On Caristi Type Maps and Generalized Distances with Applications

Abstract

We prove some new existence theorems of fixed points for Caristi type maps and some suitable generalized distances without lower semicontinuity assumptions on dominated functions. As applications of our results, some new fixed point theorems and new generalizations of the Banach contraction principle are given.

1. Introduction

In 1972, Caristi proved the following famous fixed point theorem.

Theorem 1 (Caristi [1]). Let (𝑋,𝑑) be a complete metric space and 𝑓∶𝑋→ℝ a lower semicontinuous and bounded below function. Suppose that 𝑇 is a Caristi type map on 𝑋 dominated by 𝑓; that is, 𝑇 satisfies 𝑑(𝑥,𝑇𝑥)≤𝑓(𝑥)−𝑓(𝑇𝑥)𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑥∈𝑋. (1) Then 𝑇 has a fixed point in 𝑋.

It is well-known that the Caristi’s fixed point theorem is one of the most valuable generalization of the Banach contraction principle [2], and it is equivalent to the Ekeland’s variational principle, to the Takahashi’s nonconvex minimization theorem, to the Daneš’ drop theorem, to the petal theorem, and to the Oettli-Théra’s theorem; see [3–26] and references therein for more details. A number of generalizations in various different directions of the Caristi’s fixed point theorem have been investigated by several authors; see, for example, [4–30] and references therein. An interesting direction of research is the extension of Caristi’s fixed point theorem, Ekeland’s variational principle, and Takahashi’s nonconvex minimization theorem to generalized distances, for example, 𝑤-distances [5, 10, 14, 19], 𝜏-distances [11, 12, 22], 𝜏-functions [13, 15, 18, 22–25, 31–36], weak 𝜏-functions [24, 25], 𝑃-distances [26], 𝑄-functions [21], generalized pseudodistances [22, 23], and others. For more details on these generalizations, one can refer to [5, 10–26] and references therein.

Let us recall how we can exploit Caristi’s fixed point theorem to prove the Banach contraction principle. A selfmap 𝑇 on a metric space (𝑋,𝑑) is called contractive or Banach type if there exists a real number 𝜆∈[0,1) such that 𝑑𝑇𝑥,𝑇𝑦≤𝜆𝑑𝑥,𝑦forany𝑥,𝑦∈𝑋. (2) It is obvious that if 𝑇 is a contractive map on 𝑋, then 𝑇 is continuous on 𝑋 and (2) will deduce the following inequality: 𝑑()≤1𝑥,𝑇𝑥𝑑()−11−𝜆𝑥,𝑇𝑥𝑑1−𝜆𝑇𝑥,𝑇²𝑥forany𝑥∈𝑋. (3) The inequality (3) admits that 𝑇 is a Caristi type map on 𝑋 dominated by 𝑓 defined by 𝑓(𝑥)∶=(1/(1−𝜆))𝑑(𝑥,𝑇𝑥). From the continuity of 𝑇, the function 𝑓 is continuous on 𝑋, and therefore the Caristi’s fixed point theorem is applicable to prove the Banach contraction principle. It is quite obvious that for any map 𝑇 and any generalized distances 𝑝, the function 𝑥↦𝑝(𝑥,𝑇𝑥) is not necessarily to be continuous even lower semicontinuous, so such well-known generalized versions of Caristi’s fixed point theorem with lower semicontinuity are not easily applicable to any generalized version of Banach contraction principle for generalized distances. Motivated by the reason, in the recently paper [20], the author established some new versions of Caristi type fixed point theorem such that they can be applicable to prove generalized versions of Banach contraction principle for suitable generalized distances.

This work can be considered as a continuation of the paper [20]. In this paper, we first establish some new fixed point theorems for Caristi type maps and some suitable generalized distances without assuming that the dominated functions possess lower semicontinuity property. As applications of our results, some new fixed point theorems and new generalizations of the Banach contraction principle are given. We have already succeeded in utilizing our new versions of Caristi type fixed point theorem to deal with the existence results for any map 𝑇 satisfying 𝑑𝑇𝑥,𝑇𝑦≤𝛼𝑑𝑥,𝑦𝑑𝑥,𝑦∀𝑥,𝑦∈𝑋, (4) where 𝛼∶[0,+∞)→[0,1) is a function satisfying limsup_𝑠→𝑡+𝛼(𝑠)<1 for all 𝑡∈[0,+∞).

2. Preliminaries

We recall in this section the notations, definitions, and results needed. Let (𝑋,𝑑) be a metric space. An extended real valued function 𝜙∶𝑋→(−∞,+∞] is said to be lower semicontinuous (l.s.c., for short) at 𝑤∈𝑋 if for any sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛→𝑤 as 𝑛→∞, we have 𝜙(𝑤)≤liminf_𝑛→∞𝜙(𝑥_𝑛). The function 𝜙 is called to be l.s.c. on 𝑋 if 𝜙 is l.s.c. at every point of 𝑋. The function 𝜙 is said to be proper if 𝜙≢+∞. Let 𝑇∶𝑋→𝑋 be a selfmap. 𝑇 is said to be closed if 𝐺𝑟𝑇={(𝑥,𝑦)∈𝑋×𝑋∶𝑦=𝑇𝑥}, the graph of 𝑇, is closed in 𝑋×𝑋. A point 𝑣 in 𝑋 is a fixed point of 𝑇 if 𝑇𝑣=𝑣. The set of fixed points of 𝑇 is denoted by ℱ(𝑇). Throughout this paper we denote by ℕ and ℝ, the set of positive integers and nonnegative real numbers, respectively.

