Caristi Type Coincidence Point Theorem in Topological Spaces

Abstract

A generalized Caristi type coincidence point theorem and its equivalences in the setting of topological spaces by using a kind of nonmetric type function are obtained. These results are used to establish variational principle and its equivalences in 𝑑-complete spaces, bornological vector space, seven kinds of completed quasi-semimetric spaces equipped with 𝑄-functions, uniform spaces with 𝑞-distance, generating spaces of quasimetric family, and fuzzy metric spaces.

1. Introduction

Caristi’s fixed-point theorem [1, 2] and its equivalences, Ekeland variational principle [3, 4], and Takahashi minimization theorem are forceful tools in nonlinear analysis, control theory, and global analysis; see, for example, [3–5]. In the last two decades, Caristi’s fixed-point theorem and Ekeland variational principle have been generalized and extended in several directions. About these, one can refer to, for example, [1–32] and the references therein. In particular, in [25], a very general Ekeland variational principle and Caristi’s fixed-point theorem are presented, which give a unified approach to three classes of Ekeland type variational principle: in the first class, the underlying space is a sequentially complete uniform space (or equivalently, a sequentially complete F-type topological space), and the perturbation involves a family of topology generating pseudometrics (or quasimetrics); in the second class, the underlying space is a locally complete locally convex space (resp., a locally complete locally 𝑝-convex space), and the perturbation involves a family of topology generating seminorms (resp., topology generating 𝑝-homogeneous 𝐹-pseudonorms) or involves a single Minkowski functional; in the third class, the underlying space is a complete metric space, and the perturbation involves a 𝑤-distance or a 𝜏-function. On the other hand, Banach fixed-point theorem has been extended to large class of nonmetric spaces which included 𝑑-complete topological spaces, symmetric spaces, and quasimetric spaces (see, e.g., [33–35]). But to our knowledge, neither Ekeland's variational principle nor any of its equivalents have been established in such 𝑑-complete topological spaces.

Motivated by the aforementioned works, we attempt to give a unified approach to the previous works. A generalized Caristi type coincidence point theorem in the setting of topological spaces by using a kind of nonmetric type function is proved. As an application of this Caristi's coincidence point theorem, an Ekeland type variational principle and its equivalences in the setting of topological spaces are obtained. Also, these results present Caristi type coincidence point theorem, variational principle, and its equivalences in 𝑑-complete topological spaces. Moreover, these results are used to establish variational principle and its equivalences in bornological vector space, seven kinds of completed quasi-semimetric spaces equipped with 𝑄-functions, uniform spaces with 𝑞-distance, generating spaces of quasimetric family, and fuzzy metric spaces. The results of this paper uniformly extend and generalize the corresponding results appeared in the literature [1–4, 6–13, 15, 25, 26, 28, 30, 32].

2. Caristi Type Coincidence Point Theorem

The primary goal of this section is to establish two equivalent generalized Caristi type coincidence point theorems in the setting of topological spaces by using a kind of nonmetric type function. As an application of these Caristi’s coincidence point theorems, equivalent generalized Caristi type common fixed point theorem, Caristi type fixed point theorem for set valued, Caristi type fixed point theorem for single-valued map, Ekeland type variational principles and its equivalences in the setting of topological spaces are obtained. To establish our main results, we need the following definitions.

Definition 1 (see [15]). Let (𝑋,𝜏) be a topological space. An extended real-valued function 𝑓∶𝑋→(−∞,+∞] is said to be sequentially lower monotone if for every sequence {𝑥_𝑛} converging to 𝑥 and satisfying 𝑓𝑥₁≥𝑓𝑥₂≥⋯≥𝑓𝑥_𝑛≥⋯ (1) we have 𝑓(𝑥)≤𝑓(𝑥_𝑛), for each 𝑛.

Definition 2. Let (𝑋,𝜏) be a topological space and 𝑝∶𝑋×𝑋→[0,+∞] a function. A proper function 𝑓∶𝑋→(−∞,+∞] (i.e., 𝑓(𝑥) is not identically to +∞) is said to be sequentially lower monotone with respect to 𝑝 (in short, sequentially lower monotone with respect to 𝑝) if for any sequence {𝑥_𝑛} in 𝑋 satisfying ∑^∞_𝑛=1𝑝(𝑥_𝑛,𝑥_𝑛+1)<+∞, lim_𝑛→∞𝑥_𝑛=𝑥 and 𝑓(𝑥_𝑛+1)≤𝑓(𝑥_𝑛) for each 𝑛, we have 𝑓(𝑥)≤𝑓(𝑥_𝑛) for each 𝑛∈𝑁.

Definition 3. Let (𝑋,𝜏) be a topological space and 𝑝∶𝑋×𝑋→[0,+∞] a function.(1)(𝑋,𝜏) is said to be 𝑝-complete [33, 34] if any sequence {𝑥_𝑛} with ∑^∞_𝑛=1𝑝(𝑥_𝑛,𝑥_𝑛+1)<+∞ implies that the sequence {𝑥_𝑛} is convergent to some 𝑥∈𝑋.(2)(𝑋,𝜏) is said to be sequentially lower complete with respect to 𝑝, if any sequence {𝑥_𝑛} with ∑^∞_𝑛=1𝑝(𝑥_𝑛,𝑥_𝑛+1)<+∞ implies that the sequence {𝑥_𝑛} is convergent to some 𝑥∈𝑋, and liminf_𝑛→∞𝑝𝑥_𝑛,𝑦≤𝑝𝑥,𝑦, (2) for any 𝑦∈𝑋.(3)Let 𝑓∶𝑋→(−∞,+∞] be a proper function. The topological space (𝑋,𝜏) is said to be sequentially lower complete with respect to 𝑝 and 𝑓 if any sequence {𝑥_𝑛} in 𝑋 satisfying ∑^∞_𝑛=1𝑝(𝑥_𝑛,𝑥_𝑛+1)<+∞ and 𝑓(𝑥_𝑛+1)≤𝑓(𝑥_𝑛) for all 𝑛 is convergent to some 𝑥∈𝑋, and (2) holds for any 𝑦∈𝑋.

