Next Article in Journal
A Three-Party Decision Evolution Game Analysis of Coal Companies and Miners under China’s Government Safety Special Rectification Action
Previous Article in Journal
Quasihomeomorphisms and Some Topological Properties
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Almost Nonlinear Contractions under Locally Finitely Transitive Relations with Applications to Integral Equations

by
Faizan Ahmad Khan
1,*,
Musaad S. Aldhabani
1,*,
Ahmed Alamer
1,*,
Esmail Alshaban
1,
Fahad M. Alamrani
1 and
Hamid I. A. Mohammed
1,2
1
Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
2
Department of Mathematics, University of Bahri, Khartoum 11111, Sudan
*
Authors to whom correspondence should be addressed.
Mathematics 2023, 11(23), 4749; https://doi.org/10.3390/math11234749
Submission received: 25 October 2023 / Revised: 19 November 2023 / Accepted: 22 November 2023 / Published: 24 November 2023
(This article belongs to the Section Computational and Applied Mathematics)

Abstract

:
This article consists of some new fixed point theorems verifying relation-theoretic strict almost nonlinear contractions that are preserved by a locally finitely transitive relation. Our results improve certain well-known results in the literature. To confirm the reliability of our results, some examples are delivered. We carry out our findings to solve a nonlinear Fredholm integral equation.

1. Introduction

In fact, the Banach contraction principle (BCP) is one of the most celebrated fixed point theorems ever proved in Mathematics and still remains an inspiration to the workers of metric fixed point theory. As an instance of recent developments in BCP, we merely refer the work of Younis et al. [1]. Alam and Imdad [2] explored a new variant of BCP in the framework of relation metric space. Because of creativity, the results of Alam and Imdad [2] have been expanded and developed by numerous researchers, e.g., [3,4,5]. The contraction that occurs in these outcomes needs to merely be satisfied for comparative elements. It turns out that the class of relational contractions continues to be larger than the class of usual contractions.
Berinde [6] extended the BCP by initiating the concept of almost contraction, which is further generalized by various authors. Indeed, the almost contraction map does not possess a unique fixed point but a sequence of Picard’s iterations of such a map converges to a fixed point. Babu et al. [7] investigated the corresponding uniqueness theorem by initiating a restricted class of almost contraction as follows:
Definition 1
([7]). A map ϕ : Υ Υ , where ( Υ , ς ) remains a metric space, is named as a strict almost contraction if there exists α ( 0 , 1 ) and λ [ 0 , ) such that
ς ( ϕ y , ϕ z ) α · ς ( y , z ) + λ · min { ς ( y , ϕ y ) , ς ( z , ϕ z ) , ς ( y , ϕ z ) , ς ( z , ϕ y ) } , f o r a l l y , z Υ .
Theorem 1
([7]). Every strict almost contraction map on a complete metric space provides a unique fixed point.
Through the use of a class of local finitely transitive relations, Alam et al. [5] presented a couple of fixed point results under a relational nonlinear contraction. Their choice of control functions was the following family:
Σ = { ϝ : [ 0 , ) [ 0 , ) : t ( 0 , ) t > ϝ ( t ) and t > lim sup r t ϝ ( r ) } .
The manuscript’s purpose is to encompass the concepts involved in Alam et al. [5] and Babu et al. [7] by proving the fixed point results under relational strict almost nonlinear contraction using a locally finitely ϕ -transitive relation. Our results are illustrated via certain examples. Our results allow us to come across the unique solution of a specialized nonlinear Fredholm integral equation.