Recall that a function 𝑝∶𝑋×𝑋→[0,+∞) is called a 𝑤-distance [5, 10–20, 31], first introduced and defined by Kada, Suzuki, and Takahashi, if the following are satisfied:(𝑤1)𝑝(𝑥,𝑧)≤𝑝(𝑥,𝑦)+𝑝(𝑦,𝑧) for any 𝑥,𝑦,𝑧∈𝑋;(𝑤2) for any 𝑥∈𝑋, 𝑝(𝑥,⋅)∶𝑋→[0,+∞) is l.s.c.;(𝑤3) for any 𝜀>0, there exists 𝛿>0 such that 𝑝(𝑧,𝑥)≤𝛿 and 𝑝(𝑧,𝑦)≤𝛿 imply 𝑑(𝑥,𝑦)≤𝜀.

A function 𝑝∶𝑋×𝑋→[0,+∞) is said to be a 𝜏-function [13, 15, 18, 22–25, 31–36], first introduced and studied by Lin and Du, if the following conditions hold:(𝜏1)𝑝(𝑥,𝑧)≤𝑝(𝑥,𝑦)+𝑝(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋;(𝜏2) if 𝑥∈𝑋 and {𝑦_𝑛} in 𝑋 with lim_𝑛→∞𝑦_𝑛=𝑦 such that 𝑝(𝑥,𝑦_𝑛)≤𝑀 for some 𝑀=𝑀(𝑥)>0, then 𝑝(𝑥,𝑦)≤𝑀;(𝜏3) for any sequence {𝑥_𝑛} in 𝑋 with lim_𝑛→∞sup{𝑝(𝑥_𝑛,𝑥_𝑚)∶𝑚>𝑛}=0, if there exists a sequence {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝑝(𝑥_𝑛,𝑦_𝑛)=0, then lim_𝑛→∞𝑑(𝑥_𝑛,𝑦_𝑛)=0;(𝜏4) for 𝑥,𝑦,𝑧∈𝑋, 𝑝(𝑥,𝑦)=0 and 𝑝(𝑥,𝑧)=0 imply 𝑦=𝑧.

Note that not either of the implications 𝑝(𝑥,𝑦)=0⇔𝑥=𝑦 necessarily holds and 𝑝 is nonsymmetric in general. It is well known that the metric 𝑑 is a 𝑤-distance and any 𝑤-distance is a 𝜏-function, but the converse is not true; see [13, 31] for more detail.

Example 2 (see [31, Example  A]). Let 𝑋=ℝ with the metric 𝑑(𝑥,𝑦)=|𝑥−𝑦| for 𝑥,𝑦∈𝑋, and 0<𝑎<𝑏. Define the function 𝑝∶𝑋×𝑋→[0,+∞) by 𝑝𝑥,𝑦=max𝑎𝑦−𝑥,𝑏𝑥−𝑦. (5) Then 𝑝 is a 𝜏-function.

The following result is crucial in this paper.

Theorem 3 (see [18, Lemma 2.1]). Let (𝑋,𝑑) be a metric space and 𝑝∶𝑋×𝑋→[0,+∞) a function. Assume that 𝑝 satisfies the condition (𝜏3). If a sequence {𝑥_𝑛} in 𝑋 with lim_𝑛→∞sup{𝑝(𝑥_𝑛,𝑥_𝑚)∶𝑚>𝑛}=0, then {𝑥_𝑛} is a Cauchy sequence in 𝑋.

Recently, the concepts of weak 𝜏-function and generalized pseudodistance were introduced and studied by Khanh and Quy [24, 25] and Włodarczyk and Plebaniak [22] as follows.

Definition 4. Let (𝑋,𝑑) be a metric space. A function 𝑝∶𝑋×𝑋→[0,+∞) is called(i)a weak 𝜏-function [24, 25] on 𝑋 if conditions (𝜏1), (𝜏3), and (𝜏4) hold;(ii)a generalized pseudodistance [22] on 𝑋 if conditions (𝜏1) and (𝜏3) hold.

It is obvious that any 𝜏-function is a weak 𝜏-function and every weak 𝜏-function is a generalized pseudodistance, but the converse parts are not always true. The first observation is that there exists a weak 𝜏-function which is not a 𝜏-function.

Example 5 (see [24, Example  2.5]). Let 𝑋=[0,+∞), 𝛾>0, and 𝑝∶𝑋×𝑋→[0,+∞) be defined by 𝑝𝑥,𝑦=3𝑥−𝑦+𝛾,if𝑥≠𝑦,2𝛾,if𝑥=𝑦. (6) Then 𝑝 is a weak 𝜏-function which is neither a 𝜏-function nor a 𝑤-distance.

The following example shows that there exists a generalized pseudodistance which is not a weak 𝜏-function.

Example 6 (see [22, Example  1.3]). Define a function 𝑝∶[0,2]×[0,2]→[0,+∞) by 𝑝𝑥,𝑦=0,if𝑥−𝑦=−2;𝑥−𝑦,if−2<𝑥−𝑦≤0;𝑥−𝑦+2,if0<𝑥−𝑦≤2. (7) Then 𝑝 is a generalized pseudodistance but not a weak 𝜏-function.

