Generalized 𝜶-𝝍 Contractive Type Mappings and Related Fixed Point Theorems with Applications

Abstract

We establish fixed point theorems for a new class of contractive mappings. As consequences of our main results, we obtain fixed point theorems on metric spaces endowed with a partial order and fixed point theorems for cyclic contractive mappings. Various examples are presented to illustrate our obtained results.

1. Introduction and Preliminaries

Let Ψ be the family of functions 𝜓∶[0,∞)→[0,∞) satisfying the following conditions: (Ψ₁)𝜓 is nondecreasing; (Ψ₂)∑^+∞𝑛=1𝜓^𝑛(𝑡)<∞ for all 𝑡>0, where 𝜓^𝑛 is the 𝑛th iterate of 𝜓. These functions are known in the literature as (c)-comparison functions. It is easily proved that if 𝜓 is a (c)-comparison function, then 𝜓(𝑡)<𝑡 for any 𝑡>0.

Very recently, Samet et al. [1] introduced the following concepts.

Definition 1.1. Let (𝑋,𝑑) be a metric space and 𝑇∶𝑋→𝑋 be a given mapping. We say that 𝑇 is an 𝛼-𝜓 contractive mapping if there exist two functions 𝛼∶𝑋×𝑋→[0,∞) and 𝜓∈Ψ such that 𝛼(𝑥,𝑦)𝑑(𝑇𝑥,𝑇𝑦)≤𝜓(𝑑(𝑥,𝑦)),∀𝑥,𝑦∈𝑋.(1.1)
Clearly, any contractive mapping, that is, a mapping satisfying Banach contraction, is an 𝛼-𝜓 contractive mapping with 𝛼(𝑥,𝑦)=1 for all 𝑥,𝑦∈𝑋 and 𝜓(𝑡)=𝑘𝑡, 𝑘∈(0,1).

Definition 1.2. Let 𝑇∶𝑋→𝑋 and 𝛼∶𝑋×𝑋→[0,∞). We say that 𝑇 is 𝛼-admissible if for all 𝑥,𝑦∈𝑋, and we have 𝛼(𝑥,𝑦)≥1⟹𝛼(𝑇𝑥,𝑇𝑦)≥1.(1.2) Various examples of such mappings are presented in [1].
The main results in [1] are the following fixed point theorems.

Theorem 1.3. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 be an 𝛼-𝜓 contractive mapping. Suppose that(i)𝑇 is 𝛼 admissible; (ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1; (iii)𝑇 is continuous. Then there exists 𝑢∈𝑋 such that 𝑇𝑢=𝑢.

Theorem 1.4. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 be an 𝛼-𝜓 contractive mapping. Suppose that (i)𝑇 is 𝛼 admissible; (ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1; (iii)if {𝑥_𝑛} is a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑥∈𝑋 as 𝑛→∞, then 𝛼(𝑥_𝑛,𝑥)≥1 for all 𝑛. Then there exists 𝑢∈𝑋 such that 𝑇𝑢=𝑢.

Theorem 1.5. Adding to the hypotheses of Theorem 1.3 (resp., Theorem 1.4) the condition, for all 𝑥,𝑦∈𝑋, there exists 𝑧∈𝑋 such that 𝛼(𝑥,𝑧)≥1 and 𝛼(𝑦,𝑧)≥1, and one obtains uniqueness of the fixed point.

In the present work, we introduce the concept of generalized 𝛼-𝜓 contractive type mappings, and we study the existence and uniqueness of fixed points for such mappings. Presented theorems in this paper extend and generalize the above results derived by Samet et al. in [1]. Moreover, from our fixed point theorems, we will deduce various fixed point results on metric spaces endowed with a partial order and fixed point results for cyclic contractive mappings.

2. Main Results

We introduce the concept of generalized 𝛼-𝜓 contractive type mappings as follows.

Definition 2.1. Let (𝑋,𝑑) be a metric space and 𝑇∶𝑋→𝑋 be a given mapping. We say that 𝑇 is a generalized 𝛼-𝜓 contractive mapping if there exist two functions 𝛼∶𝑋×𝑋→[0,∞) and 𝜓∈Ψ such that for all 𝑥,𝑦∈𝑋, and we have 𝛼(𝑥,𝑦)𝑑(𝑇𝑥,𝑇𝑦)≤𝜓(𝑀(𝑥,𝑦)),(2.1) where 𝑀(𝑥,𝑦)=max{𝑑(𝑥,𝑦),(𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦))/2,(𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥))/2}.

Remark 2.2. Clearly, since 𝜓 is nondecreasing, every 𝛼-𝜓 contractive mapping is a generalized 𝛼-𝜓 contractive mapping.

Our first result is the following.

Theorem 2.3. Let (𝑋,𝑑) be a complete metric space. Suppose that 𝑇∶𝑋→𝑋 is a generalized 𝛼-𝜓 contractive mapping and satisfies the following conditions: (i)𝑇 is 𝛼 admissible; (ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1; (iii)𝑇 is continuous. Then there exists 𝑢∈𝑋 such that 𝑇𝑢=𝑢.

