Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces

Wasfi Shatanawi; Ishak Altun; Hacer Dağ

doi:10.1515/math-2020-0149

Article Open Access

Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces

Wasfi Shatanawi , Ishak Altun and Hacer Dağ

Published/Copyright: May 22, 2020

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$Open Mathematics$

From the journal Open Mathematics Volume 18 Issue 1

Abstract

In this paper, we present some fixed point results for multivalued mappings with both closed values and proximinal values on left K-complete quasi metric spaces. We also provide a nontrivial example to illustrate our results.

Keywords: fixed point; multivalued mappings; quasi metric space

MSC 2010: Primary 47H10; Secondary 54H25

1 Introduction and preliminaries

In recent studies, many authors have provided new fixed point results to fill out gaps in the literature by taking into account different conditions on different spaces. Accordingly, in this paper, we want to complete one of these literature gaps on the fixed point theory. Therefore, we provide some new fixed point results by considering a recent contractive technique, which is called F-contraction, for multivalued mappings on some kind of complete quasi metric spaces.

Now, we recall some basic concepts of quasi metric spaces.

Let M be a nonempty set and ρ : M × M → ℝ + be a function. Consider the following conditions: for all ζ, η, ξ ∈ M

(ρ ₁) ρ(ζ, ζ) = 0,
(ρ ₂) ρ(ζ, η) ≤ ρ(ζ, ξ) + ρ(ξ,η),
(ρ ₃) ρ ( ζ , η ) = ρ ( η , ζ ) = 0 ⇒ ζ = η ,
(ρ ₄) ρ ( ζ , η ) = 0 ⇒ ζ = η .

If (ρ ₁),(ρ ₂) and (ρ ₃) hold, then function ρ is called a quasi metric on M. If a quasi metric ρ also satisfies (ρ ₄), then it is called a T ₁-quasi metric. We can easily see that every metric is a T ₁-quasi metric and every T ₁-quasi metric is a quasi metric. If ρ is a quasi metric on M, then ρ ⁻¹ is also a quasi metric, where

ρ − 1 ( ζ , η ) := ρ ( η , ζ )

for ζ, η ∈ M. On the other hand, if ρ is a quasi metric on M, then function ρ ^s is metric on M, where

ρ s ( ζ , η ) =: max { ρ ( ζ , η ) , ρ − 1 ( ζ , η ) }

for ζ, η ∈ M. If ρ is a quasi metric on M, then ρ generates a T ₀ topology on M. The base of this topology is the family of open balls. We will denote it by τ _ρ. If ρ is T ₁-quasi metric, then τ _ρ is T ₁ topology on M. The closure of a subset A of M with respect to τ _ρ, τ _ρ−1 and τ ρ s is denoted by c l τ ρ ( A ) , c l τ ρ − 1 ( A ) and c l τ ρ s ( A ) , respectively.

Let (M, ρ) be a quasi metric space and ζ ∈ M. The convergence of a sequence {ζ _n} to ζ with respect to τ _ρ is defined by

ζ n ⟶ ρ ζ ⇔ ρ ( ζ , ζ n ) → 0 .

We will call it ρ-convergence and denote by ζ n ⟶ ρ ζ . Similarly, we can define ρ ⁻¹-convergence and ρ ^s-convergence.

Definition 1

[1,2] Let {ζ _n} be a sequence in a quasi metric space (M, ρ). Then, {ζ _n} is called

left K-Cauchy if for every ε > 0, there exists n 0 ∈ ℕ such that
∀ n , k , n ≥ k ≥ n 0 , ρ ( ζ k , ζ n ) < ε ,
right K-Cauchy if for every ε > 0, there exists n 0 ∈ ℕ such that
∀ n , k , n ≥ k ≥ n 0 , ρ ( ζ n , ζ k ) < ε ,
ρ ^s-Cauchy if for every ε > 0, there exists n 0 ∈ ℕ such that

∀ n , k ≥ n 0 , ρ ( ζ n , ζ k ) < ε .