Very recently, the author first introduced the following concepts.

Definition 7 (see [20]). Let (𝑋,𝑑) be a metric space, and let 𝑓∶𝑋→ℝ, 𝜑∶ℝ→(0,+∞), and 𝑝∶𝑋×𝑋→[0,+∞) be functions. A single-valued selfmap 𝑇∶𝑋→𝑋 is called(i)Caristi type on 𝑋 dominated by 𝑝, 𝜑, and 𝑓 (abbreviated as (𝑝,𝜑,𝑓)-Caristi type on 𝑋) if 𝑝(𝑥,𝑇𝑥)≤𝜑𝑓(𝑥)𝑓(𝑥)−𝑓(𝑇𝑥)foreach𝑥∈𝑋; (8)(ii)Caristi type on 𝑋 dominated by 𝑝 and 𝑓 (abbreviated as (𝑝,𝑓)-Caristi type on 𝑋) if 𝑝(𝑥,𝑇𝑥)≤𝑓(𝑥)−𝑓(𝑇𝑥)foreach𝑥∈𝑋; (9)(iii)Caristi type on 𝑋 dominated by 𝜑 and 𝑓 (abbreviated as (𝜑,𝑓)-𝐶𝑎𝑟𝑖𝑠𝑡𝑖𝑡𝑦𝑝𝑒 on 𝑋) if 𝑑(𝑥,𝑇𝑥)≤𝜑𝑓(𝑥)𝑓(𝑥)−𝑓(𝑇𝑥)foreach𝑥∈𝑋; (10)(iv)Caristi type on 𝑋 dominated by 𝑓 (abbreviated as (𝑓)-Caristi type on 𝑋) if 𝑑(𝑥,𝑇𝑥)≤𝑓(𝑥)−𝑓(𝑇𝑥)foreach𝑥∈𝑋. (11)

Clearly, if 𝑇 is (𝑝,𝑓)-Caristi type (resp. (𝑓)-Caristi type) on 𝑋, then 𝑇 is (𝑝,𝜑,𝑓)-Caristi type (resp. (𝜑,𝑓)-Caristi type) on 𝑋 with 𝜑(𝑡)=1 for all 𝑡. The following example illustrates that their converse are not always true.

Example 8. Let 𝑋=[0,+∞) with the usual metric 𝑑(𝑥,𝑦)=|𝑥−𝑦|. Then (𝑋,𝑑) is a complete metric space. Let 𝑝∶𝑋×𝑋→[0,+∞) be defined by 𝑝𝑥,𝑦=max20𝑥−𝑦,40𝑦−𝑥, (12) for all 𝑥, 𝑦∈𝑋. By Example 2, we know that 𝑝 is a 𝜏-function. Let 𝑇∶𝑋→𝑋 be defined by 𝑇𝑥=𝑥², 𝑥∈𝑋. Define 𝑓∶𝑋→ℝ by 𝑓(𝑥)=[)[).4𝑥−12,if𝑥∈0,115−8𝑥,if𝑥∈1,+∞ (13) Then 𝑓 is not lower semicontinuous at 𝑥=1. For 𝑖=1,2, let 𝜑_𝑖∶ℝ→(0,+∞) be defined by 𝜑₁(𝑡)=2,𝜑₂(𝑡)=6∀𝑡∈ℝ, (14) respectively. For 𝑥∈[0,1), we have 𝑑(𝑥,𝑇𝑥)=𝑥−𝑥²<4𝑥−𝑥²(𝑥)(),𝑝(){()(=𝑓−𝑓𝑇𝑥𝑥,𝑇𝑥=max20𝑥−𝑇𝑥,40𝑇𝑥−𝑥)}=20𝑥−𝑥²<𝜑₂𝑓(𝑥)𝑓(𝑥)−𝑓(𝑇𝑥). (15) For 𝑥∈[1,+∞), we have 𝑑(𝑥,𝑇𝑥)=𝑥²−𝑥<8𝑥²(𝑥)(),𝑝()𝑥−𝑥=𝑓−𝑓𝑇𝑥𝑥,𝑇𝑥=40²−𝑥<𝜑₂𝑓(𝑥)𝑓(𝑥)−𝑓(𝑇𝑥). (16) Hence, for any 𝑥∈𝑋, we show 𝑑(𝑥,𝑇𝑥)≤𝑓(𝑥)−𝑓(𝑇𝑥)≤𝜑_𝑖𝑓(𝑥)𝑓(𝑥)(){},𝑝()−𝑓𝑇𝑥foreach𝑖∈1,2𝑥,𝑇𝑥≤𝜑₂𝑓(𝑥)𝑓(𝑥)−𝑓(𝑇𝑥). (17) So, 𝑇 is (𝑓)-Caristi type on 𝑋 as well as (𝜑_𝑖,𝑓)-Caristi type on 𝑋 for all 𝑖∈{1,2}. Moreover, we know that 𝑇 is (𝑝,𝜑₂,𝑓)-Caristi type on 𝑋, but it is neither (𝑝,𝑓)-Caristi type nor (𝑝,𝜑₁,𝑓)-Caristi type on 𝑋 based on the following fact 𝑝(𝑥,𝑇𝑥)>𝜑₁𝑓(𝑥)𝑓(𝑥)−𝑓(𝑇𝑥)>𝑓(𝑥)−𝑓(𝑇𝑥)∀𝑥∈𝑋. (18)

Definition 9 (see [31–36]). A function 𝛼∶[0,+∞)→[0,1) is said to be an ℳ𝒯-function (or ℛ-function) if limsup_𝑠→𝑡+𝛼(𝑠)<1 for all 𝑡∈[0,+∞).