Proof. Let 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1 (such a point exists from condition (ii)). Define the sequence {𝑥_𝑛} in 𝑋 by 𝑥_𝑛+1=𝑇𝑥_𝑛 for all 𝑛≥0. If 𝑥_𝑛0=𝑥_𝑛0+1 for some 𝑛₀, then 𝑢=𝑥_𝑛0 is a fixed point of 𝑇. So, we can assume that 𝑥_𝑛≠𝑥_𝑛+1 for all 𝑛. Since 𝑇 is 𝛼 admissible, we have 𝛼𝑥₀,𝑥₁=𝛼𝑥₀,𝑇𝑥₀≥1⟹𝛼𝑇𝑥₀,𝑇𝑥₁=𝛼𝑥₁,𝑥₂≥1.(2.2) Inductively, we have 𝛼𝑥_𝑛,𝑥_𝑛+1≥1,∀𝑛=0,1,….(2.3) From (2.1) and (2.3), it follows that for all 𝑛≥1, we have 𝑑𝑥_𝑛+1,𝑥_𝑛=𝑑𝑇𝑥_𝑛,𝑇𝑥_𝑛−1≤𝛼𝑥_𝑛,𝑥_𝑛−1𝑑𝑇𝑥_𝑛,𝑇𝑥_𝑛−1≤𝜓𝑀𝑥_𝑛,𝑥_𝑛−1.(2.4) On the other hand, we have 𝑀𝑥_𝑛,𝑥_𝑛−1=max𝑑𝑥_𝑛,𝑥_𝑛−1,𝑑𝑥^𝑛,𝑇𝑥_𝑛+𝑑𝑥_𝑛−1,𝑇𝑥_𝑛−12,𝑑𝑥^𝑛,𝑇𝑥_𝑛−1+𝑑𝑥_𝑛−1,𝑇𝑥_𝑛=max𝑑𝑥_𝑛2,𝑥_𝑛−1,𝑑𝑥^𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛−1,𝑥_𝑛2,𝑑𝑥^𝑛−1,𝑥_𝑛+1≤max𝑑𝑥_𝑛2,𝑥_𝑛−1,𝑑𝑥^𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛−1,𝑥_𝑛≤max𝑑𝑥_𝑛2,𝑥_𝑛−1,𝑑𝑥_𝑛,𝑥_𝑛+1.(2.5) From (2.4) and taking in consideration that 𝜓 is a nondecreasing function, we get that 𝑑𝑥_𝑛+1,𝑥_𝑛≤𝜓max𝑑𝑥_𝑛,𝑥_𝑛−1,𝑑𝑥_𝑛,𝑥_𝑛+1,(2.6) for all 𝑛≥1. If for some 𝑛≥1, we have 𝑑(𝑥_𝑛,𝑥_𝑛−1)≤𝑑(𝑥_𝑛,𝑥_𝑛+1), from (2.6), we obtain that 𝑑𝑥_𝑛+1,𝑥_𝑛≤𝜓𝑑𝑥_𝑛,𝑥_𝑛+1<𝑑𝑥_𝑛,𝑥_𝑛+1,(2.7) a contradiction. Thus, for all 𝑛≥1, we have max𝑑𝑥_𝑛,𝑥_𝑛−1,𝑑𝑥_𝑛,𝑥_𝑛+1=𝑑𝑥_𝑛,𝑥_𝑛−1.(2.8) Using (2.6) and (2.8), we get that 𝑑𝑥_𝑛+1,𝑥_𝑛≤𝜓𝑑𝑥_𝑛,𝑥_𝑛−1,(2.9) for all 𝑛≥1. By induction, we get 𝑑𝑥_𝑛+1,𝑥_𝑛≤𝜓^𝑛𝑑𝑥₁,𝑥₀,∀𝑛≥1.(2.10) From (2.10) and using the triangular inequality, for all 𝑘≥1, we have 𝑑𝑥_𝑛,𝑥_𝑛+𝑘≤≤𝑑𝑥_{𝑛𝑛+𝑘−1},𝑥_𝑛+1+⋯+𝑑𝑥_{𝑛+𝑘−1},𝑥_𝑛+𝑘_𝑝=𝑛≤𝜓^𝑛𝑑𝑥₁,𝑥₀_+∞_𝑝=𝑛𝜓^𝑛𝑑𝑥₁,𝑥₀⟶0as𝑛→∞.(2.11) This implies that {𝑥_𝑛} is a Cauchy sequence in (𝑋,𝑑). Since (𝑋,𝑑) is complete, there exists 𝑢∈𝑋 such that lim_𝑛→∞𝑑𝑥_𝑛,𝑢=0.(2.12) Since 𝑇 is continuous, we obtain from (2.12) that lim_𝑛→∞𝑑𝑥_𝑛+1,𝑇𝑢=lim_𝑛→∞𝑑𝑇𝑥_𝑛,𝑇𝑢=0.(2.13) From (2.12), (2.13) and the uniqueness of the limit, we get immediately that 𝑢 is a fixed point of 𝑇, that is, 𝑇𝑢=𝑢.

The next theorem does not require the continuity of 𝑇.

Theorem 2.4. Let (𝑋,𝑑) be a complete metric space. Suppose that 𝑇∶𝑋→𝑋 is a generalized 𝛼-𝜓 contractive mapping and the following conditions hold: (i)𝑇 is 𝛼 admissible; (ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1; (iii)if {𝑥_𝑛} is a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑥∈𝑋 as 𝑛→∞, then there exists a subsequence {𝑥_𝑛(𝑘)} of {𝑥_𝑛} such that 𝛼(𝑥_𝑛(𝑘),𝑥)≥1 for all 𝑘. Then there exists 𝑢∈𝑋 such that 𝑇𝑢=𝑢.

