Existence results for ABC-fractional BVP via new fixed point results of F-Lipschitzian mappings

Nayyar Mehmood; Israr Ali Khan; Muhammad Ayyaz Nawaz; Niaz Ahmad

doi:10.1515/dema-2022-0028

Open Access Published by De Gruyter Open Access September 21, 2022

Existence results for ABC-fractional BVP via new fixed point results of F-Lipschitzian mappings

Nayyar Mehmood , Israr Ali Khan , Muhammad Ayyaz Nawaz and Niaz Ahmad

From the journal Demonstratio Mathematica

https://doi.org/10.1515/dema-2022-0028

Abstract

In this article, fixed point results for self-mappings in the setting of two metrics satisfying F -lipschitzian conditions of rational-type are proved, where F is considered as a semi-Wardowski function with constant τ ∈ R instead of τ > 0 . Two metrics have been considered, one as an incomplete while the other is orbitally complete. The mapping is taken to be orbitally continuous from one metric to another. Some examples are provided to validate our results. For applications, we present existence results for the solutions of a new type of ABC-fractional boundary value problem.

Keywords: fixed points; F-lipschitzian; two metrics; fractional BVP

MSC 2010: 47H10; 54H25; 55M20

1 Introduction

Fixed point theory is a fundamental area of analysis with lot of applications in different branches of science and engineering. The presence or absence of fixed point is an intrinsic property of a function. Banach fixed point theorem [1] is an important tool which describes the systematic procedure to find the fixed point of a self-mapping of a complete metric space. It guarantees the existence and uniqueness of a fixed point of a given contractive self-mapping on metric spaces. There are plenty of extensions of this theorem, which can be seen in [2,3, 4,5,6, 7,8] among others.

To find the solutions of operators in function spaces, it is obvious that the space with given metric is either incomplete or with the given metric the operator may not be contractive. In both cases, we need another metric on the set, which either make the operator contractive or become complete, so that we could use the relationship of operator between two topologies to find the solution. In this regard, in 1968, Maia [9] proved a fixed point result for self-mappings of a nonempty set with two metrics. Later on, in 1980, Fisher [10] introduced the rational-type contractive conditions and Khan [11] found out the fixed points of mappings on complete metric space, satisfying the rational contractive conditions.

In 2012, Wardowski [8] generalized the Banach contraction principle using the contractive condition called F -contraction. Many articles on F -contraction are available in the literature [8,12, 13,14]. In [13], some fixed point results using F -contraction on a space with two metrics have been proved. In [15], fixed point results for sequence of mappings were presented.

Fractional calculus is a generalization of ordinary differentiation and integration to an arbitrary (non-integer) order. In 1822, Fourier suggested an integral representation [16] in order to define the fractional derivative, and his version can be considered as the first definition for the fractional derivative of arbitrary positive order. It has recently been applied in various areas of engineering, science, finance, applied mathematics, and modeling in bio-engineering. Fractional boundary value problems (BVPs) are considered by many authors [17,18,19].

Motivated by the mentioned works and keeping applications of the fractional calculus in mind, in this article, we prove fixed point results using F -lipschitzian conditions of rational-type with two metrics on a space, where F is a semi-Wardowski function. F -lipschitzian conditions of rational type allow us to choose the constant involved in F -contraction τ ∈ R , instead of τ > 0 . Our results generalize many well-known results regarding fixed points of self-mappings on a set with two metrics in the literature. Examples are given for the validation of our results and F -lipschitzian conditions of rational-type. As an application, existence results for the solution of new type of ABC-fractional BVP are proved. The remaining article is distributed in the following way.

In Section 2, some preliminaries and relevant results from the literature are given. We present our main results and examples in Section 3. The application of our main results and existence results for the solution of a new type of fractional BVP are given in Section 4.

2 Preliminaries

We recall some concepts related to Maia and Wardowski’s fixed point results for single-valued mappings on a space with two metrics.

Theorem 1

[20] Let W be a nonempty set, ρ and δ two metrics on W , and ϒ : W → W . If

∃ c > 0 : ρ ( ϒ ( υ ) , ϒ ( ω ) ) ≤ c δ ( υ , ω ) , for all υ , ω ∈ W ,
( W , ρ ) is a complete metric space,
ϒ : ( W , ρ ) → ( W , ρ ) is continuous,
∃ α ∈ ] 0 , 1 [ : δ ( ϒ ( υ ) , ϒ ( ω ) ) ≤ α δ ( υ , ω ) , for all υ , ω ∈ W ,
then ϒ has a unique fixed point υ ∗ , which can be calculated as lim n → ∞ ϒ n ( υ 0 ) for any υ 0 ∈ W .

Maia’s result was further generalized as follows.

