Common Fixed Point Theorem of Two Mappings Satisfying a Generalized Weak Contractive Condition

Abstract

Existence of common fixed point for two mappings which satisfy a generalized weak contractive condition is established. As a consequence, a common fixed point result for mappings satisfying a contractive condition of integral type is obtained. Our results generalize, extend, and unify several well-known comparable results in literature.

1. Introduction and Preliminaries

Let 𝑋 be a metric space and 𝑇∶𝐶→𝐶 a mapping. Recall that 𝑇 is contraction if 𝑑(𝑇𝑥,𝑇𝑦)≤𝑘𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋, where 0≤𝑘<1. A point 𝑥∈𝐶 is a fixed point of 𝑇 provided 𝑇𝑥=𝑥. If a map 𝑇 satisfies 𝐹(𝑇)=𝐹(𝑇^𝑛) for each 𝑛∈𝑁, where 𝐹(𝑇) denotes the set of all fixed points of 𝑇, then it is said to have property 𝑃. Banach contraction principle which gives an answer on existence and uniqueness of a solution of an operator equation 𝑇𝑥=𝑥 is the most widely used fixed point theorem in all of analysis. Branciari [1] obtained a fixed point theorem for a mapping satisfying an analogue of Banach's contraction principle for an integral type inequality. Akgun and Rhoades [2] have shown that a map satisfying a Meir- Keeler type contractive condition of integral type has a property 𝑃. Rhoades and Abbas [3] extended [4, Theorem1] for mappings satisfying contractive condition of integral type. They also studied several results for maps which have property 𝑃, defined on a metric space satisfying generalized contractive conditions of integral type. Rhoades [5] proved two fixed point theorems involving more general contractive condition of integral type (see, also [6, 7]). If maps 𝑆 and 𝑇 satisfy 𝐹(𝑆)∩𝐹(𝑇)=𝐹(𝑆^𝑛)∩𝐹(𝑇^𝑛) for each 𝑛∈𝑁, then they are said to have property 𝑄. Jeong and Rhoades [8] studied the property 𝑄 for pairs of maps satisfying a number of contractive conditions.

Recently Dutta and Choudhury [9] gave a generalization of Banach contraction principle, which in turn generalize [4, Theorem 1] and corresponding result of [10]. Sessa [11] defined the concept of weakly commuting to obtain common fixed point for pairs of maps. Jungck generalized this idea, first to compatible mappings [12] and then to weakly compatible mappings [13]. There are examples that show that each of these generalizations of commutativity is a proper extension of the previous definition. The aim of this paper is to present a common fixed point theorem for weakly compatible maps satisfying a generalized weak contractive condition which is more general than the corresponding contractive condition of integral type. Our results substantially extend, improve, and generalize comparable results in literature [3, 14, 15].

The following definitions and results will be needed in the sequel.

Definition 1.1. Let 𝑋 be a set, and 𝑓,𝑔 selfmaps of 𝑋. A point 𝑥 in 𝑋 is called a coincidence point of 𝑓 and 𝑔 if and only if 𝑓𝑥=𝑔𝑥. We will call 𝑤=𝑓𝑥=𝑔𝑥 a point of coincidence of 𝑓 and 𝑔.

Definition 1.2. Two maps 𝑓 and 𝑔 are said to be weakly compatible if they commute at their coincidence points.

Lemma 1.3 (see [16]). Let 𝑓 and 𝑔 be weakly compatible self maps of a set 𝑋. If 𝑓 and 𝑔 have a unique point of coincidence 𝑤 (say), then 𝑤 is the unique common fixed point of 𝑓 and 𝑔.

2. A Common Fixed Point Theorem

Set ϝ={𝜙∶𝑅⁺→𝑅⁺∶𝜙 is a Lebesgue integrable mapping which is summable and nonnegative and satisfies ∫^𝜀₀𝜙(𝑡)𝑑𝑡>0, for each 𝜀>0} and 𝐺={𝜓∶[0,∞]→[0,∞]∶𝜓 is continuous and nondecreasing mapping with 𝜓(𝑡)=0 if and only if 𝑡=0}.

