Common Fixed Point Theorems for Weakly Compatible Maps Satisfying a General Contractive Condition

Abstract

We introduce a new generalized contractive condition for four mappings in the framework of metric space. We give some common fixed point results for these mappings and we deduce a fixed point result for weakly compatible mappings satisfying a contractive condition of integral type.

1. Introduction and Preliminaries

The study of common fixed point of mappings satisfying contractive type conditions has been a very active field of research during recent years. The most general of the common fixed point theorems pertaining to four mappings, 𝐴,𝐵,𝑆, and 𝑇 of a metric space (𝑋,𝑑), uses either a Banach-type contractive condition of the form𝑑(𝐴𝑥,𝐵𝑦)≤𝑘𝑀(𝑥,𝑦),0≤𝑘<1, (1.1)where𝑀(𝑥,𝑦)=max𝑑(𝑆𝑥,𝑇𝑦),𝑑(𝐴𝑥,𝑆𝑥),𝑑(𝐵𝑦,𝑇𝑦),𝑑(𝑆𝑥,𝐵𝑦)+𝑑(𝐴𝑥,𝑇𝑦)2, (1.2)or a Meir-Keeler-type (𝜀,𝛿)-contractive condition, that is, given 𝜀>0, there exists a 𝛿>0 such that𝜀≤𝑀(𝑥,𝑦)<𝜀+𝛿⟹𝑑(𝐴𝑥,𝐵𝑦)<𝜀, (1.3)or a 𝜓-contractive condition of the form𝑑(𝐴𝑥,𝐵𝑦)≤𝜓𝑀(𝑥,𝑦), (1.4)involving a contractive gauge function 𝜓∶[0,+∞[→[0,+∞[ such that 𝜓(𝑡)<𝑡 for each 𝑡>0. Clearly, Banach-type contractive condition is a special case of both conditions Meir-Keeler-type (𝜀,𝛿)-contractive and 𝜓-contractive. A 𝜓-contractive condition does not guarantee the existence of a fixed point unless some additional condition is assumed. Moreover, a 𝜓-contractive condition, in general, does not imply the Meir-Keeler-type (𝜀,𝛿)-contractive condition [1, Example 1.1].

Recently, some fixed point results for mappings satisfying an integral-type contractive condition are obtained in [2–5]. Suzuki [6] showed that Meir-Keeler contractions of integral type are still Meir-Keeler contractions. Zhang [7] introduced a generalized contractive-type condition for a pair of mappings in metric space and proved common fixed point theorems that extend results in [3–5]. In this paper, we give a new generalized contractive-type condition for four mappings in metric space and prove some common fixed point results for these mappings. The results obtained extend well-known comparable results in [2–5, 7].

Lemma 1.1 (see [8]). For every function 𝜓∶[0,+∞[→[0,+∞[, let 𝜓^𝑛 be the 𝑛th iterate of 𝜓. Then the following hold: (i)if 𝜓 is nondecreasing, then for each 𝑡>0, lim_𝑛→+∞𝜓^𝑛(𝑡)=0 implies 𝜓(𝑡)<𝑡;(ii)if 𝜓 is right continuous with 𝜓(𝑡)<𝑡 for 𝑡>0, then lim_𝑛→+∞𝜓^𝑛(𝑡)=0.

2. Common Fixed Points

In this section, we give our main result. Two self-mappings 𝐴 and 𝑆 of a metric space (𝑋,𝑑) are called weakly compatible if they commute at their coincidence points. Let 𝐴,𝐵,𝑆, and 𝑇 be self mappings of a metric space (𝑋,𝑑). In the sequel, we set𝑀(𝑥,𝑦)=max𝑑(𝑆𝑥,𝑇𝑦),𝑑(𝐴𝑥,𝑆𝑥),𝑑(𝐵𝑦,𝑇𝑦),𝑑(𝑆𝑥,𝐵𝑦)+𝑑(𝐴𝑥,𝑇𝑦)2. (2.1)

