Open AccessArticle

A Relation-Theoretic Metrical Fixed Point Theorem for Rational Type Contraction Mapping with an Application

Asik Hossain

^1,*,

Faizan Ahmad Khan

^2,* and

Qamrul Haq Khan

Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia

Authors to whom correspondence should be addressed.

Axioms 2021, 10(4), 316; https://doi.org/10.3390/axioms10040316

Submission received: 18 October 2021 / Revised: 18 November 2021 / Accepted: 19 November 2021 / Published: 23 November 2021

(This article belongs to the Collection Mathematical Analysis and Applications)

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Abstract

In this article, we discuss the relation theoretic aspect of rational type contractive mapping to obtain fixed point results in a complete metric space under arbitrary binary relation. Furthermore, we provide an application to find a solution to a non-linear integral equation.

Keywords:

binary relation; ℜ-completeness; rational contraction

MSC:

47H10; 54H25

1. Introduction

In 1922, the first prosperous result was postulated by S. Banach [1] in the fixed point theory for contractive mapping. For its modesty, his work functioned as a schematic research tool in a different branch of mathematics. This theorem went in a different direction to verify its effectiveness. Such as

(i): Enlarging the ambient space;
(ii): Improving the underlying contraction condition;
(iii): Weakening the involved metrical notions.

Among the several extensions of the Banach contraction principle to various spaces, some are rectangular metric space, generalized metric space, partial metric space, b-metric space, partial b-metric space, symmetric space and quasi metric space. Partial metric space was introduced by Matthews [2] in 1994. Nowadays, there are many fixed point theories in Partial metric space.

Several researchers stated various contraction conditions [3,4,5,6,7,8] for the fixed point theorem. Inspired by Turinici’s [9] work, Ran-Reurings in 2004 formulate the result that there will be a fixed point of self-mappings that is applied only for those points which are comparable to each other by an order relation in partial metric space. Later, the work was extended by J.J. Nieto and R. Rodríguez-López [10]. In 1975, Dass and Gupta [11], came up with a new contractive condition termed as a rational type contraction. Later, Canbrera et al. [12] used the result of Dass and Gupta [11] in 2013 to obtain the fixed point results in partial ordered metric space.

Alternatively, Alam and Imdad [13] established a profound generalization of the Banach contraction principle with an amorphous binary relation. With this structure, various relation-theoretic results were proposed in different aspects of the binary relation or contractive condition.

There are too many applications of fixed point theory in the field of ordinary differential equations, systems of matrix equation, integral equations, game theory, economics, optimization models and numerical models in statistics. Moreover, for the multivalued maps in the equilibrium in the duopoly markets and in aquatic ecosystem there are also too many applications. In an ordinary differential equation, the application provided by J.J. Nieto and R. Rodríguez-López [10] and the system of matrix equations by Ran and Reurings [14], the fixed point for iteration to find optimal solution in statistics [15], for the stability problem in Intuitionistics Fuzzy Banach Space [16], and many more such as [17].

This article intends to establish some fixed point theorems under contractive mapping over a complete metric space. Ultimately, an example is provided to establish the result for our assumptions. Furthermore, we provide an application [18] in a non-linear integral equation to obtain a fixed point.

2. Preliminaries

In this section, we present some basic definitions which will be required in proving our main results. We denote

N

∪

{0}

N_{0}

throughout the paper.

Definition 1

([11]). Let

(W, d)

be a complete metric space and T a self-mapping on W. Then, T is said to be a rational type contraction if there exist

δ_{1}, δ_{2} \in [0, 1)

with

δ_{1} + δ_{2} < 1

, satisfying

d (T μ, T ν) \leq δ_{1} \frac{d (ν, T ν) [1 + d (μ, T μ)]}{[1 + d (μ, ν)]} + δ_{2} d (μ, ν) f o r a l l μ, ν \in W .

Definition 2

([19]). Let W be a nonempty set. A subset ℜ of

W^{2}

is called a binary relation on W. The subsets

W^{2}

and ∅ of

W^{2}

are in trivial relation.

Definition 3

([13]). Consider a binary relation ℜ on a nonempty set W. For

μ, ν \in W

, one may say that μ and ν are ℜ-comparative if either

(μ, ν) \in ℜ

(ν, μ) \in ℜ

. We symbolize it with

[μ, ν] \in ℜ

Definition 4

([19,20,21,22,23,24]). On a nonempty set W, a binary relation ℜ is termed as

(i): Reflexive if $(μ, μ) \in ℜ \forall μ \in W$ ;
(ii): Symmetric if $(μ, ν) \in ℜ$ then $(ν, μ) \in ℜ$ ;
(iii): Anti-symmetric if $(μ, ν) \in ℜ$ and $(ν, μ) \in ℜ$ then $μ = ν$ ;
(iv): Transitive if $(μ, ν) \in ℜ$ and $(ν, κ) \in ℜ$ then $(μ, κ) \in ℜ$ ;
(v): A partial order if ℜ is reflexive, anti-symmetric and transitive.