Proof. Following the proof of Theorem 2.3, we know that the sequence {𝑥_𝑛} defined by 𝑥_𝑛+1=𝑇𝑥_𝑛 for all 𝑛≥0, converges for some 𝑢∈𝑋. From (2.3) and condition (iii), there exists a subsequence {𝑥_𝑛(𝑘)} of {𝑥_𝑛} such that 𝛼(𝑥_𝑛(𝑘),𝑢)≥1 for all 𝑘. Applying (2.1), for all 𝑘, we get that 𝑑𝑥_𝑛(𝑘)+1,𝑇𝑢=𝑑𝑇𝑥_𝑛(𝑘),𝑇𝑢≤𝛼𝑥_𝑛(𝑘),𝑢𝑑𝑇𝑥_𝑛(𝑘),𝑇𝑢≤𝜓𝑀𝑥_𝑛(𝑘),𝑢.(2.14) On the other hand, we have 𝑀𝑥_𝑛(𝑘),𝑢=max𝑑𝑥_𝑛(𝑘),𝑢,𝑑𝑥^𝑛(𝑘),𝑥_𝑛(𝑘)+1+𝑑(𝑢,𝑇𝑢)2,𝑑𝑥^𝑛(𝑘),𝑇𝑢+𝑑𝑢,𝑥_𝑛(𝑘)+12.(2.15) Letting 𝑘→∞ in the above equality, we get that lim_𝑘→∞𝑀𝑥_𝑛(𝑘),𝑢=𝑑(𝑢,𝑇𝑢)2.(2.16) Suppose that 𝑑(𝑢,𝑇𝑢)>0. From (2.16), for 𝑘 large enough, we have 𝑀(𝑥_𝑛(𝑘),𝑢)>0, which implies that 𝜓(𝑀(𝑥_𝑛(𝑘),𝑢))<𝑀(𝑥_𝑛(𝑘),𝑢). Thus, from (2.14), we have 𝑑𝑥_𝑛(𝑘)+1,𝑇𝑢<𝑀𝑥_𝑛(𝑘),𝑢.(2.17) Letting 𝑘→∞ in the above inequality, using (2.16), we obtain that 𝑑(𝑢,𝑇𝑢)≤𝑑(𝑢,𝑇𝑢)2,(2.18) which is a contradiction. Thus we have 𝑑(𝑢,𝑇𝑢)=0, that is, 𝑢=𝑇𝑢.

With the following example, we will show that hypotheses in Theorems 2.3 and 2.4 do not guarantee uniqueness of the fixed point.

Example 2.5. Let 𝑋={(1,0),(0,1)}⊂ℝ² be endowed with the Euclidean distance 𝑑((𝑥,𝑦),(𝑢,𝑣))=|𝑥−𝑢|+|𝑦−𝑣| for all (𝑥,𝑦),(𝑢,𝑣)∈𝑋. Obviously, (𝑋,𝑑) is a complete metric space. The mapping 𝑇(𝑥,𝑦)=(𝑥,𝑦) is trivially continuous and satisfies for any 𝜓∈Ψ𝛼((𝑥,𝑦),(𝑢,𝑣))𝑑(𝑇(𝑥,𝑦),𝑇(𝑢,𝑣))≤𝜓(𝑀((𝑥,𝑦),(𝑢,𝑣))),(2.19) for all (𝑥,𝑦),(𝑢,𝑣)∈𝑋, where 𝛼((𝑥,𝑦),(𝑢,𝑣))=10if(𝑥,𝑦)=(𝑢,𝑣),if(𝑥,𝑦)≠(𝑢,𝑣).(2.20) Thus 𝑇 is a generalized 𝛼-𝜓 contractive mapping. On the other hand, for all (𝑥,𝑦),(𝑢,𝑣)∈𝑋, we have 𝛼((𝑥,𝑦),(𝑢,𝑣))≥1⟶(𝑥,𝑦)=(𝑢,𝑣)⟶𝑇(𝑥,𝑦)=𝑇(𝑢,𝑣)⟶𝛼(𝑇(𝑥,𝑦),𝑇(𝑢,𝑣))≥1.(2.21) Thus 𝑇 is 𝛼 admissible. Moreover, for all (𝑥,𝑦)∈𝑋, we have 𝛼((𝑥,𝑦),𝑇(𝑥,𝑦))≥1. Then the assumptions of Theorem 2.3 are satisfied. Note that the assumptions of Theorem 2.4 are also satisfied; indeed if {(𝑥_𝑛,𝑦_𝑛)} is a sequence in 𝑋 that converges to some point (𝑥,𝑦)∈𝑋  with 𝛼((𝑥_𝑛,𝑦_𝑛),(𝑥_𝑛+1,𝑦_𝑛+1))≥1 for all 𝑛, then, from the definition of 𝛼, we have (𝑥_𝑛,𝑦_𝑛)=(𝑥,𝑦) for all 𝑛, which implies that 𝛼((𝑥_𝑛,𝑦_𝑛),(𝑥,𝑦))=1 for all 𝑛. However, in this case, 𝑇 has two fixed points in 𝑋.

For the uniqueness of a fixed point of a generalized 𝛼-𝜓 contractive mapping, we will consider the following hypothesis. (H) For all 𝑥,𝑦∈Fix(𝑇), there exists 𝑧∈𝑋 such that 𝛼(𝑥,𝑧)≥1 and 𝛼(𝑦,𝑧)≥1.

Here, Fix(𝑇) denotes the set of fixed points of 𝑇.

Theorem 2.6. Adding condition (H) to the hypotheses of Theorem 2.3 (resp., Theorem 2.4), one has obtains that 𝑢 is the unique fixed point of 𝑇.