The following is the main result of this paper.

Theorem 2.1. Let 𝑓,𝑔 be two self maps of a metric space (𝑋,𝑑) satisfying 𝜓(𝑑(𝑓𝑥,𝑓𝑦))≤𝜓(𝑑(𝑔𝑥,𝑔𝑦))−𝜑(𝑑(𝑔𝑥,𝑔𝑦)) (2.1) for all𝑥,𝑦∈𝑋, where 𝜓,𝜑∈𝐺. If range of 𝑔 contains the range of 𝑓 and 𝑔(𝑋) is a complete subspace of 𝑋, then 𝑓 and 𝑔 have a unique point of coincidence in 𝑋. Moreover if 𝑓 and 𝑔 are weakly compatible, 𝑓 and 𝑔 have a unique common fixed point.

Proof. Let 𝑥₀ be an arbitrary point of 𝑋. Choose a point 𝑥₁in 𝑋 such that 𝑓(𝑥₀)=𝑔(𝑥₁). This can be done, since the range of 𝑔 contains the range of 𝑓. Continuing this process, having chosen 𝑥_𝑛 in 𝑋, we obtain 𝑥_𝑛+1 in 𝑋 such that 𝑓(𝑥_𝑛)=𝑔(𝑥_𝑛+1),𝑛=0,1,2,…. Suppose for any 𝑛,𝑔(𝑥_𝑛)≠𝑔(𝑥_𝑛+1), since, otherwise, 𝑓 and 𝑔 have a point of coincidence. From (2.1), we have 𝜓(𝑑(𝑔𝑥_𝑛+1,𝑔𝑥_𝑛))=𝜓(𝑑(𝑓𝑥_𝑛,𝑓𝑥_𝑛−1))≤𝜓(𝑑(𝑔𝑥_𝑛,𝑔𝑥_𝑛−1))−𝜑(𝑑(𝑔𝑥_𝑛,𝑔𝑥_𝑛−1))<𝜓(𝑑(𝑔𝑥_𝑛,𝑔𝑥_𝑛−1)), (2.2) that is, 𝜓(𝑑(𝑔𝑥_𝑛+1,𝑔𝑥_𝑛))<𝜓(𝑑(𝑔𝑥_𝑛,𝑔𝑥_𝑛−1)), and hence 𝑑(𝑔𝑥_𝑛,𝑔𝑥_𝑛+1)≤𝑑(𝑔𝑥_𝑛,𝑔𝑥_𝑛−1). (2.3) It follows that {𝑑(𝑔𝑥_𝑛,𝑔𝑥_𝑛+1)} is monotone decreasing sequence of numbers and consequently there exists 𝑟≥0 such that 𝑑(𝑔𝑥_𝑛,𝑔𝑥_𝑛+1)→𝑟 as 𝑛→∞. Suppose that 𝑟>0, then 0<𝜓(𝑟)≤𝜓(𝑑(𝑔𝑥_𝑛+1,𝑔𝑥_𝑛))=𝜓(𝑑(𝑓𝑥_𝑛,𝑓𝑥_𝑛−1))≤𝜓(𝑑(𝑔𝑥_𝑛,𝑔𝑥_𝑛−1))−𝜑(𝑑(𝑔𝑥_𝑛,𝑔𝑥_𝑛−1)), (2.4) which on taking limit as 𝑛→∞ yields 𝜓(𝑟)≤𝜓(𝑟)−𝜑(𝑟)<𝜓(𝑟), (2.5) which is a contradiction. Therefore 𝑟=0. Now we prove that {𝑔𝑥_𝑛} is a Cauchy sequence. If not, then there exist some 𝜀>0 and subsequences {𝑔𝑥_𝑛𝑘} and {𝑔𝑥_𝑚𝑘} of {𝑔𝑥_𝑛} with 𝑘<𝑛_𝑘<𝑚_𝑘 such that 𝑑(𝑔𝑥_𝑛𝑘,𝑔𝑥_𝑚𝑘)≥3𝜀 for each 𝑘. As 