Definition 5

([19]). Let W be a nonempty set and ℜ a binary relation on W.

(i): The dual relation, transpose or inverse of ℜ, signified by $ℜ^{- 1}$ is interpreted by,

$ℜ^{- 1} = {(μ, ν) \in W^{2} : (ν, μ) \in ℜ} .$
(ii): Symmetric closure $ℜ^{s}$ of ℜ, is defined to be the set ℜ∪ $ℜ^{- 1}$ (i.e., $ℜ^{s} = ℜ \cup ℜ^{- 1}$ ).

Proposition 1

([13]). For a binary relation ℜ defined on a nonempty set W,

(μ, ν) \in ℜ^{s} \Rightarrow [μ, ν] \in ℜ .

Definition 6

([13]). Consider a nonempty set W and let ℜ be a binary relation on W. A sequence

{μ_{n}}

⊂W is called ℜ-preserving if

(μ_{n}, μ_{n + 1}) \in ℜ \forall n \in N_{0} .

Definition 7

([13]). For a nonempty set W with a self-mapping T on it. Any binary relation ℜ on W is T-closed if ∀

μ, ν \in W

(μ, ν) \in ℜ \Rightarrow (T μ, T ν) \in ℜ .

Definition 8

([25]). Let

(W, d)

be a metric space and ℜ a binary relation on W. Then,

(W, d)

is ℜ-complete if every ℜ-preserving Cauchy sequence in W converges.

It is obvious that every complete metric space is ℜ-complete with respect to a binary relation ℜ but not conversely. For instance, Suppose

W = (- 2, 2]

together with the usual metric d. Notice that

(W, d)

is not complete. Now endow W with the following relation:

ℜ = {(μ, ν) \in W^{2} : μ, ν \geq 0} .

Then,

(W, d)

is a ℜ-complete metric space.

Definition 9

([22]). Let W be a nonempty set endowed with a binary relation ℜ. A subset D of W is called ℜ-directed if for each

μ, ν \in D

, there exists

κ \in W

such that

(μ, κ) \in ℜ

and

(ν, κ) \in ℜ

Definition 10

([25]). Let

(W, d)

be a metric space endowed with a binary relation ℜ with

μ \in W

. Then

T : W \to W

is called ℜ-continuous at μ if for any ℜ-preserving sequence

{μ_{n}}

with

μ_{n} \overset{d}{\to} μ,

we obtain

T (μ_{n}) \overset{d}{\to} T (μ) .

Furthermore, T is called ℜ-continuous if it is ℜ-continuous at each point of W.

Definition 11

([13]). Let

(W, d)

be a metric space. A binary relation ℜ on W is termed as d-self-closed if whenever

{μ_{n}}

is an ℜ-preserving sequence and

μ_{n} \overset{d}{\to} μ,

then there exists a subsequence

{μ_{n_{k}}}

{μ_{n}}

with

[μ_{n_{k}}, μ]

∈ℜ∀k∈

N_{0} .

Proposition 2

([13]). If

(W, d)

is a metric space, ℜ is a binary relation on W, T is a self-mapping on W and

δ_{1}, δ_{2} \in [0, 1)

with

δ_{1} + δ_{2} < 1

then the following conditions are equivalent

(i): $d (T μ, T ν) \leq δ_{1} \frac{d (ν, T ν) [1 + d (μ, T μ)]}{[1 + d (μ, ν)]} + δ_{2} d (μ, ν) \forall μ, ν \in W w i t h (μ, ν) \in ℜ,$
(ii): $d (T μ, T ν) \leq δ_{1} \frac{d (ν, T ν) [1 + d (μ, T μ)]}{[1 + d (μ, ν)]} + δ_{2} d (μ, ν) \forall μ, ν \in W w i t h [μ, ν] \in ℜ .$

The proof is followed by the symmetrycity of the metric d.

3. Main Result

In this fragment, we will introduce the fixed point theorem under rational contraction in the relation theoretic sense.

Theorem 1.

