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BY 4.0 license Open Access Published by De Gruyter Open Access September 1, 2022

Remarks on the generalized interpolative contractions and some fixed-point theorems with application

  • Muhammad Nazam , Maha M. A. Lashin EMAIL logo , Aftab Hussain and Hamed H. Al Sulami
From the journal Open Mathematics

Abstract

In this manuscript, some remarks on the papers [H. A. Hammad, P. Agarwal, S. Momani, and F. Alsharari, Solving a fractional-order differential equation using rational symmetric contraction mappings, Fractal Fract. 5 (2021), 159] and [A. Hussain, F. Jarad, and E. Karapinar, A study of symmetric contractions with an application to generalized fractional differential equations, Adv. Differ. Equ. 2021 (2021), 300] are given. In the light of remarks, we introduce a new property that makes it convenient to investigate the existence of fixed points of the interpolative contractions in the orthogonal metric spaces. We derive several new results based on known contractions from the main theorems. As an application, we resolve a Urysohn integral equation.

MSC 2010: 47H10; 46N20; 30L15; 54E50

1 Introduction

The investigation of generalized contraction principles to present new and useful fixed-point theorems is the key element in the metric fixed-point theory. The founder of this field was Banach (1922) who introduced the contraction principle. Boyd and Wong [1] generalized the well-known Banach contraction principle (BCP) [2] by introducing the control function Ψ : [ 0 , ) [ 0 , ) verifying the following conditions for each y > 0 :

  1. Ψ ( y ) < y ;

  2. lim x y + Ψ ( x ) < y .

Boyd and Wong [1] contraction principle generalizes the contraction principle introduced by Rakotch [3]. The Boyd-Wong idea has been generalized by Matkowski [4], Samet et al. [5], Karapinar and Samet [6], and Pasicki [7].

Karapinar [8] introduced interpolative contractions and presented a method to obtain fixed points of such contractions. Agarwal and Karapinar also introduced the interpolative Rus-Reich-Ćirić type contractions [9], w-interpolative Ćirić-Reich-Rus-type contractions [10], interpolative Hardy-Rogers type contractions [11], and interpolative Boyd-Wong and Matkowski type contractions [12] to ensure the existence of fixed points in (generalized) metric spaces. Gautam et al. [13] presented some fixed-point results for Chatterjea and cyclic Chatterjea interpolative contractions in complete quasi-partial b -metric spaces. Debnath et al. [14] proved some fixed-point theorems for Rus-Reich-Ćirić and Hardy-Rogers type interpolative contractions in b -metric spaces. Recently, Hussain et al. [15], introduced the following generalized interpolative contractions:

(1.1) d F ( S x , S y ) λ ( F ˇ i ( x , y ) ) , i = 1 , 2 , 3 , 4 ,

where d F is an F metric, S : A A , and for positive real numbers a , b , c , the mappings F ˇ 1 , F ˇ 2 , F ˇ 3 , F ˇ 4 : A × A [ 0 , ) are defined by

F ˇ 1 ( x , y ) = d F ( x , y ) [ d F ( x , S x ) ] 1 ( a b ) ( a c ) [ d F ( y , S y ) ] 1 ( a b ) ( a c ) × [ d F ( x , S x ) + d F ( y , S y ) ] 1 ( b a ) ( b c ) [ d F ( x , S y ) + d F ( y , S x ) ] 1 ( c a ) ( c b ) , F ˇ 2 ( x , y ) = d F ( x , y ) [ d F ( x , S x ) ] a ( a b ) ( a c ) [ d F ( y , S y ) ] a ( a b ) ( a c ) × [ d F ( x , S x ) + d F ( y , S y ) ] b ( b a ) ( b c ) [ d F ( x , S y ) + d F ( y , S x ) ] c ( c a ) ( c b ) , F ˇ 3 ( x , y ) = max d F ( x , y ) , [ d F ( x , S x ) ] a 2 ( a b ) ( a c ) [ d F ( y , S y ) ] a 2 ( a b ) ( a c ) [ d F ( x , S x ) + d ( y , S y ) ] b 2 ( b a ) ( b c ) [ d F ( x , S y ) + d F ( y , S x ) ] c 2 ( c a ) ( c b ) , F ˇ 4 ( x , y ) = d F ( x , y ) a 3 ( a b ) ( a c ) d F ( y , S y ) a 3 ( a b ) ( a c ) × [ d F ( x , S x ) + d F ( y , S y ) ] b 3 ( b a ) ( b c ) [ d F ( x , S y ) + d F ( y , S x ) ] c 3 ( c a ) ( c b ) .