Proof. Suppose that 𝑣 is another fixed point of 𝑇. From (H), there exists 𝑧∈𝑋 such that 𝛼(𝑢,𝑧)≥1,𝛼(𝑣,𝑧)≥1.(2.22) Since 𝑇 is 𝛼 admissible, from (2.22), we have 𝛼(𝑢,𝑇^𝑛𝑧)≥1,𝛼(𝑣,𝑇^𝑛𝑧)≥1,∀𝑛.(2.23) Define the sequence {𝑧_𝑛} in 𝑋 by 𝑧_𝑛+1=𝑇𝑧_𝑛 for all 𝑛≥0 and 𝑧₀=𝑧. From (2.23), for all 𝑛, we have 𝑑𝑢,𝑧_𝑛+1=𝑑𝑇𝑢,𝑇𝑧_𝑛≤𝛼𝑢,𝑧_𝑛𝑑𝑇𝑢,𝑇𝑧_𝑛≤𝜓𝑀𝑢,𝑧_𝑛.(2.24) On the other hand, we have 𝑀𝑢,𝑧_𝑛=max𝑑𝑢,𝑧_𝑛,𝑑𝑧^𝑛,𝑧_𝑛+12,𝑑𝑢,𝑧^𝑛+1+𝑑𝑧_𝑛,𝑢≤max𝑑𝑢,𝑧_𝑛2,𝑑𝑧^𝑛,𝑢+𝑑𝑢,𝑧_𝑛+1≤max𝑑𝑢,𝑧_𝑛2,𝑑𝑢,𝑧_𝑛+1.(2.25) Using the above inequality, (2.24) and the monotone property of 𝜓, we get that 𝑑𝑢,𝑧_𝑛+1≤𝜓max𝑑𝑢,𝑧_𝑛,𝑑𝑢,𝑧_𝑛+1,(2.26) for all 𝑛. Without restriction to the generality, we can suppose that 𝑑(𝑢,𝑧_𝑛)>0 for all 𝑛. If max{𝑑(𝑢,𝑧_𝑛),𝑑(𝑢,𝑧_𝑛+1)}=𝑑(𝑢,𝑧_𝑛+1), we get from (2.26) that 𝑑𝑢,𝑧_𝑛+1≤𝜓𝑑𝑢,𝑧_𝑛+1<𝑑𝑢,𝑧_𝑛+1,(2.27) which is a contradiction. Thus we have max{𝑑(𝑢,𝑧_𝑛),𝑑(𝑢,𝑧_𝑛+1)}=𝑑(𝑢,𝑧_𝑛), and 𝑑𝑢,𝑧_𝑛+1≤𝜓𝑑𝑢,𝑧_𝑛,(2.28) for all 𝑛. This implies that 𝑑𝑢,𝑧_𝑛≤𝜓^𝑛𝑑𝑢,𝑧₀,∀𝑛≥1.(2.29) Letting 𝑛→∞ in the above inequality, we obtain that lim_𝑛→∞𝑑𝑧_𝑛,𝑢=0.(2.30) Similarly, one can show that lim_𝑛→∞𝑑𝑧_𝑛,𝑣=0.(2.31) From (2.30) and (2.31), it follows that 𝑢=𝑣. Thus we proved that 𝑢 is the unique fixed point of 𝑇.

Example 2.7. Let 𝑋=[0,1] be endowed with the standard metric 𝑑(𝑥,𝑦)=|𝑥−𝑦| for all 𝑥,𝑦∈𝑋. Obviously, (𝑋,𝑑) is a complete metric space. Define the mapping 𝑇∶𝑋→𝑋 by 𝑇𝑥=140if𝑥∈[0,1),if𝑥=1.(2.32) In this case, 𝑇 is not continuous. Define the mapping 𝛼∶𝑋×𝑋→[0,∞) by 𝛼(𝑥,𝑦)=1if(𝑥,𝑦)∈0,104×14,1∪41,1×0,14,otherwise.(2.33) We will prove that (A)𝑇∶𝑋→𝑋 is a generalized 𝛼-𝜓 contractive mapping, where 𝜓(𝑡)=𝑡/2 for all 𝑡≥0; (B)𝑇 is 𝛼-admissible; (C)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1; (D)if {𝑥_𝑛} is a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑥∈𝑋 as 𝑛→+∞, then there exists a subsequence {𝑥_𝑛(𝑘)} of {𝑥_𝑛} such that 𝛼(𝑥_𝑛(𝑘),𝑥)≥1 for all 𝑘; (E)condition (H) is satisfied.

Proof of (A). To show (A), we have to prove that (2.1) is satisfied for every 𝑥,𝑦∈𝑋. If 𝑥∈[0,1/4] and 𝑦=1, we have 𝛼(𝑥,𝑦)𝑑(𝑇𝑥,𝑇𝑦)=𝑑(𝑇𝑥,𝑇𝑦)=|||14−0|||=41𝑑(𝑦,𝑇𝑦)≤𝜓(𝑀(𝑥,𝑦)).(2.34) Then (2.1) holds. If 𝑥=1 and 𝑦∈[0,1/4], we have 𝛼(𝑥,𝑦)𝑑(𝑇𝑥,𝑇𝑦)=𝑑(𝑇𝑥,𝑇𝑦)=|||0−14|||=41𝑑(𝑥,𝑇𝑥)≤𝜓(𝑀(𝑥,𝑦)).(2.35) Then (2.1) holds also in this case. The other cases are trivial. Thus (2.1) is satisfied for every 𝑥,𝑦∈𝑋.