Definition 2

[1,2] Let (M, ρ) be a quasi metric space.

If every left (right) K-Cauchy sequence is ρ-convergent, then (M, ρ) is called left (right) K -complete.
If every left (right) K-Cauchy sequence is ρ ⁻¹-convergent, then (M, ρ) is called left (right) ℳ -complete.
If every left (right) K-Cauchy sequence is ρ ^s-convergent, then (M, ρ) is called left (right) Smyth complete.

Let (M, ρ) be a quasi metric space. We will consider the following family:

P ( M ) = { A ⊆ M : A is nonempty } , C ρ ( M ) = { A ⊆ M : A is nonempty and τ ρ -closed } , K ρ ( M ) = { A ⊆ M : A is nonempty and τ ρ -compact } .

Also, we will denote the family of all subsets A of M satisfying the following property by A ρ ( M ) :

there exists a = a ( ζ ) ∈ A such that ρ ( ζ , A ) = ρ ( ζ , a ) for all ζ ∈ M .

In fact, A ρ ( M ) is the family of all τ _ρ-proximinal subsets of M. It is clear that, if ρ is a metric on M, then K ρ ( M ) ⊆ A ρ ( M ) ⊆ C ρ ( M ) ⊆ P ( M ) . If ρ is a quasi metric on M, then each one of these classes is independent from each other. However, although there is no connection between these classes on quasi metric spaces, if (M, ρ) is a T ₁-quasi metric space, then A ρ ( M ) ⊆ C ρ ( M ) (for more details see [3]).

We can find many fixed point results for both single valued and multivalued mappings on quasi metric spaces in the literature (see, for example, [2,4,5,6]).

Recently, Wardowski [7] considered the following family of functions to give more general contractive condition for the fixed point theory on metric spaces. Let W be the family of all functions F : ( 0 , ∞ ) → ℝ satisfying the following:

(W1) For all α, β ∈ (0, ∞) such that α < β, F(α) < F(β).

(W2) For each sequence {a _n} of positive numbers lim_n→∞ a _n = 0 if and only if lim_n→∞ F(a _n) = −∞.

(W3) There exists k ∈ (0, 1) such that lim_α→∞ + α ^k F(α) = 0.

Many authors have extended fixed point results on metric spaces by considering family W (see, for example, [8,9,10,11]). Then, some fixed point results for multivalued mappings which are closed values on metric spaces have been obtained by adding the following condition (W4) (see, for example, [12,13,14,15,16]):

(W4) F(inf A) = inf F(A) for all A ⊂ (0, ∞) with inf A > 0.

It is clear that, if function F satisfies (W1), then it satisfies (W4) if and only if it is right continuous. We denote by W ⁎ the set of all functions F satisfying (W1)–(W4).

Let (M, ρ) be a quasi metric space, S : M → P ( M ) be a multivalued mapping, F ∈ W and σ ≥ 0. For ζ ∈ M with ρ(ζ, Sζ) > 0, define the set F σ ζ ⊆ M as

F σ ζ = { η ∈ S ζ : F ( ρ ( ζ , η ) ) ≤ F ( ρ ( ζ , S ζ ) ) + σ } .

It is obvious that, if σ ₁ ≤ σ ₂, then F σ 1 ζ ⊆ F σ 2 ζ . There are some cases for F σ ζ that are investigated by Dağ et al. [3]: for ζ ∈ M with ρ(ζ, Sζ) > 0,

if S ζ ∈ A ρ ( M ) , then F σ ζ ≠ ∅ for all σ ≥ 0,
if S ζ ∈ K ρ ( M ) , then F σ ζ may be empty for some ζ ∈ M and σ > 0,
if S ζ ∈ C ρ ( M ) , then F σ ζ may be empty for some ζ ∈ M and σ > 0,
if S ζ ∈ C ρ ( M ) (even if S ζ ∈ P ( M ) ) and F ∈ W ⁎ , then for all σ > 0, we have F σ ζ ≠ ∅ .