Remark 4. It is clearly that if (𝑋,𝜏) is sequentially lower complete w.r.t. 𝑝, then for any proper function 𝑓∶𝑋→(−∞,+∞], (𝑋,𝜏) is sequentially lower complete w.r.t. 𝑝 and 𝑓.

Now, we can prove the following Caristi type coincidence point theorem in the setting of topological spaces.

Theorem 5. Let (𝑋,𝜏) be a topological space, 𝑝∶𝑋×𝑋→[0,+∞] a function, 𝑓∶𝑋→(−∞,+∞] a proper, bounded from below, sequentially lower monotone function with respect to 𝑝, and 𝜑∶(−∞,+∞)→(0,+∞) a nondecreasing function. Assume that (𝑋,𝜏) is sequentially lower complete with respect to 𝑝 and 𝑓. Let 𝐷 be a nonempty subset of 𝑋, 𝑔∶𝐷→𝑋 a surjective function, 𝐼 an index set, and, for each 𝑖∈𝐼, 𝑇_𝑖∶𝐷→2^𝑋 a multivalued map. Then the following conclusions hold and are equivalent.(1)Suppose that for each 𝑥∈𝐷 with 𝑔(𝑥)∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists 𝑦∈𝑋, such that𝑔(𝑥)𝑓𝑔(𝑥)𝑓𝑔(𝑥)𝑦,𝑔(𝑥)𝑓𝑔(𝑥)𝑓𝑔(𝑥)𝑦.𝑒𝑖𝑡ℎ𝑒𝑟0=𝑝,𝑦<𝜑−𝑓𝑜𝑟0<𝑝,𝑦≤𝜑−𝑓 (3)Then for any 𝑥₀∈𝑋, there exists a coincidence point 𝑢∈𝐷 of 𝑔 and {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑔(𝑢)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that 𝑓𝑔(𝑢)≤𝑓𝑥₀. (4)(2)Suppose that for each 𝑥∈𝐷 with 𝑔(𝑥)∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists an 𝑖₀∈𝐼 and 𝑦∈𝑇_𝑖0(𝑥), such that (3) holds. Then for any 𝑥₀∈𝑋, there exists a coincidence point 𝑢∈𝐷 of 𝑔 and {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑔(𝑢)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that (4) holds.