It is important to note that despite a , b , c > 0 , some exponents are negative, for example, if a > b , a > c , and b > c , then 1 ( b a ) ( b c ) < 0 . If any one of a , b , and c goes to , then F ˇ 1 ( x , y ) = d F ( x , y ) . Moreover, we have the following interesting facts about the exponents, which can be proved by using basic algebraic tools:

1 ( a b ) ( a c ) + 1 ( b a ) ( b c ) + 1 ( c a ) ( c b ) = 0 , a ( a b ) ( a c ) + b ( b a ) ( b c ) + c ( c a ) ( c b ) = 0 , a 2 ( a b ) ( a c ) + b 2 ( b a ) ( b c ) + c 2 ( c a ) ( c b ) = 1 , a 3 ( a b ) ( a c ) + b 3 ( b a ) ( b c ) + c 3 ( c a ) ( c b ) = a + b + c .

These contractions are named as “symmetric fractional contractions of type I, II, III, and IV” for i = 1 , 2 , 3 , 4 , respectively. The existence of fixed points of these contractions has been shown in [15]. In the proof of [15, Theorem 2.3], the following simplifications was involved.

F ˇ 1 ( n 1 , n ) = d F ( n 1 , n ) d F ( n 1 , S n 1 ) 1 ( a b ) ( a c ) d F ( n , S n ) 1 ( a b ) ( a c ) × [ d F ( n 1 , S n 1 ) + d F ( n , S n ) ] 1 ( b a ) ( b c ) [ d F ( n 1 , S n ) + d F ( n , S n 1 ) ] 1 ( c a ) ( c b ) d F ( n 1 , n ) d F ( n 1 , n ) 1 ( a b ) ( a c ) d F ( n , n + 1 ) 1 ( a b ) ( a c ) × [ d F ( n 1 , n ) + d F ( n , n + 1 ) ] 1 ( b a ) ( b c ) [ d F ( n 1 , n + 1 ) + d F ( n , n ) ] 1 ( c a ) ( c b ) d F ( n 1 , n ) d F ( n 1 , n ) 1 ( a b ) ( a c ) d F ( n , n + 1 ) 1 ( a b ) ( a c ) × [ d F ( n 1 , n ) + d F ( n , n + 1 ) ] 1 ( b a ) ( b c ) [ d F ( n 1 , n ) + d F ( n , n + 1 ) ] 1 ( c a ) ( c b ) = d F ( n 1 , n ) d F ( n 1 , n ) 1 ( a b ) ( a c ) d F ( n , n + 1 ) 1 ( a b ) ( a c ) × [ d F ( n 1 , n ) + d F ( n , n + 1 ) ] 1 ( b a ) ( b c ) + 1 ( c a ) ( c b ) d F ( n 1 , n ) d F ( n 1 , n ) 1 ( a b ) ( a c ) d F ( n , n + 1 ) 1 ( a b ) ( a c ) × [ d F ( n 1 , n ) d F ( n , n + 1 ) ] 1 ( b a ) ( b c ) + 1 ( c a ) ( c b ) = d F ( n 1 , n ) 1 + 1 ( a b ) ( a c ) + 1 ( b a ) ( b c ) + 1 ( c a ) ( c b ) d F ( n , n + 1 ) 1 ( a b ) ( a c ) + 1 ( b a ) ( b c ) + 1 ( c a ) ( c b ) = d F ( n 1 , n ) .

