Gregus-Type Common Fixed Point Theorems for Tangential Multivalued Mappings of Integral Type in Metric Spaces

Abstract

The concept of tangential for single-valued mappings is extended to multivalued mappings and used to prove the existence of a common fixed point theorem of Gregus type for four mappings satisfying a strict general contractive condition of integral type. Consequently, several known fixed point results generalized and improved the corresponding recent result of Pathak and Shahzad (2009) and many authors.

1. Introduction

The first important result on fixed points for contractive-type mappings was the well-known Banach contraction principle, published for the first time in 1922 in [1] (see also [2]). Banach contraction principle has been extended in many different directions, see [3–5], and so forth. Many authors in [3, 5–12] established fixed point theorems involving more general contractive conditions. In 1969, Nadler [13] combines the ideas of set-valued mapping and Lipschitz mapping and prove some fixed point theorems about multivalued contraction mappings. Afterward, the study of fixed points for multivalued contractions using the Hausdorff metric was initiated by Markin [14]. Later, an interesting and rich fixed point theory for such maps was developed (see [15–18]). The theory of multivalued maps has applications in optimization problems, control theory, differential equations, and economics.

Sessa [19] introduced the concept of weakly commuting maps. Jungck [20] defined the notion of compatible maps in order to generalize the concept of weak commutativity and showed that weakly commuting mappings are compatible but the converse is not true. This concept was further improved by Jungck and Rhoades [21] with the notion of weakly compatible mappings. In 2002, Aamri and Moutawakil [22] defined property (E.A). This concept was frequently used to prove existence theorems in common fixed point theory. Three years later, Liu et al. [23] introduced common property (E.A). The class of (E.A) maps contains the class of noncompatible maps. Branciari [3] studied contractive conditions of integral type, giving an integral version of the Banach contraction principle, that could be extended to more general contractive conditions. Recently, Pathak and Shahzad [24] introduced the new concept of weak tangent point and tangential property for single-valued mappings and established common fixed point theorems. Very recently, Vetro [25] obtained an interesting theorem for mappings satisfying a contractive condition of integral type which is a generalization of Branciari [3, Theorem 2].

The aim of this paper is to define a tangential property for multivalued mappings which generalize the concept of tangential property for single-valued mappings of Pathak and Shahzad [24] and prove a common fixed point theorem of Gregus type for four mappings satisfying a strict general contractive condition of integral type.

2. Preliminary

Throughout this paper, (𝑋,𝑑) denotes a metric space. We denote by CB(𝑋), the class of all nonempty bounded closed subsets of 𝑋. The Hausdorff metric induced by 𝑑 on CB(𝑋) is given by 𝐻(𝐴,𝐵)=maxsup_𝑎∈𝐴𝑑(𝑎,𝐵),sup_𝑏∈𝐵𝑑(𝑏,𝐴), (2.1) for every 𝐴,𝐵∈CB(𝑋), where 𝑑(𝑎,𝐵)=𝑑(𝐵,𝑎)=inf{𝑑(𝑎,𝑏)∶𝑏∈𝐵} is the distance from 𝑎 to 𝐵⊆𝑋. Let 𝑓∶𝑋→𝑋 and 𝑇∶𝑋→CB(𝑋). A point 𝑥∈𝑋 is a fixed point of 𝑓 (resp. 𝑇) if 𝑓𝑥=𝑥 (resp. 𝑥∈𝑇𝑥). The set of all fixed points of 𝑓 (resp. 𝑇) is denoted by 𝐹(𝑓) (resp. 𝐹(𝑇)). A point 𝑥∈𝑋 is a coincidence point of 𝑓 and 𝑇 if 𝑓𝑥∈𝑇𝑥. The set of all coincidence points of 𝑓 and 𝑇 is denoted by 𝐶(𝑓,𝑇). A point 𝑥∈𝑋 is a common fixed point of 𝑓 and 𝑇 if 𝑥=𝑓𝑥∈𝑇𝑥. The set of all common fixed points of 𝑓 and 𝑇 is denoted by 𝐹(𝑓,𝑇).

Definition 2.1. The maps 𝑓∶𝑋→𝑋 and 𝑔∶𝑋→𝑋 are said to be commuting if 𝑓𝑔𝑥=𝑔𝑓𝑥, for all 𝑥∈𝑋.