Lemma 2.1. Let 𝐴,𝐵,𝑆, and 𝑇 be self-mappings of a metric space (𝑋,𝑑) such that 𝐴𝑋⊂𝑇𝑋, 𝐵𝑋⊂𝑆𝑋. Assume that there exist 𝐹,𝜓∶[0,+∞[→[0,+∞[ such that (i)𝐹 is nondecreasing, continuous, and 𝐹(0)=0<𝐹(𝑡) for every 𝑡>0;(ii)𝜓 is nondecreasing, right continuous, and 𝜓(𝑡)<𝑡 for every 𝑡>0. If for all 𝑥,𝑦∈𝑋,𝐹𝑑(𝐴𝑥,𝐵𝑦)≤𝜓𝐹𝑀(𝑥,𝑦), (2.2)then for each 𝑥₀∈𝑋, the sequence (𝑦_𝑛) of points of 𝑋 defined by the rule𝑦_2𝑛=𝐴𝑥_2𝑛=𝑇𝑥_2𝑛+1,𝑦_2𝑛−1=𝐵𝑥_2𝑛−1=𝑆𝑥_2𝑛 (2.3)is a Cauchy sequence.

Proof. We have𝑀𝑥_2𝑛,𝑥_2𝑛+1=max𝑑𝑦_2𝑛−1,𝑦_2𝑛,𝑑𝑦_2𝑛,𝑦_2𝑛−1,𝑑𝑦_2𝑛+1,𝑦_2𝑛,𝑑𝑦_2𝑛−1,𝑦_2𝑛+1+𝑑𝑦_2𝑛,𝑦_2𝑛2=max𝑑𝑦_2𝑛,𝑦_2𝑛−1,𝑑𝑦_2𝑛,𝑦_2𝑛+1,𝑑𝑦_2𝑛−1,𝑦_2𝑛+12=max𝑑𝑦_2𝑛,𝑦_2𝑛−1,𝑑𝑦_2𝑛,𝑦_2𝑛+1. (2.4)
Similarly𝑀𝑥_2𝑛,𝑥_2𝑛−1=max𝑑𝑦_2𝑛,𝑦_2𝑛−1,𝑑𝑦_2𝑛−1,𝑦_2𝑛−2. (2.5)
If for some 𝑛 we have either 𝑦_2𝑛=𝑦_2𝑛−1 or 𝑦_2𝑛=𝑦_2𝑛+1, then by condition (2.2) we obtain that the sequence (𝑦_𝑛) is definitely constant and thus is a Cauchy sequence. Suppose 𝑦_𝑛≠𝑦_𝑛−1 for each 𝑛.
From𝐹𝑑𝑦_2𝑛,𝑦_2𝑛+1=𝐹𝑑𝐴𝑥_2𝑛,𝐵𝑥_2𝑛+1≤𝜓𝐹𝑀𝑥_2𝑛,𝑥_2𝑛+1=𝜓𝐹𝑑𝑦_2𝑛,𝑦_2𝑛−1<𝐹𝑑𝑦_2𝑛,𝑦_2𝑛−1,𝐹𝑑𝑦_2𝑛,𝑦_2𝑛−1=𝐹𝑑𝐴𝑥_2𝑛,𝐵𝑥_2𝑛−1≤𝜓𝐹𝑀𝑥_2𝑛,𝑥_2𝑛−1=𝜓𝐹𝑑𝑦_2𝑛−1,𝑦_2𝑛−2<𝐹𝑑𝑦_2𝑛−1,𝑦_2𝑛−2, (2.6)we deduce𝐹𝑑𝑦_𝑛+1,𝑦_𝑛<𝐹𝑑𝑦_𝑛,𝑦_𝑛−1, (2.7)for all 𝑛∈ℕ. Now, from𝐹𝑑𝑦_𝑛+1,𝑦_𝑛≤𝜓𝐹𝑑𝑦_𝑛,𝑦_𝑛−1≤⋯≤𝜓^𝑛𝐹𝑑𝑦₀,𝑦₁ (2.8)and (ii) of Lemma 1.1, we obtain lim_𝑛→+∞𝐹(𝑑(𝑦_𝑛+1,𝑦_𝑛))=0, which implieslim_𝑛→+∞𝑑𝑦_𝑛+1,𝑦_𝑛=0. (2.9)