Theorem 2

[15] Let W be a nonempty set, ρ and δ two metrics on W , and ϒ n , ϒ : W → W . If

( W , ρ ) , ( W , δ ) , and ϒ satisfy the hypotheses of Theorem 1,
the sequence ( ϒ n ) n ∈ N converges uniformly on ( W , ρ ) to ϒ ,
F ϒ n ≠ ϕ ,
∃ c 1 > 0 : δ ( υ , ω ) ≤ c 1 ρ ( υ , ω ) , for all υ , ω ∈ W ,
c c 1 α < 1 ,
then for every υ n ∈ F ϒ n , the sequence ( υ n ) n ∈ N converges in ( W , ρ ) to the unique fixed point, υ ∗ , of ϒ .

In [8], a new type of contractive mapping was introduced as F -contraction and defined as follows.

Definition 3

[8] Let ℱ denote the collection of all functions F : R + ⟶ R with the following properties:

F increases strictly, i.e. for s , t ∈ R + with s < t implies that F ( s ) < F ( t ) ;
For any sequence { α n } n ∈ N of real numbers with α n ≥ 0 , lim n ⟶ ∞ α n = 0 if and only if lim n ⟶ ∞ F ( α n ) = − ∞ ;
There exists k ∈ ( 0 , 1 ) satisfying lim α ⟶ 0 + α k F ( α ) = 0 .

A mapping ϒ : W ⟶ W is called an F -contraction if there exists τ > 0 and F ∈ ℱ satisfying

for all υ , ω ∈ W , ( ρ ( ϒ υ , ϒ ω ) > 0 ⇒ τ + F ( ρ ( ϒ υ , ϒ ω ) ) ≤ F ( ρ ( υ , ω ) ) ) .

The function F which satisfies ( F 1 ) – ( F 3 ) is called the Wardowski function. In [14], it has been given that ( F 2 ) can be replaced by ( F 2 ) ′ for any sequence ( ℓ n ) ⊂ ( 0 , ∞ ) , if lim t n ⟶ 0 + F ( ℓ n ) = − ∞ , then lim n ⟶ ∞ ℓ n = 0 .

Any function F that satisfies ( F 1 ) and ( F 2 ) ′ is called the semi-Wardowski function.

Theorem 4

[21] Let ( W , ρ ) be a complete metric space and let ϒ : W ⟶ W be a mapping. If there exist F ∈ ℱ and τ > 0 such that

∀ υ , ω ∈ W , ρ ( ϒ υ , ϒ ω ) > 0 implies that τ + F ( ρ ( ϒ υ , ϒ ω ) ) ≤ F ( M ( υ , ω ) ) ,

where

M ( υ , ω ) = max ρ ( υ , ω ) , ρ ( υ , ϒ υ ) , ρ ( ω , ϒ ω ) , 1 2 [ ρ ( υ , ϒ ω ) + ρ ( ω , ϒ υ ) ] ,

then ϒ has a unique fixed point in W, provided that ϒ or F is continuous.

The above theorem has been generalized as follows.

Theorem 5

[13] Let a complete metric space ( W , ρ ′ ) be given and ρ be another metric on W . Let ϒ : W ⟶ W be a mapping and F ∈ ℱ be continuous. If there exists τ > 0 with

∀ υ , ω ∈ W , ρ ( ϒ υ , ϒ ω ) > 0 implies that τ + F ( ρ ( ϒ υ , ϒ ω ) ) ≤ F ( M ( υ , ω ) ) ,

where

M ( υ , ω ) = max ρ ( υ , ω ) , ρ ( υ , ϒ υ ) , ρ ( ω , ϒ ω ) , 1 2 [ ρ ( υ , ϒ ω ) + ρ ( ω , ϒ υ ) ] .

Moreover, if ρ ≱ ρ ′ and ϒ is a uniformly continuous function from ( W , ρ ) into ( W , ρ ′ ) , and if ρ ≠ ρ ′ , with ϒ is a continuous function from ( W , ρ ′ ) into ( W , ρ ′ ) , then there is a point z ∈ W with z = ϒ z .

If we choose F ( α ) = ln α in the aforementioned theorem, then the following theorem holds.

Theorem 6

[22] Let a complete metric space ( W , ρ ′ ) be given and ρ be another metric on W . Let ϒ : W ⟶ W be a mapping. Suppose there exists q ∈ ( 0 , 1 ) such that for υ , ω ∈ W we have

(1) ρ ( ϒ υ , ϒ ω ) ≤ q M ( υ , ω ) .

Moreover, if ρ ≱ ρ ′ , and ϒ is a uniformly continuous function from ( W , ρ ) into ( W , ρ ′ ) , and if ρ ≠ ρ ′ , with ϒ is a continuous function from ( W , ρ ′ ) into ( W , ρ ′ ) , then there is a point z ∈ W with z = ϒ z .