It is obvious that if 𝛼∶[0,+∞)→[0,1) is a nondecreasing function or a nonincreasing function, then 𝛼 is an ℳ𝒯-function. So the set of ℳ𝒯-functions is a rich class. But it is worth to mention that there exist functions which are not ℳ𝒯-functions.

Example 10 (see [32]). Let 𝛼∶[0,+∞)→[0,1) be defined by 𝛼(𝑡)∶=sin𝑡𝑡𝜋,if𝑡∈0,20,otherwise. (19) Since limsup_𝑠→0+𝛼(𝑠)=1,𝜑 is not an ℳ𝒯-function.

Recently, Du [32] first proved the following characterizations of ℳ𝒯-functions.

Theorem 11 (see [32]). Let 𝛼∶[0,+∞)→[0,1) be a function. Then the following statements are equivalent.(a)𝛼 is an ℳ𝒯-function.(b)For each 𝑡∈[0,+∞), there exist 𝑟_𝑡⁽¹⁾∈[0,1) and 𝜀_𝑡⁽¹⁾>0 such that 𝛼(𝑠)≤𝑟_𝑡⁽¹⁾ for all 𝑠∈(𝑡,𝑡+𝜀_𝑡⁽¹⁾).(c)For each 𝑡∈[0,+∞), there exist 𝑟_𝑡⁽²⁾∈[0,1) and 𝜀_𝑡⁽²⁾>0 such that 𝛼(𝑠)≤𝑟_𝑡⁽²⁾ for all 𝑠∈[𝑡,𝑡+𝜀_𝑡⁽²⁾].(d)For each 𝑡∈[0,+∞), there exist 𝑟_𝑡⁽³⁾∈[0,1) and 𝜀_𝑡⁽³⁾>0 such that 𝛼(𝑠)≤𝑟_𝑡⁽³⁾ for all 𝑠∈(𝑡,𝑡+𝜀_𝑡⁽³⁾].(e)For each 𝑡∈[0,+∞), there exist 𝑟_𝑡⁽⁴⁾∈[0,1) and 𝜀_𝑡⁽⁴⁾>0 such that 𝛼(𝑠)≤𝑟_𝑡⁽⁴⁾ for all 𝑠∈[𝑡,𝑡+𝜀_𝑡⁽⁴⁾).(f)For any nonincreasing sequence {𝑥_𝑛}_𝑛∈ℕ in [0,+∞), one has 0≤sup_𝑛∈ℕ𝛼(𝑥_𝑛)<1.(g)𝛼 is a function of contractive factor; that is, for any strictly decreasing sequence {𝑥_𝑛}_𝑛∈ℕ in [0,+∞), one has 0≤sup_𝑛∈ℕ𝛼(𝑥_𝑛)<1.

3. New Results for Caristi Type Maps and Their Applications

We start with the following useful auxiliary result.

Theorem 12. Let (𝑋,𝑑) be a metric space, 𝑓∶𝑋→(−∞,+∞] a proper and bounded below function, 𝜑∶ℝ→(0,+∞) a nondecreasing function, 𝑝∶𝑋×𝑋→[0,∞) a function, and 𝑇∶𝑋→𝑋 a selfmap on 𝑋. Let 𝑢∈𝑋 with 𝑓(𝑢)<+∞. Define 𝑥₁=𝑢 and 𝑥_𝑛+1=𝑇𝑥_𝑛 for each 𝑛∈ℕ. If 𝑝 satisfies (𝜏1) and 𝑇 is (𝑝,𝜑,𝑓)-Caristi type on 𝑋, then lim_𝑛→∞sup𝑝𝑥_𝑛,𝑥_𝑚∶𝑚>𝑛=0. (20) Moreover, if we futher assume that 𝑝 satisfies (𝜏3), then {𝑥_𝑛}_𝑛∈ℕ is a Cauchy sequence in 𝑋.