We observe that the inequality in the red color, that is,

[ d F ( n 1 , n ) + d F ( n , n + 1 ) ] 1 ( b a ) ( b c ) + 1 ( c a ) ( c b ) [ d F ( n 1 , n ) d F ( n , n + 1 ) ] 1 ( b a ) ( b c ) + 1 ( c a ) ( c b ) ,

is not true in general due to the following fact:

( a + b ) r ( a b ) r for all a , b 2 and r 1 .

Also, in the proof of [15, Theorems 3.2, 4.2, and 5.2], the simplifications of F ˇ 2 ( n 1 , n ) , F ˇ 3 ( n 1 , n ) , and F ˇ 4 ( n 1 , n ) s have some gaps.

Similarly, Hammad et al. [16] defined the so-called rational symmetric contractions of type I, II, III and IV as follows:

(1.2) d θ ( S x , S y ) q λ ( A ˇ i ( x , y ) ) , i = 1 , 2 , 3 , 4 ,

where q 1 , λ [ 0 , 1 ) , d θ is a θ -metric, S : A A a self-mapping, and

A ˇ 1 ( x , y ) = d θ ( x , y ) q [ d θ ( x , S x ) ] q ( a b ) ( a c ) [ d θ ( y , S y ) ] q ( a b ) ( a c ) 1 + d θ ( x , y ) q × [ d θ ( x , S x ) + d θ ( y , S y ) ] q ( b a ) ( b c ) 1 + d θ ( x , y ) q [ d θ ( x , S y ) + d θ ( y , S x ) ] q ( c a ) ( c b ) 1 + d θ ( x , y ) q . A ˇ 2 ( x , y ) = d θ ( x , y ) q [ d θ ( x , S x ) ] q a ( a b ) ( a c ) [ d θ ( y , S y ) ] q a ( a b ) ( a c ) 1 + d θ ( x , y ) q × [ d θ ( x , S x ) + d θ ( y , S y ) ] q b ( b a ) ( b c ) 1 + d θ ( x , y ) q [ d θ ( x , S y ) + d θ ( y , S x ) ] q c ( c a ) ( c b ) 1 + d θ ( x , y ) q . A ˇ 3 ( x , y ) = max d θ ( x , y ) q , [ d θ ( x , S x ) ] q a 2 ( a b ) ( a c ) [ d θ ( y , S y ) ] q a 2 ( a b ) ( a c ) 1 + d θ ( x , y ) q [ d θ ( x , S x ) + d θ ( y , S y ) ] q b 2 ( b a ) ( b c ) 1 + d θ ( x , y ) q [ d θ ( x , S y ) + d θ ( y , S x ) ] q c 2 ( c a ) ( c b ) 1 + d θ ( x , y ) q . A ˇ 4 ( x , y ) = d θ ( x , y ) q a 3 ( a b ) ( a c ) [ d θ ( y , S y ) ] q a 3 ( a b ) ( a c ) 1 + d θ ( x , y ) q × [ d θ ( x , S x ) + d θ ( y , S y ) ] q b 3 ( b a ) ( b c ) 1 + d θ ( x , y ) q [ d θ ( x , S y ) + d θ ( y , S x ) ] q c 3 ( c a ) ( c b ) 1 + d θ ( x , y ) q .

We note that in the proof of [16, Theorem 2], the following inequality:

(1.3) [ d θ ( n 1 , n ) + d θ ( n , n + 1 ) ] q ( b a ) ( b c ) + q ( c a ) ( c b ) [ d θ ( n 1 , n ) d θ ( n , n + 1 ) ] q ( b a ) ( b c ) + q ( c a ) ( c b ) ,

considered true to obtain the desired result; however, it is not true in general due to the following fact:

( a + b ) r ( a b ) r for all a , b 2 and for any positive real number r .

We deduce that the inequality (1.3) is true for d θ ( i , i + 1 ) 2 for all i { 0 , 1 , 2 , 3 , , n } and not true if d θ ( i , i + 1 ) < 2 .

Also, the proof of [16, Theorems 4, 6, and 8] contains some gaps.

To obtain the refinements of the proofs given in [15,16], we begin with the following observation.