Consider

(W, d)

as a metric space together with a binary relation ℜ and a self-mapping T on it. Assume that the following conditions hold:

(i): $(W, d)$ is ℜ-complete;
(ii): $W (T; ℜ)$ is non-empty;
(iii): ℜ is T-closed;
(iv): Either T is continuous or ℜ is d-self closed;
(v): There exist $δ_{1}, δ_{2} \in [0, 1)$ with $δ_{1} + δ_{2} < 1$ such that

$d (T μ, T ν) \leq δ_{1} \frac{d (ν, T ν) [1 + d (μ, T μ)]}{1 + d (μ, ν)} + δ_{2} d (μ, ν) f o r μ, ν \in W w i t h (μ, ν) \in ℜ .$

Then T has a fixed point.

Proof.

From the condition

(i i)

, we always have a

μ_{0} \in W

such that

(μ_{0}, T μ_{0}) \in ℜ

, then define a Picard sequence of iterates

μ_{n + 1} = T μ_{n}

. If

T μ_{0} = μ_{0}

then nothing to prove.

T μ_{0} \neq μ_{0}

then by condition ℜ is T-closed we obtain

(T μ_{0}, T^{2} μ_{0}), ((T^{2} μ_{0}, T^{3} μ_{0})), ((T^{3} μ_{0}, T^{4} μ_{0})) \dots ((T^{n} μ_{0}, T^{n + 1} μ_{0})) \dots \in ℜ .

So we have

(μ_{n}, μ_{n + 1}) \in ℜ \forall n \in N_{0}

i . e ., {μ_{n}}

is a ℜ preserving sequence.

μ_{n_{0} + 1} = T μ_{n_{0}} = μ_{n_{0}}

then

μ_{n_{0}}

is a fixed point of T; then the proof is complete.

μ_{n + 1} \neq μ_{n}

for

n \geq 1

then by condition

(v)

for

(μ_{n}, μ_{n + 1}) \in ℜ

, we have

\begin{matrix} d (μ_{n}, μ_{n + 1}) = d (T μ_{n - 1}, T μ_{n}) & \leq & δ_{1} \frac{d (μ_{n}, T μ_{n}) [1 + d (μ_{n - 1}, T μ_{n - 1})]}{[1 + d (μ_{n - 1}, μ_{n})]} + δ_{2} d (μ_{n - 1}, μ_{n}) \\ = & δ_{1} \frac{d (μ_{n}, μ_{n + 1}) [1 + d (μ_{n - 1}, μ_{n})]}{[1 + d (μ_{n - 1}, μ_{n})]} + δ_{2} d (μ_{n - 1}, μ_{n}) \\ = & δ_{1} d (μ_{n}, μ_{n + 1}) + δ_{2} d (μ_{n - 1}, μ_{n}) \\ (1 - δ_{1}) d (μ_{n}, μ_{n + 1}) & \leq & δ_{2} d (μ_{n - 1}, μ_{n}) \\ d (μ_{n}, μ_{n + 1}) & \leq & \frac{δ_{2}}{1 - δ_{1}} d (μ_{n - 1}, μ_{n}) . \end{matrix}

Then, by an induction process, we will obtain

d (μ_{n}, μ_{n + 1}) \leq {(\frac{δ_{2}}{1 - δ_{1}})}^{n} d (μ_{0}, μ_{1}) for any n \in N_{0} .

(1)

Denote

γ = \frac{δ_{2}}{1 - δ_{1}} < 1

, then Equation (1) can be rewritten as

d (μ_{n}, μ_{n + 1}) \leq {(γ)}^{n} d (μ_{0}, μ_{1}) for any n \in N_{0} .

Next, for

{μ_{n}}

to be a Cauchy sequence, let

m > n

then

\begin{matrix} d (μ_{n}, μ_{m}) & \leq & d (μ_{n}, μ_{n + 1}) + d (μ_{n + 1}, μ_{n + 2}) + \dots \dots + d (μ_{m - 1}, μ_{m}) \\ \leq & γ^{n} d (μ_{0}, μ_{1}) + γ^{n + 1} d (μ_{0}, μ_{1}) + \dots \dots + γ^{m - 1} d (μ_{0}, μ_{1}) \\ \leq & (γ^{n} + γ^{n + 1} + \dots \dots + γ^{m - 1}) d (μ_{0}, μ_{1}) \\ \leq & γ^{n} (1 + γ + γ^{2} + \dots \dots + γ^{m - n - 1}) d (μ_{0}, μ_{1}) \\ \leq & γ^{n} (\frac{1 - γ^{m - n}}{1 - γ}) d (μ_{0}, μ_{1}) . \end{matrix}

For

m, n \to \infty

and as

γ < 1

, then we obtain

lim_{n, m \to \infty} d (μ_{n}, μ_{m}) = 0

. So, we have proved that

{μ_{n}}

is a Cauchy sequence.