𝑑(𝑔𝑥_𝑛𝑘+1,𝑔𝑥_𝑛𝑘)→0 as 𝑘→∞, for large enough 𝑘, we have 𝑑(𝑔𝑥_𝑛𝑘+1,𝑔𝑥_𝑛𝑘)<𝜀 and 𝑑(𝑔𝑥_𝑚𝑘+1,𝑔𝑥_𝑚𝑘)<𝜀. Thus we obtain 𝑑(𝑔𝑥_𝑛𝑘+1,𝑔𝑥_𝑚𝑘)≥𝑑(𝑔𝑥_𝑛𝑘,𝑔𝑥_𝑚𝑘)−𝑑(𝑔𝑥_𝑛𝑘+1,𝑔𝑥_𝑛𝑘)>𝜀,𝑑(𝑔𝑥_𝑛𝑘+1,𝑔𝑥_𝑚𝑘−1)≥𝑑(𝑔𝑥_𝑛𝑘,𝑔𝑥_𝑚𝑘)−𝑑(𝑔𝑥_𝑚𝑘−1,𝑔𝑥_𝑚𝑘)−𝑑(𝑔𝑥_𝑛𝑘+1,𝑔𝑥_𝑛𝑘)>𝜀. (2.6) We may assume that 𝑛_𝑘 are even and 𝑚_𝑘 are odd and that 𝑑(𝑔𝑥_𝑛𝑘,𝑔𝑥_𝑚𝑘)>𝜀 for all 𝑘. Put 𝑟_𝑘=min{𝑚_𝑘∶𝑑(𝑔𝑥_𝑛𝑘,𝑔𝑥_𝑚𝑘)>𝜀}. (2.7) Now, 𝜀<𝑑(𝑔𝑥_𝑛𝑘,𝑔𝑥_𝑟𝑘)≤𝑑(𝑔𝑥_𝑛𝑘,𝑔𝑥_𝑟𝑘−2)+𝑑(𝑔𝑥_𝑟𝑘−2,𝑔𝑥_𝑟𝑘−1)+𝑑(𝑔𝑥_𝑟𝑘−1,𝑔𝑥_𝑟𝑘) (2.8) implies that 𝑑(𝑔𝑥_𝑛𝑘,𝑔𝑥_𝑟𝑘)→𝜀 as 𝑘→∞. Furthermore 𝑑(𝑔𝑥_𝑛𝑘,𝑔𝑥_𝑟𝑘)−𝑑(𝑔𝑥_𝑛𝑘,𝑔𝑥_𝑛𝑘+1)−𝑑(𝑔𝑥_𝑟𝑘,𝑔𝑥_𝑟𝑘+1)≤𝑑(𝑔𝑥_𝑛𝑘+1,𝑔𝑥_𝑟𝑘+1)≤𝑑(𝑔𝑥_𝑛𝑘,𝑔𝑥_𝑟𝑘)+𝑑(𝑔𝑥_𝑛𝑘,𝑔𝑥_𝑛𝑘+1)+𝑑(𝑔𝑥_𝑟𝑘,𝑔𝑥_𝑟𝑘+1) (2.9) gives 𝑑(𝑔𝑥_𝑛𝑘+1,𝑔𝑥_𝑟𝑘+1)→𝜀 , as 𝑘→∞. Therefore 𝜓(𝑑(𝑔𝑥_𝑛𝑘+1,𝑔𝑥_𝑟𝑘+1))=𝜓(𝑑(𝑓𝑥_𝑛𝑘,𝑓𝑥_𝑟𝑘))≤𝜓(𝑑(𝑔𝑥_𝑛𝑘,𝑔𝑥_𝑟𝑘))−𝜑(𝑑(𝑔𝑥_𝑛𝑘,𝑔𝑥_𝑟𝑘)). (2.10) Taking limit as 𝑘→∞ yields 𝜓(𝜀)≤𝜓(𝜀)−𝜑(𝜀), (2.11) which is a contradiction. Hence {𝑔𝑥_𝑛} is a Cauchy sequence. From completeness of 𝑔(𝑋), there exists a point 𝑞 in 𝑔(𝑋) such that 𝑔𝑥_𝑛→𝑞 as 𝑛→∞. Consequently, we can find 𝑝 in 𝑋 such that 𝑔(𝑝)=𝑞. Now 𝜓(𝑑(𝑔𝑥_𝑛+1,𝑓𝑝))=𝜓(𝑑(𝑓𝑥_𝑛,𝑓𝑝))≤𝜓(𝑑(𝑔𝑥_𝑛,𝑔𝑝))−𝜑(𝑑(𝑔𝑥_𝑛,𝑔𝑝)) (2.12) on taking limit as 𝑛→∞ implies 𝜓(𝑑(𝑞,𝑓𝑝))≤𝜓(0)−𝜑(0), (2.13)𝜓(𝑑(𝑞,𝑓𝑝))=0, and 𝑓(𝑝)=𝑞. Hence 𝑞 is the point of coincidence of 𝑓 and 𝑔. Assume that there is another point of coincident 𝑟 in 𝑋 such that 𝑟≠𝑞. Then there exists 𝑠 in 𝑋 such that 𝑓(𝑠)=𝑔(𝑠)=𝑟.Using (2.1), we have 𝜓(𝑑(𝑔𝑝,𝑔𝑠))=𝜓(𝑑(𝑓𝑝,𝑓𝑠))≤𝜓(𝑑(𝑔𝑝,𝑔𝑠))−𝜑(𝑑(𝑔𝑝,𝑔𝑠))<𝜓(𝑑(𝑔𝑝,𝑔𝑠)), (2.14) which is a contradiction which proves the uniqueness of point of coincidence; the result now follows from Lemma 1.3