We prove that (𝑦_𝑛) is a Cauchy sequence. Suppose not, then there exists 𝜀>0 such that 𝑑(𝑦_𝑛,𝑦_𝑚)≥2𝜀 for infinite values of 𝑚 and 𝑛 with 𝑚<𝑛. This assures that there exist two sequences (𝑚_𝑘),(𝑛_𝑘) of natural numbers, with 𝑚_𝑘<𝑛_𝑘, such that𝑑𝑦_2𝑚𝑘,𝑦_2𝑛𝑘+1>𝜀∀𝑘. (2.10)It is not restrictive to suppose that 𝑛_𝑘 is the least positive integer exceeding 𝑚_𝑘 and satisfying (2.10). We have𝜀<𝑑𝑦_2𝑚𝑘,𝑦_2𝑛𝑘+1≤𝑑𝑦_2𝑚𝑘,𝑦_{2𝑛𝑘−1}+𝑑𝑦_{2𝑛𝑘−1},𝑦_2𝑛𝑘+𝑑𝑦_2𝑛𝑘,𝑦_2𝑛𝑘+1≤𝜀+𝑑𝑦_{2𝑛𝑘−1},𝑦_2𝑛𝑘+𝑑𝑦_2𝑛𝑘,𝑦_2𝑛𝑘+1. (2.11)Then 𝑑(𝑦_2𝑚𝑘,𝑦_2𝑛𝑘+1)→𝜀. We note𝑑𝑦_2𝑚𝑘,𝑦_2𝑛𝑘+1−𝑑𝑦_2𝑚𝑘,𝑦_2𝑚𝑘+1−𝑑𝑦_2𝑛𝑘+2,𝑦_2𝑛𝑘+1≤𝑑𝑦_2𝑚𝑘+1,𝑦_2𝑛𝑘+2≤𝑑𝑦_2𝑚𝑘,𝑦_2𝑛𝑘+1+𝑑𝑦_2𝑚𝑘,𝑦_2𝑚𝑘+1+𝑑𝑦_2𝑛𝑘+2,𝑦_2𝑛𝑘+1, (2.12)and thus 𝑑(𝑦_2𝑚𝑘+1,𝑦_2𝑛𝑘+2)→𝜀 as 𝑘→+∞. We have𝑀𝑥_2𝑛𝑘+2,𝑥_2𝑚𝑘+1=max𝑑𝑦_2𝑚𝑘,𝑦_2𝑛𝑘+1,𝑑𝑦_2𝑛𝑘+1,𝑦_2𝑛𝑘+2,𝑑𝑦_2𝑚𝑘,𝑦_2𝑚𝑘+1,𝑑𝑦_2𝑚𝑘+1,𝑦_2𝑛𝑘+1+𝑑𝑦_2𝑚𝑘,𝑦_2𝑛𝑘+22=𝑑𝑦_2𝑚𝑘,𝑦_2𝑛𝑘+1+𝑑_𝑘, (2.13)where 𝑑_𝑘→0 as 𝑘→+∞ and 𝑑_𝑘≥0 for all 𝑘. Then from𝐹𝑑𝑦_2𝑚𝑘+1,𝑦_2𝑛𝑘+2=𝐹𝑑𝐴𝑥_2𝑛𝑘+2,𝐵𝑥_2𝑚𝑘+1≤𝜓𝐹𝑀𝑥_2𝑛𝑘+2,𝑥_2𝑚𝑘+1=𝜓𝐹𝑑𝑦_2𝑚𝑘,𝑦_2𝑛𝑘+1+𝑑_𝑘, (2.14)as 𝑘→+∞, 𝐹 being continuous and 𝜓 right continuous, we get𝐹(𝜀)≤𝜓𝐹(𝜀)<𝐹(𝜀). (2.15)This is a contradiction. Therefore (𝑦_𝑛) is a Cauchy sequence.