Observation 1.1

The following inequality holds for all a , b 2 , and r 1 ,

( a + b ) r ( a b ) r .

Proof

We note that the equality holds for a = b = 2 . We can assume that a b , then a = η b ; η 1 . Let b = t , so that a = η t , t 2 . Define the function f : [ 2 , ) ( , ) by

f ( t ) = ( η t 2 ) r ( η t + t ) r , t [ 2 , ) .

This implies that

f ( t ) = d d t ( f ( t ) ) = r t r 1 ( η + 1 ) r 2 t r η η + 1 r 1 .

Since 2 t r η η + 1 r > 1 (otherwise t < 1 ), we have f ( t ) > 0 . This implies that f ( t ) 0 , and hence, ( η t 2 ) r ( η t + t ) r 0 , that is, ( a + b ) r ( a b ) r .□

Observation 1.2

Let K 2 . For any nonempty set A , define the mapping d : A × A [ 0 , ) by

d ( u , v ) = K if u v , 0 if u = v .

Then the pair ( A , d ) is a metric space.

Now, we introduce a new property (P), which plays a key role in the refinement of the results given in [15,16].

Definition 1.3

Let ( A , d ) be a metric space. A mapping T : A A is said to have property (P), if for any real number r , it satisfies the following inequality:

( d ( x , T ( x ) ) + d ( T ( x ) , T 2 ( x ) ) ) r ( d ( x , T ( x ) ) d ( T ( x ) , T 2 ( x ) ) ) r , x A .

Example 1.4

Let A = [ 1 , ) and consider the metric d defined by d ( u , v ) = u v for all u , v A . The mapping T : A A defined by T ( x ) = K x for all x A and K 5 2 satisfies the property (P). Indeed,

( d ( x , T ( x ) ) + d ( T ( x ) , T 2 ( x ) ) ) r = [ ( K 1 ) x + ( K 1 ) K x ] r [ ( K 1 ) ( K + 1 ) x ] r [ ( K 1 ) 2 K x 2 ] r = ( d ( x , T ( x ) ) d ( T ( x ) , T 2 ( x ) ) ) r .

Example 1.5

Every identity mapping satisfies the property (P). The constant mapping does not satisfy the property (P). The mapping T : A A defined by T ( x ) = 0 for all x A satisfies the property P only for x = 0 .

Example 1.6

Let A = ( , ) . The mapping T : A A defined by T ( x ) = 2 3 x for all x A satisfies the property (P). In fact, the mapping T : A A defined by T ( x ) = a b x for all x A for b > a satisfies the property (P).

Example 1.7

Let A = [ 2.5 , ) . The mapping T : A A defined by T ( x ) = 2 x 1 for all x A satisfies the property (P).

Example 1.8

Let A = [ 1 , ) . The mapping T : A A defined by T ( x ) = 1 x 2 for all x A satisfies the property (P).

Remark 1.9

The corrected proof of [15, Theorems 2.3, 3.2, 4.2, and 5.2] and [16, Theorems 2, 4, 6, and 8] can be obtained by assuming either “Observation 1.1 and Observation 1.2” or “Property (P).”

The main contribution of this paper is the introduction of property (P) that proved to be a good tool to address the gaps found in [15,16]. Second, we show that interpolative contraction implies orthogonal interpolative contraction, and orthogonal interpolative contraction implies ( Ψ , Φ ) -orthogonal interpolative contraction but converse is not true. Thus, presented theorems generalize the results in [9,10,11, 12,17].

2 Orthogonal relations

In this section, we define orthogonal set (a non-empty set whose elements obey the orthogonal relation), -regular space, and O-sequence (a sequence whose terms are pairwise orthogonal). The binary relation (orthogonal relation) is a generalization of a partial order, α -admissible function, and directed graph. It also contains the notion of orthogonality in the inner product spaces. The following definition is one of the key notions of this paper.

Definition 2.1

[18] Let be a binary relation defined on a nonempty set A (i.e., A × A ). If satisfies the property (O), then we call it orthogonal relation, and the pair ( A , ) is called orthogonal set (in short, O-set).