2. Preliminaries

As usual, N , N 0 and R will symbolize the set of natural numbers, whole numbers and real numbers. A subset of Υ 2 is termed as a relation on the set Υ . In the sequel, let Υ be ambient set, ϕ : Υ Υ a map, £ a relation on Υ and ς a metric on Υ . We say that:
Definition 2
([8]). The relation £ 1 : = { ( y , z ) Υ 2 : ( z , y ) £ } is inverse of £.
Definition 3
([8]). The symmetric relation £ s : = £ £ 1 is symmetric closure of £.
Definition 4
([2]). Elements y , z Υ are £-comparative if ( y , z ) £ or ( z , y ) £ . Such a pair is usually denoted by [ y , z ] £ . Obviously, ( y , z ) £ s [ y , z ] £ .
Definition 5
([2]). £ is ϕ-closed if for every y , z Υ with ( y , z ) £ , we have
( ϕ y , ϕ z ) £ .
Definition 6
([2]).  { y η } Υ is a £-preserving sequence if ( y η , y η + 1 ) £ for all η N .
Definition 7
([3]).  ( Υ , ς ) is £-complete metric space if all the £-preserving Cauchy sequences in  Υ  converge.
Definition 8
([3]). ϕ is £-continuous at y Υ if for each £-preserving sequence { y η } Υ with y η ς y ,
ϕ ( y η ) ς ϕ ( y ) .
Definition 9
([3]). ϕ is £-continuous if it remains £-continuous function at every point.
Definition 10
([2]). £ is ς-self-closed if every £-preserving convergent sequence { y η } Υ verifying y η ς y Υ admits a subsequence { y η k } with [ y η k , y ] £ .
Definition 11
([9]). Given Ƶ Υ , the set £ | Ƶ : = £ Ƶ 2 being a relation on Ƶ is restriction of £ on Ƶ .
Definition 12
([4]). £ is locally ϕ-transitive if for each £-preserving sequence { z η } ϕ ( Υ ) with range Ƶ = { z η : η N } , £ | Ƶ is transitive.
Definition 13
([10]). Given n N { 1 } , £ is n-transitive if for every y 0 , y 1 , , y n Υ ,
( y i 1 , y i ) £ for each i ( 1 i n ) ( y 0 , y n ) £ .
Thus far, 2-transitivity coincides with transitivity.
Definition 14
([11]). £ is finitely transitive if there exists n N { 1 } such that £ is n-transitive.
Definition 15
([5]). £ is locally finitely ϕ-transitive if for each £-preserving sequence { z η } ϕ ( Υ ) with range Ƶ = { z η : η N } , £ | Ƶ is finitely transitive.
Remark 1.
The following implications are straightforward:
finitely transitivity⟹ locally finitely ϕ-transitivity;
locally ϕ-transitivity⟹ locally finitely ϕ-transitivity.
Definition 16
([12]).  Ƶ Υ is termed as £-directed if for every y , z Ƶ , there exists w Υ which satisfies ( y , w ) £ and ( z , w ) £ .
Now, we indicate the following result of Alam et al. [5].
Theorem 2
([5]). Consider ( Υ , ς ) serves as a metric space, ϕ : Υ Υ as a map, and £ as a relation on Υ . Also,
(i)
There exists y 0 Υ satisfying ( y 0 , ϕ y 0 ) £ ;
(ii)
£ is locally finitely ϕ-transitive and ϕ-closed;
(iii)
( Υ , ς ) remains £-complete;
(iv)
Y remains £-continuous, or £ remains ς-self-closed;
(v)
There exists ϝ Σ verifying.
ς ( ϕ y , ϕ z ) ϝ ( ς ( y , z ) ) , f o r a l l y , z Υ w i t h ( y , z ) £ .
Then, ϕ admits a fixed point. Moreover, if ϕ ( Υ ) is £ s -directed, then ϕ admits a unique fixed point.
Proposition 1
([4]). If £ is ϕ-closed, then £ is ϕ η -closed, for each η N 0 .
Lemma 1
([11]). Assume that  Υ  is a set endowed with a relation £. If { y η } Υ is £-preserving sequence and £ is to be continued an n-transitive on Ƶ = { y η : η N 0 } , then
( y η , y η + 1 + υ ( n 1 ) ) £ , f o r a l l η , υ N 0 .
Lemma 2
([10]). In a metric space ( Υ , ς ) , if a sequence { y η } is not Cauchy, then there exist ϵ > 0 and subsequences { y η k } and { y μ k } of { y η } enjoying the properties:
(i)
k μ k < η k f o r a l l k N ,
(ii)
ς ( y μ k , y η k ) ϵ f o r a l l k N ,
(iii)
ς ( y μ k , y ν k ) < ϵ for all ν k { μ k + 1 , μ k + 2 , , η k 2 , η k 1 } .
Moreover, if lim η ς ( y η , y η + 1 ) = 0 , then
lim k ς ( y μ k , y η k + υ ) = ϵ , f o r a l l υ N 0 .
Proposition 2.
Given ϝ Σ and λ 0 , (A) and (B) are equivalent:
(A)
ς ( ϕ y , ϕ z ) ϝ ( ς ( y , z ) ) + λ · min { ς ( y , ϕ y ) , ς ( z , ϕ z ) , ς ( y , ϕ z ) , ς ( z , ϕ y ) } , f o r a l l y , z Υ w i t h ( y , z ) £ .
(B)
ς ( ϕ y , ϕ z ) ϝ ( ς ( y , z ) ) + λ · min { ς ( y , ϕ y ) , ς ( z , ϕ z ) , ς ( y , ϕ z ) , ς ( z , ϕ y ) } , f o r a l l y , z Υ w i t h [ y , z ] £ .
Proof. 
The conclusion (B)⇒(A) holds trivially. Conversely, let (A) holds. Assume that y , z Υ with [ y , z ] £ . Then, in case ( y , z ) £ , (A) implies (B). Otherwise, in case ( z , y ) £ , by symmetric character of metric ς and (A), we obtain
ς ( ϕ y , ϕ z ) = ς ( ϕ z , ϕ y ) ϝ ( ς ( z , y ) ) + λ · min { ς ( z , ϕ z ) , ς ( y , ϕ y ) , ς ( z , ϕ y ) , ς ( y , ϕ z ) } = ϝ ( ς ( y , z ) ) + λ · min { ς ( y , ϕ y ) , ς ( z , ϕ z ) , ς ( y , ϕ z ) , ς ( z , ϕ y ) } .
It follows that (A)⇒(B). □