Lemma 2.2. Let (𝑋,𝑑) be a metric space and let 𝐴,𝐵,𝑆,𝑇,𝐹, and 𝜓 be as in Lemma 2.1. If one of 𝐴𝑋,𝑇𝑋,𝐵𝑋, and 𝑆𝑋 is a complete subspace of 𝑋, then the following hold: (i)𝐴 and 𝑆 have a coincidence point;(ii)𝑇 and 𝐵 have a coincidence point.

Proof. Fix 𝑥₀∈𝑋 and let (𝑦_𝑛) be the sequence defined in Lemma 2.1. If 𝑦_2𝑛=𝑦_2𝑛−1 for some 𝑛, then 𝐴𝑥_2𝑛=𝑇𝑥_2𝑛+1=𝐵𝑥_2𝑛−1=𝑆𝑥_2𝑛, and 𝐴 and 𝑆 have a coincidence point. If 𝑦_2𝑛=𝑦_2𝑛+1 for some 𝑛, then 𝐴𝑥_2𝑛=𝑇𝑥_2𝑛+1=𝐵𝑥_2𝑛+1=𝑆𝑥_2𝑛+2, and 𝑇 and 𝐵 have a coincidence point. Assume that 𝑦_𝑛≠𝑦_𝑛+1 for every 𝑛 and 𝑇𝑋 is complete. By Lemma 2.1, the sequence (𝑦_𝑛) is Cauchy; as (𝑦_2𝑛)⊂𝑇𝑋, there exists 𝑢∈𝑇𝑋 such that 𝑦_𝑛→𝑢. Let 𝑣∈𝑋 be such that 𝑇𝑣=𝑢. To prove that 𝐵𝑣=𝑢. We have𝑀(𝑥_2𝑛,𝑣)=max𝑑𝑦_2𝑛−1,𝑢,𝑑𝑦_2𝑛,𝑦_2𝑛−1,𝑑𝑦,𝑑𝐵𝑣,𝑢^2𝑛−1,𝐵𝑣)+𝑑𝑦_2𝑛,𝑢2. (2.16)If 𝐵𝑣≠𝑢, then 𝑀(𝑥_2𝑛,𝑣)=𝑑(𝑢,𝐵𝑣) definitely and consequently for large 𝑛,𝐹𝑑𝐴𝑥_2𝑛,𝐵𝑣≤𝜓𝐹𝑀𝑥_2𝑛,𝑣=𝜓𝐹𝑑(𝑢,𝐵𝑣). (2.17)𝐹 being continuous, as 𝑛→+∞, we obtain𝐹𝑑(𝑢,𝐵𝑣)≤𝜓𝐹𝑑(𝑢,𝐵𝑣)<𝐹𝑑(𝑢,𝐵𝑣). (2.18)This is a contradiction, therefore 𝐵𝑣=𝑢 and 𝑣 is a coincidence point for 𝑇 and 𝐵. From 𝐵𝑋⊂𝑆𝑋, which gives 𝑢∈𝑆𝑋, we deduce that there exists 𝑤∈𝑋 such that 𝑆𝑤=𝑢. To prove that 𝐴𝑤=𝑢. We have𝑀(𝑤,𝑣)=max𝑑(𝑢,𝑢),𝑑(𝐴𝑤,𝑢),𝑑(𝑢,𝑢),𝑑(𝑢,𝑢)+𝑑(𝐴𝑤,𝑢)2=𝑑(𝐴𝑤,𝑢), (2.19)and hence𝐹𝑑(𝐴𝑤,𝐵𝑣)≤𝜓𝐹𝑀(𝑤,𝑢)=𝜓𝐹𝑑(𝐴𝑤,𝑢)<𝐹𝑑(𝐴𝑤,𝑢), (2.20)which gives 𝐴𝑤=𝑢.
The same result holds if we suppose that one of 𝑆𝑋,𝐴𝑋,𝐵𝑋 is complete.