Proof of (B). Let (𝑥,𝑦)∈𝑋×𝑋 such that 𝛼(𝑥,𝑦)≥1. From the definition of 𝛼, we have two cases.
Case  1 (if (𝑥,𝑦)∈[0,1/4]×[1/4,1]). In this case, we have (𝑇𝑥,𝑇𝑦)∈[1/4,1]×[0,1/4], which implies that 𝛼(𝑇𝑥,𝑇𝑦)=1.
Case  2 (if (𝑥,𝑦)∈[1/4,1]×[0,1/4]). In this case, we have (𝑇𝑥,𝑇𝑦)∈[0,1/4]×[1/4,1], which implies that 𝛼(𝑇𝑥,𝑇𝑦)=1.
So, in all cases, we have 𝛼(𝑇𝑥,𝑇𝑦)≥1. Thus 𝑇 is 𝛼 admissible.

Proof of (C). Taking 𝑥₀=0, we have 𝛼(𝑥₀,𝑇𝑥₀)=𝛼(0,1/4)=1.

Proof of (D). Let {𝑥_𝑛} be a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑥 as 𝑛→+∞ for some 𝑥∈𝑋. From the definition of 𝛼, for all 𝑛, we have 𝑥_𝑛,𝑥_𝑛+1∈0,14×14,1∪14,1×0,14.(2.36) Since ([0,1/4]×[1/4,1])∪([1/4,1]×[0,1/4]) is a closed set with respect to the Euclidean metric, we get that (𝑥,𝑥)∈0,14×14,1∪14,1×0,14,(2.37) which implies that 𝑥=1/4. Thus we have 𝛼(𝑥_𝑛,𝑥)≥1 for all 𝑛.

Proof of (E). Let (𝑥,𝑦)∈𝑋×𝑋. It is easy to show that, for 𝑧=1/4, we have 𝛼(𝑥,𝑧)=𝛼(𝑦,𝑧)=1. So, condition (H) is satisfied.

Conclusion. Now, all the hypotheses of Theorem 2.6 are satisfied; thus 𝑇 has a unique fixed point 𝑢∈𝑋. In this case, we have 𝑢=1/4.

3. Consequences

Now, we will show that many existing results in the literature can be deduced easily from our Theorem 2.6.

3.1. Standard Fixed Point Theorems

Taking in Theorem 2.6, 𝛼(𝑥,𝑦)=1 for all 𝑥,𝑦∈𝑋, we obtain immediately the following fixed point theorem.

Corollary 3.1. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 be a given mapping. Suppose that there exists a function 𝜓∈Ψ such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜓(𝑀(𝑥,𝑦)),(3.1) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point.

The following fixed point theorems follow immediately from Corollary 3.1.

Corollary 3.2 (see Berinde [2]). Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 be a given mapping. Suppose that there exists a function 𝜓∈Ψ such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜓(𝑑(𝑥,𝑦)),(3.2) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point.

Corollary 3.3 (see Ćirić [3]). Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 be a given mapping. Suppose that there exists a constant 𝜆∈(0,1) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜆max𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦)2,𝑑(𝑥,𝑇𝑦)+2𝑑(𝑦,𝑇𝑥),(3.3) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point.

Corollary 3.4 (see Hardy and Rogers [4]). Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 be a given mapping. Suppose that there exist constants 𝐴,𝐵,𝐶≥0 with (𝐴+2𝐵+2𝐶)∈(0,1) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝐴𝑑(𝑥,𝑦)+𝐵[𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦)]+𝐶[𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥)],(3.4) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point.

Corollary 3.5 (see Banach Contraction Principle [5]). Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 be a given mapping. Suppose that there exists a constant 𝜆∈(0,1) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜆𝑑(𝑥,𝑦),(3.5) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point.

Corollary 3.6 (see Kannan [6]). Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 be a given mapping. Suppose that there exists a constant 𝜆∈(0,1/2) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜆[𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦)],(3.6) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point.

Corollary 3.7 (see Chatterjea [7]). Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 be a given mapping. Suppose that there exists a constant 𝜆∈(0,1/2) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜆[𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥)],(3.7) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point.

3.2. Fixed Point Theorems on Metric Spaces Endowed with a Partial Order

Recently there have been so many exciting developments in the field of existence of fixed point on metric spaces endowed with partial orders. This trend was started by Turinici [8] in 1986. Ran and Reurings in [9] extended the Banach contraction principle in partially ordered sets with some applications to matrix equations. The obtained result in [9] was further extended and refined by many authors (see, e.g., [10–15] and the references cited therein). In this section, from our Theorem 2.6, we will deduce very easily various fixed point results on a metric space endowed with a partial order. At first, we need to recall some concepts.

Definition 3.8. Let (𝑋,⪯) be a partially ordered set and 𝑇∶𝑋→𝑋 be a given mapping. We say that 𝑇 is nondecreasing with respect to ⪯ if 𝑥,𝑦∈𝑋,𝑥⪯𝑦⟹𝑇𝑥⪯𝑇𝑦.(3.8)

Definition 3.9. Let (𝑋,⪯) be a partially ordered set. A sequence {𝑥_𝑛}⊂𝑋 is said to be nondecreasing with respect to ⪯ if 𝑥_𝑛⪯𝑥_𝑛+1 for all 𝑛.

Definition 3.10. Let (𝑋,⪯) be a partially ordered set and 𝑑 be a metric on 𝑋. We say that (𝑋,⪯,𝑑) is regular if for every nondecreasing sequence {𝑥_𝑛}⊂𝑋 such that 𝑥_𝑛→𝑥∈𝑋 as 𝑛→∞, there exists a subsequence {𝑥_𝑛(𝑘)} of {𝑥_𝑛} such that 𝑥_𝑛(𝑘)⪯𝑥 for all 𝑘.

We have the following result.