Since the space

(W, d)

is a ℜ-complete metric space, then there always exists

μ \in W

such that

μ_{n} \to μ

Then by continuity of T, we have

T μ = T (lim_{n \to \infty} μ_{n}) = lim_{n \to \infty} T μ_{n} = lim_{n \to \infty} μ_{n + 1} = μ .

So,

μ

is a fixed point of T.

If otherwise, ℜ is d-self closed then for the ℜ-preserving sequence

{μ_{n}} \overset{d}{\to} μ

there exists a subsequence

{μ_{n_{k}}}

{μ_{n}}

with

[μ_{n_{k}}, μ] \in ℜ

\forall k \in N_{0} .

Then, by condition

(v), [μ_{n_{k}}, μ] \in ℜ

and Proposition 2 and

μ_{n_{k}} \to μ

\begin{matrix} d (μ_{n_{k} + 1}, T μ) = d (T μ_{n_{k}}, T μ) & \leq & δ_{1} \frac{d (μ, T μ) [1 + d (μ_{n_{k}}, T μ_{n_{k}})]}{1 + d (μ_{n_{k}}, μ)} + δ_{2} d (μ_{n_{k}}, μ) \\ \leq & δ_{1} \frac{d (μ, T μ) [1 + d (μ_{n_{k}}, μ_{n_{k} + 1})]}{1 + d (μ_{n_{k}}, μ)} + δ_{2} d (μ_{n_{k}}, μ) . \end{matrix}

Taking

n \to \infty

, we obtain

\begin{matrix} d (μ, T μ) & \leq & δ_{1} d (μ, T μ) \\ (1 - δ_{1}) d (μ, T μ) & \leq & 0 . \end{matrix}

Since

0 \leq δ_{1} < 1

then, the only possibility is

d (μ, T μ) = 0

. Hence,

μ = T μ

. Then,

μ

is a fixed point of T. □

Theorem 2.

If in addition to Theorem 1 we have the condition:

(v i)

T(W) is

ℜ^{s}

-directed.

Then T has a unique fixed point.

Proof.

Let us suppose that

μ, ν

are two fixed points, i.e.,

T μ = μ

and

T ν = ν

then we have the two cases,

Case I: if

(μ, ν) \in ℜ

then

\begin{matrix} d (μ, ν) = d (T μ, T ν) & \leq & δ_{1} \frac{d (ν, T ν) [1 + d (μ, T μ)]}{[1 + d (μ, ν)]} + δ_{2} d (μ, ν) \\ d (μ, ν) & \leq & δ_{2} d (μ, ν) \end{matrix}

then

(1 - δ_{2}) d (μ, ν) \leq 0

implies that

d (μ, ν) = 0

δ_{2} > 0 .

Case II: if

(μ, ν) \notin ℜ

then by

T (W)

ℜ^{s}

-directed then there exists

κ \in W

such that

(μ, κ) \in ℜ

and

(κ, ν) \in ℜ

. Since ℜ is T-closed

T^{n} κ

will be related to

T^{n} μ

i . e ., (T^{n} κ, T^{n} μ = μ) \in ℜ

for any

n \in N_{0}

. Then, by contractive condition

(v)

of Theorem 1, for any

n \in N_{0}

, we have

\begin{matrix} d (T^{n} κ, μ) = d (T^{n} κ, T^{n} μ) & \leq & δ_{1} \frac{d (T^{n - 1} μ, T^{n} μ) [1 + d (T^{n - 1} κ, T^{n} κ)]}{1 + d (T^{n - 1} κ, T^{n - 1} μ)} + δ_{2} d (T^{n - 1} κ, T^{n - 1} μ) \\ = & δ_{1} \frac{d (μ, μ) [1 + d (T^{n - 1} κ, T^{n} κ)]}{1 + d (T^{n - 1} κ, μ)} + δ_{2} d (T^{n - 1} κ, μ) \\ = & δ_{2} d (T^{n - 1} κ, μ) . \end{matrix}

Then by mathematical induction, we obtain

d (T^{n} κ, μ) \leq {(δ_{2})}^{n} d (κ, μ) .