3. Main Results

This section includes the results on existence and uniqueness of fixed point under relational strict almost nonlinear contraction.
Theorem 3.
Consider ( Υ , ς ) serves as a metric space, ϕ : Υ Υ as a map, and £ as a relation on Υ . Also,
(i)
There exists y 0 Υ satisfying ( y 0 , ϕ y 0 ) £ ;
(ii)
£ is locally finitely ϕ-transitive and ϕ-closed;
(iii)
( Υ , ς ) remains £-complete;
(iv)
Y remains £-continuous, or £ remains ς-self-closed;
(v)
There exist ϝ Σ and λ 0 verifying.
ς ( ϕ y , ϕ z ) ϝ ( ς ( y , z ) ) + λ · min { ς ( y , ϕ y ) , ς ( z , ϕ z ) , ς ( y , ϕ z ) , ς ( z , ϕ y ) } , f o r a l l y , z Υ w i t h ( y , z ) £ .
Then, ϕ admits a fixed point.
Proof. 
Initiating with the point y 0 Υ , we construct a sequence { y η } Υ satisfying
y η : = ϕ n ( y 0 ) = ϕ ( y η 1 ) , for all η N .
Utilizing (i), (ii) and Proposition 1, we get
( ϕ n y 0 , ϕ n + 1 y 0 ) £ ,
which using (1) becomes
( y η , y η + 1 ) £ , for all η N 0 .
It yields that { y η } is a £-preserving sequence.
Define the following sequence of positive real numbers:
ς η : = ς ( y η , y η + 1 ) .
Whenever ς η 0 = 0 for some η 0 N 0 , we have ϕ ( y η 0 ) = y η 0 so that y η 0 is a fixed point of ϕ and hence the proof is finished.
Otherwise, we have ς η > 0 for all η N 0 . In this case, by (1), (2) and assumption (v), we obtain
ς η = ς ( y η , y η + 1 ) = ς ( ϕ y η 1 , ϕ y η ) ϝ ( ς ( y η 1 , y η ) ) + λ · min { ς ( y η 1 , y η ) , ς ( y η , y η + 1 ) , ς ( y η 1 , y η + 1 ) , 0 }
so that
ς η ϝ ( ς η 1 ) , for all η N 0 .
Using the property of ϝ in (3), we have
ς η ϝ ( ς η 1 ) < ς η 1 , for all η N .
It yields that { ς η } is a decreasing sequence. Now, { ς η } being bounded below ensures the existence of ς ¯ 0 such that
lim η ς η = ς ¯ .
Our claim is that ς ¯ = 0 . Otherwise, in case ς ¯ > 0 using the upper limit in (3) and owing to (4) and the definition of Σ , we find
ς ¯ = lim sup η ς η lim sup η ϝ ( ς η 1 ) = lim sup ς η l + ϝ ( ς η 1 ) < ς ¯ ,
which is not possible. Hence,
lim η ς η = 0 .
On contrary, assume that { y η } is not Cauchy. Using Lemma 2, there exist ϵ > 0 and subsequences { y η k } and { y μ k } of { y η } satisfying
k μ k < n k , ς ( y μ k , y η k ) ϵ > ς ( y μ k , y ν k ) ,
for all k N , ν k { μ k + 1 , μ k + 2 , , η k 2 , η k 1 } .
By (5) and Lemma 2, we get
lim k ς ( y μ k , y η k + υ ) = ϵ , for all υ N 0 .
By (1), we have Ƶ : = { y η : η N 0 } ϕ ( Υ ) . Thus, using locally finitely ϕ -transitivity of £, there exists n { 2 , 3 , } such that £ | Ƶ is n-transitive.
Using the fact: μ k < η k and n 1 > 0 and applying division algorithm, we have
η k μ k = ( n 1 ) ( α k 1 ) + ( n β k )
α k 1 0 , 0 n β k < n 1
η k + β k = μ k + 1 + ( n 1 ) α k α k 1 , 1 < β k n .
Clearly, β k ( 1 , n ] . Consequently the subsequences { y η k } and { y μ k } of { y n } (verifying (6)) can be such chosen for which β k = β (a constant) such that
μ k = η k + β = μ k + 1 + ( n 1 ) α k .
By (6) and (7), we get
lim k ς ( y μ k , y μ k ) = lim k ς ( y μ k , y η k + β ) = ϵ .
Use of the triangular inequality yields that
ς ( y μ k + 1 , y μ k + 1 ) ς ( y μ k + 1 , y μ k ) + ς ( y μ k , y μ k ) + ς ( y μ k , y μ k + 1 )
and
ς ( y μ k , y μ k ) ς ( y μ k , y μ k + 1 ) + ς ( y μ k + 1 , y μ k + 1 ) + ς ( y μ k + 1 , y μ k ) .
Therefore, we have
ς ( y μ k , y μ k ) ς ( y μ k , y μ k + 1 ) ς ( y μ k + 1 , y μ k ) ς ( y μ k + 1 , y μ k + 1 ) ς ( y μ k + 1 , y μ k ) + ς ( y μ k , y μ k ) + ς ( y μ k , y μ k + 1 )
which one (by letting k and with the help of (5) and (8) reduces to