Corollary 3.11. Let (𝑋,⪯) be a partially ordered set and 𝑑 be a metric on 𝑋 such that (𝑋,𝑑) is complete. Let 𝑇∶𝑋→𝑋 be a nondecreasing mapping with respect to ⪯. Suppose that there exists a function 𝜓∈Ψ such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜓(𝑀(𝑥,𝑦)),(3.9) for all 𝑥,𝑦∈𝑋 with 𝑥≽𝑦. Suppose also that the following conditions hold: (i)there exists 𝑥₀∈𝑋 such that 𝑥₀⪯𝑇𝑥₀; (ii)𝑇 is continuous or (𝑋,⪯,𝑑) is regular. Then 𝑇 has a fixed point. Moreover, if for all 𝑥,𝑦∈𝑋 there exists 𝑧∈𝑋 such that 𝑥⪯𝑧 and 𝑦⪯𝑧, one has uniqueness of the fixed point.

Proof. Define the mapping 𝛼∶𝑋×𝑋→[0,∞) by 𝛼(𝑥,𝑦)=10if𝑥⪯𝑦or𝑥⪰𝑦,otherwise.(3.10) Clearly, 𝑇 is a generalized 𝛼-𝜓 contractive mapping, that is, 𝛼(𝑥,𝑦)𝑑(𝑇𝑥,𝑇𝑦)≤𝜓(𝑀(𝑥,𝑦)),(3.11) for all 𝑥,𝑦∈𝑋. From condition (i), we have 𝛼(𝑥₀,𝑇𝑥₀)≥1. Moreover, for all 𝑥,𝑦∈𝑋, from the monotone property of 𝑇, we have 𝛼(𝑥,𝑦)≥1⟹𝑥⪰𝑦or𝑥⪯𝑦⟹𝑇𝑥⪰𝑇𝑦or𝑇𝑥⪯𝑇𝑦⟹𝛼(𝑇𝑥,𝑇𝑦)≥1.(3.12) Thus 𝑇 is 𝛼 admissible. Now, if 𝑇 is continuous, the existence of a fixed point follows from Theorem 2.3. Suppose now that (𝑋,⪯,𝑑) is regular. Let {𝑥_𝑛} be a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑥∈𝑋 as 𝑛→∞. From the regularity hypothesis, there exists a subsequence {𝑥_𝑛(𝑘)} of {𝑥_𝑛} such that 𝑥_𝑛(𝑘)⪯𝑥 for all 𝑘. This implies from the definition of 𝛼 that 𝛼(𝑥_𝑛(𝑘),𝑥)≥1 for all 𝑘. In this case, the existence of a fixed point follows from Theorem 2.4. To show the uniqueness, and let 𝑥,𝑦∈𝑋. By hypothesis, there exists 𝑧∈𝑋 such that 𝑥⪯𝑧 and 𝑦⪯𝑧, which implies from the definition of 𝛼 that 𝛼(𝑥,𝑧)≥1 and 𝛼(𝑦,𝑧)≥1. Thus we deduce the uniqueness of the fixed point by Theorem 2.6.

The following results are immediate consequences of Corollary 3.11.

Corollary 3.12. Let (𝑋,⪯) be a partially ordered set and 𝑑 be a metric on 𝑋 such that (𝑋,𝑑) is complete. Let 𝑇∶𝑋→𝑋 be a nondecreasing mapping with respect to ⪯. Suppose that there exists a function 𝜓∈Ψ such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜓(𝑑(𝑥,𝑦)),(3.13) for all 𝑥,𝑦∈𝑋  with 𝑥≽𝑦. Suppose also that the following conditions hold: (i)there exists 𝑥₀∈𝑋 such that 𝑥₀⪯𝑇𝑥₀; (ii)𝑇 is continuous or (𝑋,⪯,𝑑) is regular.
Then 𝑇 has a fixed point. Moreover, if for all 𝑥,𝑦∈𝑋 there exists 𝑧∈𝑋 such that 𝑥⪯𝑧 and 𝑦⪯𝑧, one has uniqueness of the fixed point.

Corollary 3.13. Let (𝑋,⪯) be a partially ordered set and 𝑑 be a metric on 𝑋 such that (𝑋,𝑑) is complete. Let 𝑇∶𝑋→𝑋 be a nondecreasing mapping with respect to ⪯. Suppose that there exists a constant 𝜆∈(0,1) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜆max𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦)2,𝑑(𝑥,𝑇𝑦)+2𝑑(𝑦,𝑇𝑥),(3.14) for all 𝑥,𝑦∈𝑋  with 𝑥≽𝑦. Suppose also that the following conditions hold: (i)there exists 𝑥₀∈𝑋 such that 𝑥₀⪯𝑇𝑥₀; (ii)𝑇 is continuous or (𝑋,⪯,𝑑) is regular.
Then 𝑇 has a fixed point. Moreover, if for all 𝑥,𝑦∈𝑋 there exists 𝑧∈𝑋 such that 𝑥⪯𝑧 and 𝑦⪯𝑧, one has uniqueness of the fixed point.

Corollary 3.14. Let (𝑋,⪯) be a partially ordered set and 𝑑 be a metric on 𝑋 such that (𝑋,𝑑) is complete. Let 𝑇∶𝑋→𝑋 be a nondecreasing mapping with respect to ⪯. Suppose that there exist constants 𝐴,𝐵,𝐶≥0 with (𝐴+2𝐵+2𝐶)∈(0,1) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝐴𝑑(𝑥,𝑦)+𝐵[𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦)]+𝐶[𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥)],(3.15) for all 𝑥,𝑦∈𝑋 with 𝑥≽𝑦. Suppose also that the following conditions hold: (i)there exists 𝑥₀∈𝑋 such that 𝑥₀⪯𝑇𝑥₀; (ii)𝑇 is continuous or (𝑋,⪯,𝑑) is regular.
Then 𝑇 has a fixed point. Moreover, if for all 𝑥,𝑦∈𝑋 there exists 𝑧∈𝑋 such that 𝑥⪯𝑧 and 𝑦⪯𝑧, one has uniqueness of the fixed point.