Proof. (1) We take an 𝑥₀∈𝑋, since 𝑓≢+∞; without loss of generality, we can assume that 𝑓(𝑥₀)<+∞. Since 𝑔∶𝐷→𝑋 is a surjective function, there exists 𝑢₀∈𝐷, such that 𝑔(𝑢₀)=𝑥₀. If 𝑔(𝑢₀)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢₀), then the conclusion holds. Otherwise, by the supposition, there exists 𝑦∈𝑋, such thateither0=𝑝𝑔𝑢₀,𝑦<𝜑𝑓𝑔𝑢₀𝑓𝑔𝑢₀𝑦,𝑔𝑢−𝑓or0<𝑝₀,𝑦≤𝜑𝑓𝑔𝑢₀𝑓𝑔𝑢₀−𝑓𝑦. (5)Thus, 𝑝𝑥₀,𝑦=𝑝𝑔𝑢₀𝑓𝑔𝑢,𝑦<2𝜑₀𝑓𝑔𝑢₀𝑦𝑓𝑥−𝑓=2𝜑₀𝑓𝑥₀−𝑓𝑦. (6) Hence, 𝑆₁=𝑦∈𝑋∶𝑝𝑥₀,𝑦<2𝜑𝑓𝑥₀𝑓𝑥₀−𝑓𝑦≠∅. (7) Obviously, for any 𝑦∈𝑆₁, 𝑓(𝑥₀)>𝑓(𝑦). Thus, we can take 𝑦₁∈𝑆₁ such that 𝑓𝑦₁<12𝑓𝑥₀+inf_𝑥∈𝑆1𝑓(𝑥)<𝑓𝑥₀. (8) Assume that 𝑦_𝑛 has been taken, and 𝑦_𝑛=𝑔(𝑢_𝑛) for 𝑢_𝑛∈𝐷. If 𝑔(𝑢_𝑛)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢_𝑛), then the conclusion holds. Otherwise, 𝑆_𝑛+1=𝑦∈𝑋∶𝑝𝑦_𝑛,𝑦<2𝜑𝑓𝑦_𝑛𝑓𝑦_𝑛−𝑓𝑦≠∅. (9) Note that for any 𝑦∈𝑆_𝑛+1, we have 𝑓(𝑦_𝑛)>𝑓(𝑦). Thus, we can take 𝑦_𝑛+1∈𝑆_𝑛+1 such that 𝑓𝑦_𝑛+1<12𝑓𝑦_𝑛+inf_{𝑥∈𝑆𝑛+1}𝑓(𝑥)<𝑓𝑦_𝑛,𝑝𝑦_𝑛,𝑦_𝑛+1<2𝜑𝑓𝑦_𝑛𝑓𝑦_𝑛−𝑓𝑦_𝑛+1. (10)(11) It remains to consider the case that there is an infinite sequence {𝑦_𝑛} which satisfies (10) and (11). From (10), we know that {𝑓(𝑦_𝑛)} is a decreasing sequence. Since 𝜑 is nondecreasing, it follows from (11) that ^∞_𝑛=1𝑝𝑦_𝑛,𝑦_𝑛+1≤^∞_𝑛=12𝜑𝑓𝑦_𝑛𝑓𝑦_𝑛−𝑓𝑦_𝑛+1≤2𝜑𝑓𝑥₀^∞_𝑛=1𝑓𝑦_𝑛−𝑓𝑦_𝑛+1≤2𝜑𝑓𝑥₀𝑓𝑥₀−inf_𝑥∈𝑋𝑓(𝑥)<+∞. (12) Since (𝑋,𝜏) is sequentially lower complete w.r.t. 𝑝 and 𝑓, there exists 𝑣∈𝑋 such that lim_𝑛→∞𝑦_𝑛=𝑣 and liminf_𝑛→∞𝑝(𝑦_𝑛,𝑥)≤𝑝(𝑣,𝑥) for any 𝑥∈𝑋. Assume that 𝑣=𝑔(𝑢), 𝑢∈𝐷. We claim that the conclusion holds for 𝑢. Since 𝑓 is sequentially lower monotone, we have 𝑓𝑔(𝑢)=𝑓(𝑣)≤𝑓𝑦_𝑛<𝑓𝑥₀. (13) That is (4) holds. If 𝑔(𝑢)∉⋂_𝑖∈𝐼𝑇_𝑖(𝑢), then there exists 𝑦^′∈𝑋, such that 𝑝𝑣,𝑦^′=𝑝𝑔(𝑢),𝑦^′<2𝜑𝑓𝑔(𝑢)𝑓𝑔(𝑢)−𝑓𝑦^′=2𝜑𝑓(𝑣)𝑓(𝑣)−𝑓𝑦^′. (14) It follows from liminf_𝑛→∞𝑝(𝑦_𝑛,𝑦^′)≤𝑝(𝑣,𝑦^′) that there exists a subsequence {𝑦_𝑛𝑘}⊂{𝑦_𝑛} such that lim_𝑘→∞𝑝𝑦_𝑛𝑘,𝑦^′=liminf_𝑛→∞𝑝𝑦_𝑛,𝑦^′. (15) From this and (14) we know that there exists an 𝑚 such that for all 𝑘≥𝑚, 𝑝𝑦_𝑛𝑘,𝑦^′<2𝜑𝑓(𝑣)𝑓(𝑣)−𝑓𝑦^′. (16) From (13) we have 𝑝𝑦_𝑛𝑘,𝑦^′<2𝜑𝑓𝑦_𝑛𝑘𝑓𝑦_𝑛𝑘−𝑓𝑦^′, (17) that is, 𝑦^′∈𝑆_𝑛𝑘+1, 𝑘≥𝑚. From (10) we have 2𝑓𝑦_𝑛𝑘+1−𝑓𝑦_𝑛𝑘≤inf_{𝑦∈𝑆𝑛𝑘+1}𝑓𝑦≤𝑓𝑦^′. (18) By letting 𝑘→+∞, we get that lim_𝑘→∞𝑓𝑦_𝑛𝑘=lim_𝑛→∞𝑓𝑦_𝑛≤𝑓𝑦^′. (19) Combing with (13) we have 𝑓(𝑣)≤𝑓(𝑦^′) which contradicts (14). Thus the conclusion of Theorem 5(1) holds.
(2) It is clear that Theorem 5(1) ⇒ Theorem 5(2). Now, we prove that Theorem 5(2) ⇒ Theorem 5(1). Assume that the conditions of Theorem 5(1) are satisfied; then, for each 𝑥∈𝐷, if 𝑔(𝑥)∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists 𝑦∈𝑋, such that (3) holds. Then we get that 𝐺(𝑥)=𝑦∈𝑋∶𝑔(𝑥),𝑦satisfy(3)≠∅ (20) and 𝑔(𝑥)∉𝐺(𝑥). For each 𝑖∈𝐼, we define 𝑇_𝑖∶𝐷→2^𝑋 by 𝑇_𝑖(𝑥)=𝑇_𝑖(𝑥),if𝑔(𝑥)∈_𝑖∈𝐼𝑇_𝑖(𝑥);𝑇_𝑖(𝑥)∪𝐺(𝑥),if𝑔(𝑥)∉_𝑖∈𝐼𝑇_𝑖(𝑥). (21) It is clear that 𝑔(𝑥)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑥) if and only if 𝑔(𝑥)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑥). Also, {𝑇_𝑖} satisfies the condition of Theorem 5(2). Then by the conclusion of Theorem 5(2), there exists a coincidence point 𝑢∈𝐷 of 𝑔 and {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑔(𝑢)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that (4) holds. Therefore, there exists a coincidence point 𝑢∈𝐷 of 𝑔 and {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑔(𝑢)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that (4) holds. That is Theorem 5(1) holds. The proof is completed.

Remark 6. If for any 𝑥,𝑦∈𝑋 with 𝑥≠𝑦 implies that 𝑝(𝑥,𝑦)>0, then the conclusion (1) of Theorem 5 can be rewritten as: for each 𝑥∈𝐷 with 𝑔(𝑥)∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥) there exists 𝑦∈𝑋, such that 0<𝑝𝑔(𝑥),𝑦≤𝜑𝑓𝑔(𝑥)𝑓𝑔(𝑥)−𝑓𝑦. (22)

In particular, if 𝐷=𝑋 and 𝑔∶𝐷→𝑋 is the identity map in Theorem 5, then we obtain the following generalized Caristi type common fixed point theorem, Caristi type fixed point theorem for set-valued, and single-valued map.

Theorem 7. Let (𝑋,𝜏) be a topological space, 𝑝∶𝑋×𝑋→[0,+∞] a function, 𝑓∶𝑋→(−∞,+∞] a proper, bounded from below, sequentially lower monotone function with respect to 𝑝, and 𝜑∶(−∞,+∞)→(0,+∞) a nondecreasing function. Assume that (𝑋,𝜏) is sequentially lower complete w.r.t. 𝑝 and 𝑓. Then the following conclusions hold and are all equivalent to Theorem 5.(1)Let 𝐼 be an index set, and, for each 𝑖∈𝐼, let 𝑇_𝑖∶𝑋→2^𝑋 be a multivalued map.Suppose further that for each 𝑥∈𝑋 with 𝑥∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists 𝑦∈𝑋, such that 𝑓(𝑥)𝑓(𝑥)𝑦,𝑓(𝑥)𝑓(𝑥)𝑦.𝑒𝑖𝑡ℎ𝑒𝑟0=𝑝𝑥,𝑦<𝜑−𝑓𝑜𝑟0<𝑝𝑥,𝑦≤𝜑−𝑓 (23) Then for any 𝑥₀∈𝑋, there exists a common fixed point 𝑢∈𝑋 of {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑢∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that 𝑓(𝑢)≤𝑓𝑥₀. (24)(2)Let 𝑇∶𝑋→2^𝑋 be a multivalued map.Suppose further that for each 𝑥∈𝑋 with 𝑥∉𝑇(𝑥), there exists 𝑦∈𝑋, such that (23) holds. Then for any 𝑥₀∈𝑋, 𝑇 has a fixed point 𝑢∈𝑋, such that (24) holds.(3)Let 𝑇∶𝑋→𝑋 be a map.Suppose further that for each 𝑥∈𝑋 with 𝑥≠𝑇(𝑥), there exists 𝑦∈𝑋, such that (23) holds. Then for any 𝑥₀∈𝑋, 𝑇 has a fixed point 𝑢∈𝑋, such that (24) holds.