Definition 2.2 (see [19]). The maps 𝑓∶𝑋→𝑋 and 𝑔∶𝑋→𝑋 are said to be weakly commuting if 𝑑(𝑓𝑔𝑥,𝑔𝑓𝑥)≤𝑑(𝑓𝑥,𝑔𝑥), for all 𝑥∈𝑋.

Definition 2.3 (see [20]). The maps 𝑓∶𝑋→𝑋 and 𝑔∶𝑋→𝑋 are said to be compatible if lim_𝑛→∞𝑑(𝑓𝑔𝑥_𝑛,𝑔𝑓𝑥_𝑛)=0 whenever {𝑥_𝑛} is a sequence in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑧, for some 𝑧∈𝑋.

Definition 2.4 (see [26]). The maps 𝑓∶𝑋→𝑋 and 𝑔∶𝑋→𝑋 are said to be weakly compatible 𝑓𝑔𝑥=𝑔𝑓𝑥, for all 𝑥∈𝐶(𝑓,𝑔).

Definition 2.5 (see [22]). Let 𝑓∶𝑋→𝑋 and 𝑔∶𝑋→𝑋. The pair (𝑓,𝑔) satisfies property (E.A) if there exist the sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑧∈𝑋. (2.2)

See example of property (E.A) in Kamran [27, 28] and Sintunavarat and Kumam [11].

Definition 2.6 (see [23]). Let 𝑓,𝑔,𝐴,𝐵∶𝑋→𝑋. The pair (𝑓,𝑔) and (𝐴,𝐵) satisfy a common property (E.A) if there exist sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=lim_𝑛→∞𝐴𝑦_𝑛=lim_𝑛→∞𝐵𝑦_𝑛=𝑧∈𝑋. (2.3)

Remark 2.7. If 𝐴=𝑓, 𝐵=𝑔, and {𝑥_𝑛}={𝑦_𝑛} in (2.3), then we get the definition of property (E.A).

Definition 2.8 (see [24]). Let 𝑓,𝑔∶𝑋→𝑋. A point 𝑧∈𝑋 is said to be a weak tangent point to (𝑓,𝑔) if there exist sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑦_𝑛=𝑧∈𝑋. (2.4)

Remark 2.9. If {𝑥_𝑛}={𝑦_𝑛} in (2.4), we get the definition of property (E.A).

Definition 2.10 (see [24]). Let 𝑓,𝑔,𝐴,𝐵∶𝑋→𝑋. The pair (𝑓,𝑔) is called tangential with respect to the pair (𝐴,𝐵) if there exist sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑦_𝑛=lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝐵𝑦_𝑛=𝑧∈𝑋. (2.5)

3. Main Results

In this section, we first introduce the notion of tangential property for two single-valued and two multivalued mappings. Throughout this section, ℝ₊ denotes the set of nonnegative real numbers.

Definition 3.1. Let 𝑓,𝑔∶𝑋→𝑋 and 𝐴,𝐵∶𝑋→CB(𝑋). The pair (𝑓,𝑔) is called tangential with respect to the pair (𝐴,𝐵) if lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝐵𝑦_𝑛=𝐷∈CB(𝑋), (3.1) whenever sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑦_𝑛=𝑧∈𝐷, (3.2) for some 𝑧∈𝑋.

Example 3.2. Let (ℝ₊,𝑑) be a metric space with usual metric 𝑑. Let 𝑓,𝑔∶ℝ₊→ℝ₊ and 𝐴,𝐵∶ℝ₊→CB(ℝ₊) be mappings defined by 𝑓𝑥=𝑥+1, 𝑔𝑥=𝑥+2, 𝐴𝑥={𝑥²/2+1}, and 𝐵𝑥={𝑥²+2}, for all 𝑥∈ℝ₊. Clearly, there exists two sequences {𝑥_𝑛=2+1/𝑛} and {𝑦_𝑛=1+1/𝑛} such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝐵𝑦_𝑛={3}∈CBℝ₊ (3.3) whenever lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑦_𝑛=3∈ℝ₊. (3.4) So, the pair (𝑓,𝑔) is tangential with respect to the pair (𝐴,𝐵).