Corollary 2.2. Let 𝑓,𝑔 be two self maps of a metric space (𝑋,𝑑) satisfying ₀^{𝜓(𝑑(𝑓𝑥,𝑓𝑦))}𝜙(𝑡)𝑑𝑡≤₀^{𝜓(𝑑(𝑔𝑥,𝑔𝑦))}𝜙(𝑡)𝑑𝑡−₀^{𝜑(𝑑(𝑔𝑥,𝑔𝑦))}𝜙(𝑡)𝑑𝑡 (2.15) for all 𝑥,𝑦∈𝑋, where 𝜙∈ϝ and 𝜓,𝜑∈𝐺. If range of 𝑔 contains the range of 𝑓 and 𝑔(𝑋) is a complete subspace of 𝑋, then 𝑓 and 𝑔 have a unique point of coincidence in 𝑋. Moreover if 𝑓 and 𝑔 are weakly compatible, 𝑓 and 𝑔 have a unique common fixed point.

Proof. Define Φ∶𝑅⁺→𝑅⁺ by Φ(𝑥)=∫^𝑥₀𝜙(𝑡)𝑑𝑡, then Φ∈𝐺 and (2.15) becomes Φ(𝜓(𝑑(𝑓𝑥,𝑓𝑦)))≤Φ(𝜓(𝑑(𝑔𝑥,𝑔𝑦)))−Φ(𝜑(𝑑(𝑔𝑥,𝑔𝑦))), (2.16) which further can be written as 𝜓₁(𝑑(𝑓𝑥,𝑓𝑦))≤𝜓₁(𝑑(𝑔𝑥,𝑔𝑦))−𝜑₁(𝑑(𝑔𝑥,𝑔𝑦)), (2.17) where 𝜓₁=Φ∘𝜓 and 𝜑₁=Φ∘𝜑∈𝐺. Clearly 𝜓₁,𝜑₁∈𝐺. Hence by Theorem 2.1𝑓 and 𝑔 have unique common fixed point.

Now we present two examples in the support of Theorem 2.1.