Proof. For 𝑥₁=𝑢, 𝑓(𝑥₁)<+∞. Since 𝑇 is (𝑝,𝜑,𝑓)-Caristi type on 𝑋, we get 𝑝𝑥₁,𝑥₂=𝑝𝑥₁,𝑇𝑥₁≤𝜑𝑓𝑥₁𝑓𝑥₁−𝑓𝑇𝑥₁=𝜑𝑓𝑥₁𝑓𝑥₁−𝑓𝑥₂, (21) which implies 𝑓𝑥₂≤𝑓𝑥₁<+∞. (22) Similarly, we have 𝑝𝑥₂,𝑥₃=𝑝𝑥₂,𝑇𝑥₂≤𝜑𝑓𝑥₂𝑓𝑥₂−𝑓𝑥₃,𝑓𝑥₃≤𝑓𝑥₂≤𝑓𝑥₁<+∞. (23) Hence, by induction, we can obtain the following inequalities 𝑝𝑥_𝑛,𝑥_𝑛+1≤𝜑𝑓𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛+1,𝑓𝑥_𝑛+1≤𝑓𝑥_𝑛<+∞foreach𝑛∈ℕ. (24)(25) Since 𝑓 is bounded below, 𝑟∶=lim_𝑛→∞𝑓𝑥_𝑛=inf_𝑛∈ℕ𝑓𝑥_𝑛exists. (26) By (25), since 𝜑 is nondecreasing, we have 𝜑𝑓𝑥_𝑛≤𝜑𝑓𝑥₁∀𝑛∈ℕ. (27) For 𝑚>𝑛 with 𝑚,𝑛∈ℕ, taking into account (𝜏1), (24), (26), and (27), we get 𝑝𝑥_𝑛,𝑥_𝑚≤^𝑚−1_𝑗=𝑛𝑝𝑥_𝑗,𝑥_𝑗+1≤𝜑𝑓𝑥₁𝑓𝑥_𝑛−𝑟. (28) Let 𝛼_𝑛=𝜑(𝑓(𝑥₁))(𝑓(𝑥_𝑛)−𝑟), 𝑛∈ℕ. Then sup{𝑝(𝑥_𝑛,𝑥_𝑚)∶𝑚>𝑛}≤𝛼_𝑛 for each 𝑛∈ℕ. Since lim_𝑛→∞𝑓(𝑥_𝑛)=𝑟, we obtain lim_𝑛→∞𝛼_𝑛=0, and hence lim_𝑛→∞sup{𝑝(𝑥_𝑛,𝑥_𝑚)∶𝑚>𝑛}=0. Moreover, if 𝑝 satisfies (𝜏3), then the desired conclusion follows from Theorem 3 immediately. The proof is completed.

Applying Theorem 12, we prove a new fixed point theorem for Caristi type maps and generalized pseudodistances. It is worth to mention that in Theorem 13 we pose some suitable assumptions on the map 𝑇 without assuming that the dominated functions possess lower semicontinuity property.

Theorem 13. Let (𝑋,𝑑) be a complete metric space, 𝑓∶𝑋→(−∞,+∞] a proper and bounded below function, 𝜑∶ℝ→(0,+∞) a nondecreasing function, and 𝑝 a generalized pseudodistance on 𝑋. Suppose that 𝑇∶𝑋→𝑋 is a (𝑝,𝜑,𝑓)-Caristi type selfmap on 𝑋 and one of the following conditions is satisfied:(H1)𝑇 is continuous; (H2)𝑇 is closed;(H3)𝑝(𝑥,𝑦)=0 implies 𝑥=𝑦 for all 𝑥,𝑦∈𝑋 and the map 𝑔∶𝑋→[0,∞) defined by 𝑔(𝑥)=𝑝(𝑥,𝑇𝑥) is l.s.c.;(H4) the map ℎ∶𝑋→[0,∞) defined by ℎ(𝑥)=𝑑(𝑥,𝑇𝑥) is l.s.c.;(H5) for any sequence {𝑧_𝑛} in 𝑋 with 𝑧_𝑛+1=𝑇𝑧_𝑛, 𝑛∈ℕ and lim_𝑛→∞𝑧_𝑛=𝑎, we have lim_𝑛→∞𝑝(𝑧_𝑛,𝑇𝑎)=0.

Then 𝑇 admits a fixed point in 𝑋. Moreover, for any 𝑤∈𝑋 with 𝑓(𝑤)<+∞, the sequence {𝑇^𝑛𝑤}_𝑛∈ℕ converges to a fixed point of 𝑇.