The following definitions are essential for proving our main results.

Definition 7

[23] Let W be a nonempty set and ϒ : W ⟶ W . The orbit of an element υ ∈ W is defined by O ( υ ) = { υ , ϒ υ , ϒ 2 υ , … } .

Definition 8

[23] Let ( W , ρ ) be a metric space and ϒ : W ⟶ W be a mapping, ϒ is called orbitally continuous if

lim k ⟶ ∞ ϒ n k υ = u

implies

lim k ⟶ ∞ ϒ ϒ n k υ = ϒ u

for each υ ∈ W .

A space ( W , ρ ) is ϒ -orbitally complete if every Cauchy sequence from orbit { ϒ n k υ } k = 1 ∞ , υ ∈ W converges in W .

Definition 9

[24] Let ϒ be a self-mapping on a metric space ( W , ρ ) . We say that ϒ is called a Lipschitz mapping with constant k ≥ 0 if

(2) ρ ( ϒ υ , ϒ ω ) ≤ k ρ ( υ , ω ) for all υ , ω ∈ W .

Clearly every Lipschitz mapping is uniformly continuous on W . A Lipschitz mapping with k < 1 is called strict (or Banach) contractions with k = 1 called nonexpansive. Lipschitz mappings are generally also known as Lipschitzian.

Lemma 10

[25] For a given semi-Wardowski function F, there exists a countable subset Ω ⊂ ( 0 , ∞ ) satisfying

F ( t − 0 ) = F ( t ) = F ( t + 0 )

for each t ∈ ( 0 , ∞ ) \ Ω .

Lemma 11

[25] Let F be a given semi-Wardowski function. Then

F ( t ) < F ( s ) ⇒ t < s

for all t , s ∈ ( 0 , ∞ ) .

3 Main results

In this section, we prove our main results of fixed points using F -lipschitzian conditions of rational-type with two metrics on a space, where F is a semi-Wardowski function. An example to validate our main result is also provided.

Theorem 12

Let ρ 1 and ρ 2 be two metrics on a nonempty set W , let ϒ : W → W be an orbitally ρ 2 -continuous and ( W , ρ 2 ) be an orbitally complete metric space, if there exists τ such that

(3) τ + F ( ρ 1 ( ϒ υ , ϒ ω ) ) ≤ F ( M ( υ , ω ) ) ,

where the function F is a semi-Wardowski function and

M ( υ , ω ) = max { ρ 1 ( υ , ϒ υ ) , ρ 1 ( ω , ϒ ω ) , ρ 1 ( υ , ω ) } , if τ > 0 max ρ 1 ( ϒ υ , ω ) ρ 1 2 ( υ , ϒ ω ) + ρ 1 ( υ , ϒ ω ) ρ 1 2 ( ϒ υ , ω ) ρ 1 2 ( υ , ϒ ω ) + ρ 1 2 ( ϒ υ , ω ) , ρ 1 ( υ , ϒ ω ) ρ 1 ( ϒ υ , ω ) ρ 1 ( υ , ϒ ω ) + ρ 1 ( ϒ υ , ω ) , if τ ∈ R and with F is right continuous at 0 ,

provided ρ 1 2 ( υ , ϒ ω ) + ρ 1 2 ( ϒ υ , ω ) ≠ 0 and ρ 1 ( υ , ϒ ω ) + ρ 1 ( ϒ υ , ω ) ≠ 0 , for all υ , ω ∈ W .

If ρ 1 2 ( υ , ϒ ω ) + ρ 1 2 ( ϒ υ , ω ) = 0 or ρ 1 ( υ , ϒ ω ) + ρ 1 ( ϒ υ , ω ) = 0 , then M ( υ , ω ) = 0 .

In addition,

(4) if ρ 1 ≱ ρ 2 , assume ϒ : ( W , ρ 1 ) → ( W , ρ 2 ) is orbitally uniformly continuous

and

(5) if ρ 1 ≠ ρ 2 , assume ϒ : ( W , ρ 2 ) → ( W , ρ 2 ) is orbitally continuous .

Then ϒ has a fixed point.

Proof

For υ 0 ∈ W , consider the iterative sequence

υ n + 1 = ϒ υ n , for n = 0 , 1 , 2 , …

and substituting υ = υ n − 1 and ω = υ n in ( 3 ) , we have

τ + F ( ρ 1 ( ϒ υ n − 1 , ϒ υ n ) ) ≤ F ( M ( υ n − 1 , υ n ) ) ,

where

M ( υ n − 1 , υ n ) = max { ρ 1 ( υ n − 1 , ϒ υ n − 1 ) , ρ 1 ( υ n , ϒ υ n ) , ρ 1 ( υ n − 1 , υ n ) } .