(O): x 0 A : either ( y , x 0 y ) or ( y , y x 0 ) .

To illustrate the orthogonal set, we have the following examples.

Example 2.2

Let A be the set of integers and define the relation a θ if and only if a 1 ( mod θ ) . Then ( A , ) is an O-set. Indeed, 1 θ for each θ .

Example 2.3

Let A be the set of all persons in the word. Define x y if x can give blood to y . According to the blood transfusion protocol, if x 0 is a person such that his (her) blood type is O , then we have x 0 y for all y A . This means that ( A , ) is an O-set. In this O-set, x 0 is not unique. Note that x 0 may be a person with blood type AB+. In this case, we have y x 0 for all y A .

Example 2.4

In the graph theory, a wheel graph W n is a graph with n vertices for each n 4 , formed by connecting a single vertex to all vertices of an ( n 1 ) -cycle. Let A be the set of all vertices of W n for each n 4 . Define x y if there is a connection from x to y . Then, ( A , ) is an O-set.

Example 2.5

Let A be a inner product space with the inner product , . Define x y if x , y = 0 . It is easy to see that 0 y for all y A . Hence, ( A , ) is an O-set.

Definition 2.6

[18] A sequence { n : n N } is said to be an O-sequence if either x n x n + 1 or x n + 1 x n for all n .

Definition 2.7

[18] The O-set ( A , ) endowed with a metric d is called an O-metric space (in short, OMS) denoted by ( A , , d ) .

Definition 2.8

[18] The O-sequence { x n } A is said to be O-Cauchy if lim n , m d ( x n , x m ) = 0 . If each O-Cauchy sequence converges in A , then ( A , , d ) is called O-complete.

Definition 2.9

[18] Let ( A , , d ) be an orthogonal metric space. A mapping f : A A is said to be an orthogonal contraction if there exists k [ 0 , 1 ) such that

d ( f x , f y ) k d ( x , y ) x , y A with x y .

In the following, we give some comparisons between fundamental notions.

  1. The continuity implies orthogonal continuity, but converse is not true. If f : R R is defined by f ( x ) = [ x ] , x R and the relation R × R is defined by

    x y if x , y i + 1 3 , i + 2 3 , i Z or x = 0 .

    Then f is -continuous, while f is discontinuous on R .

  2. The completeness of the metric space implies O-completeness, but the converse is not true. We know that A = [ 0 , 1 ) with Euclidean metric d is not complete metric space. If we define the relation A × A by

    x y x y 1 2 or x = 0 .

    Then, ( A , , d ) is an O-complete.

  3. The Banach contraction implies orthogonal contraction, but converse is not true. Let A = [ 0 , 10 ) with Euclidean metric d , so that ( A , d ) is a metric space. If we define the relation A × A by

    x y if x y x y .

    Then ( A , , d ) is an O-metric space. Define f : A A by f ( x ) = x 2 (if x 2 ) and f ( x ) = 0 (if x > 2 ). Since d ( f ( 3 ) , f ( 2 ) ) > k d ( 3 , 2 ) , so, f is not a contraction while it is an orthogonal contraction.

We will use the following lemma to support the proofs.

Lemma 2.10

[17] Let ( X , d ) be a metric space and { x n } X be a sequence verifying lim n d ( x n , x n + 1 ) = 0 . If the sequence { x n } is not Cauchy, then there are { x n k } , { x m k } and ξ > 0 such that

(2.1) lim k d ( x n k + 1 , x m k + 1 ) = ξ + .

(2.2) lim k d ( x n k , x m k ) = d ( x n k + 1 , x m k ) = d ( x n k , x m k + 1 ) = ξ .

Let ( A , , d ) be an OMS and P ( A ) form the set of nonempty subsets of A , C B ( A ) form the set of all nonempty bounded and closed subsets of A . The set of nonempty compact subsets of A is denoted by K ( A ) . For Λ C B ( A ) and χ A , consider d ( χ , Λ ) = inf ς Λ d ( χ , ς ) . Let H : C B ( A ) × C B ( A ) [ 0 , ) be given as follows:

(2.3) H ( Λ 1 , Λ 2 ) = max { sup q Λ 1 d ( q , Λ 2 ) , sup b Λ 2 d ( b , Λ 1 ) } for all Λ 1 , Λ 2 C B ( A ) .