Definition 3.3. Let 𝑓∶𝑋→𝑋 and 𝐴∶𝑋→CB(𝑋). The mapping 𝑓 is called tangential with respect to the mapping 𝐴 if lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝐴𝑦_𝑛=𝐷∈CB(𝑋), (3.5) whenever sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑓𝑦_𝑛=𝑧∈𝐷, (3.6) for some 𝑧∈𝑋.

Example 3.4. Let (ℝ₊,𝑑) be a metric space with usual metric 𝑑. Let 𝑓∶ℝ₊→ℝ₊ and 𝐴∶ℝ₊→CB(ℝ₊) be mappings defined by 𝑓𝑥=𝑥+1,𝐴𝑥=𝑥²+1. (3.7) Clearly, there exist two sequences {𝑥_𝑛=1+1/𝑛} and {𝑦_𝑛=1−1/𝑛} such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝐴𝑦_𝑛={2}∈CBℝ₊ (3.8) whenever lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑓𝑦_𝑛=2∈ℝ₊. (3.9) So, the mapping 𝑓 is tangential with respect to the mapping 𝐴.

Now, we state and prove our main result.

Theorem 3.5. Let 𝑓,𝑔∶𝑋→𝑋 and 𝐴,𝐵∶𝑋→CB(𝑋) satisfy 1+𝛼₀^{𝑑(𝑓𝑥,𝑔𝑦)}𝜓(𝑡)𝑑𝑡₀^{𝐻(𝐴𝑥,𝐵𝑦)}𝜓(𝑡)𝑑𝑡<𝛼₀^{𝑑(𝐴𝑥,𝑓𝑥)}𝜓(𝑡)𝑑𝑡₀^{𝑑(𝐵𝑦,𝑔𝑦)}𝜓(𝑡)𝑑𝑡+₀^{𝑑(𝐴𝑥,𝑔𝑦)}𝜓(𝑡)𝑑𝑡₀^{𝑑(𝑓𝑥,𝐵𝑦)}𝜓(𝑡)𝑑𝑡+𝑎₀^{𝑑(𝑓𝑥,𝑔𝑦)}𝜓(𝑡)𝑑𝑡+(1−𝑎)max₀^{𝑑(𝐴𝑥,𝑓𝑥)}𝜓(𝑡)𝑑𝑡,₀^{𝑑(𝐵𝑦,𝑔𝑦)}𝜓(𝑡)𝑑𝑡,_{0𝑑(𝐴𝑥,𝑓𝑥)}𝜓(𝑡)𝑑𝑡^1/2₀^{𝑑(𝐴𝑥,𝑔𝑦)}𝜓(𝑡)𝑑𝑡^1/2,_{0𝑑(𝑓𝑥,𝐵𝑦)}𝜓(𝑡)𝑑𝑡^1/2₀^{𝑑(𝐴𝑥,𝑔𝑦)}𝜓(𝑡)𝑑𝑡^1/2, (3.10) for all 𝑥,𝑦∈𝑋 for which the right-hand side of (3.10) is positive, where 0<𝑎<1, 𝛼≥0 and 𝜓∶ℝ₊→ℝ₊ is a Lebesgue integrable mapping which is a summable nonnegative and such that ^𝜖₀𝜓(𝑡)𝑑𝑡>0, (3.11) for each 𝜖>0. If the following conditions (a)–(d) hold: (a)there exists a point 𝑧∈𝑓(𝑋)∩𝑔(𝑋) which is a weak tangent point to (𝑓,𝑔), (b)(𝑓,𝑔) is tangential with respect to (𝐴,𝐵), (c)𝑓𝑓𝑎=𝑓𝑎, 𝑔𝑔𝑏=𝑔𝑏, and 𝐴𝑓𝑎=𝐵𝑔𝑏 for 𝑎∈𝐶(𝑓,𝐴) and 𝑏∈𝐶(𝑔,𝐵), (d)the pairs (𝑓,𝐴) and (𝑔,𝐵) are weakly compatible. Then, 𝑓, 𝑔, 𝐴, and 𝐵 have a common fixed point in 𝑋.

