \daleth$ is orbitally continuous; or

$iii_2) \daleth$ is m-continuous for $m\geq1.$

Corollary 2.2. Suppose $(Q, b_\vartheta)$ be a complete extended $b$-metric space. Suppose that there exists a sequence $\{p_j\}_{j\in \mathbb{N}}, \ p_j > 1,$ for all $j\in \mathbb{N}$ such that $\vartheta(s_j, s_m) < p_j,$ for all $m > j$ and $\daleth$: $Q\rightarrow Q$ be a mapping such that

For any $s, v\in Q\backslash Fix(Q),$ where $A_\daleth^l, l = 1, 2,$ are defined by (2.2) and (2.3), and $\psi\in \Theta.$ Then, $\daleth$ has a fixed point $\varpi \in Q,$ provided that either $\daleth$ is orbitally continuous or $\daleth$ is m-continuous for $m\geq 1$.

Proof. Plug $\alpha(s, v) = 1$ in Theorems 2.1 and 2.2, respectively. □

By replacing the continuity of the function $\daleth$ with the continuity of $b_\vartheta$, we will have the following result.

Theorem 2.3. Suppose $(Q, b_\vartheta)$ be a complete, $\alpha$-regular extended-$bMS$, where $b_\vartheta$ is continuous, and $\daleth$: $Q\rightarrow Q$ is such that

where $\psi \in \Theta$ and $A_\daleth^l,$ for $l = 1, 2$ are given by (2.2) and (2.3). Consider that:

(1) There exists $a_o\in Q$ such that $\alpha(a_o, \daleth a_o)\geq1;$

(2) $\daleth$ is $\alpha$-orbital admissible.

Then, $\daleth$ contains a fixed point $\varpi \in Q,$ and the sequence $\{\daleth^ma_o\}$ converges to this point $\varpi.$

Proof. As we know from the proof of Theorem 2.1, the sequence $\{a_j\}$ where

converges to a point $\varpi\in Q,$ and this point $\varpi$ is claimed to be a fixed point of the mapping $\daleth$. For this reason, we can declare that

or

Indeed, supposing on contrary

and

we get that

which leads to contradiction and then (2.7) and (2.8) holds. Keeping the regularity condition of the space $(Q, b_\vartheta)$ in mind, we have that $\alpha(a_j, \varpi)\geq 1$ for any $j\in N$.

Case 1. When $l = 1,$ if (2.7) holds, we get

we can distinguish the following two situations:

(1) $p_1+p_2 > 0$, letting $j\rightarrow +\infty$ above, we obtain $b_\vartheta(\varpi, \daleth \varpi) = 0$, thus $\daleth \varpi = \varpi$.

(2) $p_1 = p_2 = 0$, when $j\rightarrow \infty$ above, and keeping in mind the continuity of extended-$bMS$ we obtain

which is a contradiction. So, we have $\daleth \varpi = \varpi,$ i.e., $\varpi$ is a fixed point of the mapping $\daleth$.

Case 2. When $l = 2$. If (2.7) holds, we obtain

if (2.8) holds,

we can distinguish the following two situations:

(1) $p_1+p_2+p_4+p_5 > 0$, letting $j\rightarrow \infty$ above, we obtain $b_\vartheta(\varpi, \daleth \varpi) = 0$, thus $\daleth \varpi = \varpi$.

(2) $p_1 = p_2 = p_4 = p_5 = 0$, in this case, when $j\rightarrow \infty$ above, we get

which is a contradiction.

So, we get $\daleth \varpi = \varpi$, i.e., $\varpi$ is a fixed point of the mapping $\daleth$. □

This result possesses the below corollaries.

Corollary 2.3. Let $(Q, b_\vartheta)$ be a complete extended $bMS$. Suppose $\{p_j\}_{j\in \mathbb{N}}$ be a sequence, $p_j > 1$ for all $j\in \mathbb{N}$ such that $\vartheta(s_j, s_m) < p_j$ for all $m > j$ and $\daleth$: $Q\rightarrow Q$ be a mapping such that $\exists$ $k\in [0, 1)$ such that

for any $s, v\in Q_{-Fie(Q)}$ where $A_\daleth^l, l = 1, 2$ are defined by (2.2) and (2.3). Then, $\daleth$ contains a fixed point $\varpi\in Q,$ provided that either $\daleth$ is orbitally continuous or $\daleth$ is m-continuous for $m\geq1$.

Proof. Plug $\psi(t) = kt$ in the above corollary. □

Corollary 2.4. Suppose $(Q, b_\vartheta)$ be a complete extended-$bMS$ such that $b_\vartheta$ is continuous. Suppose there exist a sequence $\{p_j\}_{j\in \mathbb{N}}, p_j > 1$ for all $j\in N$ such that $\vartheta(s_j, s_m) < p_j$ for all $m > j,$ and $\daleth$: $Q\rightarrow Q$ be a self-mapping. Then $\daleth$ has a fixed point provided that

where $\psi\in \digamma$ and $A_\daleth^l, l = 1, 2$ are given by (2.2) and (2.3).

Proof. Put $\alpha(s, v) = 1$ in Theorem 2.3. □

Corollary 2.5. Consider $(Q, b_\vartheta)$ be a complete extended-$bMS$ such that $b_\vartheta$ is continuous. Suppose that exists $\{p_j\}_{j\in \mathbb{N}}, p_j > 1$ for all $j\in \mathbb{N}$ such that $\vartheta(s_j, s_m) < p_j$ for all $m > j$ and $\daleth$: $Q\rightarrow Q,$ self-mapping. Then $\daleth$ will have a fixed point in Q provided that there exist $k\in [0, 1)$ such that

where $A_\daleth^l, l = 1, 2$ are given by (2.2) and (2.3).