Proof. Let 𝑆={𝑥∈𝑋∶𝑓(𝑥)<+∞}. Since 𝑓 is proper, 𝑆≠∅. Let 𝑤∈𝑆. Define 𝑥₁=𝑤 and 𝑥_𝑛+1=𝑇𝑥_𝑛=𝑇^𝑛𝑤 for each 𝑛∈ℕ. Since 𝑝 is a generalized pseudodistance on 𝑋, by applying Theorem 12, we know that {𝑥_𝑛}_𝑛∈ℕ is a Cauchy sequence in 𝑋 and lim_𝑛→∞sup𝑝𝑥_𝑛,𝑥_𝑚∶𝑚>𝑛=0. (29) The last equality implies lim_𝑛→∞𝑝𝑥_𝑛,𝑥_𝑛+1=0. (30) By the completeness of 𝑋, there exists 𝑣_𝑤∈𝑋 such that 𝑥_𝑛→𝑣_𝑤 as 𝑛→∞.
Now, we verify 𝑣_𝑤∈ℱ(𝑇). If (H1) holds, since 𝑇 is continuous on 𝑋, 𝑥_𝑛+1=𝑇𝑥_𝑛 for each 𝑛∈ℕ and 𝑥_𝑛→𝑣_𝑤 as 𝑛→∞, we get 𝑣_𝑤=lim_𝑛→∞𝑥_𝑛=lim_𝑛→∞𝑥_𝑛+1=lim_𝑛→∞𝑇𝑥_𝑛=𝑇lim_𝑛→∞𝑥_𝑛=𝑇𝑣_𝑤, (31) which means 𝑣_𝑤∈ℱ(𝑇). If (H2) holds, since 𝑇 is closed, 𝑥_𝑛+1=𝑇𝑥_𝑛 for each 𝑛∈ℕ and 𝑥_𝑛→𝑣_𝑤 as 𝑛→∞, we have 𝑇𝑣_𝑤=𝑣_𝑤. Suppose that (H3) holds. By the lower semicontinuity of 𝑔, 𝑥_𝑛→𝑣_𝑤 as 𝑛→∞ and (30), we obtain 𝑝𝑣_𝑤,𝑇𝑣_𝑤=𝑔𝑣_𝑤≤liminf_𝑛→∞𝑔𝑥_𝑛=lim_𝑛→∞𝑝𝑥_𝑛,𝑥_𝑛+1=0, (32) which implies 𝑝(𝑣_𝑤,𝑇𝑣_𝑤)=0. By the hypothesis in (H3), we get 𝑣_𝑤∈ℱ(𝑇). Suppose that (H4) holds. Since {𝑥_𝑛}_𝑛∈ℕ is convergent in 𝑋, lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=0. (33) Since 𝑑𝑣_𝑤,𝑇𝑣_𝑤=ℎ𝑣_𝑤≤liminf_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=0, (34) we obtain 𝑑(𝑣_𝑤,𝑇𝑣_𝑤)=0, and hence 𝑣_𝑤∈ℱ(𝑇). Finally, assume (H5) holds. Since lim_𝑛→∞sup{𝑝(𝑥_𝑛,𝑥_𝑚)∶𝑚>𝑛}=0 and lim_𝑛→∞𝑝(𝑥_𝑛,𝑇𝑣_𝑤)=0, there exists {𝑎_𝑛}⊂{𝑥_𝑛} with lim_𝑛→∞sup{𝑝(𝑎_𝑛,𝑎_𝑚)∶𝑚>𝑛}=0 and 𝑏_𝑛=𝑇𝑣_𝑤 for all 𝑛∈ℕ, such that lim_𝑛→∞𝑝(𝑎_𝑛,𝑏_𝑛)=0. By (𝜏3), lim_𝑛→∞𝑑(𝑎_𝑛,𝑏_𝑛)=0. Since 𝑎_𝑛→𝑣_𝑤 as 𝑛→∞ and 𝑑(𝑏_𝑛,𝑣_𝑤)≤𝑑(𝑏_𝑛,𝑎_𝑛)+𝑑(𝑎_𝑛,𝑣_𝑤), we get 𝑏_𝑛→𝑣_𝑤 as 𝑛→∞. So 𝑇𝑣_𝑤=𝑣_𝑤 or 𝑣_𝑤∈ℱ(𝑇). Therefore, in any case, we prove 𝑣_𝑤∈ℱ(𝑇). Since 𝑤∈𝑆 is arbitrary, the sequence {𝑇^𝑛𝑤}_𝑛∈ℕ converges to a fixed point 𝑣_𝑤 of 𝑇. This completes the proof.

Here, we give an example illustrating Theorem 13. This example also gives a negative answer to the uniqueness of fixed point.

Example 14. Let 𝑋=[0,1] with the usual metric 𝑑(𝑥,𝑦)=|𝑥−𝑦|. Then (𝑋,𝑑) is a complete metric space. Define 𝑝∶𝑋×𝑋→[0,+∞) by 𝑝𝑥,𝑦=max2𝑥−𝑦,3𝑦−𝑥, (35) for all 𝑥, 𝑦∈𝑋. Then 𝑝 is a generalized pseudodistance on 𝑋. Let 𝑓∶𝑋→ℝ and 𝜑∶ℝ→(0,+∞) be defined by 𝑓(𝑥)=131𝑥−25,if𝑥∈0,2121𝑥−3,if𝑥∈2,𝜑(𝑡),1=10∀𝑡∈ℝ, (36) respectively. So 𝑓(𝑥)<+∞ for all 𝑥∈𝑋. Note that 𝑓 is not lower semicontinuous at 𝑥=1/2, so 𝑓 is not lower semicontinuous on 𝑋. Since 𝑓(𝑥)≥−25 for all 𝑥∈𝑋, 𝑓 is a bounded below function on 𝑋. Let 𝑇∶𝑋→𝑋 be defined by 𝑇𝑥=𝑥²∀𝑥∈𝑋. (37) Then 𝑇 is continuous on 𝑋 and ℱ(𝑇)={0,1}. It is also easy to see that 𝑇 is closed and the map 𝑥↦𝑑(𝑥,𝑇𝑥) is l.s.c. Hence (H1), (H2), and (H4) as in Theorem 13 hold. We deduce that for any 𝑥∈𝑋, 𝑝(𝑥,𝑇𝑥)=max{2(𝑥−𝑇𝑥),3(𝑇𝑥−𝑥)}=2𝑥−𝑥²<𝜑𝑓(𝑥)𝑓(𝑥)−𝑓(𝑇𝑥), (38) so 𝑇 is (𝑝,𝜑,𝑓)-Caristi type on 𝑋. Let 𝑤∈𝑋. Since lim_𝑛→∞𝑇^𝑛𝑤=lim_𝑛→∞𝑤^2𝑛=0,if𝑤∈[0,1),1,if𝑤=1, (39) we know that {𝑇^𝑛𝑤}_𝑛∈ℕ converges and the limit of {𝑇^𝑛𝑤}_𝑛∈ℕ belongs to ℱ(𝑇)={0,1}. On the other hand, since all the assumptions of Theorem 13 are satisfied, by applying Theorem 13, we also prove that 𝑇 has a fixed point in 𝑋 and for any 𝑤∈𝑋, the sequence {𝑇^𝑛𝑤}_𝑛∈ℕ converges to a fixed point of 𝑇. It is worth noticing that any well-known generalized version of Caristi’s fixed point theorem is not applicable here.