Proof. It is clear that the following implications hold: Theorem 5(1) ⇒ Theorem 7(1) ⇒ Theorem 7(2) ⇒ Theorem 7(3).
Now, we prove that Theorem 7(3) ⇒ Theorem 5(1). Assume that the conditions of Theorem 5 hold. It is similar to the proof of Lemma 2.1 in [36] that, by using the axiom of choice, we can prove that there exists a subset 𝐸⊆𝐷 such that 𝑔(𝐸)=𝑔(𝐷)=𝑋 and 𝑔∶𝐸→𝑋 is one-to-one. Define a map 𝑇∶𝑋→𝑋 by 𝑇(𝑥)=𝑥,if𝑔(𝑢)=𝑥∈_𝑖∈𝐼𝑇_𝑖(𝑢)(𝑢),𝑢∈𝐸;𝑦,if𝑔=𝑥∉_𝑖∈𝐼𝑇_𝑖(𝑢),𝑢∈𝐸, (25) where 𝑦∈𝑋 such that either 𝑔(𝑢)𝑓𝑔(𝑢)𝑓𝑔(𝑢)𝑦𝑓(𝑥)𝑓(𝑥)𝑦,0=𝑝,𝑦=𝑝𝑥,𝑦<𝜑−𝑓=𝜑−𝑓 (26) or 𝑔(𝑢)𝑓𝑔(𝑢)𝑓𝑔(𝑢)𝑦𝑓(𝑥)𝑓(𝑥)𝑦.0<𝑝,𝑦=𝑝𝑥,𝑦≤𝜑−𝑓=𝜑−𝑓 (27) Then 𝑇 satisfies the condition of Theorem 7(3); thus for any 𝑥₀∈𝑋, 𝑇 has a fixed point 𝑣∈𝑋, such that 𝑓(𝑣)≤𝑓(𝑥₀) holds. Since 𝑔(𝐸)=𝑋, there exists 𝑢∈𝐸⊂𝐷, such that 𝑔(𝑢)=𝑣. Then by the definition of 𝑇, we get that 𝑔(𝑢)=𝑣∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢). That is, the conclusion of Theorem 5 holds. The proof is completed.

The following corollary is an extension of the results in [19, 20]. In Corollary 8, we remove the condition that 𝜂 is nondecreasing, which is used in [19, 20].

Corollary 8. Let (𝑋,𝑑) be a completed metric space. Suppose that 𝜂∶[0,+∞)→(−∞,+∞) satisfies 𝜂(0)=0 and that 𝑓∶𝑋→(−∞,+∞) is lower semicontinuous on 𝑋, and there exist 𝑥₀∈𝑋 and two real numbers 𝑎,𝑏∈(−∞,+∞), such that 𝑓(𝑥)≥𝑎𝑑𝑥,𝑥₀+𝑏, (28) and one of the following conditions is satisfied:(i)𝑎≥0, 𝜂 is nonnegative on 𝑊={𝑑(𝑥,𝑦)∶𝑥,𝑦∈𝑋}, and there exist 𝑐>0 and 𝜀>0 such that 𝜂(𝑡)≥𝑐𝑡,∀𝑡∈𝑡≥0∶𝜂(𝑡)≤𝜀∩𝑊; (29)(ii)𝑎<0, 𝜂(𝑡)+𝑎𝑡 is nonnegative on 𝑊, and there exist 𝑐>0 and 𝜀>0 such that 𝜂(𝑡)+𝑎𝑡≥𝑐𝑡,∀𝑡∈𝑡≥0∶𝜂(𝑡)+𝑎𝑡≤𝜀∩𝑊. (30) Then each Caristi type mapping 𝑇∶𝑋→𝑋 (i.e., satisfying 𝜂(𝑑(𝑥,𝑇𝑥))≤𝑓(𝑥)−𝑓(𝑇𝑥),∀𝑥∈𝑋) has a fixed point 𝑣∈𝑋.

Proof. Case (i). It follows from 𝑎≥0 and (28) that 𝑓 is a bounded from below, lower semicontinuous function on 𝑋. Let 𝛼=inf_𝑥∈𝑋𝑓(𝑥),𝑄=𝑥∈𝑋∶𝑓(𝑥)≤𝛼+𝜀. (31) Then, as in the proof of the Theorem 1 in [20] that (𝑄,𝑑) is a complete metric space, 𝑇𝑄⊂𝑄, and 0≤𝜂(𝑑(𝑥,𝑇𝑥))≤𝑓(𝑥)−𝑓(𝑇𝑥)≤𝜀,∀𝑥∈𝑄. (32) Thus we have, by (29), 𝑐𝑑(𝑥,𝑇𝑥)≤𝜂(𝑑(𝑥,𝑇𝑥))≤𝑓(𝑥)−𝑓(𝑇𝑥)≤𝜀,∀𝑥∈𝑄. (33) Define a function 𝑝∶𝑄×𝑄→[0,+∞] by 𝑝(𝑥,𝑦)=𝑐𝑑(𝑥,𝑦). Let {𝑥_𝑛} be a sequence in 𝑄, such that ^∞_𝑛=1𝑝𝑥_𝑛,𝑥_𝑛+1=^∞_𝑛=1𝑐𝑑𝑥_𝑛,𝑥_𝑛+1<+∞. (34) This implies that {𝑥_𝑛} is a Cauchy sequence in 𝑄. Since (𝑄,𝑑) is a complete metric space, {𝑥_𝑛} is a convergent sequence in 𝑄. If lim_𝑛→∞𝑥_𝑛=𝑥, then for any 𝑦∈𝑄, lim_𝑛→∞𝑝(𝑥_𝑛,𝑦)=lim_𝑛→∞𝑐𝑑(𝑥_𝑛,𝑦)=𝑐𝑑(𝑥,𝑦)=𝑝(𝑥,𝑦). Thus, (𝑄,𝑑) is sequentially lower complete w.r.t. 𝑝. Clearly, all conditions of Theorem 7 are satisfied. Therefore, 𝑇 has a fixed point in 𝑄.
Case (ii). Let 𝜓(𝑥)=𝑓(𝑥)−𝑎𝑑𝑥,𝑥₀𝜂,∀𝑥∈𝑋,₁(𝑡)=𝜂(𝑡)+𝑎𝑡,∀𝑡∈[0,+∞). (35) Then we have 𝜂₁(𝑑(𝑥,𝑇𝑥))≤𝜓(𝑥)−𝜓(𝑇𝑥),∀𝑥∈𝑋. (36) Thus, the conclusion can be deduced by Case (i). The proof is completed.