lim k ς ( y μ k + 1 , y μ k + 1 ) = ϵ .
By (7) and Lemma 2, we get ( y μ k , y μ k ) £ . Denote δ k : = ς ( y μ k , y μ k ) . Using the contractivity condition (v), we obtain
ς ( y μ k + 1 , y μ k + 1 ) = ς ( ϕ y μ k , ϕ y μ k ) ϝ ( ς ( y μ k , y μ k ) ) + λ · min { ς ( y μ k , ϕ y μ k ) , ς ( y μ k , ϕ y μ k ) , ς ( y μ k , ϕ y μ k ) , ς ( y μ k , ϕ y μ k ) }
so that
ς ( y μ k + 1 , y μ k + 1 ) ϝ ( δ k ) + λ · min { ς μ k , ς μ k , ς ( y μ k , y μ k + 1 ) , ς ( y μ k , y μ k + 1 ) } .
Using upper limit in (10) and by Lemma 2 and the property of ϝ , we get
ϵ = lim sup k ς ( y μ k + 1 , y μ k + 1 ) lim sup k ϝ ( δ k ) + λ · min { 0 , 0 , ϵ , ϵ } = lim sup s ϵ + ϝ ( s ) < ϵ ,
which arises a contradiction.
Therefore, { y η } remains Cauchy, which is also £-preserving. It turns out by £-completeness of Υ that there exists y ¯ Υ such that y η ς y ¯ .
Finally, the proof will be concluded using (iv). Suppose firstly that ϕ is £-continuous. Since { y η } is a £-preserving sequence such that y η ς y ¯ ; therefore, by £-continuity of ϕ , we get
y η + 1 = ϕ ( y η ) ς ϕ ( y ¯ ) .
This implies that ϕ ( y ¯ ) = y ¯ .
Secondly, assume that £ is ς -self-closed. Then we determine a subsequence { y η k } of { y η } such that [ y η k , y ¯ ] £ , for all k N . By condition (v), Proposition 2 and [ y η k , y ¯ ] £ , we get
ς ( y η k + 1 , ϕ y ¯ ) = ς ( ϕ y η k , ϕ y ¯ ) ϝ ( ς ( y η k , y ¯ ) ) + λ · min { ς ( y η k , y η k + 1 ) , 0 , ς ( y η k , y ¯ ) , ς ( y ¯ , y η k + 1 ) } = ϝ ( ς ( y η k , y ¯ ) ) .
We claim that
ς ( y η k + 1 , ϕ y ¯ ) ς ( y η k , y ¯ ) , for all k N .
If for some k 0 N , ς ( y η k 0 , y ¯ ) = 0 , then we get ς ( ϕ y η k 0 , ϕ y ¯ ) = 0 , i.e., ς ( y η k 0 + 1 , ϕ y ¯ ) = 0 and hence (11) holds for this k 0 N . If ς ( y η k , y ¯ ) > 0 for all k N , then using property of ϝ , we obtain ς ( y η k + 1 , ϕ y ¯ ) ϝ ( ς ( y η k , y ¯ ) ) < ς ( y η k , y ¯ ) for all k N . Therefore, (11) remains valid for every k N . Using limit of (11) and by y η k ς y ¯ , we obtain y η k + 1 ς ϕ ( y ¯ ) . This implies that ϕ ( y ¯ ) = y ¯ . Thus, y ¯ remains a fixed point of ϕ . □
Theorem 4.
In addition to the assertions of Theorem 3, if ϕ ( Υ ) is £ s -directed, then ϕ admits a unique fixed point.
Proof. 
Let y ¯ , z ¯ Υ be two fixed points of ϕ . We have
ϕ ( y ¯ ) = y ¯ and ϕ ( z ¯ ) = z ¯ .
As y ¯ , z ¯ ϕ ( Υ ) , by our hypothesis there exists w Υ verifying
[ y ¯ , w ] £ and [ z ¯ , w ] £ .
Set ϱ η : = ς ( y ¯ , ϕ η w ) . By (12), (13), (v) and Proposition 2, we conclude
ϱ η = ς ( y ¯ , ϕ η w ) = ς ( ϕ y ¯ , ϕ ( ϕ η 1 w ) ) ϝ ( ς ( y ¯ , ϕ η 1 w ) ) + λ · min { 0 , ς ( ϕ η 1 w , ϕ η w ) , ς ( y ¯ , ϕ η w ) , ς ( ϕ η 1 w , y ¯ ) } = ϝ ( ϱ η 1 )
so that
ϱ η ϝ ( ϱ η 1 ) .
Assuming first that for some η 0 N , ϱ η 0 = 0 . Then ϱ η 0 ϱ η 0 1 . Secondly, we have ϱ η > 0 , for all η N . Then, using the property of ϝ , (14) becomes ϱ η < ϱ η 1 . Thus, in both cases, we get
ϱ η ϱ η 1 .
As a result, similar to Theorem 3, the above relation becomes
lim η ϱ η = lim η ς ( y ¯ , ϕ η w ) = 0 .
In similar manner, we find
lim η ς ( z ¯ , ϕ η w ) = 0 .
Combining (15) and (16), we get
ς ( y ¯ , z ¯ ) = ς ( y ¯ , ϕ η w ) + ς ( ϕ η w , z ¯ ) 0 as η
i.e., y ¯ = z ¯ . Thus, uniqueness of fixed point is concluded. □
Remark 2.
Particularly, for λ = 0 , Theorems 3 and 4 reduce to corresponding results of Alam et al. [5] (i.e., Theorem 2).
Remark 3.
Under trivial relation £ = Υ 2 and for ϝ ( t ) = α · t , Theorem 4 deduces the main result of Babu et al. [7] (i.e., Theorem 1).