Proof. Substituted $\psi(t) = kt$ in the above corollary. □

Now, we are going to present some examples of the above results.

Example 2.1. Let $Q = [0, +\infty)$ and $b_\vartheta$: $Q\times Q\rightarrow [0, +\infty)$ be an extended-$bMS$ defined as

and $\vartheta$: $Q\times Q\rightarrow [1, +\infty)$ be defined as $\vartheta(s, v) = 1+s+v$ for all $s, v\in Q.$ Let the mapping $\daleth$: $Q\rightarrow Q$ be defined by

and a function $\alpha:Q\times Q\rightarrow [0, +\infty),$ where

Let also the comparison function $\psi$: $[0, \infty)\rightarrow [0, \infty)$, $\psi(s) = s/3$, and we choose $p_1 = p_5 = 1/5$, $p_2 = p_4 = 1/10$, and $p_3 = 2/5.$ Therefore, we can clearly see that conditions (i) and (ii) are verified, and since $\daleth^2(s) = 1/5$ is continuous, condition (iv) is also satisfied.

Case (1). For $s, v\in[0, 1],$ we have $b_\vartheta(\daleth s, \daleth v) = 0,$ so inequality (2.1) holds.

Case (2). For $s = 0$ and $v = 2,$ we have $\frac{1}{2}b_\vartheta(0, 1/2) = 1/4\leq 2 = b_\vartheta(0, 2)$ and $b_\vartheta(\daleth s, \daleth v) = 0.$ Thus, the inequality (2.1) holds.

Case(3). For $s = 1/4$ and $v = 3,$ we have

hence (2.1) holds.

Case(4). For $s = 1/4$ and $v = 9$, we have

All other cases are true because $\alpha(s, v) = 0$. Hence, the mapping $\daleth$ is an $A_\daleth^1$-admissible interpolative contraction. So, as all the conditions of Theorem 2.1 are verified, we obtained that there exists a fixed point of the mapping $\daleth$, that is $u = 1/5.$

Example 2.2. Let $Q = \{1, 2, 3, 5\}$ and the extended-$bMS$ defined $b_\vartheta$: $Q\times Q\rightarrow R^+$ as $b_\vartheta(s, v) = |s-v|^4$ with $\vartheta(s, v) = 1+x+y$ and $\daleth$: $Q\rightarrow Q$ such that $\daleth(1) = \daleth(5) = 1 \ \ and \ \ \daleth(2) = \daleth(3) = 2.$ Taking $\alpha$: $Q\times Q\rightarrow R^+, \alpha(s, v) = 3 \ for \ all \ s, v\in Q,$ and $\psi(t) = t/2.$ The constants here are all equal, i.e., $p_i = 1/5 \ \ \forall \ i = \{1, 2, 3, 4, 5\},$ we have

which implies

Therefore, all the requirements of Theorem 2.3 are satisfied and it is clear that $\daleth$ has (at least) a fixed point.

3.   Application

In this segment, we apply one of the observed results to study the existence of a solution for the Fredholm integral equation. Suppose $Q = C([a, b], R)$ be the space of all continuous real-valued functions defined on [a, b]. Note that the space $Q$ is complete by considering the extended-$bMS$

with

where $\vartheta(s, v)$: $Q\times Q\rightarrow [1, +\infty)$ and $\psi\in \Theta$ be the $b$-comparison function defined as $\psi(e) = e/2.$ Consider the Fredholm integral equation as:

Define a mapping $\daleth$: $Q\rightarrow Q,$ as

Theorem 3.1. Consider that the following conditions hold:

(1) Suppose $M$: $[p, q]\times[p, q]\times R\rightarrow R$ and $g$: $[p, q]\rightarrow R$ be continuous.

(2)$\daleth$ is $A_\daleth^l$-admissible interpolative contraction, $A_\daleth^l, l = 1, 2$ is defined in (2.2) and (2.3), respectively.

(3)

for each $e, i\in[p, q]$ and $s, v\in Q.$

Then, the integral Eq (3.1) has a solution.

Proof. Suppose $(Q, b_\vartheta)$ be a complete extended-$bMS$ and $\alpha(s, v) = 1.$ Then as

we have

All the conditions of Theorem 2.3 are fulfilled. Therefore, the integral Eq (3.1) has a solution. □

4.   Conclusions and future directions

Many physical problems can be described by various Fredholm integral equations. There are several methods available in the literature for the establishment of solutions to these equations. One powerful method is the fixed-point method. Therefore, in the current work, some new fractional interpolative contractions were introduced. With the help of these fractional interpolative contractions, some fixed-point results were studied in extended $bMS$. For the validity of the presented results, certain examples were given. Lastly, as a practical application, an existence theorem for the solution of the Fredholm integral equation was provided. This work generalizes some well-known results from the existing literature. In the future, one can explore the established work for multi-valued mapping and investigating the existence of solutions for integral inclusions.

Use of AI tools declaration

The authors declare they have not used artificial intelligence (AI) tools in the creation of this article.

Acknowledgments

Authors are thankful to Prince Sultan University for the support of this work through TAS research lab.

This work is funded by Natural Science Basic Research Plan in Shaanxi Province of China (No. 2022JQ-040).

Conflicts of interest

The authors declare no conflicts of interest.