Proof. Since 𝑧∈𝑓(𝑋)∩𝑔(𝑋), 𝑧=𝑓𝑢=𝑔𝑣 for some 𝑢,𝑣∈𝑋. It follows from a point 𝑧 which is a weak tangent point to (𝑓,𝑔) that there exist sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑦_𝑛=𝑧. (3.12) Because the pair (𝑓,𝑔) is tangential with respect to the pair (𝐴,𝐵), we get lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝐵𝑦_𝑛=𝐷, (3.13) for some 𝐷∈CB(𝑋). Since 𝑧=𝑓𝑢=𝑔𝑣 and (3.12) and (3.13) are true, we have 𝑧=𝑓𝑢=𝑔𝑣=lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑦_𝑛∈lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝐵𝑦_𝑛=𝐷. (3.14)
We claim that 𝑧∈𝐵𝑣. If not, then condition (3.10) implies1+𝛼^{𝑑(𝑓𝑥𝑛}₀^,𝑔𝑣)𝜓(𝑡)𝑑𝑡^{𝐻(𝐴𝑥𝑛}₀^,𝐵𝑣)𝜓(𝑡)𝑑𝑡<𝛼^{𝑑(𝐴𝑥𝑛,𝑓𝑥𝑛)}₀𝜓(𝑡)𝑑𝑡₀^{𝑑(𝐵𝑣,𝑔𝑣)}𝜓(𝑡)𝑑𝑡+^{𝑑(𝐴𝑥𝑛}₀^,𝑔𝑣)𝜓(𝑡)𝑑𝑡^{𝑑(𝑓𝑥𝑛}₀^,𝐵𝑣)𝜓(𝑡)𝑑𝑡+𝑎^{𝑑(𝑓𝑥𝑛}₀^,𝑔𝑣)𝜓(𝑡)𝑑𝑡+(1−𝑎)max^{𝑑(𝐴𝑥𝑛,𝑓𝑥𝑛)}₀𝜓(𝑡)𝑑𝑡,₀^{𝑑(𝐵𝑣,𝑔𝑣)}𝜓(𝑡)𝑑𝑡,_{𝑑(𝐴𝑥𝑛,𝑓𝑥𝑛)0}𝜓(𝑡)𝑑𝑡^{1/2𝑑(𝐴𝑥𝑛}₀^,𝑔𝑣)𝜓(𝑡)𝑑𝑡^1/2,_{𝑑(𝑓𝑥𝑛0,𝐵𝑣)}𝜓(𝑡)𝑑𝑡^{1/2𝑑(𝐴𝑥𝑛}₀^,𝑔𝑣)𝜓(𝑡)𝑑𝑡^1/2. (3.15) Letting 𝑛→∞, we get ₀^{𝐻(𝐷,𝐵𝑣)}𝜓(𝑡)𝑑𝑡≤(1−𝑎)₀^{𝑑(𝐵𝑣,𝑧)}𝜓(𝑡)𝑑𝑡. (3.16) Since ₀^{𝑑(𝑧,𝐵𝑣)}𝜓(𝑡)𝑑𝑡<₀^{𝐻(𝐷,𝐵𝑣)}𝜓(𝑡)≤()𝑑𝑡1−𝑎₀^{𝑑(𝐵𝑣,𝑧)}𝜓(𝑡)𝑑𝑡<₀^{𝑑(𝑧,𝐵𝑣)}𝜓(𝑡)𝑑𝑡, (3.17) which is a contradiction, then 𝑧∈𝐵𝑣.
Again, we claim that 𝑧∈𝐴𝑢. If not, then condition (3.10) implies1+𝛼^{𝑑(𝑓𝑢,𝑔𝑦𝑛)}₀𝜓(𝑡)𝑑𝑡^{𝐻(𝐴𝑢,𝐵𝑦𝑛)}₀𝜓(𝑡)𝑑𝑡<𝛼₀^{𝑑(𝐴𝑢,𝑓𝑢)}𝜓(𝑡)𝑑𝑡^{𝑑(𝐵𝑦𝑛,𝑔𝑦𝑛)}₀𝜓(𝑡)𝑑𝑡+^{𝑑(𝐴𝑢,𝑔𝑦𝑛)}₀𝜓(𝑡)𝑑𝑡^{𝑑(𝑓𝑢,𝐵𝑦𝑛)}₀𝜓(𝑡)𝑑𝑡+𝑎^{𝑑(𝑓𝑢,𝑔𝑦𝑛)}₀𝜓(𝑡)𝑑𝑡+(1−𝑎)max₀^{𝑑(𝐴𝑢,𝑓𝑢)}𝜓(𝑡)𝑑𝑡,^{𝑑(𝐵𝑦𝑛,𝑔𝑦𝑛)}₀𝜓(𝑡)𝑑𝑡,_{0𝑑(𝐴𝑢,𝑓𝑢)}𝜓(𝑡)𝑑𝑡^{1/2𝑑(𝐴𝑢,𝑔𝑦𝑛)}₀𝜓(𝑡)𝑑𝑡^1/2,_{𝑑(𝑓𝑢,𝐵𝑦𝑛)0}𝜓(𝑡)𝑑𝑡^{1/2𝑑(𝐴𝑢,𝑔𝑦𝑛)}₀𝜓(𝑡)𝑑𝑡^1/2. (3.18) Letting 𝑛→∞, we get ₀^{𝐻(𝐴𝑢,𝐷)}𝜓(𝑡)𝑑𝑡≤(1−𝑎)₀^{𝑑(𝐴𝑢,𝑧)}𝜓(𝑡)𝑑𝑡. (3.19) Since ₀^{𝑑(𝑧,𝐴𝑢)}𝜓(𝑡)𝑑𝑡<₀^{𝐻(𝐴𝑢,𝐷)}𝜓(𝑡)≤()𝑑𝑡1−𝑎₀^{𝑑(𝐴𝑢,𝑧)}𝜓(𝑡)𝑑𝑡<₀^{𝑑(𝑧,𝐴𝑢)}𝜓(𝑡)𝑑𝑡, (3.20) which is a contradiction, then 𝑧∈𝐴𝑢.
Now, we conclude 𝑧=𝑔𝑣∈𝐵𝑣 and 𝑧=𝑓𝑢∈𝐴𝑢. It follows from 𝑣∈𝐶(𝑔,𝐵), 𝑢∈𝐶(𝑓,𝐴) that 𝑔𝑔𝑣=𝑔𝑣, 𝑓𝑓𝑢=𝑓𝑢, and 𝐴𝑓𝑢=𝐵𝑔𝑣. Hence, 𝑔𝑧=𝑧, 𝑓𝑧=𝑧 and 𝐴𝑧=𝐵𝑧.
Since the pair (𝑔,𝐵) is weakly compatible, 𝑔𝐵𝑣=𝐵𝑔𝑣. Thus 𝑔𝑧∈𝑔𝐵𝑣=𝐵𝑔𝑣=𝐵𝑧. Similarly, we can prove that 𝑓𝑧∈𝐴𝑧. Consequently, 𝑧=𝑓𝑧=𝑔𝑧∈𝐴𝑧=𝐵𝑧. Therefore the maps 𝑓,𝑔,𝐴, and 𝐵 have a common fixed point.