Now, we recall one of the aspects of development of the multivalued fixed point theory. In 1969, Nadler [17] obtained the first fixed point result for contractive-type multivalued mappings thus and so every multivalued contraction on a complete metric space has a fixed point. This result have lighted the way for new fixed point theorems for multivalued mappings. For example, Feng and Liu [18] proved the following:

Theorem 1

[18] Let (M, ρ) be a complete metric space and S : M → C ρ ( M ) . Assume that there exist b, c ∈ (0,1) such that for any ζ ∈ M there is η ∈ Sζ satisfying

b ρ ( ζ , η ) ≤ ρ ( ζ , S ζ )

and

ρ ( η , S η ) ≤ c ρ ( ζ , η ) .

If c < b and ζ → ρ(ζ, Sζ) is lower semi-continuous, then S has a fixed point in M.

Now, we will order some studies, which are on fixed point results of multivalued contractive maps in the light of Feng and Liu’s [18] theorem, to show their development from a complete metric space to a complete quasi metric space.

First, Minak et al. [15] extended Feng and Liu’s theorem as follows:

Theorem 2

[15, Theorem 2.6] Let (M, ρ) be a complete metric space and S : M → C ρ ( M ) be a multivalued map and F ∈ W ⁎ . Assume that there exists τ > 0 such that for any ζ ∈ M with ρ(ζ, Sζ) > 0, there is η ∈ F σ ζ satisfying

τ + F ( ρ ( η , S η ) ) ≤ F ( ρ ( ζ , η ) ) .

If σ < τ and ζ → ρ(ζ, Sζ) is lower semi-continuous, then S has a fixed point in M.

In Theorem 2.5 of [15], Minak et al. showed that by taking K ρ ( M ) instead of C ρ ( M ) , condition (F4) can be removed in Theorem 2.

Then, these results are carried out on a complete quasi metric space by Dağ et al. [3] as follows:

Theorem 3

[3, Theorem 3 (resp. Theorem 1)] Let (M, ρ) be a left K -complete quasi (resp. T ₁ -quasi) metric space, S : M → C ρ ( M ) (resp. S : M → A ρ ( M ) ) be a multivalued mapping and F ∈ W ⁎ (resp. F ∈ W ). Assume that there exists τ > 0 such that for any ζ ∈ M with ρ(ζ, Sζ) > 0, there is η ∈ F σ ζ satisfying

τ + F ( ρ ( η , S η ) ) ≤ F ( ρ ( ζ , η ) ) .

If σ < τ and ζ → ρ(ζ, Sζ) is lower semi-continuous with respect to τ _ρ, then S has a fixed point in M.

In the same study, Dağ et al. [3] obtained some other results by taking into account left ℳ -completeness of quasi metric spaces (see Theorems 2 and 4 in [3]).

On the other hand, a different kind of generalization of Theorem 1 was presented by Klim and Wardowski [19] as follows:

Theorem 4

[19] Let (M, ρ) be a complete metric space and S : M → C ρ ( M ) . Assume that the following conditions hold:

there exists b ∈ (0, 1) and a function φ: [0, ∞) → [0, b) satisfying
lim sup t → s + φ ( t ) < b , ∀ s ≥ 0 ,
for any ζ ∈ M, there is η ∈ Sζ satisfying

b ρ ( ζ , η ) ≤ ρ ( ζ , S ζ )

and

ρ ( η , S η ) ≤ φ ( ρ ( ζ , η ) ) ρ ( ζ , η ) .

Then, S has a fixed point in M provided that ζ → ρ(ζ, Sζ) is lower semi-continuous.

Then, in the following theorem Altun et al. [20] provided a proper generalization of Theorem 4 by taking into account the F-contractive technique.