Since

δ_{2} < 1

then

lim_{n \to \infty} d (T^{n} κ, μ) = 0

, which provides us

lim_{n \to \infty} T^{n} κ = μ

In a similar fashion we also obtain

lim_{n \to \infty} T^{n} κ = ν

Then, by the unity of limits we obtain

μ = ν

So our supposition that

μ

and

ν

are two different fixed points is wrong. Hence, the mapping T has a unique fixed point. □

Corollary 1.

If we substitute

δ_{2} = 0

into Theorems 1 and 2, we have the following fixed point theorem.

Consider

(W, d)

as a metric space together with a binary relation ℜ and a self mapping T on it. Assume that the following conditions holds:

(i’): $W (T; ℜ)$ is non-empty;
(ii’): ℜ is T-closed;
(iii’): $(W, d)$ is ℜ-complete;
(iv’): Either T is ℜ-continuous or ℜ is d-self closed;
(v’): There exist $δ_{1} \in [0, 1)$ such that

$d (T μ, T ν) \leq δ_{1} \frac{d (ν, T ν) [1 + d (μ, T μ)]}{[1 + d (μ, ν)]} f o r (μ, ν) \in ℜ and μ, ν \in W,$
(vi’): $T (W)$ is $ℜ^{s}$ -directed.

Then, T has a unique fixed point.

Remark 1.

If we put

δ_{1} = 0

into Theorems 1 and 2, then under the setting of

ℜ = ⪯

as the partial order, we obtain Theorems

(2.1), (2.2) a n d (2.3)

of [26].

Remark 2.

If we substitute

δ_{1} = 0

and

0 < δ_{2} < \frac{1}{3}

into Theorems 1 and 2, then the condition

(v)

reduces to the Kannan contraction [27]

(δ_{3} \in (0, \frac{1}{2}))

d (T μ, T ν) \leq δ_{3} [d (μ, T μ) + d (ν, T ν)] f o r μ, ν \in W w i t h (μ, ν) \in ℜ .

Proof.

δ_{1} = 0

and

0 \leq δ_{2} < \frac{1}{3}

, then the condition

(v)

of Theorem 1 reduces to the form

\begin{matrix} d (T μ, T ν) \leq δ_{2} d (μ, ν) & \leq & δ_{2} [d (μ, T μ) + d (T μ, T ν) + d (T ν, ν)] \\ \leq & δ_{2} [d (μ, T μ) + d (ν, T ν)] + δ_{2} d (T μ, T ν) \\ (1 - δ_{2}) d (T μ, T ν) & \leq & [d (μ, T μ) + d (ν, T ν)] \\ d (T μ, T ν) & \leq & \frac{δ_{2}}{1 - δ_{2}} [d (μ, T μ) + d (ν, T ν)] \\ d (T μ, T ν) & \leq & δ_{3} [d (μ, T μ) + d (ν, T ν)] f o r δ_{3} = \frac{δ_{2}}{1 - δ_{2}} < \frac{1}{2} . \end{matrix}

□

Remark 3.

If diameter

(W) \leq 1

and

2 δ_{1} + δ_{2} < 1

, then conditions

(v)

of Theorem 1 reduces to the Riech [28] type conditions:

d (T μ, T ν) \leq δ_{1} d (μ, T μ) + δ_{1} d (ν, T ν) + δ_{2} d (μ, ν) .

Proof.

For any

μ, ν \in W

with

(μ, ν) \in ℜ

\begin{matrix} d (T μ, T ν) & \leq & δ_{1} \frac{d (ν, T ν) [1 + d (μ, T μ)]}{[1 + d (μ, ν)]} + δ_{2} d (μ, ν) \\ \leq & δ_{1} d (ν, T ν) + δ_{1} d (ν, T ν) d (μ, T μ) + δ_{2} d (μ, ν) . \end{matrix}

As the diameter of W is less than equal to one then

d (ν, T ν) < 1

. Then, we have

d (T μ, T ν) \leq δ_{1} d (ν, T ν) + δ_{1} d (μ, T μ) + δ_{2} d (μ, ν) .

□

Finally, we produce an illustrative example to substantiate the utility of our result, which does not satisfy the hypotheses of the existing results [1,11,12,13,18], but satisfies the hypotheses of our result, and hence has a fixed point.

Example 1.

Consider the metric space

W = (- 1, 1]

with the usual metric d and a binary relation

ℜ = {(μ, ν) \in W^{2} : ν > μ \geq 0}

together with a mapping

T : W \to W

defined by

T (x) = \{\begin{matrix} \frac{1}{2}, i f - 1 < μ < 0, \\ \frac{μ}{4}, i f 0 \leq μ \leq 1 . \end{matrix}

It is clear that ℜ is T-closed and T is not a continuous function.