So we have

τ + F ( ρ 1 ( υ n , υ n + 1 ) ) ≤ F ( max { ρ 1 ( υ n , υ n + 1 ) , ρ 1 ( υ n − 1 , υ n ) } ) , if τ > 0 .

max { ρ 1 ( υ n , υ n + 1 ) , ρ 1 ( υ n − 1 , υ n ) } = ρ 1 ( υ n , υ n + 1 ) ,

then, we have

τ + F ( ρ 1 ( υ n , υ n + 1 ) ) ≤ F ( ρ 1 ( υ n , υ n + 1 ) ) , if τ > 0 F ( ρ 1 ( υ n , υ n + 1 ) ) < F ( ρ 1 ( υ n , υ n + 1 ) )

if and only if

ρ 1 ( υ n , υ n + 1 ) < ρ 1 ( υ n , υ n + 1 ) ,

which is a contradiction. So

max { ρ 1 ( υ n , υ n + 1 ) , ρ 1 ( υ n − 1 , υ n ) } = ρ 1 ( υ n − 1 , υ n ) .

Then, we have

(6) τ + F ( ρ 1 ( υ n , υ n + 1 ) ) ≤ F ( ρ 1 ( υ n − 1 , υ n ) ) , if τ > 0 F ( ρ 1 ( υ n , υ n + 1 ) ) < F ( ρ 1 ( υ n − 1 , υ n ) )

if and only if

(7) ρ 1 ( υ n , υ n + 1 ) < ρ 1 ( υ n − 1 , υ n )

and

M ( υ n − 1 , υ n ) = max ρ 1 ( ϒ υ n − 1 , υ n ) ρ 1 2 ( υ n − 1 , ϒ υ n ) + ρ 1 ( υ n − 1 , ϒ υ n ) ρ 1 2 ( ϒ υ n − 1 , υ n ) ρ 1 2 ( υ n − 1 , ϒ υ n ) + ρ 1 2 ( ϒ υ n − 1 , υ n ) , ρ 1 ( υ n − 1 , ϒ υ n ) ρ 1 ( ϒ υ n − 1 , υ n ) ρ 1 ( υ n − 1 , ϒ υ n ) + ρ 1 ( ϒ υ n − 1 , υ n ) = max 0 ρ 1 2 ( υ n − 1 , υ n + 1 ) , 0 ρ 1 ( υ n − 1 , υ n + 1 ) = 0 , for τ ∈ R .

So we have

(8) τ + lim n → ∞ F ( ρ 1 ( ϒ υ n − 1 , ϒ υ n ) ) ≤ lim s n → 0 + F ( s n ) , if τ ∈ R and F is right continuous at 0 ,

where

s n = lim n → ∞ M ( υ n − 1 , υ n ) , for τ ∈ R .

lim n → ∞ F ( ρ 1 ( υ n , υ n + 1 ) ) ≤ lim s n → 0 + F ( s n ) − τ lim n → ∞ F ( ρ 1 ( υ n , υ n + 1 ) ) ≤ − ∞ , since τ ∈ R .

Then, by ( F 2 ) ′

lim n → ∞ ρ 1 ( υ n , υ n + 1 ) = 0 .

From (7), we have { ρ 1 ( υ n − 1 , υ n ) } is nonincreasing and bounded from below so converges to some α ≥ 0 . We claim α = 0 , if not then there exists some k ∈ N such that ρ 1 ( υ n − 1 , υ n ) > 0 for all n ≥ k . Using (7), we have α < α , which is a contradiction, therefore α = 0 . Hence, in both cases ( τ > 0 and τ ∈ R ) we have

lim n → ∞ ρ 1 ( υ n , υ n + 1 ) = 0 .

Now to prove { υ n } is a Cauchy sequence, we use Lemma ( 10 ) , since F is strictly increasing. Let Ω be the set of all points of discontinuities of F , clearly Ω can be at most countable. Now suppose contrary that { υ n } is not Cauchy, and there exists ε > 0 , ε ∉ Ω such that for every k ≥ 0 , we can choose natural numbers n k and m k such that

ρ 1 ( υ m k , υ n k ) > ε , for n k > m k ≥ k .