Theorem 5

[20, Theorem 10] Let (M, ρ) be a complete metric space and S : M → C ρ ( M ) and F ∈ W ⁎ . Assume that the following conditions hold:

there exists σ > 0 and a function τ: (0, ∞) → (σ, ∞) such that

lim inf t → s + τ ( t ) > σ for all s ≥ 0 ,

for any ζ ∈ M with ρ(ζ, S ζ) > 0, there exists η ∈ F σ ζ satisfying

τ ( ρ ( ζ , η ) ) + F ( ρ ( η , S η ) ) ≤ F ( ρ ( ζ , η ) ) .

Then, S has a fixed point in M provided that ζ → ρ(ζ, Sζ) is lower semi-continuous.

In the same study, Altun et al. [20] also presented Theorem 11 by taking K ρ ( M ) instead of C ρ ( M ) and they removed condition (W4) on F. Then, these results are also carried out on a complete quasi metric space by Altun and Dağ [4] as follows:

Theorem 6

[4, Theorem 7 (resp. Theorem 5)] Let (M, ρ) be a left K -complete quasi (resp. T ₁-quasi) metric space and S : M → C ρ ( M ) (resp. S : M → A ρ ( M ) ) be a multivalued mapping and F ∈ W ⁎ (resp. F ∈ W ). Assume that the following conditions hold:

there exists σ > 0 and function τ: (0, ∞) → (σ, ∞) such that
lim inf t → s + τ ( t ) > σ for all s ≥ 0
for any ζ ∈ M with ρ(ζ, Sζ ) > 0, there exists η ∈ F σ ζ satisfying

τ ( ρ ( ζ , η ) ) + F ( ρ ( η , S η ) ) ≤ F ( ρ ( ζ , η ) ) .

Then, S has a fixed point in M provided that ζ → ρ(ζ, Sζ) is lower semi-continuous with respect to τ _ρ.

Considering the same direction, in 2009, Ćirić [21] introduced new multivalued nonlinear contractions and established a few nice fixed point theorems for such mappings, one of them is as follows:

Theorem 7

[21] Let (M, ρ) be a complete metric space and S : M → C ρ ( M ) . Assume that the following conditions hold:

there exists a function φ: [0, ∞) → [a, 1), 0 < a < 1, satisfying
lim sup t → s + φ ( t ) < 1 , ∀ s ≥ 0 ,
for any ζ ∈ M, there is η ∈ Sζ satisfying

φ ( ρ ( ζ , S ζ ) ) ρ ( ζ , η ) ≤ ρ ( ζ , S ζ )

and

ρ ( η , S η ) ≤ φ ( ρ ( ζ , S ζ ) ) ρ ( ζ , η ) .

Then, S has a fixed point in M provided that ζ → ρ(ζ, Sζ ) is lower semi-continuous.

Then, Altun et al. [14] gave a generalization of Ćirić’s theorem as follows:

Theorem 8

[14, Theorem 13] Let (M, ρ) be a complete metric space, S : M → C ρ ( M ) be a mapping and F ∈ W ⁎ . Assume that the following conditions hold:

there exists a function τ: (0, ∞) → (0, σ], σ > 0 such that
lim inf t → s + τ ( t ) > 0 , ∀ s ≥ 0 ;
for any ζ ∈ M with ρ(ζ, Sζ ) > 0, there is η ∈ Sζ satisfying

F ( ρ ( ζ , η ) ) ≤ F ( ρ ( ζ , S ζ ) ) + τ ( ρ ( ζ , S ζ ) ) 2

and

τ ( ρ ( ζ , S ζ ) ) + F ( ρ ( η , S η ) ) ≤ F ( ρ ( ζ , η ) ) .

If ζ → ρ(ζ, Sζ ) is lower semi-continuous, then S has a fixed point in M.

In the same study, Altun et al. [14] also gave Theorem 14 by considering K ρ ( M ) instead of C ρ ( M ) and so they relaxed family W ⁎ by taking into account family W .

At this point, we shall provide some quasi metric versions of Theorem 8.