If 𝛼=0 in Theorem 3.5, we get the following corollary.

Corollary 3.6. Let 𝑓,𝑔∶𝑋→𝑋 and 𝐴,𝐵∶𝑋→CB(𝑋) satisfy ₀^{𝐻(𝐴𝑥,𝐵𝑦)}𝜓(𝑡)𝑑𝑡<𝑎₀^{𝑑(𝑓𝑥,𝑔𝑦)}𝜓(𝑡)+()𝑑𝑡1−𝑎max₀^{𝑑(𝐴𝑥,𝑓𝑥)}𝜓(𝑡)𝑑𝑡,₀^{𝑑(𝐵𝑦,𝑔𝑦)}𝜓(𝑡)𝑑𝑡,_{0𝑑(𝐴𝑥,𝑓𝑥)}𝜓(𝑡)𝑑𝑡^1/2₀^{𝑑(𝐴𝑥,𝑔𝑦)}𝜓(𝑡)𝑑𝑡^1/2,_{0𝑑(𝑓𝑥,𝐵𝑦)}𝜓(𝑡)𝑑𝑡^1/2₀^{𝑑(𝐴𝑥,𝑔𝑦)}𝜓(𝑡)𝑑𝑡^1/2, (3.21) for all 𝑥,𝑦∈𝑋 for which the right-hand side of (3.21) is positive, where 0<𝑎<1 and 𝜓∶ℝ₊→ℝ₊ is a Lebesgue integrable mapping which is a summable nonnegative and such that ^𝜖₀𝜓(𝑡)𝑑𝑡>0, (3.22) for each 𝜖>0. If the following conditions (a)–(d) hold: (a)there exists a point 𝑧∈𝑓(𝑋)∩𝑔(𝑋) which is a weak tangent point to (𝑓,𝑔), (b)(𝑓,𝑔) is tangential with respect to (𝐴,𝐵), (c)𝑓𝑓𝑎=𝑓𝑎, 𝑔𝑔𝑏=𝑔𝑏 and 𝐴𝑓𝑎=𝐵𝑔𝑏 for 𝑎∈𝐶(𝑓,𝐴) and 𝑏∈𝐶(𝑔,𝐵), (d)the pairs (𝑓,𝐴) and (𝑔,𝐵) are weakly compatible. Then, 𝑓, 𝑔, 𝐴, and 𝐵 have a common fixed point in 𝑋.