4. Examples

With a view of demonstrating Theorems 3 and 4, we attempt the subsequent examples.
Example 1.
Assuming that Υ = [ 0 , ) with standard metric ς and a relation £ : = { ( y , z ) Υ 2 : y z > 0 } . Define the map ϕ : Υ Υ by ϕ ( y ) = y y + 1 . Clearly, the relation £ is locally finitely ϕ-transitive, the metric space ( Υ , ς ) is £-complete and the map ϕ is £-continuous.
Take ( y , z ) £ . Then, we have y z > 0 . Now, we have
ϕ ( y ) ϕ ( z ) = y z ( y + 1 ) ( z + 1 ) > 0
so that ( ϕ y , ϕ z ) £ . It follows that £ ϕ-closed.
Let the function ϝ : [ 0 , ) [ 0 , ) be defined by ϝ ( t ) = t t + 1 and λ 0 be arbitrarily. Now, for all ( y , z ) £ , we have
ς ( ϕ y , ϕ z ) = | y y + 1 z z + 1 | = | y z 1 + y + z + y z | y z 1 + ( y z ) = ς ( y , z ) 1 + ς ( y , z ) ϕ ( ς ( y , z ) ) + λ · min { ς ( y , ϕ y ) , ς ( z , ϕ z ) , ς ( y , ϕ z ) , ς ( z , ϕ y ) } .
It means that the presumption (v) of Theorem 3 is fulfilled. Also, y 0 = 1 verifies the presumption (i) of Theorem 3. Finally, ϕ ( Υ ) is £ s -directed as for any arbitrary y , z ϕ ( Υ ) , the element w : = ( y + z ) / 2 satisfies [ y , w ] £ and [ z , w ] £ . Therefore, all the assumptions of Theorems 3 and 4 hold and hence ϕ has a unique fixed point, y ¯ = 0 .
Example 2.
Assuming that Υ = [ 0 , 1 ] with standard metric ς and a relation £ : = . Define the map ϕ : Υ Υ by
ϕ ( y ) = y 2 , if 0 y < 1 / 4 0 , if 1 / 4 y 1 .
Then ( 0 , ϕ 0 ) £ . Clearly, the relation £ is locally finitely ϕ-transitive and ϕ-closed. Also, the metric space ( Υ , ς ) is £-complete.
Let { y η } Υ be £-preserving convergent sequence verifying y η ς y . Then { y η } being monotone increasing convergent sequence satisfies y η y so that ( y η , y ) £ for each η N . It follows that £ is ς-self-closed.
Let the function ϝ : [ 0 , ) [ 0 , ) be defined by ϝ ( t ) = t / 2 and chose λ = 1 . Then one can easily verify the presumption (v) of Theorem 3. Finally, ϕ ( Υ ) is £ s -directed as for any arbitrary y , z ϕ ( Υ ) , the element w : = max { y , z } satisfies ( y , w ) £ and ( z , w ) £ . Therefore, all the assumptions of Theorems 3 and 4 hold and hence ϕ has a unique fixed point, y ¯ = 0 .