2 Main results

At the beginning, we want to emphasize the following: Let (M, ρ) be a quasi metric space, S : M → P ( M ) be a multivalued mapping and τ: (0, ∞) → (0, σ], (σ > 0) be a function. Then,

if ρ(ζ, Sζ ) > 0, S ζ ∈ A ρ ( M ) for ζ ∈ M and F ∈ W , then there exists η ∈ Sζ satisfying
(2.1) F ( ρ ( ζ , η ) ) ≤ F ( ρ ( ζ , S ζ ) ) + τ ( ρ ( ζ , S ζ ) ) 2 ,
if ρ(ζ, Sζ ) > 0, S ζ ∈ C ρ ( M ) (even if S ζ ∈ P ( M ) ) for ζ ∈ M and F ∈ W ⁎ , then there exists η ∈ Sζ satisfying (2.1).

Theorem 9

Let (M, ρ) be a left K -complete quasi metric space, S : M → C ρ ( M ) and F ∈ W ⁎ . Assume that the following conditions hold:

the map ζ → ρ(ζ, Sζ) is lower semi-continuous with respect to τ _ρ,
there exists a function τ: (0, ∞) → (0, σ], σ > 0 such that
(2.2) lim inf t → s + τ ( t ) > 0 , ∀ s ≥ 0 ,
for any ζ ∈ M with ρ(ζ, Sζ) > 0, there is η ∈ Sζ satisfying (2.1) and

(2.3) τ ( ρ ( ζ , S ζ ) ) + F ( ρ ( η , S η ) ) ≤ F ( ρ ( ζ , η ) ) .

Then, S has a fixed point in M.

Proof

First, assume that S has no fixed point in M. Then, ρ(ζ, Sζ ) > 0 for all ζ ∈ M. (Note that, if ρ(ζ, Sζ ) = 0 for some ζ ∈ M, then ζ ∈ c l τ ρ ( S ζ ) = S ζ = Sζ since S ζ ∈ C ρ ( M ) .) Therefore, since τ(t) > 0 for all t > 0 and F ∈ W ⁎ , then for any ζ ∈ M there exists η ∈ Sζ such that (2.1) holds. Let ζ ₀ ∈ M be an initial point. By assumptions (2.1) and (2.3), we can choose ζ ₁ ∈ Sζ ₀ such that

(2.4) F ( ρ ( ζ 0 , ζ 1 ) ) ≤ F ( ρ ( ζ 0 , S ζ 0 ) ) + τ ( ρ ( ζ 0 , S ζ 0 ) ) 2

and

(2.5) τ ( ρ ( ζ 0 , S ζ 0 ) ) + F ( ρ ( ζ 1 , S ζ 1 ) ) ≤ F ( ρ ( ζ 0 , ζ 1 ) ) .

From (2.4) and (2.5), we get

(2.6) τ ( ρ ( ζ 0 , S ζ 0 ) ) 2 + F ( ρ ( ζ 1 , S ζ 1 ) ) ≤ F ( ρ ( ζ 0 , S ζ 0 ) ) .

Now, we choose ζ ₂ ∈ Sζ ₁ such that

F ( ρ ( ζ 1 , ζ 2 ) ) ≤ F ( ρ ( ζ 1 , S ζ 1 ) ) + τ ( ρ ( ζ 1 , S ζ 1 ) ) 2

and

τ ( ρ ( ζ 1 , S ζ 1 ) ) + F ( ρ ( ζ 2 , S ζ 2 ) ) ≤ F ( ρ ( ζ 1 , ζ 2 ) ) .

Hence, we get

τ ( ρ ( ζ 1 , S ζ 1 ) ) 2 + F ( ρ ( ζ 2 , S ζ 2 ) ) ≤ F ( ρ ( ζ 1 , S ζ 1 ) ) .

Continuing this process, we can choose a sequence {ζ _n} such that ζ _n+1 ∈ Sζ _n satisfying

(2.7) F ( ρ ( ζ n , ζ n + 1 ) ) ≤ F ( ρ ( ζ n , S ζ n ) ) + τ ( ρ ( ζ n , S ζ n ) ) 2

and

(2.8) τ ( ρ ( ζ n , S ζ n ) ) 2 + F ( ρ ( ζ n + 1 , S ζ n + 1 ) ) ≤ F ( ρ ( ζ n , S ζ n ) )

for all n ≥ 0.