Denote by m ¯ k and n ¯ k the least of m k and n k , respectively, such that

ρ 1 ( υ m ¯ k , υ n ¯ k ) > ε

for all k ≥ 0 . Choose k 0 ∈ N such that ε k < ε for all k ≥ k 0 , we have

ε < ρ 1 ( υ m ¯ k , υ n ¯ k ) ≤ ρ 1 ( υ m ¯ k , υ n ¯ k − 1 ) + ρ 1 ( υ n ¯ k − 1 , υ n ¯ k ) ≤ ε + ε n ¯ k

for all k ≥ k 0 . Taking lim as k → ∞ , we have

ε < lim k → ∞ ρ 1 ( υ m ¯ k , υ n ¯ k ) ≤ ε + lim k → ∞ ε n ¯ k ⇒ lim k → ∞ ρ 1 ( υ m ¯ k , υ n ¯ k ) → ε .

Now for all k ≥ 0 , we have

(A) ρ 1 ( υ m ¯ k + 1 , υ n ¯ k + 1 ) ≤ ρ 1 ( υ m ¯ k , υ m ¯ k + 1 ) + ρ 1 ( υ m ¯ k , υ n ¯ k ) + ρ 1 ( υ n ¯ k , υ n ¯ k + 1 ) ρ 1 ( υ m ¯ k + 1 , υ n ¯ k + 1 ) ≤ ε m ¯ k + 1 + ρ 1 ( υ m ¯ k , υ n ¯ k ) + ε n ¯ k + 1 .

Also as

(B) ρ 1 ( υ m ¯ k , υ n ¯ k ) ≤ ρ 1 ( υ m ¯ k , υ m ¯ k + 1 ) + ρ 1 ( υ m ¯ k + 1 , υ n ¯ k + 1 ) + ρ 1 ( υ n ¯ k , υ n ¯ k + 1 ) ρ 1 ( υ m ¯ k , υ n ¯ k ) ≤ ε m ¯ k + 1 + ρ 1 ( υ m ¯ k + 1 , υ n ¯ k + 1 ) + ε n ¯ k + 1 ρ 1 ( υ m ¯ k , υ n ¯ k ) − ε m ¯ k + 1 − ε n ¯ k + 1 ≤ ρ 1 ( υ m ¯ k + 1 , υ n ¯ k + 1 ) .

From (A) and (B), we have

ρ 1 ( υ m ¯ k , υ n ¯ k ) − ε m ¯ k + 1 − ε n ¯ k + 1 ≤ ρ 1 ( υ m ¯ k + 1 , υ n ¯ k + 1 ) ≤ ε m ¯ k + 1 + ρ 1 ( υ m ¯ k , υ n ¯ k ) + ε n ¯ k + 1 .

Taking lim as k → ∞ , we have

ε ≤ lim k → ∞ ρ 1 ( υ m ¯ k + 1 , υ n ¯ k + 1 ) ≤ ε

lim k → ∞ ρ 1 ( υ m ¯ k + 1 , υ n ¯ k + 1 ) → ε .

Now from ( 6 ) , we have

τ ≤ F ( ρ 1 ( υ m ¯ k , υ n ¯ k ) ) − F ( ρ 1 ( υ m ¯ k + 1 , υ n ¯ k + 1 ) ) , k ≥ 0 , and for τ > 0 .

Taking again lim as k → ∞ and using the fact that F is continuous at ε , we have

τ ≤ F ( ε ) − F ( ε ) = 0 ,

which is a contradiction. Also from ( 8 ) , we have

τ + lim k → ∞ F ( ρ 1 ( υ m ¯ k + 1 , υ n ¯ k + 1 ) ) ≤ lim s → 0 + F ( s ) , if τ ∈ R and F is right continuous at 0 .

Using the fact that F is continuous at ε , we have

τ + F ( ε ) ≤ − ∞ τ ≤ − ∞ ,

which is a contradiction.

Hence, { υ n } is a Cauchy sequence in ( W , ρ 1 ) .

Now we claim that { υ n } is a Cauchy sequence in ( W , ρ 2 ) . The case ρ 1 ≥ ρ 2 is trivial. Suppose ρ 1 ≱ ρ 2 and by using the orbitally uniform continuity of ϒ , there exists δ ( ε ) > 0 , such that ρ 2 ( ϒ υ , ϒ ω ) < ε whenever ρ 1 ( υ , ω ) < δ ( ε ) . This implies ρ 2 ( υ m , υ n ) < ε for all m , n ≥ k , whenever ρ 1 ( υ m − 1 , υ n − 1 ) < δ . Hence, { υ n } is a Cauchy sequence in ( W , ρ 2 ) . As ( W , ρ 2 ) is ϒ orbitally complete, so there exists υ ∈ W such that υ n → υ . Also ϒ is ρ 2 -orbitally continuous, so υ n → υ implies ϒ υ n → ϒ υ . Then for n → ∞ , we have

υ = ϒ υ .