Now, for

(μ, ν) \in ℜ

\begin{matrix} d (T μ, T ν) & = & | T μ - T ν | \\ = & | \frac{μ}{4} - \frac{ν}{4} | \\ = & \frac{1}{4} | μ - ν | \\ \leq & \frac{1}{2} \frac{| μ - ν |}{[1 + | μ - ν |]} \\ < & \frac{1}{2} \frac{1}{[1 + | μ - ν |]} \times \frac{8}{5} |\frac{3 ν}{4}| [1 + |\frac{3 μ}{4}|] \\ < & \frac{1}{2} \times \frac{8}{5} \frac{| \frac{3 ν}{4} | [1 + | \frac{3 μ}{4} |]}{1 + | μ - ν |} \\ < & \frac{4}{5} \frac{| ν - \frac{ν}{4} | [1 + | μ - \frac{μ}{4} |]}{1 + | μ - ν |} + \frac{1}{10} | μ - ν | \\ = & δ_{1} \frac{d (ν, T ν) [1 + d (μ, T μ)]}{[1 + d (μ, ν)]} + δ_{2} d (μ, ν) f o r δ_{1} = \frac{4}{5} < 1 a n d δ_{2} = \frac{1}{10} < 1 . \end{matrix}

So,

δ_{1} + δ_{2} = \frac{4}{5} + \frac{1}{10} = \frac{9}{10} < 1 .

Then T has fixed point

μ = 0

Notice that condition

(v)

of Theorem 1 does not hold for the whole space (for example, take

μ = 1 and ν = 0

). Therefore, this example cannot be solved by the existing results, which establishes the importance of our result.

4. Application to Non-Linear Integral Equations

Consider

W = C [a, b]

the class of all continuous functions from

[a, b]

[a, b]

with metric

d (μ, ν) = sup_{τ \in [a, b]} \{| μ (τ) - ν (τ) |\}

then

(W, d)

is a complete metric space.

Theorem 3.

Consider the non-linear integral equation

μ (τ) = f (τ) + \int_{a}^{b} Q (τ, r, μ (r)) d r

(2)

where

τ \in [a, b]

f : [a, b] \to R

and

Q : {[a, b]}^{2} \times R \to R

. Suppose the following conditions holds:

(i): f is continuous and $Q (τ, r, μ (r))$ is integrable w.r.t r on $[a, b]$ ;
(ii): $T μ \in C [a, b]$ for all $μ \in C [a, b]$ where

$T μ (τ) = f (τ) + \int_{a}^{b} Q (τ, r, μ (r)) d r,$
(iii): For all $r, τ \in [a, b]$ and $μ, ν \in C [a, b]$ with $(μ, ν) \in ℜ$

$Q (τ, r, ν (r)) - Q (τ, r, μ (r)) < η (t, r) | μ (r) - ν (r) |$

where $η : {[a, b]}^{2} \to R^{+}$ is a continuous function satisfying

$sup_{τ \in [a, b]} (\int_{a}^{b} η (τ, r) d r) < 1,$
(iv): There exist $μ_{0} (τ) \leq f (τ) + \int_{a}^{b} Q (τ, r, μ_{0} (r)) d r$ for all $τ \in [a, b] .$

Then, the non-linear integral Equation (2) has a unique solution

μ \in C [a, b]

Proof.

For the proof, let us define a binary relation on W

ℜ = \{(μ, ν) \in W^{2} if μ (τ) < ν (τ) for all τ \in [a, b]\} .

By assumption

(i v)

we have

μ_{0} \in C [a, b]

such that

\begin{matrix} μ_{0} (τ) & \leq & f (τ) + \int_{a}^{b} Q (τ, r, μ_{0} (r)) d r \\ \leq & T μ_{0} (τ) \end{matrix}

this implies that

μ_{0} ℜ T μ_{0}

, then

W (T; ℜ)

is non-empty.

Now, to prove that the relation ℜ is T-closed, choose

μ, ν \in C [a, b]

such that

μ ℜ ν

, then

\begin{matrix} μ (r) & < & ν (r) \\ | μ (r) - ν (r) | & > & 0 \\ η (τ, r) | μ (r) - ν (r) | & > & 0 \end{matrix}

then by condition (iii) of Theorem 3, we have

\begin{matrix} Q (τ, r, μ (r)) & < & Q (τ, r, ν (r)) \\ \int_{a}^{b} Q (τ, r, μ (r)) d r & < & \int_{a}^{b} Q (τ, r, ν (r)) d r \\ f (τ) + \int_{a}^{b} Q (τ, r, μ (r)) d r & < & f (τ) + \int_{a}^{b} Q (τ, r, ν (r)) d r \\ T μ (r) & < & T ν (r) \end{matrix}

then

(T μ, T ν) \in ℜ

. Hence, ℜ is T-closed.