Now, we will show that ρ(ζ _n, Sζ _n) → 0 as n → ∞. From (2.8), we conclude that {ρ(ζ _n, Sζ _n)} is a decreasing sequence of positive real numbers. Therefore, there exists δ ≥ 0 such that

lim n → ∞ ρ ( ζ n , S ζ n ) = δ .

Suppose δ > 0. Then, since F is right continuous, taking the limit inferior on both sides of (2.8) and having in mind assumption (2.2), we have

lim inf ρ ( ζ n , S ζ n ) → δ + τ ( ρ ( ζ n , S ζ n ) ) 2 + F ( δ ) ≤ F ( δ ) ,

which is a contradiction. Thus, δ = 0, that is,

(2.9) lim n → ∞ ρ ( ζ n , S ζ n ) = 0 .

Now, we shall show that {ζ _n} is a left K-Cauchy sequence in ζ. Let

α = lim inf ρ ( ζ n , S ζ n ) → δ + τ ( ρ ( ζ n , S ζ n ) ) 2 > 0 .

and 0 < q < α. Then, there exists n 0 ∈ ℕ such that τ ( ρ ( ζ n , S ζ n ) ) 2 > q for all n ≥ n ₀. Thus, from (2.8),

q + F ( ρ ( ζ n + 1 , S ζ n + 1 ) ) ≤ F ( ρ ( ζ n , S ζ n ) )

for each n ≥ n ₀. Hence, by induction, for all n ≥ n ₀

(2.10) F ( ρ ( ζ n + 1 , S ζ n + 1 ) ) ≤ F ( ρ ( ζ n , S ζ n ) ) − q ⋮ ≤ F ( ρ ( ζ n 0 , S ζ n 0 ) ) − ( n + 1 − n 0 ) q .

Since 0 < τ(t) ≤ σ for all t > 0. From (2.7), we get

F ( ρ ( ζ n , ζ n + 1 ) ) ≤ F ( ρ ( ζ n , S ζ n ) ) + σ .

Thus, by (2.10), for all n ≥ n ₀

(2.11) F ( ρ ( ζ n , ζ n + 1 ) ) ≤ F ( ρ ( ζ n , S ζ n ) ) + σ ≤ F ( ρ ( ζ n 0 , S ζ n 0 ) ) − ( n − n 0 ) q + σ .

From (2.11), we get lim_n→∞ F(ρ(ζ _n,ζ _n+1)) = −∞. Thus, from (W2) we have lim_n→∞ ρ(ζ _n,ζ _n+1)) = 0. Therefore, from (W3) there exists k ∈ (0, 1) such that

lim n → ∞ [ ρ ( ζ n , ζ n + 1 ) ] k F ( ρ ( ζ n , ζ n + 1 ) ) = 0 .

By (2.11), for all n ≥ n ₀

(2.12) [ ρ ( ζ n , ζ n + 1 ) ] k F ( ρ ( ζ n , ζ n + 1 ) ) − [ ρ ( ζ n , ζ n + 1 ) ] k F ( ρ ( ζ n 0 , S ζ n 0 ) ) ≤ − [ ρ ( ζ n , ζ n + 1 ) ] k [ ( n − n 0 ) q + σ ] ≤ 0 .

Letting n → ∞ in (2.12), we obtain that

(2.13) lim n → ∞ [ ρ ( ζ n , ζ n + 1 ) ] k [ ( n − n 0 ) q + σ ] = 0 .

From (2.13), there exits n 1 ∈ ℕ with n ₁ > n ₀ such that

[ ρ ( ζ n , ζ n + 1 ) ] k [ ( n − n 0 ) q + σ ] ≤ 1

for all n ≥ n ₁. So, we have, for all n ≥ n ₁

(2.14) ρ ( ζ n , ζ n + 1 ) ≤ 1 [ ( n − n 0 ) q + σ ] 1 k .

In order to show that {ζ _n} is a left K-Cauchy sequence consider m , n ∈ ℕ such that m > n ≥ n ₁. Using the triangular inequality for the quasi metric and from (2.14), we have

ρ ( ζ n , ζ m ) ≤ ρ ( ζ n , ζ n + 1 ) + ρ ( ζ n + 1 , ζ n + 2 ) + ⋯ + ρ ( ζ m − 1 , ζ m ) = ∑ i = n m − 1 ρ ( ζ i , ζ i + 1 ) ≤ ∑ i = n ∞ ρ ( ζ i , ζ i + 1 ) ≤ ∑ i = n ∞ 1 [ ( i − n 0 ) q + σ ] 1 / k .

By the convergence of the series

∑ i > n 0 − σ q 1 [ ( i − n 0 ) q + σ ] 1 / k ,

passing to limit n, m → ∞, we get ρ(ζ _n, ζ _m) → 0. This yields that {ζ _n} is a left K-Cauchy sequence in (M, ρ). Since (M, ρ) is a left K-complete quasi metric space, the sequence {ζ _n} is ρ-convergent to a point ξ ∈ M, that is, ρ(ξ, ζ _n) → 0 as n → ∞.

On the other hand, since lim_n→∞ ρ(ζ _n, Sζ _n) = −0 and function ζ → ρ(ζ, Sζ) is lower semi-continuous with respect to τ _ρ, then

0 ≤ ρ ( ξ , S ξ ) ≤ lim inf n → ∞ ρ ( ζ n , S ζ n ) = 0 .

Hence, ρ(ξ, Sξ) = 0, which is a contradiction. Therefore, S has a fixed point in M.□

The following result is left ℳ -complete version of Theorem 9.

Theorem 10

Let (M, ρ) be a left ℳ -complete quasi metric space, S : M → C ρ ( M ) and F ∈ W ⁎ . If we replace

(i′) the map ζ → ρ(ζ, Sζ) is lower semi-continuous with respect to τ _ρ−1 , instead of condition (i) at Theorem 9, then S has a fixed point in M.

Proof

Assume that S has no fixed point. Like in the proof of Theorem 9, we can construct a sequence {ζ _n}, which is left K-Cauchy. As (M, ρ) is left ℳ -complete, there is ξ ∈ M such that {ζ _n} is τ _ρ−1-convergent to ξ, that is, ρ(ζ _n, ξ ) → 0 as n → ∞. Since ζ → ρ(ζ, Sζ ) is lower semi-continuous with respect to τ _ρ−1, we have

0 ≤ ρ ( ξ , S ξ ) ≤ lim inf n → ∞ ρ ( ζ n , S ζ n )=0 .

Therefore, ρ(ξ, Sξ ) = 0, which is a contradiction. Therefore, S has a fixed point in M.□

If we take A ρ ( M ) instead of C ρ ( M ) in the aforementioned theorems, although we do not need condition (W4) on F, we need the space to be a T ₁-quasi metric space. Notice that, if ρ(ζ, Sζ ) = 0, then since S ζ ∈ A ρ ( M ) , there exists a ∈ Sζ such that ρ(ζ, a) = ρ(ζ, Sζ ) = 0. So, a = ζ ∈ Sζ because ρ is a T ₁-quasi metric. Hence, the proofs of the following theorems are obvious.

Theorem 11

Let (M, ρ) be a left K -complete T ₁ -quasi metric space, S : M → A ρ ( M ) and F ∈ W . Assume that conditions (i), (ii), and (iii) of Theorem 9 hold, then S has a fixed point in M.

Theorem 12

Let (M, ρ) be a left ℳ -complete T ₁-quasi metric space, S : M → A ρ ( M ) and F ∈ W . Assume that conditions (i′), (ii), and (iii) of Theorem 10 hold, then S has a fixed point in M.

Now, we present a nontrivial example.

Example 1

Let M = { 1 n 2 : n ∈ ℕ \ { 0 } } ∪ { 0 } and

ρ ( ζ , η ) = { ζ − η , ζ ≥ η , η − ζ 2 , ζ < η .

It is clear that (M, ρ) is a left K -complete. Let S : M → C ρ ( M ) be defined by

S ζ = { { 0 , 1 ( n + 1 ) 2 } , ζ = 1 n 2 , n > 1 , { ζ } , ζ ∈ { 0 , 1 } .

In this case,

f ( ζ ) = ρ ( ζ , S ζ ) = { 0 , ζ ∈ { 0 , 1 } , 2 n + 1 n 2 ( n + 1 ) 2 , ζ = 1 n 2 .

Since f − 1 ( ( − ∞ , α ] ) ∈ C ρ ( M ) for all α ∈ ℝ , then f is lower-semicontinuous with respect to τ _ρ. For τ(t) = ln 2 and σ = 1, condition (ii) is also satisfied. Now, we claim that condition (iii) is satisfied with

F ( α ) = { ln α α , α ≤ 1 , α − 1 , α > 1 .

It can be seen that F ∈ W ⁎ . If ρ(ζ, Sζ) > 0, then ζ = 1 n 2 , n > 1. Therefore, we choose η = 1 ( n + 1 ) 2 ∈ S ζ and so (2.1) is clearly satisfied since ρ(ζ, η) = ρ(ζ, Sζ). Also, by standard calculation we can see that

ρ ( η , S η ) 1 ρ ( η , S η ) ρ ( ζ , η ) − 1 ρ ( ζ , η ) ≤ 1 2

and so we have

ln 2 + F ( ρ ( η , S η ) ) ≤ F ( ρ ( ζ , η ) )

since the diameter of M with respect to ρ is not greater than 1. Thus, all conditions of Theorem 9 are satisfied and so S has a fixed point in M.

Now, we show that Theorem 7 cannot be applied to this example even if we consider the usual metric on M. Suppose that there exist a constant a ∈ (0, 1) and a function φ: [0, ∞) → a, 1) satisfying the assumptions in Theorem 7. Take ζ = 1 n 2 , then S ζ = { 0 , 1 ( n + 1 ) 2 } . If η = 0, then

φ ( ρ ( ζ , S ζ ) ) ρ ( ζ , η ) ≤ ρ ( ζ , S ζ ) ⇔ φ ( 2 n + 1 n 2 ( n + 1 ) 2 ) ≤ ( 2 n + 1 ) 2 ( n + 1 ) 4 .

Taking limit as n → ∞, we have the following contradiction:

0 < a ≤ lim n → ∞ φ ( 2 n + 1 n 2 ( n + 1 ) 2 ) ≤ 0 .

If η = 1 ( n + 1 ) 2 , then

ρ ( η , S η ) ≤ φ ( ρ ( ζ , S ζ ) ) ρ ( ζ , η ) ⇔ 2 n + 3 ( n + 1 ) 2 ( n + 2 ) 2 ≤ φ ( 2 n + 1 n 2 ( n + 1 ) 2 ) 2 n + 1 n 2 ( n + 1 ) 2 ⇔ ( 2 n + 3 ) n 2 ( 2 n + 1 ) ( n + 2 ) 2 ≤ φ ( 2 n + 1 n 2 ( n + 1 ) 2 ) .

Taking limit supremum as n → ∞, we have the following contradiction:

1 ≤ lim sup n → ∞ φ ( 2 n + 1 n 2 ( n + 1 ) 2 ) ≤ lim sup t → 0 + φ ( t ) < 1 .

ishak.altun@tdtu.edu.vn

Acknowledgement

The authors are thankful to the referees for making valuable suggestions leading to the better presentations of the paper. This work was supported by the Prince Sultan University through the Research Group NAMAM under Grant RG-DES-2017-01-17.

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Received: 2019-10-08

Revised: 2020-02-15

Accepted: 2020-03-27

Published Online: 2020-05-22

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2020-0149

Keywords for this article

fixed point; multivalued mappings; quasi metric space

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