5. An Application to Nonlinear Integral Equations

This section covers an application of earlier theorems to compute a unique solution of the following nonlinear integral equation:
y ( X ) = H ( X ) + a b ¥ ( X , τ ) ( τ , y ( τ ) ) d τ , X I : = [ a , b ]
where H : I R , ¥ : I 2 R and : I × R R are functions.
Definition 17.
y ¯ C ( I ) is viewed as a lower solution of (17) if
y ¯ ( X ) H ( X ) + a b ¥ ( X , τ ) ( τ , y ¯ ( τ ) ) d τ , f o r a l l X I .
Definition 18.
y ̲ C ( I ) is viewed as an upper solution of (17) if
y ̲ ( X ) H ( X ) + a b ¥ ( X , τ ) ( τ , y ̲ ( τ ) ) d τ , f o r a l l X I .
We now proceed to present our main results concerning this section.
Theorem 5.
Assume, in the course of Problem (17), that
(a)
H, ℏ and ¥ are continuous;
(b)
¥ ( X , τ ) > 0 , f o r a l l X , τ I ;
(c)
There exist 0 < 1 and an increasing function ϝ Σ such that
0 ( X , u ) ( X , v ) 1 ϝ ( u v ) , f o r a l l X I a n d u , v R w i t h u v ,
(d)
sup X I a b ¥ ( X , τ ) d τ .
If there exists a lower solution of (17), then the problem possesses a unique solution.
Proof. 
Indicate Υ : = C ( I ) . Undertake metric ς on Υ by
ς ( y , z ) = sup X I | y ( X ) z ( X ) | , for all y , z Υ .
Undertake a relation £ on Υ by
£ = { ( y , z ) Υ 2 : y ( X ) z ( X ) , X I } .
Define a map ϕ : Υ Υ by
( ϕ y ) ( X ) = H ( X ) + a b ¥ ( X , τ ) ( τ , y ( τ ) ) d τ , for all X Υ .
Obviously, y 0 Υ will form a solution of Problem (17) if and only if y 0 remains a fixed point of ϕ .
Now, we are going to validate all premises of Theorem 3.
(i)
Let y ¯ Υ be a lower solution of (17). Then
y ¯ ( X ) H ( X ) + a b ¥ ( X , τ ) ( τ , y ¯ ( τ ) ) d τ = ( ϝ y ¯ ) ( X )
yielding thereby ( y ¯ , ϕ y ¯ ) £ .
(ii)
Take y , z Υ such that ( y , z ) £ . Employing item (c), we obtain
( X , y ( τ ) ) ( X , z ( τ ) ) 0 , for all X , τ I .
Using (20), (21) and item (b), we obtain
( ϕ y ) ( X ) ( ϕ z ) ( X ) = a b ¥ ( X , τ ) [ ( τ , y ( τ ) ) ( τ , z ( τ ) ) ] d τ 0 ,
following that ( ϕ y ) ( X ) ( ϕ z ) ( X ) . Consequently, we have ( ϕ y , ϕ z ) £ so that £ remains ϕ -closed.
(iii)
( Υ , ς ) is £-complete as it remains complete.
(iv)
Assuming that { y η } Υ is a £-preserving sequence, which converges to y ˜ Υ . Then for every X I , { y η ( X ) } is an increasing sequence (in R ) which converges to y ˜ ( X ) . This implies that y η ( X ) y ˜ ( X ) , for all η N and X I . Using (19), we conclude ( y η , y ˜ ) £ , for all η N ; thereby, £ will be is ς -self-closed.
(v)
Take y , z Υ such that ( y , z ) £ . Using item (c), (18) and (20), we find
ς ( ϕ y , ϕ z ) = sup X I | ( ϕ y ) ( X ) ( ϕ z ) ( X ) | = sup X I [ ( ϕ z ) ( X ) ( ϕ y ) ( X ) ] = sup X I a b ¥ ( X , τ ) [ ( τ , z ( τ ) ) ( τ , y ( τ ) ) ] d τ sup X I a b ¥ ( X , τ ) 1 ϝ ( z ( τ ) y ( τ ) ) d τ .