In Theorem 9, by using Theorem 5, we present a generalized Ekeland type variational principle, maximal element theorem for a family of multivalued maps, equilibrium theorem, and a generalized Takahashi minimization theorem in topological spaces and prove the equivalence among these results.

Theorem 9. Let (𝑋,𝜏) be a topological space, 𝑝∶𝑋×𝑋→[0,+∞] a function, 𝑓∶𝑋→(−∞,+∞] a proper, bounded from below, sequentially lower monotone function with respect to 𝑝, and 𝜑∶(−∞,+∞)→(0,+∞) a nondecreasing function. Assume that (𝑋,𝜏) is sequentially lower complete with respect to 𝑝 and 𝑓. Then for any 𝑥₀∈𝑋, the following conclusions hold, and they are equivalent to Theorem 5.(I)(Ekeland type variational principle in topological spaces) There exists 𝑣∈𝑋, such that 𝑓(𝑣)≤𝑓(𝑥₀) and 𝑝()𝑓(𝑣)𝑓(𝑣)(𝑥),()𝑝()𝑓(𝑣)𝑓(𝑣)(𝑥),()𝑣,𝑥=𝜑−𝑓∀𝑥∈𝑋𝑤𝑖𝑡ℎ𝑝𝑣,𝑥=0,𝑣,𝑥>𝜑−𝑓∀𝑥∈𝑋𝑤𝑖𝑡ℎ𝑝𝑣,𝑥>0. (37)(II) (Maximal element for a family of multivalued maps in topological spaces) Let 𝐼 be any index set, and, for each 𝑖∈𝐼, let 𝑇_𝑖∶𝑋→2^𝑋 be a multivalued map. Assume that for each (𝑥,𝑖)∈𝑋×𝐼 with 𝑇_𝑖(𝑥)≠∅, there exists 𝑦=𝑦(𝑥,𝑖)∈𝑋, such that (23) holds. Then there exists 𝑣∈𝑋, such that 𝑓(𝑣)≤𝑓(𝑥₀) and 𝑇_𝑖(𝑣)=∅ for each 𝑖∈𝐼.(III)(Equilibrium theorem in topological spaces) Let 𝐹∶𝑋×𝑋→(−∞,+∞] be a proper, bounded from below, sequentially lower monotone function in the first argument. Suppose that there exists 𝑤∈𝑋 such that for each 𝑥 with {𝑢∈𝑋∶𝐹(𝑥,𝑢)>0}≠∅, (38) there exists 𝑦=𝑦(𝑥)∈𝑋 such that (𝐹(𝐹(),(𝐹(𝐹().𝑒𝑖𝑡ℎ𝑒𝑟0=𝑝𝑥,𝑦<𝜑𝑥,𝑤))𝑥,𝑤−𝐹𝑦,𝑤𝑜𝑟0<𝑝𝑥,𝑦≤𝜑𝑥,𝑤))𝑥,𝑤−𝐹𝑦,𝑤 (39) If (𝑋,𝜏) is sequentially lower complete w.r.t. 𝑝 and 𝐹(⋅,𝑤), then there exists 𝑣∈𝑋, such that 𝐹(𝑣,𝑤)≤𝐹(𝑥₀,𝑤) and 𝐹(𝑣,𝑦)≤0,𝑓𝑜𝑟𝑎𝑙𝑙𝑦∈𝑋.(IV)(Generalized Takahashi minimization theorem in topological spaces) Suppose that for any 𝑥 with 𝑓(𝑥)>inf_𝑥∈𝑋𝑓(𝑥), there exists 𝑦∈𝑋 such that (23) holds. Then there exists 𝑣∈𝑋, such that 𝑓(𝑣)≤𝑓(𝑥₀) and 𝑓(𝑣)=inf_𝑥∈𝑋𝑓(𝑥).