Theorem 2.3. Let 𝐴,𝐵,𝑆, and 𝑇 be self-mappings of a metric space (𝑋,𝑑) such that 𝐴𝑋⊂𝑇𝑋, 𝐵𝑋⊂𝑆𝑋. Assume that there exist 𝐹,𝜓∶[0,+∞[→[0,+∞[ such that (i)𝐹 is nondecreasing, continuous, and 𝐹(0)=0<𝐹(𝑡) for every 𝑡>0;(ii)𝜓 is nondecreasing, right continuous, and 𝜓(𝑡)<𝑡 for every 𝑡>0;(iii)𝐹(𝑑(𝐴𝑥,𝐵𝑦))≤𝜓(𝐹(𝑀(𝑥,𝑦))) for all 𝑥,𝑦∈𝑋. If one of 𝐴𝑋,𝑇𝑋,𝐵𝑋, and 𝑆𝑋 is a complete subspace of 𝑋, then the following hold: (iv)𝐴 and 𝑆 have a coincidence point;(v)𝑇 and 𝐵 have a coincidence point.
Further, if 𝐴 and 𝑆 as well as 𝐵 and 𝑇 are weakly compatible, then 𝐴,𝐵,𝑆, and 𝑇 have a unique common fixed point.

Proof. Fix 𝑥₀∈𝑋 and let (𝑦_𝑛) be the sequence defined in Lemma 2.1. Assume that 𝑇𝑋 is complete and let 𝑢,𝑣, and 𝑤 be as in Lemma 2.2. If 𝐴 and 𝑆 are weakly compatible, then𝐴𝑢=𝐴𝑆𝑤=𝑆𝐴𝑤=𝑆𝑢, (2.21)therefore 𝑢 is a coincidence point of 𝐴 and 𝑆. To prove that 𝑑(𝐴𝑢,𝑢)=0. Suppose that 𝑑(𝐴𝑢,𝑢)≠0. We have𝑀(𝑢,𝑣)=max𝑑(𝑆𝑢,𝑢),𝑑(𝐴𝑢,𝑆𝑢),𝑑(𝑢,𝑢),𝑑(𝑆𝑢,𝑢)+𝑑(𝐴𝑢,𝑢)2𝐹𝐹𝐹.=𝑑(𝐴𝑢,𝑢)𝑑(𝐴𝑢,𝐵𝑣)=𝐹𝑑(𝐴𝑢,𝑢)≤𝜓𝑀(𝑢,𝑣)=𝜓𝑑(𝐴𝑢,𝑢)<𝐹𝑑(𝐴𝑢,𝑢) (2.22)This is a contradiction, and thus 𝐴𝑢=𝑢. Since 𝐴𝑢=𝑆𝑢=𝑢, we obtain that 𝑢 is a common fixed point for 𝐴 and 𝑆.
Similarly, if 𝐵 and 𝑇 are weakly compatible, we deduce that 𝑢 is a common fixed point for 𝐵 and 𝑇. Now if 𝐴 and 𝑆 as well as 𝐵 and 𝑇 are weakly compatible, then 𝑢 is a common fixed point for 𝐴,𝐵,𝑆, and 𝑇. If 𝑧∈𝑋 is also a common fixed point for 𝐴,𝐵,𝑆, and 𝑇 with 𝑢≠𝑧, then𝐹𝑑(𝐴𝑢,𝐵𝑧)≤𝜓𝐹𝑀(𝑢,𝑧)=𝜓𝐹𝑑(𝐴𝑢,𝐵𝑣)<𝐹𝑑(𝐴𝑢,𝐵𝑣), (2.23)which gives 𝑢=𝑧.