As 0 z ( τ ) y ( τ ) ς ( y , z ) , using increasing property of ϝ , we get
ϝ ( z ( τ ) y ( τ ) ) ϝ ( ς ( y , z ) ) .
Therefore, (22) becomes
ς ( ϕ y , ϕ z ) 1 ϝ ( ς ( y , z ) ) sup X I a b ¥ ( X , τ ) d τ 1 ϝ ( ς ( y , z ) ) . = ϝ ( ς ( y , z ) )
yielding thereby
ς ( ϕ y , ϕ z ) ϝ ( ς ( y , z ) ) + λ · min { ς ( y , ϕ y ) , ς ( z , ϕ z ) , ς ( y , ϕ z ) , ς ( z , ϕ y ) } , for all y , z Υ such that ( y , z ) £ ,
where λ 0 is arbitrary.
Finally, we will verify the assumption of Theorem 4. Let y , z be chosen arbitrarily. Write w : = max { ϕ y , ϕ z } Υ . We conclude that ( ϕ y , w ) £ and ( ϕ z , w ) £ . This yields that ϕ ( Υ ) is £ s -directed and hence by Theorem 4, ϕ possesses a unique fixed point, which constitutes a unique solution for (17). □
Theorem 6.
In the collaboration to assertions (a)–(d) of Theorem 5, if there exists an upper solution of (17), then the problem possesses a unique solution.
Proof. 
Consider Υ : = C ( I ) with a metric ς and a map ϝ : Υ Υ to Theorem 5. Take the following relation £ on Υ :
£ = { ( y , z ) Υ 2 : y ( X ) z ( X ) , X I } .
If y ̲ Υ is an upper solution of (17), then we conclude
y ̲ ( X ) H ( X ) + a b ¥ ( X , τ ) ( τ , y ̲ ( τ ) ) d τ = ( ϕ y ̲ ) ( X )
yielding thereby ( y ̲ , ϕ y ̲ ) £ .
Take y , z Υ such that ( y , z ) £ . Employing item (c), we obtain
( X , y ( τ ) ) ( X , z ( τ ) ) 0 , for all X , τ I .
Using (20), (24) and item (b), we obtain
( ϕ y ) ( X ) ( ϕ z ) ( X ) = a b ¥ ( X , τ ) [ ( τ , y ( τ ) ) ( τ , z ( τ ) ) ] d τ 0 ,
which yields that ( ϕ y ) ( X ) ( ϕ z ) ( X ) so that ( ϕ y , ϕ z ) £ . Therefore, £ is ϕ -closed.
Let { y η } Υ be a £ -preserving sequence, which converges to y Υ . Then for every X I , { y η ( X ) } is a decreasing sequence (in R ) which converges to y ( X ) . Consequently, we have y η ( X ) y ( X ) , for all η N and for all X I . Using (23), we conclude that ( y η , y ) £ , for all m N . Thus £ is ς -self-closed.
Thus in all, we have verified all assertions of Theorems 3 and 4 for the metric space ( Υ , ς ) , the map ϕ and the relation £ . This completes the proof. □

6. Conclusions

Very recently, Alharbi and Khan [13] have established new fixed point theorems for relational almost strict Boyd–Wong contractions via a locally ϕ -transitive relation. In proving our results, although the class of contractions is relatively strong, we employed a locally finitely ϕ -transitive relation, which remains more general than what is utilized in [13]. Hence, our results are completely different from those proved by Alharbi and Khan [13]. In the near term, learners may extend our results to a pair of mappings.