The following conclusions are immediate from Theorem 13.

Corollary 15. Let (𝑋,𝑑) be a complete metric space, 𝑓∶𝑋→(−∞,+∞] a proper and bounded below function, and 𝑝 a generalized pseudodistance on 𝑋. Suppose that 𝑇∶𝑋→𝑋 is a (𝑝,𝑓)-Caristi type selfmap on 𝑋 and one of the conditions (H1), (H2), (H3), (H4), and (H5) as in Theorem 13 holds. Then 𝑇 admits a fixed point in 𝑋. Moreover, for any 𝑤∈𝑋 with 𝑓(𝑤)<+∞, the sequence {𝑇^𝑛𝑤}_𝑛∈ℕ converges to a fixed point of  T.

Corollary 16. Let (𝑋,𝑑) be a complete metric space, 𝑓∶𝑋→(−∞,+∞] a proper and bounded below function, and 𝜑∶ℝ→(0,+∞) a nondecreasing function. Suppose that 𝑇∶𝑋→𝑋 is a (𝜑,𝑓)-Caristi type selfmap on 𝑋 and one of the following conditions is satisfied:(D1)𝑇 is continuous; (D2)𝑇 is closed;(D3)the map ℎ∶𝑋→[0,∞) defined by ℎ(𝑥)=𝑑(𝑥,𝑇𝑥) is l.s.c.

Then 𝑇 admits a fixed point in 𝑋. Moreover, for any 𝑤∈𝑋 with 𝑓(𝑤)<+∞, the sequence {𝑇^𝑛𝑤}_𝑛∈ℕ converges to a fixed point of 𝑇.

Corollary 17. Let (𝑋,𝑑) be a complete metric space and 𝑓∶𝑋→(−∞,+∞] a proper and bounded below function. Suppose that 𝑇∶𝑋→𝑋 is a (𝑓)-Caristi type selfmap on 𝑋 and one of the conditions (D1), (D2), and (D3) as in Corollary 16 holds. Then 𝑇 admits a fixed point in 𝑋. Moreover, for any 𝑤∈𝑋 with 𝑓(𝑤)<+∞, the sequence {𝑇^𝑛𝑤}_𝑛∈ℕ converges to a fixed point of 𝑇.

Now, we give another quite useful auxiliary theorem for our applications.

Theorem 18. Let (𝑋,𝑑) be a metric space, 𝑝∶𝑋×𝑋→[0,+∞) a function, and 𝑇∶𝑋→𝑋 a selfmap. Suppose that there exists an ℳ𝒯-function 𝛼∶[0,+∞)→[0,1) such that 𝑝𝑇𝑥,𝑇𝑦≤𝛼𝑝𝑥,𝑦𝑝𝑥,𝑦∀𝑥,𝑦∈𝑋. (40) Then there exists a function 𝛽∶𝑋→[0,1) such that for each 𝑥∈𝑋, 𝛽()(𝑥),𝛼𝑝𝑇𝑇𝑥≤𝛽_𝑛−1𝑥,𝑇^𝑛𝑥≤𝛽(𝑥)∀𝑛∈ℕ. (41)
Here, we denote 𝑇⁰=𝐼 (the identity map).

Proof. Let 𝑥∈𝑋 be given. From our hypothesis, we have 𝑝𝑇^𝑛𝑥,𝑇^𝑛+1𝑥≤𝛼𝑝𝑇^𝑛−1𝑥,𝑇^𝑛𝑥𝑝𝑇^𝑛−1𝑥,𝑇^𝑛𝑥<𝑝𝑇^𝑛−1𝑥,𝑇^𝑛𝑥 (42) for each 𝑛∈ℕ. So the sequence {𝑝(𝑇^𝑛−1𝑥,𝑇^𝑛𝑥)}_𝑛∈ℕ is strictly decreasing in [0,+∞). Since 𝛼 is an ℳ𝒯-function, by (g) of Theorem 11, we obtain 0≤sup_𝑛∈ℕ𝛼𝑝𝑇^𝑛−1𝑥,𝑇^𝑛𝑥<1. (43) Since 𝑥∈𝑋 is arbitrary, we can define a new function 𝛽∶𝑋→[0,1) by 𝛽(𝑥)∶=sup_𝑛∈ℕ𝛼𝑝𝑇^𝑛−1𝑥,𝑇^𝑛𝑥for𝑥∈𝑋. (44) It is obvious that for each 𝑥∈𝑋, we have 𝛽()(𝑥),𝛼𝑝𝑇𝑇𝑥≤𝛽_𝑛−1𝑥,𝑇^𝑛𝑥≤𝛽(𝑥)∀𝑛∈ℕ. (45)

As an interesting application of Theorem 13, we prove the following new fixed point theorems for Banach type maps.

Theorem 19. Let (𝑋,𝑑) be a complete metric space, 𝑝 a generalized pseudodistance on 𝑋 with 𝑝(𝑥,𝑦)=0 implies 𝑥=𝑦 for all 𝑥,𝑦∈𝑋, and 𝑇∶𝑋→𝑋 a selfmap. Suppose that (a)there exists an ℳ𝒯-function 𝛼∶[0,∞)→[0,1) such that 𝑝𝑇𝑥,𝑇𝑦≤𝛼𝑝𝑥,𝑦𝑝𝑥,𝑦∀𝑥,𝑦∈𝑋; (46)(b)one of the conditions (H1), (H2), (H3), (H4), and (H5) as in Theorem 13 holds.