Let 𝜑∶[0,+∞[→[0,+∞[ be a Lebesgue integrable function which is nonnegative and such that^𝜀₀𝜑(𝑡)𝑑𝑡>0,forevery𝜀>0. (2.24)The function 𝐹∶[0,+∞[→[0,+∞[, with 𝐹(𝑠)=∫^𝑠₀𝜑(𝑡)𝑑𝑡 satisfies condition (i) of Lemma 2.1 and from Theorem 2.3 we deduce the following theorem.

Theorem 2.4 (see [2, Theorem 2.4 2.1]). Let 𝐴,𝐵,𝑆, and 𝑇 be self-mappings of a metric space (𝑋,𝑑) such that 𝐴𝑋⊂𝑇𝑋, 𝐵𝑋⊂𝑆𝑋. Assume that there exists a nondecreasing right continuous function 𝜓∶[0,+∞[→[0,+∞[, with 𝜓(𝑡)<𝑡 for all 𝑡>0, such that₀^{𝑑(𝐴𝑥,𝐵𝑦)}𝜑(𝑡)𝑑𝑡≤𝜓₀^{𝑀(𝑥,𝑦)}𝜑(𝑡)𝑑𝑡, (2.25)where 𝜑∶[0,+∞[→[0,+∞[ is a Lebesgue integrable function which is nonnegative and such that^𝜀₀𝜑(𝑡)𝑑𝑡>0,forevery𝜀>0. (2.26)If one of 𝐴𝑋,𝑇𝑋,𝐵𝑋, and 𝑆𝑋 is a complete subspace of 𝑋, then the following hold: (i)𝐴 and 𝑆 have a coincidence point;(ii)𝑇 and 𝐵 have a coincidence point.
Further, if 𝐴 and 𝑆 as well as 𝐵 and 𝑇 are weakly compatible, then 𝐴,𝐵,𝑆, and 𝑇 have a unique common fixed point.

Remark 2.5. Theorem 2.4 is a generalization of the main theorem in [3], of [4, Theorem 2], and of [5, Theorem 2].

If in Theorem 2.3, we assume 𝑆=𝑇=𝐼_𝑋, where 𝐼_𝑋 is the identity map on 𝑋, we obtain the following theorem.

Theorem 2.6. Let 𝐴 and 𝐵 be self-mappings of a metric space (𝑋,𝑑). Assume that there exist 𝐹,𝜓∶[0,+∞[→[0,+∞[ such that (i)𝐹 is nondecreasing, continuous, and 𝐹(0)=0<𝐹(𝑡) for every 𝑡>0;(ii)𝜓 is nondecreasing, right continuous, and 𝜓(𝑡)<𝑡 for every 𝑡>0;(iii)𝐹(𝑑(𝐴𝑥,𝐵𝑦))≤𝜓(𝐹(𝑚(𝑥,𝑦))) for all 𝑥,𝑦∈𝑋, where𝑚(𝑥,𝑦)=max𝑑(𝑥,𝑦),𝑑(𝐴𝑥,𝑦),𝑑(𝐵𝑦,𝑦),𝑑(𝐴𝑥,𝑦)+𝑑(𝑥,𝐵𝑦)2. (2.27)If one of 𝐴𝑋 and 𝐵𝑋 is a complete subspace of 𝑋, then 𝐴 and 𝑆 have a unique common fixed point. Moreover, for each 𝑥₀∈𝑋, the iterated sequence (𝑥_𝑛) with 𝑥_2𝑛+1=𝐴𝑥_2𝑛 and 𝑥_2𝑛+2=𝐵𝑥_2𝑛+1 converges to the common fixed point of 𝐴 and 𝐵.

Theorem 2.6 includes [7, Theorem 1].

Acknowledgment

The authors are supported by the University of Palermo (R.S. ex 60%).