If 𝛼=0, 𝑔=𝑓, and 𝐵=𝐴 in Theorem 3.5, we get the following corollary.

Corollary 3.7. Let 𝑓∶𝑋→𝑋 and 𝐴∶𝑋→CB(𝑋) satisfy ₀^{𝐻(𝐴𝑥,𝐴𝑦)}𝜓(𝑡)𝑑𝑡<𝑎₀^{𝑑(𝑓𝑥,𝑓𝑦)}𝜓(𝑡)+()𝑑𝑡1−𝑎max₀^{𝑑(𝐴𝑥,𝑓𝑥)}𝜓(𝑡)𝑑𝑡,₀^{𝑑(𝐴𝑦,𝑓𝑦)}𝜓(𝑡)𝑑𝑡,_{0𝑑(𝐴𝑥,𝑓𝑥)}𝜓(𝑡)𝑑𝑡^1/2₀^{𝑑(𝐴𝑥,𝑓𝑦)}𝜓(𝑡)𝑑𝑡^1/2,_{0𝑑(𝑓𝑥,𝐴𝑦)}𝜓(𝑡)𝑑𝑡^1/2₀^{𝑑(𝐴𝑥,𝑓𝑦)}𝜓(𝑡)𝑑𝑡^1/2, (3.23) for all 𝑥,𝑦∈𝑋 for which the right-hand side of (3.23) is positive, where 0<𝑎<1 and 𝜓∶ℝ₊→ℝ₊ is a Lebesgue integrable mapping which is a summable nonnegative and such that ^𝜖₀𝜓(𝑡)𝑑𝑡>0 (3.24) for each 𝜖>0. If the following conditions (a)–(d) hold: (a)there exists sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛∈𝑋, (b)𝑓 is tangential with respect to 𝐴, (c)𝑓𝑓𝑎=𝑓𝑎 for 𝑎∈𝐶(𝑓,𝐴), (d)the pair (𝑓,𝐴) is weakly compatible. Then, 𝑓 and 𝐴 have a common fixed point in 𝑋.

If 𝜓(𝑡)=1 in Theorem 3.5, we get the following corollary.

Corollary 3.8. Let 𝑓,𝑔∶𝑋→𝑋 and 𝐴,𝐵∶𝑋→CB(𝑋) satisfy 𝐻𝑑𝑑𝑑+()𝑑,𝑑1+𝛼𝑑𝑓𝑥,𝑔𝑦𝐴𝑥,𝐵𝑦<𝛼𝐴𝑥,𝑓𝑥𝐵𝑦,𝑔𝑦+𝑑𝐴𝑥,𝑔𝑦𝑓𝑥,𝐵𝑦+𝑎𝑑𝑓𝑥,𝑔𝑦1−𝑎max𝐴𝑥,𝑓𝑥,𝑑𝐵𝑦,𝑔𝑦𝐴𝑥,𝑓𝑥^1/2𝑑𝐴𝑥,𝑔𝑦^1/2,𝑑𝑓𝑥,𝐵𝑦^1/2𝑑𝐴𝑥,𝑔𝑦^1/2 (3.25) for all 𝑥,𝑦∈𝑋 for which the right-hand side of (3.25) is positive, where 0<𝑎<1 and 𝛼≥0. If the following conditions (a)–(d) holds: (a)there exists a point 𝑧∈𝑓(𝑋)∩𝑔(𝑋) which is a weak tangent point to (𝑓,𝑔), (b)(𝑓,𝑔) is tangential with respect to (𝐴,𝐵), (c)𝑓𝑓𝑎=𝑓𝑎, 𝑔𝑔𝑏=𝑔𝑏 and 𝐴𝑓𝑎=𝐵𝑔𝑏 for 𝑎∈𝐶(𝑓,𝐴) and 𝑏∈𝐶(𝑔,𝐵), (d)the pairs (𝑓,𝐴) and (𝑔,𝐵) are weakly compatible. Then, 𝑓, 𝑔, 𝐴, and 𝐵 have a common fixed point in 𝑋.