If ρ 1 ≠ ρ 2 , then

0 ≤ ρ 2 ( υ , ϒ υ ) ≤ ρ 2 ( υ , υ n ) + ρ 2 ( υ n , ϒ υ ) = ρ 2 ( υ , υ n ) + ρ 2 ( ϒ υ n − 1 , ϒ υ ) .

For n → ∞ and using ( 5 ) , we have

0 ≤ ρ 2 ( υ , ϒ υ ) ≤ 0 ⇒ υ = ϒ υ .

Now if ρ 1 = ρ 2 and υ ≠ ϒ υ , then there exists a natural number k 1 and a subsequence { υ n k } of { υ n } such that ρ 1 ( υ n k , ϒ υ ) > 0 for all k ≥ k 1 . (If not then clearly υ n k → ϒ υ , and we are done.) From ( 3 ) , we have

(9) τ + F ( ρ 1 ( ϒ υ n k , ϒ υ ) ) ≤ F ( M ( υ n k , υ ) ) ,

where

M ( υ n k , υ ) = max { ρ 1 ( υ n k , ϒ υ n k ) , ρ 1 ( υ , ϒ υ ) , ρ 1 ( υ n k , υ ) } , for τ > 0 .

So (9) becomes

τ + F ( ρ 1 ( ϒ υ n k , ϒ υ ) ) ≤ F ( max { ρ 1 ( υ n k , ϒ υ n k ) , ρ 1 ( υ , ϒ υ ) , ρ 1 ( υ n k , υ ) } ) , if τ > 0 F ( ρ 1 ( υ n k + 1 , ϒ υ ) ) < F ( max { ρ 1 ( υ n k , υ n k + 1 ) , ρ 1 ( υ , ϒ υ ) , ρ 1 ( υ n k , υ ) } )

if and only if

ρ 1 ( υ n k + 1 , ϒ υ ) < max { ρ 1 ( υ n k , υ n k + 1 ) , ρ 1 ( υ , ϒ υ ) , ρ 1 ( υ n k , υ ) } , if τ > 0 .

For k → ∞ , we have

ρ 1 ( υ , ϒ υ ) < max { 0 , ρ 1 ( υ , ϒ υ ) } , if τ > 0 .

max { 0 , ρ 1 ( υ , ϒ υ ) } = ρ 1 ( υ , ϒ υ ) ,

then we have

ρ 1 ( υ , ϒ υ ) < ρ 1 ( υ , ϒ υ ) ,

which is a contradiction. If

max { 0 , ρ 1 ( υ , ϒ υ ) } = 0 ,

then we have

ρ 1 ( υ , ϒ υ ) < 0 ,

which is a contradiction. Hence,

υ = ϒ υ

and

M ( υ n k , υ ) = max ρ 1 ( ϒ υ n k , υ ) ρ 1 2 ( υ n k , ϒ υ ) + ρ 1 ( υ n k , ϒ υ ) ρ 1 2 ( ϒ υ n k , υ ) ρ 1 2 ( υ n k , ϒ υ ) + ρ 1 2 ( ϒ υ n k , υ ) , ρ 1 ( υ n k , ϒ υ ) ρ 1 ( ϒ υ n k , υ ) ρ 1 ( υ n k , ϒ υ ) + ρ 1 ( ϒ υ n k , υ ) , for τ ∈ R .

So (9) becomes

τ + F ( ρ 1 ( ϒ υ n k , ϒ υ ) ) ≤ F max ρ 1 ( ϒ υ n k , υ ) ρ 1 2 ( υ n k , ϒ υ ) + ρ 1 ( υ n k , ϒ υ ) ρ 1 2 ( ϒ υ n k , υ ) ρ 1 2 ( υ n k , ϒ υ ) + ρ 1 2 ( ϒ υ n k , υ ) , ρ 1 ( υ n k , ϒ υ ) ρ 1 ( ϒ υ n k , υ ) ρ 1 ( υ n k , ϒ υ ) + ρ 1 ( ϒ υ n k , υ ) , if τ ∈ R and F is right continuous at 0 .

Now for k → ∞ , we have

τ + lim k → ∞ F ( ρ 1 ( υ n k + 1 , ϒ υ ) ) ≤ lim k → ∞ F max ρ 1 ( υ n k + 1 , υ ) ρ 1 2 ( υ n k , ϒ υ ) + ρ 1 ( υ n k , ϒ υ ) ρ 1 2 ( υ n k + 1 , υ ) ρ 1 2 ( υ n k , ϒ υ ) + ρ 1 2 ( υ n k + 1 , υ ) , ρ 1 ( υ n k , ϒ υ ) ρ 1 ( υ n k + 1 , υ ) ρ 1 ( υ n k , ϒ υ ) + ρ 1 ( υ n k + 1 , υ ) .