Example 2.3. Let 𝑋=[0,1]∪{2,3,4,…}, 𝑑(𝑥,𝑦)=|𝑥−𝑦|,if𝑥,𝑦∈[0,1],𝑥≠𝑦,𝑥+𝑦,ifatleastoneof𝑥or𝑦∉[0,1],𝑥≠𝑦,0,if𝑥=𝑦. (2.18) Then (𝑋,𝑑) is a complete metric space [17]. Consider 𝑓∶𝑋→𝑋, and 𝜓,𝜑∈𝐺 as given in [9]: 1𝑓𝑥=𝑥−2𝑥²𝑡,if0≤𝑥≤1,𝑥−1,if𝑥>1,𝜓(𝑡)=𝑡,if0≤𝑡≤1,₂1,if𝑡≥1,𝜑(𝑡)=2𝑡²1,if0≤𝑡≤1,2,if𝑡≥1. (2.19)(2.20)(2.21) Let 𝑔∶𝑋→𝑋 be defined as 𝑔𝑥=𝑥,if0≤𝑥≤1,𝑥+1,if𝑥>1. (2.22) Assume that 𝑥>𝑦 and discuss the following cases.
When 𝑥∈[0,1], then 1𝜓(𝑑(𝑓𝑥,𝑓𝑦))=(𝑥−2𝑥²1)−(𝑦−2𝑦²)1≤(𝑥−𝑦)−2(𝑥−𝑦)²=𝜓(𝑑(𝑔𝑥,𝑔𝑦))−𝜑(𝑑(𝑔𝑥,𝑔𝑦)). (2.23) Taking 𝑥 in {3,4,…}, and 𝑦 in [0,1], we obtain 1𝜓(𝑑(𝑓𝑥,𝑓𝑦))=(𝑥−1+𝑦−2𝑦²)²≤(𝑥+𝑦−1)²,𝜓(𝑑(𝑔𝑥,𝑔𝑦))=(𝑥+𝑦+1)²,1𝜑(𝑑(𝑔𝑥,𝑔𝑦))=2. (2.24) Hence 𝜓(𝑑(𝑓𝑥,𝑓𝑦))≤𝜓(𝑑(𝑔𝑥,𝑔𝑦))−𝜑(𝑑(𝑔𝑥,𝑔𝑦)). (2.25) Now, when 𝑥∈{3,4,…}, and 𝑦∉[0,1], then 𝜓(𝑑(𝑓𝑥,𝑓𝑦))=(𝑥−1+𝑦−1)²<(𝑥+𝑦−1)²,𝜓(𝑑(𝑔𝑥,𝑔𝑦))=(𝑥+𝑦+2)²,1𝜑(𝑑(𝑔𝑥,𝑔𝑦))=2. (2.26) Obviously (2.31) holds. Finally when 𝑥=2, we have 𝑦∈[0,1], 𝑓𝑥=1, and 1𝑑(𝑓𝑥,𝑓𝑦)=1−(𝑦−2𝑦²)≤1, (2.27) so that 𝜓(𝑑(𝑓𝑥,𝑓𝑦))≤1, then 𝜓(𝑑(𝑔𝑥,𝑔𝑦))−𝜑(𝑑(𝑔𝑥,𝑔𝑦))=(3+𝑦)²−12>1≥𝜓(𝑑(𝑓𝑥,𝑓𝑦)). (2.28) Thus all conditions of Theorem 2.1 are satisfied. Moreover 𝑓and𝑔 have a unique common fixed point.

Example 2.4. Let 𝑋=[0,1] and  𝑓,𝑔∶𝑋→𝑋 be given as 2𝑓(𝑥)=5𝑥²+352,𝑔(𝑥)=3𝑥²+13. (2.29) Consider 𝜓,𝜑∈𝐺 as 𝜓(𝑡)=(1/2)𝑡 and 𝜑(𝑡)=(1/10)𝑡. Then we have 2𝜓(𝑑(𝑓𝑥,𝑓𝑦))=10|𝑥²−𝑦²|≤1223|𝑥²−𝑦²1|−2103|𝑥²−𝑦²|=𝜓(𝑑(𝑔𝑥,𝑔𝑦))−𝜑(𝑑(𝑔𝑥,𝑔𝑦)). (2.30) Note that 𝑥=1 is the unique coincidence point of 𝑓 and 𝑔, and 𝑓 and 𝑔 are commuting at 𝑥=1. Hence all conditions of Theorem 2.1 are satisfied. Moreover, 𝑥=1 is the unique common fixed point of 𝑓 and 𝑔.