Author Contributions

Conceptualization, M.S.A., A.A., E.A. and F.M.A.; formal analysis, A.A., E.A. and H.I.A.M.; investigation, A.A. and F.M.A.; methodology, F.A.K. and H.I.A.M.; writing—original draft, F.A.K., F.M.A. and H.I.A.M.; writing—review & editing, F.A.K., M.S.A. and E.A.; funding acquisition, M.S.A., A.A., E.A. and F.M.A.; visualization, H.I.A.M.; supervision, F.A.K. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

No data were created for this study.

Acknowledgments

All authors are grateful to an academic editor and three referees for their lucrative remarks and pertinent comments that helped us improve the depth of present paper.

Conflicts of Interest

The authors assert that they possess no competing interests.

References

  1. Younis, M.; Ahmad, H.; Chen, L.; Han, M. Computation and convergence of fixed points in graphical spaces with an application to elastic beam deformations. J. Geom. Phys. 2023, 192, 104955. [Google Scholar] [CrossRef]
  2. Alam, A.; Imdad, M. Relation-theoretic contraction principle. J. Fixed Point Theory Appl. 2015, 17, 693–702. [Google Scholar] [CrossRef]
  3. Alam, A.; Imdad, M. Relation-theoretic metrical coincidence theorems. Filomat 2017, 31, 4421–4439. [Google Scholar] [CrossRef]
  4. Alam, A.; Imdad, M. Nonlinear contractions in metric spaces under locally T-transitive binary relations. Fixed Point Theory 2018, 19, 13–24. [Google Scholar] [CrossRef]
  5. Alam, A.; Arif, M.; Imdad, M. Metrical fixed point theorems via locally finitely T-transitive binary relations under certain control functions. Miskolc Math. Notes 2019, 20, 59–73. [Google Scholar] [CrossRef]
  6. Berinde, V. Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum. 2004, 9, 43–53. [Google Scholar]
  7. Babu, G.V.R.; Sandhy, M.L.; Kameshwari, M.V.R. A note on a fixed point theorem of Berinde on weak contractions. Carpathian J. Math. 2008, 24, 8–12. [Google Scholar]
  8. Lipschutz, S. Schaum’s Outlines of Theory and Problems of Set Theory and Related Topics; McGraw-Hill: New York, NY, USA, 1964. [Google Scholar]
  9. Kolman, B.; Busby, R.C.; Ross, S. Discrete Mathematical Structures, 6th ed.; Pearson/Prentice Hall: Hoboken, NJ, USA, 2009. [Google Scholar]
  10. Berzig, M.; Karapinar, E. Fixed Point results for (αψ,βϕ)-contractive mappings for a generalized altering distance. Fixed Point Theory Appl. 2013, 2013, 205. [Google Scholar] [CrossRef]
  11. Turinici, M. Contractive maps in locally transitive relational metric spaces. Sci. World J. 2014, 2014, 169358. [Google Scholar] [CrossRef]
  12. Samet, B.; Turinici, M. Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications. Commun. Math. Anal. 2012, 13, 82–97. [Google Scholar]
  13. Alharbi, A.F.; Khan, F.A. Almost Boyd-Wong type contractions under binary relations with applications to boundary value problems. Axioms 2023, 12, 896. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Khan, F.A.; Aldhabani, M.S.; Alamer, A.; Alshaban, E.; Alamrani, F.M.; Mohammed, H.I.A. Almost Nonlinear Contractions under Locally Finitely Transitive Relations with Applications to Integral Equations. Mathematics 2023, 11, 4749. https://doi.org/10.3390/math11234749

AMA Style

Khan FA, Aldhabani MS, Alamer A, Alshaban E, Alamrani FM, Mohammed HIA. Almost Nonlinear Contractions under Locally Finitely Transitive Relations with Applications to Integral Equations. Mathematics. 2023; 11(23):4749. https://doi.org/10.3390/math11234749

Chicago/Turabian Style

Khan, Faizan Ahmad, Musaad S. Aldhabani, Ahmed Alamer, Esmail Alshaban, Fahad M. Alamrani, and Hamid I. A. Mohammed. 2023. "Almost Nonlinear Contractions under Locally Finitely Transitive Relations with Applications to Integral Equations" Mathematics 11, no. 23: 4749. https://doi.org/10.3390/math11234749

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop