Quasi-Contractive Mappings in Modular Metric Spaces

Abstract

We prove the existence of fixed point and uniqueness of quasi-contractive mappings in modular metric spaces which was introduced by Ćirić

1. Introduction and Preliminaries

In this paper, we prove the existence and uniqueness of fixed points of quasi-contractive mappings in modular metric spaces which develop the theory of metric spaces generated by modulars. Throughout the paper 𝔛 is a nonempty set and 𝜆>0. The notion of a metric modular was introduced by Chistyakov [1] as follows.

Definition 1.1. A function 𝜔∶(0,∞)×𝔛×𝔛→[0,∞] is said to be a metric modular on 𝔛 (or, simply, a modular if no ambiguity arises) if it satisfies three axioms:(i)for any 𝑥,𝑦∈𝔛, 𝜔_𝜆(𝑥,𝑦)=0 for all 𝜆>0 if and only if 𝑥=𝑦;(ii)𝜔_𝜆(𝑥,𝑦)=𝜔_𝜆(𝑦,𝑥) for all 𝜆>0, and 𝑥,𝑦∈𝔛;(iii)𝜔_𝜆+𝜇(𝑥,𝑦)≤𝜔_𝜆(𝑥,𝑧)+𝜔_𝜇(𝑧,𝑦) for all 𝜆,𝜇>0 and 𝑥,𝑦∈𝔛.

Definition 1.2. Let (𝔛,𝜔) be a metric modular space.(1)A sequence {𝑥_𝑛} in 𝔛_𝜔 is said to be 𝜔-convergent to a point 𝑥∈𝔛 if, for all 𝜆>0, 𝜔_𝜆𝑥_𝑛,𝑥⟶0(1.1) as 𝑛→∞.(2)A subset ℭ of 𝔛_𝜔 is said to be 𝜔-closed if the 𝜔-limit of a 𝜔-convergent sequence of ℭ always belongs to ℭ.(3)A subset ℭ of 𝔛_𝜔 is said to be 𝜔-complete if every 𝜔-Cauchy sequence in ℭ is 𝜔-convergent and its 𝜔-limit is in ℭ.

Definition 1.3. The metric modular 𝜔 is said to have the Fatou property if 𝜔_𝜆(𝑥,𝑦)≤liminf_𝑛→∞𝜔𝑥_𝑛,𝑦(1.2) for all 𝑦∈𝔛_𝜔 and 𝜆∈(0,∞), where {𝑥_𝑛}𝜔-converges to 𝑥.

2. Main Results

Definition 2.1. Let (𝔛,𝜔) be a metric modular space, and let ℭ be a nonempty subset of 𝔛_𝜔. The self-mapping 𝑇∶ℭ→ℭ is said to be quasi-contraction if there exists 0<𝑘<1 such that 𝜔_𝜆(𝑇(𝑥),𝑇(𝑦))≤𝑘max𝜔_𝜆(𝑥,𝑦),𝜔_𝜆(𝑥,𝑇(𝑥)),𝜔_𝜆(𝑦,𝑇(𝑦)),𝜔_𝜆(𝑥,𝑇(𝑦)),𝜔_𝜆(𝑇(𝑥),𝑦)(2.1) for any 𝑥,𝑦∈𝔛 and 𝜆∈(0,∞).

Let 𝑇∶ℭ→ℭ be a mapping, and let ℭ be a nonempty subset of 𝔛_𝜔. For any 𝑥∈ℭ, define the orbit𝒪(𝑥)=𝑥,𝑇(𝑥),𝑇²(𝑥),…(2.2) and its 𝜔-diameter by𝛿_𝜔(𝑥)=diam(𝒪(𝑥))=sup𝜔_𝜆(𝑇^𝑛(𝑥),𝑇^𝑚(𝑥))∶𝑛,𝑚∈ℕ).(2.3)

Lemma 2.2. Let (𝔛,𝜔) be a metric modular space, and let ℭ be a nonempty subset of 𝔛_𝜔. Let 𝑇∶ℭ→ℭ be a quasi-contractive mapping, and let 𝑥∈ℭ be such that 𝛿_𝜔(𝑥)<∞. Then, for any 𝑛≥1, one has 𝛿_𝜔(𝑇(𝑥))≤𝑘^𝑛𝛿_𝜔(𝑥),(2.4) where 𝑘 is the constant associated with the mapping of 𝑇. Moreover, one has 𝜔_𝜆𝑇^𝑛(𝑥),𝑇^𝑛+𝑚(𝑥)≤𝑘^𝑛𝛿_𝜔(𝑥)(2.5) for any 𝑛,𝑚≥1 and 𝜆∈(0,∞).

Proof. For each 𝑛,𝑚≥1, we have 𝜔_𝜆(𝑇^𝑛(𝑥),𝑇^𝑚(𝑦))≤𝑘max𝜔𝜔_𝜆𝑇^𝑛−1(𝑥),𝑇^𝑚−1(𝑦),𝜔_𝜆𝑇^𝑛−1(𝑥),𝑇^𝑛(𝑥)_𝜆,𝑇^𝑚−1(𝑦),𝑇^𝑚(𝑦),𝜔_𝜆𝑇^𝑛−1(𝑥),𝑇^𝑚(𝑦),𝜔_𝜆𝑇^𝑛(𝑥),𝑇^𝑚−1(𝑦)(2.6) for any 𝑥,𝑦∈ℭ and 𝜆∈(0,∞). This obviously implies that 𝛿_𝜔(𝑇^𝑛(𝑥))≤𝑘𝛿_𝜔𝑇^𝑛−1(𝑥)(2.7) for any 𝑛≥1. Hence, for any 𝑛≥1, we have 𝛿_𝜔(𝑇^𝑛(𝑥))≤𝑘^𝑛𝛿_𝜔(𝑥).(2.8) Moreover, for any 𝑛,𝑚≥1, we have 𝜔_𝜆(𝑇^𝑛(𝑥),𝑇^𝑛+𝑚(𝑥)≤𝛿_𝜔(𝑇^𝑛(𝑥))≤𝑘^𝑛𝛿_𝜔(𝑥).(2.9) This completes the proof.

The next lemma is helpful to prove the main result in this paper.

Lemma 2.3. Let (𝔛,𝜔) be a modular metric space, and let ℭ be a 𝜔-complete nonempty subset of 𝔛_𝜔. Let 𝑇∶ℭ→ℭ be quasi-contractive mapping, and let 𝑥∈ℭ be such that 𝛿_𝜔(𝑥)<∞. Then {𝑇^𝑛(𝑥)}𝜔-converges to a point 𝜈∈ℭ. Moreover, one has 𝜔_𝜆(𝑇^𝑛(𝑋)−𝜈)≤𝑘^𝑛𝛿_𝜔(𝑥)(2.10) for all 𝑛≥1 and 𝜆∈(0,∞).

Proof. From Lemma 2.2, we know that {𝑇^𝑛(𝑥)} is a 𝜔-Cauchy sequence in ℭ. Since ℭ is 𝜔-complete, then there exists 𝜈∈ℭ such that {𝑇^𝑛(𝑥)}𝜔-converges to 𝜈. Since 𝜔_𝜆𝑇^𝑛(𝑥),𝑇^𝑛+𝑚(𝑥)≤𝑘^𝑛𝛿_𝜔(𝑥)(2.11) for any 𝑛,𝑚≥1 and 𝜔 satisfies the Fatou property, and letting 𝑚→∞, we have 𝜔_𝜆(𝑇^𝑛(𝑥),𝜈)≤liminf_𝑚→∞𝜔_𝜆𝑇^𝑛(𝑥),𝑇^𝑛+𝑚(𝑥)≤𝑘^𝑛𝛿_𝜔(𝑥).(2.12) This completes the proof.

Next, we prove that 𝜈 is, in fact, a fixed point of 𝑇 and it is unique provided some extra assumptions.

Theorem 2.4. Let 𝑇,ℭ, and 𝑥 be as in Lemma 2.3. Suppose that 𝜔_𝜆(𝜈,𝑇(𝜈))<∞ and 𝜔_𝜆(𝑥,𝑇(𝑥))<∞ for all 𝜆∈(0,∞). Then the 𝜔-limit of {𝑇^𝑛(𝑥)} is a fixed point of 𝑇, that is, 𝑇(𝜈)=𝜈. Moreover, if 𝜈^∗ is any fixed point of 𝑇 in ℭ such that 𝜔_𝜆(𝜈,𝜈^∗)<∞ for all 𝜆∈(0,∞), then one has 𝜈=𝜈^∗.

Proof. We have 𝜔_𝜆(𝑇(𝑥),𝑇(𝜈))≤𝑘max𝜔_𝜆(𝑥,𝜈),𝜔_𝜆(𝑥,𝑇(𝑥)),𝜔_𝜆(𝜈,𝑇(𝜈)),𝜔_𝜆(𝑥,𝑇(𝜈)),𝜔_𝜆(𝑇(𝑥),𝜈).(2.13) From Lemma 2.3, it follows that 𝜔_𝜆(𝑇(𝑥),𝑇(𝜈))≤𝑘max𝛿_𝜔(𝑥),𝜔_𝜆(𝜈,𝑇(𝜈)),𝜔_𝜆(𝑥,𝑇(𝜈)).(2.14) Suppose that, for each 𝑛≥1, 𝜔_𝜆(T^𝑛(𝑥),𝑇(𝜈))≤max𝑘^𝑛𝛿_𝜔(𝑥),𝑘𝜔_𝜆(𝜈,𝑇(𝜈)),𝑘^𝑛𝜔_𝜆(𝑥,𝑇(𝜈)).(2.15) Then we have 𝜔_𝜆≤𝑇𝑘m^𝑛+1a(x𝑥),𝜔𝑇_𝜆(𝜈)(𝑇^𝑛(𝑥),𝜈),𝜔_𝜆𝑇^𝑛(𝑥),𝑇^𝑛+1(𝑥),𝜔_𝜆(𝜈,𝑇(𝜈)),𝜔_𝜆(𝑇^𝑛(𝑥),𝑇(𝜈)),𝜔_𝜆𝑇^𝑛+1(𝑥),𝜈.(2.16) Hence we have 𝜔_𝜆𝑇^𝑛+1(𝑥),𝑇(𝜈)≤𝑘max𝑘^𝑛𝛿_𝜔(𝑥),𝑘𝜔_𝜆(𝜈,𝑇(𝜈)),𝜔_𝜆(𝑇^𝑛(𝑥),𝑇(𝜈)).(2.17) Using our previous assumption, we get 𝜔_𝜆𝑇^𝑛+1(𝑥),𝑇(𝜈)≤max𝑘^𝑛+1𝛿_𝜔(𝑥),𝑘𝜔_𝜆(𝜈,𝑇(𝜈)),𝑘^𝑛+1𝜔_𝜆(𝑥,𝑇(𝜈)).(2.18) Thus, by induction, we have 𝜔_𝜆(𝑇^𝑛(𝑥),𝑇(𝜈))≤max𝑘^𝑛𝛿_𝜔(𝑥),𝑘𝜔_𝜆(𝜈,𝑇(𝜈)),𝑘^𝑛𝜔_𝜆(𝑥,𝑇(𝜈))(2.19) for any 𝑛≥1 and 𝜆∈(0,∞). Therefore, we have limsup_𝑛→∞𝜔_𝜆(𝑇^𝑛(𝑥),𝑇(𝑥))≤𝜔(𝜈,𝑇(𝜈))(2.20) for all 𝜆∈(0,∞). Using the Fatou property for the metric modular 𝜔, we get 𝜔_𝜆(𝜈,𝑇(𝜈))liminf_𝑛→∞𝜔_𝜆(𝑇^𝑛(𝑥),𝑇(𝜈))≤𝑘𝜔(𝜈,𝑇(𝜈))(2.21) for all 𝜆∈(0,∞). Since 𝑘<1, we get 𝜔_𝜆(𝜈,𝑇(𝜈))=0 for all 𝜆∈(0,∞), and so 𝑇(𝜈)=𝜈.
Let 𝜈^∗ be another fixed point of 𝑇 such that 𝜔_𝜆(𝜈,𝜈^∗)<∞ for all 𝜆∈(0,∞). Then we have 𝜔_𝜆𝜈,𝜈^∗=𝜔_𝜆𝑇(𝜈),𝑇𝜈^∗≤𝑘𝜔_𝜆𝜈,𝜈^∗,(2.22) which implies that 𝜔_𝜆𝜈,𝜈^∗=0(2.23) for all 𝜆∈(0,∞). Hence 𝜈=𝜈^∗. This complete the proof.

Acknowledgments

The authors are thankful to the anonymous referees and the area editor Professor Rudong Chen for their critical remarks which helped greatly to improve the presentation of this paper. The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011-0021821).