Then 𝑇 admits a unique fixed point in 𝑋. Moreover, for each 𝑥∈𝑋, the sequence {𝑇^𝑛𝑥}_𝑛∈ℕ converges to the unique fixed point of 𝑇.

Proof. Denote 𝑇⁰=𝐼 (the identity map). Applying Theorem 18, there exists a function 𝛽∶𝑋→[0,1) such that for each 𝑥∈𝑋, 𝛽()(𝑥),𝛼𝑝𝑇𝑇𝑥≤𝛽_𝑛−1𝑥,𝑇^𝑛𝑥≤𝛽(𝑥)∀𝑛∈ℕ. (47) For each 𝑥∈𝑋, by (46), we get 𝑝()𝑝()𝑝()()𝑥,𝑇𝑥−𝛼𝑥,𝑇𝑥𝑥,𝑇𝑥≤𝑝𝑥,𝑇𝑥−𝑝𝑇𝑥,𝑇²𝑥. (48) By exploiting the inequalities (47) and (48), we obtain 𝑝()≤1𝑥,𝑇𝑥𝑝()𝑝()−11−𝛼𝑥,𝑇𝑥𝑥,𝑇𝑥𝑝()𝑝1−𝛼𝑥,𝑇𝑥𝑇𝑥,𝑇²𝑥≤1(𝑥)𝑝()−11−𝛽𝑥,𝑇𝑥()𝑝1−𝛽𝑇𝑥𝑇𝑥,𝑇²𝑥. (49) Let 𝜑∶ℝ→(0,+∞) and 𝑓∶𝑋→ℝ be defined by 𝜑(𝑡)𝑓(𝑥)=1=1for𝑡∈ℝ,(𝑥)𝑝()1−𝛽𝑥,𝑇𝑥for𝑥∈𝑋, (50) respectively. Then 𝜑 is a nondecreasing function, and 𝑓 is a bounded below function. Clearly, 𝑓(𝑥)<+∞ for all 𝑥∈𝑋. By (49), we obtain 𝑝(𝑥,𝑇𝑥)≤𝜑𝑓(𝑥)𝑓(𝑥)−𝑓(𝑇𝑥)∀𝑥∈𝑋. (51) Hence we prove that 𝑇∶𝑋→𝑋 is a (𝑝,𝜑,𝑓)-Caristi type selfmap on 𝑋. Applying Theorem 13, we know ℱ(𝑇)≠∅. We claim that ℱ(𝑇) is a singleton set. Let 𝑢,𝑣∈ℱ(𝑇). Then 𝑇𝑢=𝑢 and 𝑇𝑣=𝑣. From (46), we have 𝑝(𝑢,𝑣)=𝑝(𝑇𝑢,𝑇𝑣)≤𝛼𝑝(𝑢,𝑣)𝑝(𝑢,𝑣) (52) or 1−𝛼𝑝(𝑢,𝑣)𝑝(𝑢,𝑣)≤0, (53) which implies 𝑝(𝑢,𝑣)=0. By our hypothesis, we get 𝑢=𝑣 and our claim is proved. By the uniqueness of fixed point of 𝑇 and applying Theorem 13 again, the sequence {𝑇^𝑛𝑥}_𝑛∈ℕ converges to the unique fixed point of 𝑇 for any 𝑥∈𝑋.

As a direct consequence of Theorem 19, we obtain the following result.

Corollary 20. Let (𝑋,𝑑) be a complete metric space, 𝑝 be a generalized pseudodistance on 𝑋 with 𝑝(𝑥,𝑦)=0 implies 𝑥=𝑦 for all 𝑥,𝑦∈𝑋, and 𝑇∶𝑋→𝑋 a selfmap. Suppose that(a)there exists 𝛾∈[0,1) such that 𝑝𝑇𝑥,𝑇𝑦≤𝛾𝑝𝑥,𝑦∀𝑥,𝑦∈𝑋; (54)(b)one of the conditions (H1), (H2), (H3), (H4), and (H5) as in Theorem 13 holds.

Then 𝑇 admits a unique fixed point in 𝑋. Moreover, for each 𝑥∈𝑋, the sequence {𝑇^𝑛𝑥}_𝑛∈ℕ converges to the unique fixed point of 𝑇.

Applying Theorem 19, we obtain a generalization of the celebrated Banach contraction principle.

Corollary 21. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 a selfmap. Suppose that there exists an ℳ𝒯-function 𝛼∶[0,∞)→[0,1) such that 𝑑𝑇𝑥,𝑇𝑦≤𝛼𝑑𝑥,𝑦𝑑𝑥,𝑦∀𝑥,𝑦∈𝑋. (55)

Then 𝑇 admits a unique fixed point in 𝑋. Moreover, for each 𝑥∈𝑋, the sequence {𝑇^𝑛𝑥}_𝑛∈ℕ converges to the unique fixed point of 𝑇.

Proof. By (55), we know that 𝑇 is continuous on 𝑋. Hence the conclusion follows from Theorem 19 immediately.

Remark 22. Theorems 13 and 19, Corollaries 15–21 all generalize and improve the celebrated Banach contraction principle.

Acknowledgment

In this research, the author was supported by Grant no. NSC 102–2115-M-017-001 of the National Science Council of the Republic of China.