Proof. “Theorem 5(1) ⇒Theorem 9(II).” If the conclusion of (II) does not hold, then for any 𝑥∈𝑋, there exists 𝑖, such that 𝑇_𝑖(𝑥)≠∅. By the hypotheses of (II), there exists 𝑦=𝑦(𝑥,𝑖)∈𝑋, such that (23) holds; thus, 𝑦≠𝑥. Let 𝐷=𝑋, 𝑔=𝐼_𝑑 (the identical map of 𝑋), 𝐻_𝑖(𝑥)=𝑇_𝑖(𝑥)∪𝑦(𝑥,𝑖)⧵{𝑥},if𝑇_𝑖(𝑥){𝑥}≠∅,,if𝑇_𝑖(𝑥)=∅. (40) Then the conditions of Theorem 5(1) are satisfied for {𝐻_𝑖}_𝑖∈𝐼, 𝐷=𝑋 and 𝑔=𝐼_𝑑. Thus, from Theorem 5(1) there exists 𝑣∈𝑋 such that 𝑣∈⋂_𝑙∈𝐼𝐻_𝑙(𝑣). This is a contradiction with the definition of 𝐻_𝑖. Therefore, there exists 𝑣∈𝑋 with 𝑓(𝑣)≤𝑓(𝑥₀)such that 𝑇_𝑖(𝑣)=∅ for any 𝑖∈𝐼.
“(II) ⇒ (III)” Let 𝑇_𝑦(𝑥)={𝑥},if𝐹𝑥,𝑦>0,∅,if𝐹𝑥,𝑦≤0. (41) From this we know that if 𝑇_𝑦(𝑥)≠∅, then 𝐹(𝑥,𝑦)>0. By the hypotheses of (III) there exists 𝑧=𝑧(𝑥)∈𝑋, such that ()(𝐹(𝐹()(,()(𝐹(𝐹()(.either0=𝑝𝑥,𝑧<𝜑𝑥,𝑤))(𝑥,𝑤−𝐹𝑧,𝑤))or0<𝑝𝑥,𝑧≤𝜑𝑥,𝑤))(𝑥,𝑤−𝐹𝑧,𝑤)) (42) By using (II) for 𝑓(𝑥)=𝐹(𝑥,𝑤), there exists 𝑣∈𝑋, such that 𝐹(𝑣,𝑤)≤𝐹(𝑥₀,𝑤) and 𝑇_𝑦(𝑣)=∅ for any 𝑦∈𝑋; that is 𝐹(𝑣,𝑦)≤0, for any 𝑦∈𝑋.
“(III) ⇒ (IV)” Let 𝐹(𝑥,𝑦)=𝑓(𝑥)−inf_𝑥∈𝑋𝑓(𝑥). If {𝑢∈𝑋∶𝐹(𝑥,𝑢)>0}≠∅, (43) then 𝑓(𝑥)>inf_𝑥∈𝑋𝑓(𝑥). Fix 𝑤∈𝑋; then 𝐹(𝑥,𝑤)≤𝐹(𝑥₀,𝑤) if and only if 𝑓(𝑥)≤𝑓(𝑥₀). By the hypothesis of (IV), there exists 𝑦=𝑦(𝑥)∈𝑋, such that either 𝑓(𝑥)𝑓(𝑥)𝑦𝑓(𝑥)𝐹(),0=𝑝𝑥,𝑦<𝜑−𝑓=𝜑𝑥,𝑤−𝐹𝑦,𝑤 (44) or 𝑓(𝑥)𝑓(𝑥)𝑦𝑓(𝑥)𝐹().0<𝑝𝑥,𝑦≤𝜑−𝑓=𝜑𝑥,𝑤−𝐹𝑦,𝑤 (45) Define 𝜓∶(−∞,+∞]→(0,+∞) by 𝜓(𝑡)=𝜑(𝑡+inf_𝑥∈𝑋𝑓(𝑥)). Then 𝜓(𝐹(𝑥,𝑦))=𝜑(𝑓(𝑥)); thus, the hypotheses of (III) are satisfied for 𝜓 and 𝐹. It follows from (III) that there exists 𝑣∈𝑋, such that 𝐹(𝑣,𝑤)≤𝐹(𝑥₀,𝑤) and 𝐹(𝑣,𝑦)≤0, for any 𝑦∈𝑋. This implies that 𝑓(𝑣)≤𝑓(𝑥₀) and 𝑓(𝑣)≤inf_𝑥∈𝑋𝑓(𝑥), that is, 𝑓(𝑣)=inf_𝑥∈𝑋𝑓(𝑥).
“(IV) ⇒ (I)” If (I) does not hold, then for any 𝑥∈𝑋 with 𝑓(𝑥)≤𝑓(𝑥₀), there exists 𝑦=𝑦(𝑥), such that 𝑓(𝑥)𝑓(𝑥)𝑦,𝑓(𝑥)𝑓(𝑥)𝑦.either0=𝑝𝑥,𝑦<𝜑−𝑓or0<𝑝𝑥,𝑦≤𝜑−𝑓 (46) This implies that condition of (IV) holds on 𝑋₁={𝑥∈𝑋∶𝑓(𝑥)≤𝑓(𝑥₀)}. Then, by (IV) there exists 𝑣∈𝑋₁, such that 𝑓(𝑣)≤𝑓(𝑥₀) and 𝑓(𝑣)=inf_𝑥∈𝑋1𝑓(𝑥)=inf_𝑥∈𝑋𝑓(𝑥). This is a contradiction with (46). Thus, (I) holds.
“(I) ⇒ Theorem 5(1)” From (I), there exists 𝑣∈𝑋, such that 𝑓(𝑣)≤𝑓(𝑥₀) and (37) holds. Since 𝑔 is a surjective mapping, there exists a 𝑢∈𝐷, such that 𝑔(𝑢)=𝑣. We claim that 𝑔(𝑢)∈⋂_𝑙∈𝐼𝑇_𝑙(𝑢). If 𝑔(𝑢)∉⋂_𝑙∈𝐼𝑇_𝑙(𝑢), by the hypotheses of Theorem 5(1), there exists 𝑦∈𝑋, such that (3) holds. This is a contradiction with (37). Thus, 𝑔(𝑢)∈⋂_𝑙∈𝐼𝑇_𝑙(𝑢) and 𝑓(𝑔(𝑢))≤𝑓(𝑥₀). That is, Theorem 5(1) holds.
The proof is completed.

Remark 10. Theorem 5–Theorem 9 also present Caristi type coincidence point theorem, Ekeland type variational principle, and their equivalences in 𝑝-complete topological spaces. Moreover, from [34] we know that 𝑑-complete topological spaces include 𝑑-complete symmetric (semimetric) spaces and complete quasimetric spaces.

3. Applications to Some Non-Metric Spaces

In this section, we show that our results in section two can be used with many nonmetric spaces. The reader may refer to the references [6, 13, 15, 25, 28, 37] for the notions and symbols in this section.