Following theorem can be viewed as generalization and extension of [3, Theorem 3].

Theorem 2.5. Let 𝑓 be a self map of a complete metric space (𝑋,𝑑) satisfying ₀^{𝜓(𝑑(𝑓𝑥,𝑓𝑦))}𝜙(𝑡)𝑑𝑡≤₀^{𝜓(𝑑(𝑥,𝑦))}𝜙(𝑡)𝑑𝑡−₀^{𝜑(𝑑(𝑥,𝑦))}𝜙(𝑡)𝑑𝑡 (2.31) for all 𝑥,𝑦∈𝑋, where 𝜙∈ϝ and 𝜓,𝜑∈𝐺. Then 𝑓 has a unique fixed point. Moreover 𝑓 has property 𝑃.

Proof. Existence and uniqueness of fixed point of 𝑓 follows from Corollary 2.2. Now we prove that 𝑓 has property 𝑃. Let 𝑢∈𝐹(𝑓^𝑛). We shall always assume that 𝑛>1, since the statement for 𝑛=1 is trivial. We claim that 𝑓𝑢=𝑢. If not, then, by (2.31), ₀^{𝜓(𝑑(𝑢,𝑓𝑢))}𝜙(𝑡)𝑑𝑡=^{𝜓(𝑑(𝑓𝑛𝑢,𝑓(𝑓𝑛}₀^𝑢)))𝜙(𝑡)𝑑𝑡=^{𝜓(𝑑(𝑓(𝑓𝑛−1𝑢),𝑓(𝑓𝑛}₀^𝑢)))𝜙(𝑡)𝑑𝑡≤^{𝜓(𝑑(𝑓𝑛−1𝑢,𝑓𝑛}₀^𝑢))𝜙(𝑡)𝑑𝑡−^{𝜑(𝑑(𝑓𝑛−1𝑢,𝑓𝑛}₀^𝑢))≤𝜙(𝑡)𝑑𝑡_{𝜓(𝑑(𝑓𝑛−1𝑢,𝑓𝑛0𝑢))}𝜙(𝑡)𝑑𝑡=^{𝜓(𝑑(𝑓(𝑓𝑛−2𝑢),𝑓(𝑓𝑛−1}₀^𝑢)))≤𝜙(𝑡)𝑑𝑡_{𝜓(𝑑(𝑓𝑛−2𝑢,𝑓𝑛−10𝑢))}𝜙(𝑡)𝑑𝑡−^{𝜑(𝑑(𝑓𝑛−2𝑢,𝑓𝑛−1}₀^𝑢))≤𝜙(𝑡)𝑑𝑡_{𝜓(𝑑(𝑓𝑛−2𝑢,𝑓𝑛−10𝑢))}𝜙(𝑡)𝑑𝑡. (2.32) Continuing this process we arrive at ₀^{𝜓(𝑑(𝑢,𝑓𝑢))}𝜙(𝑡)𝑑𝑡≤₀^{𝜓(𝑑(𝑢,𝑓𝑢))}𝜙(𝑡)𝑑𝑡−₀^{𝜑(𝑑(𝑢,𝑓𝑢))}<𝜙(𝑡)𝑑𝑡_{0𝜓(𝑑(𝑢,𝑓𝑢))}𝜙(𝑡)𝑑𝑡, (2.33) which is a contradiction. Hence the result follows.

Remark 2.6 s. Existence and uniqueness of fixed point of 𝑓 in above theorem also follows from [9, Theorem  1].

Remark 2.7 s. (a) It is noted that if maps 𝑓 and 𝑔 involved in Theorem 2.1 are commuting, then they have property 𝑄.
(b) Suzuki [18] observed that Branciari [1, Theorem  1] is a particular case of Meir-Keeler fixed point theorem [19]. We pose an open problem to see if a link exists between the contractive conditions (2.15) and the Meir-Keeler condition.

Acknowledgment

The authors are thankful to referees for their precise remarks to improve the presentation of the paper.