Corollary 3.15 (see Ran and Reurings [9], Nieto and López [16]). Let (𝑋,⪯) be a partially ordered set and 𝑑 be a metric on 𝑋 such that (𝑋,𝑑) is complete. Let 𝑇∶𝑋→𝑋 be a nondecreasing mapping with respect to ⪯. Suppose that there exists a constant 𝜆∈(0,1) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜆𝑑(𝑥,𝑦),(3.16) for all 𝑥,𝑦∈𝑋 with 𝑥≽𝑦. Suppose also that the following conditions hold: (i)there exists 𝑥₀∈𝑋 such that 𝑥₀⪯𝑇𝑥₀; (ii)𝑇 is continuous or (𝑋,⪯,𝑑) is regular.
Then 𝑇 has a fixed point. Moreover, if for all 𝑥,𝑦∈𝑋 there exists 𝑧∈𝑋 such that 𝑥⪯𝑧 and 𝑦⪯𝑧, one has uniqueness of the fixed point.

Corollary 3.16. Let (𝑋,⪯) be a partially ordered set and 𝑑 be a metric on 𝑋 such that (𝑋,𝑑) is complete. Let 𝑇∶𝑋→𝑋 be a nondecreasing mapping with respect to ⪯. Suppose that there exists a constant 𝜆∈(0,1/2) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜆[𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦)],(3.17) for all 𝑥,𝑦∈𝑋 with 𝑥≽𝑦. Suppose also that the following conditions hold: (i)there exists 𝑥₀∈𝑋 such that 𝑥₀⪯𝑇𝑥₀; (ii)𝑇 is continuous or (𝑋,⪯,𝑑) is regular.
Then 𝑇 has a fixed point. Moreover, if for all 𝑥,𝑦∈𝑋 there exists 𝑧∈𝑋 such that 𝑥⪯𝑧 and 𝑦⪯𝑧, one has uniqueness of the fixed point.

Corollary 3.17. Let (𝑋,⪯) be a partially ordered set and 𝑑 be a metric on 𝑋 such that (𝑋,𝑑) is complete. Let 𝑇∶𝑋→𝑋 be a nondecreasing mapping with respect to ⪯. Suppose that there exists a constant 𝜆∈(0,1/2) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜆[𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥)],(3.18) for all 𝑥,𝑦∈𝑋 with 𝑥≽𝑦. Suppose also that the following conditions hold: (i)there exists 𝑥₀∈𝑋 such that 𝑥₀⪯𝑇𝑥₀; (ii)𝑇 is continuous or (𝑋,⪯,𝑑) is regular.
Then 𝑇 has a fixed point. Moreover, if for all  𝑥,𝑦∈𝑋 there exists 𝑧∈𝑋 such that 𝑥⪯𝑧 and 𝑦⪯𝑧, one has uniqueness of the fixed point.

3.3. Fixed Point Theorems for Cyclic Contractive Mappings

One of the remarkable generalizations of the Banach Contraction Mapping Principle was reported by Kirk et al. [17] via cyclic contraction. Following the paper [17], many fixed point theorems for cyclic contractive mappings have appeared (see, e.g., [18–23]). In this section, we will show that, from our Theorem 2.6, we can deduce some fixed point theorems for cyclic contractive mappings.

We have the following result.

Corollary 3.18. Let {𝐴_𝑖}^2𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and 𝑇∶𝑌→𝑌 be a given mapping, where 𝑌=𝐴₁∪𝐴₂. Suppose that the following conditions hold: (I)𝑇(𝐴₁)⊆𝐴₂ and 𝑇(𝐴₂)⊆𝐴₁; (II)there exists a function 𝜓∈Ψ such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜓(𝑀(𝑥,𝑦)),∀(𝑥,𝑦)∈𝐴₁×𝐴₂.(3.19)Then 𝑇 has a unique fixed point that belongs to 𝐴₁∩𝐴₂.

Proof. Since 𝐴₁ and 𝐴₂ are closed subsets of the complete metric space (𝑋,𝑑), then (𝑌,𝑑) is complete. Define the mapping 𝛼∶𝑌×𝑌→[0,∞) by 𝛼(𝑥,𝑦)=10if(𝑥,𝑦)∈𝐴₁×𝐴₂∪𝐴₂×𝐴₁,otherwise.(3.20) From (II) and the definition of 𝛼, we can write 𝛼(𝑥,𝑦)𝑑(𝑇𝑥,𝑇𝑦)≤𝜓(𝑀(𝑥,𝑦)),(3.21) for all 𝑥,𝑦∈𝑌. Thus 𝑇 is a generalized 𝛼-𝜓 contractive mapping.
Let (𝑥,𝑦)∈𝑌×𝑌 such that 𝛼(𝑥,𝑦)≥1. If (𝑥,𝑦)∈𝐴₁×𝐴₂, from (I), (𝑇𝑥,𝑇𝑦)∈𝐴₂×𝐴₁, which implies that 𝛼(𝑇𝑥,𝑇𝑦)≥1. If (𝑥,𝑦)∈𝐴₂×𝐴₁, from (I), (𝑇𝑥,𝑇𝑦)∈𝐴₁×𝐴₂, which implies that 𝛼(𝑇𝑥,𝑇𝑦)≥1. Thus in all cases, we have 𝛼(𝑇𝑥,𝑇𝑦)≥1. This implies that 𝑇 is 𝛼-admissible.
Also, from (I), for any 𝑎∈𝐴₁, we have (𝑎,𝑇𝑎)∈𝐴₁×𝐴₂, which implies that 𝛼(𝑎,𝑇𝑎)≥1.
Now, let {𝑥_𝑛} be a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑥∈𝑋 as 𝑛→∞. This implies from the definition of 𝛼 that 𝑥_𝑛,𝑥_𝑛+1∈𝐴₁×𝐴₂∪𝐴₂×𝐴₁,∀𝑛.(3.22) Since (𝐴₁×𝐴₂)∪(𝐴₂×𝐴₁) is a closed set with respect to the Euclidean metric, we get that (𝑥,𝑥)∈𝐴₁×𝐴₂∪𝐴₂×𝐴₁,(3.23) which implies that 𝑥∈𝐴₁∩𝐴₂. Thus we get immediately from the definition of 𝛼 that 𝛼(𝑥_𝑛,𝑥)≥1 for all 𝑛.
Finally, let 𝑥,𝑦∈Fix(𝑇). From (I), this implies that 𝑥,𝑦∈𝐴₁∩𝐴₂. So, for any 𝑧∈𝑌, we have 𝛼(𝑥,𝑧)≥1 and 𝛼(𝑦,𝑧)≥1. Thus condition (H) is satisfied.
Now, all the hypotheses of Theorem 2.6 are satisfied, and we deduce that 𝑇 has a unique fixed point that belongs to 𝐴₁∩𝐴₂ (from (I)).