Now for

(μ, ν) \in ℜ

means that

μ (τ) < ν (τ)

\begin{matrix} d (T μ, T ν) = | T μ - T ν | & = & | f (τ) + \int_{a}^{b} Q (τ, r, μ (r)) d r - f (τ) - \int_{a}^{b} Q (τ, r, ν (r)) d r | \\ = & | \int_{a}^{b} Q (τ, r, μ (r)) d r - \int_{a}^{b} Q (τ, r, ν (r)) d r | \\ < & \int_{a}^{b} η (τ, r) | μ (r) - ν (r) | d r from condition (iii) \\ < & sup_{τ \in [a, b]} (\int_{a}^{b} η (τ, r) d r) | μ (r) - ν (r) | \\ < & δ_{2} | μ (r) - ν (r) | f o r δ_{2} = sup_{τ \in [a, b]} (\int_{a}^{b} η (τ, r) d r) < 1 \\ < & \frac{δ_{1} | ν (r) - T ν (r) | [1 + | μ (r) - T μ (r) |]}{[1 + | μ (r) - ν (r) |]} + δ_{2} | μ (r) - ν (r) | f o r δ_{1} = 1 - δ_{2} \\ < & \frac{δ_{1} d (ν, T ν) [1 + d (μ, T μ)]}{[1 + d (μ, ν)]} + δ_{2} d (μ, ν) . \end{matrix}

So, the contractive condition also satisfied.

For ℜ to be d-self closed, consider

{μ_{n}}

a ℜ-preserving Cauchy sequence converging to

μ \in C [a, b]

. As

{μ_{n}}

is ℜ preserving, we have

μ_{0} (τ) \leq μ_{1} (τ) \leq μ_{2} (τ) \leq μ_{3} (τ) \leq \dots \dots \dots \dots \leq μ_{n} (τ) \leq μ_{n + 1} (τ) \leq \dots \dots \leq μ (τ) τ \in [a, b]

then we have

μ_{n} ℜ μ

\forall n \in N

. Therefore, ℜ is d-self closed.

Now, let us assume that

κ (τ) = m a x {μ (τ), ν (τ)}

then

κ (τ) \in C [a, b]

\Rightarrow μ (τ) \leq κ (τ) a n d ν (τ) \leq κ (τ)

. So,

ℜ^{s}

- directed.

Hence, we observe that the integral Equation (2) satisfies all the conditions of Theorem 1 and Theorem 2 under the given assumption of Theorem 3, which amounts to saying that the integral Equation (2) has a unique solution. □

Now to show the guarantees, the existence of the function

Q (τ, r, μ (r))

satisfies all the assumptions of the above application.

Example 2.

Consider

W = C [0, \frac{π}{4}]

together with the metric

d (μ, ν) = sup_{τ \in [0, \frac{π}{4}]} \{| μ (τ) - ν (τ) |\} .

Define a binary relation

ℜ = \{μ ℜ ν if μ (τ) < ν (τ) f o r a l l τ \in [0, \frac{π}{4}]\} .

Consider the non-linear integral equation as

μ (τ) = 3 τ - 4 \int_{0}^{\frac{π}{4}} (τ r) s i n (μ (r)) d r f o r μ \in C [0, \frac{π}{4}],

and

T μ (τ) = 3 τ - 4 \int_{0}^{\frac{π}{4}} (τ r) s i n (μ (r)) d r f o r μ \in C [0, \frac{π}{4}] .

Proof.

Since

f (τ) = 3 τ

, which is continuous on

[0, \frac{π}{4}]

and

Q (τ, r, μ (r)) = 4 (τ r) s i n (μ (r))

is integrable w.r.t r on

[0, \frac{π}{4}]

Now for every

τ \in [0, \frac{π}{4}]

and the sequence

{τ_{n}} \subset [0, \frac{π}{4}]

with

lim_{n \to \infty} τ_{n} = τ

Then, for any

μ \in C [0, \frac{π}{4}]

\begin{matrix} | T μ (τ_{n}) - T μ (τ) | & = & | f (τ_{n}) - f (τ) - 4 \int_{0}^{\frac{π}{4}} r (τ_{n} - τ) s i n (μ (r)) d r | \\ \leq & |f (τ_{n}) - f (τ)| + 4 \int_{0}^{\frac{π}{4}} r |τ_{n} - τ| | s i n (μ (r)) | d r \\ \leq & |3 τ_{n} - 3 τ| + 4 |τ_{n} - τ| \int_{0}^{\frac{π}{4}} r d r \\ \leq & 3 |τ_{n} - τ| + \frac{π^{2}}{8} |τ_{n} - τ| . \end{matrix}