In [6], the authors introduce the concept of 𝑄-function in quasimetric spaces which generalizes the notion of the 𝜏-function and 𝑤-distance, and they also prove an Ekeland variational principle as well as its equivalences in such spaces.

For the convenience of the reader we present the main concept of quasimetric space in the following (refer to [38]).

Let 𝑋 be a nonempty set. A real valued function 𝑑∶𝑋×𝑋→[0,+∞) is said to be a quasi-semimetric on 𝑋 if the following conditions are satisfied:  (QM1)𝑑(𝑥,𝑦)≥0 and 𝑑(𝑥,𝑥)=0 for all 𝑥,𝑦∈𝑋; (QM2)𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦) for all 𝑥,𝑦,𝑧∈𝑋.

If further (QM3)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥)=0 implies 𝑥=𝑦 for all 𝑥,𝑦∈𝑋, then 𝑑∶𝑋×𝑋→[0,+∞) is said to be a quasimetric on 𝑋. A nonempty set 𝑋 together with a quasimetric 𝑑 (or quasi-semimetric 𝑑) is called a quasimetric space (or quasi-semimetric space), and it is denoted by (𝑋,𝑑). If (𝑋,𝑑) is a quasi-semimetric space, for 𝑥∈𝑋 and 𝑟>0, we define the balls in 𝑋 by the formula 𝐵(𝑥,𝑟)={𝑦∈𝑋∶𝑑(𝑥,𝑦)<𝑟}—the open ball, and  𝐵[𝑥,𝑟]={𝑦∈𝑋∶𝑑(𝑥,𝑦)≤𝑟}—the closed ball.

The topology 𝜏 of a quasi-semimetric (𝑋,𝑑) can be defined starting from the family 𝑉(𝑥) of neighborhoods of an arbitrary point 𝑥∈𝑋: 𝑉∈𝑉(𝑥)⇔∃𝑟>0 such that 𝐵(𝑥,𝑟)⊂𝑉⇔∃𝑟^′>0 such that 𝐵[𝑥,𝑟^′]⊂𝑉.

The convergence of a sequence {𝑥_𝑛} to 𝑥 with respect to 𝜏 can be characterized by 𝑑(𝑥,𝑥_𝑛)→0.

Definition 11. Let (𝑋,𝑑) be a quasi-semimetric space. A sequence {𝑥_𝑛} in 𝑋 is said to be(i)left 𝑑-Cauchy if for each 𝜀>0 there is a point 𝑥 in 𝑋 and an integer 𝑘 such that 𝑑(𝑥,𝑥_𝑚)<𝜀 for all 𝑚≥𝑘;(ii)right 𝑑-Cauchy if for each 𝜀>0 there is a point 𝑥 in 𝑋 and an integer 𝑘 such that 𝑑(𝑥_𝑚,𝑥)<𝜀 for all 𝑚≥𝑘;(iii)𝑑-Cauchy if for each 𝜀>0 there is an integer 𝑘 such that 𝑑(𝑥_𝑟,𝑥_𝑠)<𝜀 for all 𝑟,𝑠≥𝑘;(iv)right 𝐾-Cauchy if for each 𝜀>0 there is an integer 𝑘 such that 𝑑(𝑥_𝑟,𝑥_𝑠)<𝜀 for all 𝑟≥𝑠≥𝑘;(v)left 𝐾-Cauchy if for each 𝜀>0 there is an integer 𝑘 such that 𝑑(𝑥_𝑟,𝑥_𝑠)<𝜀 for all 𝑠≥𝑟≥𝑘;(vi)weakly left (right) 𝐾-Cauchy if for each 𝜀>0 there is an integer 𝑘 such that 𝑑(𝑥_𝑘,𝑥_𝑚)<𝜀 (𝑑(𝑥_𝑚,𝑥_𝑘)<𝜀) for all 𝑚≥𝑘;(vii)corresponding to the seven definitions of Cauchy sequence in a quasi-semimetric space, we have seven notions of completeness: 𝑋 is said to be left (right) 𝑑-, [weakly] left (right) 𝐾-, or 𝑑-sequentially complete if every left (right) 𝑑-, [weakly] left (right) 𝐾-, or 𝑑- (resp.) Cauchy sequence in 𝑋 converges to some point in 𝑋 (with respect to the topology 𝜏 induced on 𝑋 by 𝑑).

Remark 12. The implications between the seven notions of Cauchyness (refer to [38]) are as follows: 𝑑-Cauchy ⇒ left and right 𝐾-Cauchy, left (right) 𝐾-Cauchy ⇒ weakly left (right) 𝐾-Cauchy ⇒ left (right) 𝑑-Cauchy.

Definition 13 (see [6]). Let (𝑋,𝑑) be a quasi-semimetric space. A function 𝑞∶𝑋×𝑋→[0,+∞) is called a 𝑄-function on 𝑋 if the following conditions are satisfied: (Q1) for all 𝑥,𝑦,𝑧∈𝑋, 𝑞(𝑥,𝑧)≤𝑞(𝑥,𝑦)+𝑞(𝑦,𝑧); (Q2) if 𝑥∈𝑋, {𝑦_𝑛}_𝑛∈𝑁 is a sequence in 𝑋 such that it converges to a point 𝑦 (with respect to the quasi-semimetric) and 𝑞(𝑥,𝑦_𝑛)≤𝑀 for some 𝑀=𝑀(𝑥)>0, then 𝑞(𝑥,𝑦)≤𝑀; (Q3) for any 𝜀>0, there exists 𝛿>0 such that 𝑞(𝑥,𝑦)≤𝛿 and 𝑞(𝑥,𝑧)≤𝛿 imply 𝑑(𝑦,𝑧)≤𝜀.

Lemma 14. Let (𝑋,𝑑) be a quasi-semimetric space with one of seven completeness defined in Definition 11(vii). If 𝑞 is a 𝑄-function on 𝑋, then (𝑋,𝑑) is sequentially lower complete w.r.t. 𝑞.