The following results are immediate consequences of Corollary 3.18.

Corollary 3.19 (see Pacurar and Rus [21]). Let {𝐴_𝑖}^2𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and 𝑇∶𝑌→𝑌 be a given mapping, where 𝑌=𝐴₁∪𝐴₂. Suppose that the following conditions hold: (I)𝑇(𝐴₁)⊆𝐴₂ and 𝑇(𝐴₂)⊆𝐴₁; (II)there exists a function 𝜓∈Ψ such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜓(𝑑(𝑥,𝑦)),∀(𝑥,𝑦)∈𝐴₁×𝐴₂.(3.24)Then 𝑇 has a unique fixed point that belongs to 𝐴₁∩𝐴₂.

Corollary 3.20. Let {𝐴_𝑖}^2𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and 𝑇∶𝑌→𝑌 be a given mapping, where 𝑌=𝐴₁∪𝐴₂. Suppose that the following conditions hold: (I)𝑇(𝐴₁)⊆𝐴₂ and 𝑇(𝐴₂)⊆𝐴₁; (II)there exists a constant 𝜆∈(0,1) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜆max𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦)2∀(𝑥,𝑦,)𝑑∈(𝑥𝐴,₁𝑇𝑦)+2𝑑(𝑦,𝑇𝑥),×𝐴₂.(3.25)Then 𝑇 has a unique fixed point that belongs to 𝐴₁∩𝐴₂.

Corollary 3.21. Let {𝐴_𝑖}^2𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and 𝑇∶𝑌→𝑌 be a given mapping, where 𝑌=𝐴₁∪𝐴₂. Suppose that the following conditions hold: (I)𝑇(𝐴₁)⊆𝐴₂ and 𝑇(𝐴₂)⊆𝐴₁; (II)there exist constants 𝐴,𝐵,𝐶≥0 with (𝐴+2𝐵+2𝐶)∈(0,1) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝐴𝑑(𝑥,𝑦)+𝐵[𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦)∀]+(𝑥𝐶,𝑦[)𝑑∈(𝑥𝐴,₁𝑇𝑦)+𝑑(𝑦,𝑇𝑥)],×𝐴₂.(3.26)Then 𝑇 has a unique fixed point that belongs to 𝐴₁∩𝐴₂.

Corollary 3.22 (see Kirk et al. [17]). Let {𝐴_𝑖}^2𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and 𝑇∶𝑌→𝑌 be a given mapping, where 𝑌=𝐴₁∪𝐴₂. Suppose that the following conditions hold: (I)𝑇(𝐴₁)⊆𝐴₂ and 𝑇(𝐴₂)⊆𝐴₁; (II)there exists a constant 𝜆∈(0,1) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜆𝑑(𝑥,𝑦),∀(𝑥,𝑦)∈𝐴₁×𝐴₂.(3.27)Then 𝑇 has a unique fixed point that belongs to 𝐴₁∩𝐴₂.

Corollary 3.23. Let {𝐴_𝑖}^2𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and 𝑇∶𝑌→𝑌 be a given mapping, where 𝑌=𝐴₁∪𝐴₂. Suppose that the following conditions hold: (I)𝑇(𝐴₁)⊆𝐴₂ and 𝑇(𝐴₂)⊆𝐴₁; (II)there exists a constant 𝜆∈(0,1/2) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜆[𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦)],∀(𝑥,𝑦)∈𝐴₁×𝐴₂.(3.28)Then 𝑇 has a unique fixed point that belongs to 𝐴₁∩𝐴₂.

Corollary 3.24. Let {𝐴_𝑖}^2𝑖=1 be nonempty closed subsets of a complete metric space (𝑋,𝑑) and 𝑇∶𝑌→𝑌 be a given mapping, where 𝑌=𝐴₁∪𝐴₂. Suppose that the following conditions hold: (I)𝑇(𝐴₁)⊆𝐴₂ and 𝑇(𝐴₂)⊆𝐴₁; (II)there exists a constant 𝜆∈(0,1/2) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝜆[𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥)],∀(𝑥,𝑦)∈𝐴₁×𝐴₂.(3.29)Then 𝑇 has a unique fixed point that belongs to 𝐴₁∩𝐴₂.