Taking as a limit

n \to \infty

|T μ (τ_{n}) - T μ (τ)| = 0 .

which implies that

T μ (τ_{n}) = T μ (τ)

. Hence,

T μ \in C [0, \frac{π}{4}]

for all

μ \in C [0, \frac{π}{4}]

Now for all

r, τ \in [0, \frac{π}{4}]

and

μ, ν \in C [0, \frac{π}{4}]

with

μ ℜ ν

we have

\begin{matrix} Q (τ, r, ν (r)) - Q (τ, r, μ (r)) & \leq & 4 \times τ r |s i n (ν (r)) - s i n (ν (r))| \\ \leq & η (τ, r) |μ (r) - ν (r)| for η (τ, r) = 4 τ r . \end{matrix}

Consequently,

η (τ, r) = 4 τ r

is a continuous function from

[0, \frac{π}{4}] \times [0, \frac{π}{4}] \to [0, \infty)

and

\begin{matrix} sup_{τ \in [0, \frac{π}{4}]} (\int_{0}^{\frac{π}{4}} η (τ, r) d r) & = & sup_{τ \in [0, \frac{π}{4}]} \int_{0}^{\frac{π}{4}} 4 τ r d r \\ = & sup_{τ \in [0, \frac{π}{4}]} 4 τ {[\frac{r^{2}}{2}]}_{0}^{\frac{π}{4}} \\ = & sup_{τ \in [0, \frac{π}{4}]} 4 τ (\frac{π^{2}}{32}) \\ = & \frac{π^{2}}{8} sup_{τ \in [0, \frac{π}{4}]} τ \\ = & \frac{π^{2}}{8} \times \frac{π}{4} \\ = & \frac{π^{3}}{32} < 1 . \end{matrix}

Now choosing

μ_{0} (τ) = 2 τ

for

τ \in [0, \frac{π}{4}]

, we have

T μ_{0} (τ) = μ_{0} (τ) = 2 τ

Hence

μ_{0} (τ) = 2 τ

is a fixed point of T. □

5. Conclusions

In this article, we have established the relation theoretical fixed point results for the rational type contraction. One may observe that, for the uniqueness of the fixed point, the

ℜ^{s}

-directed condition can be replaced by other conditions. Here, we also included some contractions that can be obtained on restriction to the rational contraction. Our results deduce some well known fixed point results if the binary relation is universal. The example we provided is unique in that it will satisfy all the relational elements but fails for many elements outside of the relation. Moreover, we provide an abstract version of an application to a non-linear integral equation. Lastly, we include an example that guarantees the existence of such a non-linear integral equation.

Author Contributions

All the authors contributed equally and significantly in writing this article. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

All the authors are grateful to the anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper.

Conflicts of Interest

The authors declare no conflict of interest.

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Hossain, A.; Khan, F.A.; Khan, Q.H. A Relation-Theoretic Metrical Fixed Point Theorem for Rational Type Contraction Mapping with an Application. Axioms 2021, 10, 316. https://doi.org/10.3390/axioms10040316

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Hossain A, Khan FA, Khan QH. A Relation-Theoretic Metrical Fixed Point Theorem for Rational Type Contraction Mapping with an Application. Axioms. 2021; 10(4):316. https://doi.org/10.3390/axioms10040316

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Hossain, Asik, Faizan Ahmad Khan, and Qamrul Haq Khan. 2021. "A Relation-Theoretic Metrical Fixed Point Theorem for Rational Type Contraction Mapping with an Application" Axioms 10, no. 4: 316. https://doi.org/10.3390/axioms10040316

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Hossain, A., Khan, F. A., & Khan, Q. H. (2021). A Relation-Theoretic Metrical Fixed Point Theorem for Rational Type Contraction Mapping with an Application. Axioms, 10(4), 316. https://doi.org/10.3390/axioms10040316

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A Relation-Theoretic Metrical Fixed Point Theorem for Rational Type Contraction Mapping with an Application

Abstract

1. Introduction

2. Preliminaries

3. Main Result

4. Application to Non-Linear Integral Equations

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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