If 𝜓(𝑡)=1 and 𝛼=0 in Theorem 3.5, we get the following corollary.

Corollary 3.9. Let 𝑓,𝑔∶𝑋→𝑋 and 𝐴,𝐵∶𝑋→CB(𝑋) satisfy 𝐻+()𝑑,𝑑𝐴𝑥,𝐵𝑦<𝑎𝑑𝑓𝑥,𝑔𝑦1−𝑎max𝐴𝑥,𝑓𝑥,𝑑𝐵𝑦,𝑔𝑦𝐴𝑥,𝑓𝑥^1/2𝑑𝐴𝑥,𝑔𝑦^1/2,𝑑𝑓𝑥,𝐵𝑦^1/2𝑑𝐴𝑥,𝑔𝑦^1/2, (3.26) for all 𝑥,𝑦∈𝑋 for which the right-hand side of (3.26) is positive, where 0<𝑎<1. If the following conditions (a)–(d) hold: (a)there exists a point 𝑧∈𝑓(𝑋)∩𝑔(𝑋) which is a weak tangent point to (𝑓,𝑔), (b)(𝑓,𝑔) is tangential with respect to (𝐴,𝐵), (c)𝑓𝑓𝑎=𝑓𝑎, 𝑔𝑔𝑏=𝑔𝑏 and 𝐴𝑓𝑎=𝐵𝑔𝑏 for 𝑎∈𝐶(𝑓,𝐴) and 𝑏∈𝐶(𝑔,𝐵), (d)the pairs (𝑓,𝐴) and (𝑔,𝐵) are weakly compatible. Then, 𝑓, 𝑔, 𝐴, and 𝐵 have a common fixed point in 𝑋.

If 𝜓(𝑡)=1, 𝛼=0, 𝑔=𝑓 and 𝐵=𝐴 in Theorem 3.5, we get the following corollary.

Corollary 3.10. Let 𝑓∶𝑋→𝑋 and 𝐴∶𝑋→CB(𝑋) satisfy 𝐻+()𝑑,𝑑𝐴𝑥,𝐴𝑦<𝑎𝑑𝑓𝑥,𝑓𝑦1−𝑎max𝐴𝑥,𝑓𝑥,𝑑𝐴𝑦,𝑓𝑦𝐴𝑥,𝑓𝑥^1/2𝑑𝐴𝑥,𝑓𝑦^1/2,𝑑𝑓𝑥,𝐴𝑦^1/2𝑑𝐴𝑥,𝑓𝑦^1/2 (3.27) for all 𝑥,𝑦∈𝑋 for which the right-hand side of (3.27) is positive, where 0<𝑎<1. If the following conditions (a)–(d) holds: (a)there exists sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛∈𝑋, (b)𝑓 is tangential with respect to 𝐴, (c)𝑓𝑓𝑎=𝑓𝑎 for 𝑎∈𝐶(𝑓,𝐴), (d)the pairs (𝑓,𝐴) is weakly compatible. Then, 𝑓 and 𝐴 have a common fixed point in 𝑋.

Acknowledgments

W. Sintunavarat would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the Faculty of Science, KMUTT for financial support during the preparation of this manuscript for Ph.D. Program at KMUTT. Moreover, the authors also would like to thank the National Research University Project of Thailand's Office of the Higher Education Commission for financial support (under the CSEC project no. 54000267). Finally, the authors would like to thank Professor Frank Werner for your help and encouragement. Special thanking are also due to the reviewers, who have made a number of valuable comments and suggestions which have improved the manuscript greatly.