Such a function H verifies all the axioms of O-metric. It is known as a Pompieu-Hausdorff O-metric induced by the O-metric d .

Definition 2.11

Let T : A P ( A ) be a set-valued mapping. An element υ A is said to be a fixed point of T if υ T υ .

Definition 2.12

Let ( A , , d ) be an OMS and A × A be a binary relation. The space ( A , , d ) is called -regular if for each sequence { x n } A so that x n x n + 1 for each n 0 and x n x as n , we have either x n x , or x x n for all n 0 .

Definition 2.13

A mapping T : A A is said to be asymptotically regular at a point υ of A if

lim n d ( T n υ , T n + 1 υ ) = 0 .

If T is asymptotically regular at each point in A , then it is named as an asymptotically regular mapping.

The following, lemma is useful in the sequel.

Lemma 2.14

[19] Let Λ 1 and Λ 2 be nonempty bounded and closed subsets of a metric space ( A , d ) and ν > 1 . Then, for all x Λ 1 , there is y Λ 2 so that d ( x , y ) ν H ( Λ 1 , Λ 2 ) .

In the following, we define -admissible mapping, -preserving mapping and illustrate them with examples. Let Λ = { ( x , y ) A × A : x y } .

Definition 2.15

A mapping f : A × A [ 1 , ) is said to be strictly -admissible if f ( a , θ ) > 1 for all a , θ A with a θ and f ( a , θ ) = 1 otherwise.

Example 2.16

Let A = [ 0 , 1 ) and define the relation A × A by

a θ if a θ { a , θ } A .

Then, A is an O-set. Define f : A × A [ 1 , ) by

f ( a , θ ) = a + 2 1 + θ if a θ , 1 otherwise .

Then, f is -admissible.

Definition 2.17

Let T : A P ( A ) and A × A be an orthogonal relation. The mapping T is called -preserving if for each q A and p T ( q ) such that q p or p q , there is ω T ( p ) satisfying p ω or ω p .

Example 2.18

Let A = [ 0 , 1 ) and define the relation A × A by

a θ if a θ { a , θ } A .

Then, A is an O-set. Define S : A P ( A ) by

S ( a ) = a 15 , a + 1 7 if a Q A , { 0 } if a Q c A .

Then, S is a -preserving mapping. Indeed, for a = 0 , there is θ S ( 0 ) = 0 , 1 7 such that either a θ or θ a , and then there is x S ( θ ) such that either x θ or θ x .

3 GIFC of type I, II, III, and IV

In this section, we generalize contractions (1.1) and (1.2) by employing two functions Ψ , Φ : ( 0 , ) ( , ) and the property (P) with some necessary conditions and hence obtaining some new fixed-point theorems in the orthogonal metric spaces. These fixed-point theorems generalize some results presented by Proinov [17], Nazam et al. [20], and Karapinar et al. [9,10, 11,12].

Let d be the metric on A and S : A C B ( A ) be a set-valued mapping. Let the mappings I ˇ 1 , I ˇ 2 , I ˇ 3 , I ˇ 4 : A × A [ 0 , ) be defined by