For k → ∞ , and using the right continuity of F at 0, we have

τ + lim k → ∞ F ( ρ 1 ( υ n k + 1 , ϒ υ ) ) ≤ lim s n k ∗ → 0 + F ( s n k ∗ ) ,

where

s n k ∗ = lim k → ∞ M ( υ n k , υ ) , for τ ∈ R .

Then we have

lim k → ∞ F ( ρ 1 ( υ n k + 1 , ϒ υ ) ) ≤ lim s n k ∗ → 0 + F ( s n k ∗ ) − τ lim k → ∞ F ( ρ 1 ( υ n k + 1 , ϒ υ ) ) ≤ − ∞ , since τ ∈ R .

By ( F 2 ) ′ ,

lim k → ∞ ρ 1 ( υ n k + 1 , ϒ υ ) = 0 υ = ϒ υ .

This proves the result.□

Now we provide special cases of our main result.

If we take F ( t ) = ln t , in Theorem ( 12 ) , we obtain the following interesting result.

Corollary 13

Let ρ 1 and ρ 2 be two metrics on a nonempty set W , let ϒ : W → W be an orbitally ρ 2 -continuous, and ( W , ρ 2 ) be orbitally complete metric space, if there exists τ such that

ρ 1 ( ϒ υ , ϒ ω ) ≤ e − τ M ( υ , ω ) ,

where

M ( υ , ω ) = max { ρ 1 ( υ , ϒ υ ) , ρ 1 ( ω , ϒ ω ) , ρ 1 ( υ , ω ) } , if τ > 0 max ρ 1 ( ϒ υ , ω ) ρ 1 2 ( υ , ϒ ω ) + ρ 1 ( υ , ϒ ω ) ρ 1 2 ( ϒ υ , ω ) ρ 1 2 ( υ , ϒ ω ) + ρ 1 2 ( ϒ υ , ω ) , ρ 1 ( υ , ϒ ω ) ρ 1 ( ϒ υ , ω ) ρ 1 ( υ , ϒ ω ) + ρ 1 ( ϒ υ , ω ) , if τ ∈ R ,

provided ρ 1 2 ( υ , ϒ ω ) + ρ 1 2 ( ϒ υ , ω ) ≠ 0 and ρ 1 ( υ , ϒ ω ) + ρ 1 ( ϒ υ , ω ) ≠ 0 , for all υ , ω ∈ W .

If ρ 1 2 ( υ , ϒ ω ) + ρ 1 2 ( ϒ υ , ω ) = 0 or ρ 1 ( υ , ϒ ω ) + ρ 1 ( ϒ υ , ω ) = 0 , then M ( υ , ω ) = 0 . In addition, if ρ 1 ≱ ρ 2 , assume ϒ : ( W , ρ 1 ) → ( W , ρ 2 ) is orbitally uniformly continuous and if ρ 1 ≠ ρ 2 , assume ϒ : ( W , ρ 2 ) → ( W , ρ 2 ) is orbitally continuous. Then ϒ has a fixed point.

Corollary 14

Let ρ 1 and ρ 2 be two metrics on a nonempty set W , let ϒ : W → W be an orbitally ρ 2 -continuous and ( W , ρ 2 ) be orbitally complete metric space, if there exists τ such that

ρ 1 ( ϒ υ , ϒ ω ) ≤ M ( υ , ω ) ,

where

M ( υ , ω ) = α max { ρ 1 ( υ , ϒ υ ) , ρ 1 ( ω , ϒ ω ) , ρ 1 ( υ , ω ) } , if α ∈ ( 0 , 1 ) β max ρ 1 ( ϒ υ , ω ) ρ 1 2 ( υ , ϒ ω ) + ρ 1 ( υ , ϒ ω ) ρ 1 2 ( ϒ υ , ω ) ρ 1 2 ( υ , ϒ ω ) + ρ 1 2 ( ϒ υ , ω ) , ρ 1 ( υ , ϒ ω ) ρ 1 ( ϒ υ , ω ) ρ 1 ( υ , ϒ ω ) + ρ 1 ( ϒ υ , ω ) , if β > 0 ,

provided ρ 1 2 ( υ , ϒ ω ) + ρ 1 2 ( ϒ υ , ω ) ≠ 0 and ρ 1 ( υ , ϒ ω ) + ρ 1 ( ϒ υ , ω ) ≠ 0 , for all υ , ω ∈ W .

If we assume ρ 1 = ρ 2 , in Theorem ( 12 ) , then we obtain the following corollary.