I ˇ 1 ( x , y ) = d ( x , y ) [ d ( x , S x ) ] 1 ( a b ) ( a c ) [ d ( y , S y ) ] 1 ( a b ) ( a c ) × [ d ( x , S x ) + d ( y , S y ) ] 1 ( b a ) ( b c ) [ d ( x , S y ) + d ( y , S x ) ] 1 ( c a ) ( c b ) ( for some a , b , c > 0 ) . I ˇ 2 ( x , y ) = d ( x , y ) [ d ( x , S x ) ] a ( a b ) ( a c ) [ d ( y , S y ) ] a ( a b ) ( a c ) × [ d ( x , S x ) + d ( y , S y ) ] b ( b a ) ( b c ) [ d ( x , S y ) + d ( y , S x ) ] c ( c a ) ( c b ) ( for some a , b , c > 0 ) . I ˇ 3 ( x , y ) = max d ( x , y ) , [ d ( x , S x ) ] a 2 ( a b ) ( a c ) [ d ( y , S y ) ] a 2 ( a b ) ( a c ) [ d ( x , S x ) + d ( y , S y ) ] b 2 ( b a ) ( b c ) [ d ( x , S y ) + d ( y , S x ) ] c 2 ( c a ) ( c b ) ( for some a , b , c > 0 ) . I ˇ 4 ( x , y ) = d ( x , y ) a 3 ( a b ) ( a c ) d ( y , S y ) a 3 ( a b ) ( a c ) × [ d ( x , S x ) + d ( y , S y ) ] b 3 ( b a ) ( b c ) [ d ( x , S y ) + d ( y , S x ) ] c 3 ( c a ) ( c b ) ( for some a , b , c > 0 ) .

We begin with the following definition. The novelty of generalized interpolative fractional contraction (GIFC) is that it allows the exponents of the distances to be fractions. The classical contractions contain sum of distances, while GIFC contains the product of distances with fractional exponents. The studies on GIFC are independents of the results appeared in [21,22,23].

Definition 3.1

Let ( A , , d ) be an OMS. A mapping S : A C B ( A ) is said to be a GIFC of type I, II, III, and IV for i = 1 , 2 , 3 , 4 respectively, if there exist a strictly -admissible mapping f and a , b , c ( 0 , ] for i = 1 and a , b , c ( 0 , ) for i = 2 , 3 , 4 such that

(3.1) Ψ ( f ( x , y ) H ( S x , S y ) ) Φ ( I ˇ i ( x , y ) ) ,

for all ( x , y ) Λ , H ( S x , S y ) > 0 , x S x and y S y .

If either a = or b = or c = in GIFC of type I, we receive the recently announced ( ψ , ϕ ) -contraction by Proinov [17] provided ( x , y ) Λ .

We also note that for Φ ( x ) = Ψ ( x ) τ and Ψ ( x ) = ln ( x ) for all x ( 0 , ) , τ > 0 , the contraction (3.1 i = 1 ) can be written as follows:

τ + ln ( f ( x , y ) H ( S x , S y ) ) ln ( d ( x , y ) ) + 1 ( a b ) ( a c ) ln ( d ( x , S x ) ) + 1 ( a b ) ( a c ) ln ( d ( y , S y ) ) + 1 ( a b ) ( a c ) ln [ d ( x , S x ) + d ( y , S y ) ] + 1 ( a b ) ( a c ) ln [ d ( x , S y ) + d ( y , S x ) ] ,

and then, we have

τ + Ψ ( f ( x , y ) H ( S x , S y ) ) Ψ ( d ( x , y ) ) + 1 ( a b ) ( a c ) Ψ ( d ( x , S x ) ) + 1 ( a b ) ( a c ) Ψ ( d ( y , S y ) ) + 1 ( a b ) ( a c ) Ψ [ d ( x , S x ) + d ( y , S y ) ] + 1 ( a b ) ( a c ) Ψ [ d ( x , S y ) + d ( y , S x ) ] .

This represents a general version of the contraction introduced in [24], and if either a = or b = or c = and ( x , y ) Λ , we have exactly same contraction that was introduced in [24].

Remark 3.2

It is very important to note that the set of self-mappings satisfying property P and contraction (3.1) is not empty. For example, the mappings S ( x ) = 2 3 x for all x ( , ) and S ( x ) = 2 x 1 for all x [ 2.5 , ) satisfy both the property P and contraction (3.1) with Φ ( x ) = Ψ ( x ) τ and Ψ ( x ) = ln ( x ) for all x ( 0 , ) , where τ > 0 .

The following example explains (3.1 i = 1 ).

Example 3.3

Let A = [ 1 , 7 ) and define the relation on A by

x y if x y { x , y } .

Then is an orthogonal relation, and so ( A , ) is an O-set. Let d be the Euclidean metric on A , and then, ( A , d ) is an incomplete metric space. Define Ψ , Φ : R + R by

Ψ ( x ) = x + 1 if x 10 3 , 6 , 16 3 , 28 5 x + 10 if x R + 10 3 , 6 , 16 3 , 28 5 Φ ( x ) = x 2 if x 10 3 , 6 , 16 3 , 28 5 x + 5 if x R + 10 3 , 6 , 16 3 , 28 5 .

Let S : A C B ( A ) be defined by

S ( x ) = { 5 , 6 } if 1 x < 2 { 3 , 4 } if 2 x < 3 { 1 , 6 } if 3 x < 7 .

Define f : A × A [ 1 , ) by

f ( x , y ) = x + 2 1 + y if x y , 1 otherwise .

Then f is -admissible. Our calculations show that H ( S x , S y ) = 2 , f ( x , y ) = 5 3 , d ( x , S x ) = 4 , d ( y , S y ) = 1 , d ( x , S y ) = 2 , and d ( y , S x ) = 3 if x = 1 , and y = 2 ( 1 2 ) . Thus, there exist a , b , c ( 0 , ] and λ ( 0 , 1 ) such that

10 3 = f ( x , y ) H ( S x , S y ) > λ I ˇ 1 ( x , y ) = λ 4 1 ( a b ) ( a c ) 5 1 ( b a ) ( b c ) + 1 ( c a ) ( c b ) .

Thus, S is not a -admissible orthogonal interpolative set-valued contraction. However, S is a GIFC. Indeed,

13 3 = Ψ ( f ( x , y ) H ( S x , S y ) ) Φ ( I ˇ 1 ( x , y ) ) = 4 1 ( a b ) ( a c ) 5 1 ( b a ) ( b c ) + 1 ( c a ) ( c b ) + 5 .

We obtain the same conclusions for x = 1 , y = 3 ( 1 3 ) ; x = 1 , y = 4 ( 1 4 ) , and x = 1 , y = 5 .

The following main theorem states some conditions that guarantee the existence of a fixed point of a mapping S satisfying (3.1).

Theorem 3.4

Let ( A , , d ) be an -regular O-complete metric space (in short, OCMS). Let S : A C B ( A ) be an -preserving mapping verifying (3.1) for i = 1 and property (P). Suppose the relation is transitive and the functions Ψ , Φ : ( 0 , ) ( , ) are so that:

  1. for each 0 A , there is 1 S ( 0 ) such that 1 0 or 0 1 ;

  2. Ψ , Φ are nondecreasing and Φ ( y ) < Ψ ( y ) for all y > 0 ;

  3. limsup y δ + Φ ( y ) < Ψ ( δ + ) for all δ > 0 ;

  4. limsup a 0 Φ ( a ) lim inf a ξ + Ψ ( a ) .

Then, S has a fixed point in A .

Proof

By (i), for an arbitrary 0 A , there is 1 S ( 0 ) so that 0 1 or 1 0 . Since S is -preserving, there is 2 S ( 1 ) so that 1 2 or 2 1 and then 3 S ( 2 ) so that 2 3 or 3 2 . In general, there is n + 1 S ( n ) in order that n n + 1 or n + 1 n for all n 0 . Hence, f ( n , n + 1 ) > 1 . Note that if n S ( n ) , then n is a fixed point of S . Suppose that n S ( n ) for all n 0 . Thus, H ( S n 1 , S n ) > 0 (otherwise n S n ). Since f ( n , n + 1 ) > 1 and S ( n ) , S ( n + 1 ) are bounded and closed sets, by Lemma 2.14, there is n + 1 S ( n ) ( n n + 1 ) so that d ( n , n + 1 ) f ( n 1 , n ) H ( S ( n 1 ) , S ( n ) ) for all n 1 . By (ii) and (3.1), one writes

Ψ ( h n ) < Ψ ( f ( n 1 , n ) H ( S ( n 1 ) , S ( n ) ) ) Φ ( I ˇ 1 ( n