Corollary 15

Let ( W , ρ ) be a ϒ -orbitally complete metric space and ϒ : W → W be an orbitally continuous mapping. If there exists τ such that

τ + F ( ρ ( ϒ υ , ϒ ω ) ) ≤ F ( M ( υ , ω ) ) ,

where the function F is a semi-Wardowski function and

M ( υ , ω ) = max { ρ ( υ , ϒ υ ) , ρ ( ω , ϒ ω ) , ρ ( υ , ω ) } , if τ > 0 max ρ ( ϒ υ , ω ) ρ 2 ( υ , ϒ ω ) + ρ ( υ , ϒ ω ) ρ 2 ( ϒ υ , ω ) ρ 2 ( υ , ϒ ω ) + ρ 2 ( ϒ υ , ω ) , ρ ( υ , ϒ ω ) ρ ( ϒ υ , ω ) ρ ( υ , ϒ ω ) + ρ ( ϒ υ , ω ) , if τ ∈ R and F is right continuous at 0 ,

provided ρ 2 ( υ , ϒ ω ) + ρ 2 ( ϒ υ , ω ) ≠ 0 and ρ ( υ , ϒ ω ) + ρ ( ϒ υ , ω ) ≠ 0 , for all υ , ω ∈ W .

If ρ 2 ( υ , ϒ ω ) + ρ 2 ( ϒ υ , ω ) = 0 or ρ ( υ , ϒ ω ) + ρ ( ϒ υ , ω ) = 0 , then M ( υ , ω ) = 0 . Then ϒ has a fixed point.

Example 16

Let W = 1 2 σ : σ ∈ N ∪ { 0 } with

ρ 1 ( υ , ω ) = 0 , if υ = ω 1 + ∣ υ − ω ∣ , if υ ≠ ω

and

ρ 2 ( υ , ω ) = ∣ υ − ω ∣ .

Define a map ϒ : W → W by

ϒ υ = 0 , if υ = 0 1 2 σ + 1 , if υ = 1 2 σ , σ ∈ N .

Since

sup σ ∈ N ρ 1 ϒ 1 2 σ , ϒ ( 0 ) M 1 2 σ , 0 = sup σ ∈ N ρ 1 1 2 σ + 1 , 0 max ρ 1 1 2 σ , ϒ 1 2 σ , ρ 1 ( 0 , ϒ ( 0 ) ) , ρ 1 1 2 σ , 0 , 1 2 ρ 1 1 2 σ , ϒ ( 0 ) + ρ 1 ϒ 1 2 σ , 0 ,

where

M ( υ , ω ) = max ρ 1 ( υ , ω ) , ρ 1 ( υ , ϒ υ ) , ρ 1 ( ω , ϒ ω ) , 1 2 [ ρ 1 ( υ , ϒ ω ) + ρ 1 ( ω , ϒ υ ) ] .

sup σ ∈ N ρ 1 ϒ 1 2 σ , ϒ ( 0 ) M 1 2 σ , 0 = sup σ ∈ N 1 + 1 2 σ + 1 max 1 + 1 2 σ − 1 2 σ + 1 , 0 , 1 + 1 2 σ − 0 , 1 2 1 + 1 2 σ − 0 + 1 + 1 2 σ + 1 − 0 = sup σ ∈ N 1 + 1 2 σ + 1 max 1 + 1 2 σ − 1 2 σ + 1 , 1 + 1 2 σ , 1 + 3 2 σ + 2 ⇒ sup σ ∈ N ρ 1 ϒ 1 2 σ , ϒ ( 0 ) M 1 2 σ , 0 = sup σ ∈ N 1 + 1 2 σ + 1 1 + 1 2 σ = 1 .

We cannot find q ∈ ( 0 , 1 ) satisfying the inequality (1). Therefore, Theorem (6) cannot be applied to this example.

Now consider for α > 0 , F ( α ) = ln α + α ,

then lipschitzian condition of Theorem ( 12 ) is equivalent to the following:

(10) τ + ln ( ρ 1 ( ϒ υ , ϒ ω ) ) + ρ 1 ( ϒ υ , ϒ ω ) ≤ ln ( M ( υ , ω ) ) + M ( υ , ω ) ρ 1 ( ϒ υ , ϒ ω ) M ( υ , ω ) e ρ 1 ( ϒ υ , ϒ ω ) − M ( υ , ω ) ≤ e − τ .

First observe that for all κ , σ ∈ N ,

ρ 1 ϒ 1 2 κ , ϒ ( 0 ) > 0 and ρ 1 ϒ 1 2 κ , ϒ 1 2 σ > 0 ⇔ ( κ > σ ) .

Thus, we must consider the following two cases:

Case 1: We have

ρ 1 ϒ 1 2 κ , ϒ ( 0 ) = ρ 1 1 2 κ + 1 , 0 = 1 + 1 2 κ + 1

and for τ > 0 , we have

M 1 2 κ , 0 = max ρ 1 1