Proof. Assume that {𝑥_𝑛} is a sequence in 𝑋 and ∑^∞_𝑛=1𝑞(𝑥_𝑛,𝑥_𝑛+1)<+∞. Let 𝜆_𝑛=∑^∞_𝑘=𝑛+1𝑞(𝑥_𝑛,𝑥_𝑛+1), then we have lim_𝑛→∞𝜆_𝑛=0 and for any 𝑚>𝑛, 𝑞𝑥_𝑛,𝑥_𝑚≤𝜆_𝑛. (47) By (Q3), for any 𝜀>0, there exists 𝛿>0 such that 𝑞(𝑥,𝑦)≤𝛿 and 𝑞(𝑥,𝑧)≤𝛿 imply 𝑑(𝑦,𝑧)≤𝜀. For the 𝛿>0, there exists 𝐾>0, such that 𝜆_𝑛<𝛿 for any 𝑛≥𝐾. It follows from (47) that for any 𝑟,𝑠≥𝑛>𝐾, we have 𝑞𝑥_𝑛,𝑥_𝑟≤𝜆_𝑛<𝛿,𝑞𝑥_𝑛,𝑥_𝑠≤𝜆_𝑛<𝛿. (48) Thus (Q3) implies that 𝑑(𝑥_𝑟,𝑥_𝑠)<𝜀. That is, {𝑥_𝑛} is a 𝑑-Cauchy sequence. Therefore, by Remark 12, {𝑥_𝑛} is any one of seven Cauchy sequences in Definition 11. Thus, {𝑥_𝑛} converges to some 𝑥∈𝑋. Equation (47) and (Q2) imply that 𝑞(𝑥_𝑛,𝑥)≤𝜆_𝑛. This shows that lim_𝑛→∞𝑞(𝑥_𝑛,𝑥)=0. For any 𝑦∈𝑋, it follows from (Q1) that 𝑞𝑥_𝑛,𝑦≤𝑞𝑥_𝑛,𝑥+𝑞𝑥,𝑦. (49) Therefore liminf_𝑛→∞𝑞𝑥_𝑛,𝑦≤lim_𝑛→∞𝑞𝑥_𝑛,𝑥+𝑞𝑥,𝑦=𝑞𝑥,𝑦. (50) Thus, (𝑋,𝑑) is sequentially lower complete w.r.t. 𝑞. The proof is completed.

From Lemma 14 and Theorems 5, 7, and 9, we can get the following Ekeland type variational principle and its equivalences in quasi-semimetric spaces equipped with 𝑄-functions, which also generalize the results in [6, 12].

Theorem 15. Let (𝑋,𝑑) be a complete quasi-semimetric space with one of seven completeness defined in Definition 11(vii) and 𝑞∶𝑋×𝑋→[0,+∞) a 𝑄-function on 𝑋. Let 𝑓∶𝑋→(−∞,+∞] be a proper, bounded from below, sequentially lower monotone function and 𝜑∶(−∞,+∞)→(0,+∞) a nondecreasing function. Let 𝐷 be a nonempty subset of 𝑋, 𝑔∶𝐷→𝑋 a surjective function, 𝐼 an index set and for each 𝑖∈𝐼, 𝑇_𝑖∶𝐷→2^𝑋 a multivalued map. Then the following conclusions hold and are equivalent.(1)Suppose that for each 𝑥∈𝐷 with 𝑔(𝑥)∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists 𝑦∈𝑋, such that𝑔(𝑥)𝑓𝑔(𝑥)𝑓𝑔(𝑥)𝑦,𝑔(𝑥)𝑓𝑔(𝑥)𝑓𝑔(𝑥)𝑦.𝑒𝑖𝑡ℎ𝑒𝑟0=𝑞,𝑦<𝜑−𝑓𝑜𝑟0<𝑞,𝑦≤𝜑−𝑓 (51)Then for any 𝑥₀∈𝑋, there exists a coincidence point 𝑢∈𝐷 of 𝑔 and {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑔(𝑢)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that 𝑓𝑔(𝑢)≤𝑓𝑥₀. (52)(2)Suppose that for each 𝑥∈𝐷 with 𝑔(𝑥)∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists an 𝑖₀∈𝐼 and 𝑦∈𝑇_𝑖0(𝑥), such that (51) holds. Then for any 𝑥₀∈𝑋, there exists a coincidence point 𝑢∈𝐷 of 𝑔 and {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑔(𝑢)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that (52) holds.(3)Let 𝐼 be an index set, and for each 𝑖∈𝐼, let 𝑇_𝑖∶𝑋→2^𝑋 be a multivalued map.Suppose further that for each 𝑥∈𝑋 with 𝑥∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists 𝑦∈𝑋, such that 𝑓(𝑥)𝑓(𝑥)𝑦,𝑓(𝑥)𝑓(𝑥)𝑦.𝑒𝑖𝑡ℎ𝑒𝑟0=𝑞𝑥,𝑦<𝜑−𝑓𝑜𝑟0<𝑞𝑥,𝑦≤𝜑−𝑓 (53) Then for any 𝑥₀∈𝑋, there exists a common fixed point 𝑢∈𝑋 of {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑢∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that 𝑓(𝑢)≤𝑓𝑥₀. (54)(4) (Ekeland type variational principle in quasi-semimetric spaces) For any 𝑥₀∈𝑋, there exists 𝑣∈𝑋, such that 𝑓(𝑣)≤𝑓(𝑥₀) and 𝑞()𝑓(𝑣)𝑓(𝑣)(𝑥),()𝑞()𝑓(𝑣)𝑓(𝑣)(𝑥),()𝑣,𝑥=𝜑−𝑓∀𝑥∈𝑋𝑤𝑖𝑡ℎ𝑞𝑣,𝑥=0,𝑣,𝑥>𝜑−𝑓∀𝑥∈𝑋𝑤𝑖𝑡ℎ𝑞𝑣,𝑥>0. (55) Moreover, the rest of corresponding equivalent principles in Theorem 9 hold.

In particularly, if (𝑋,𝑑) is a complete quasi-metric space, then from Theorem 15, we have the following results.

Theorem 16. Let (𝑋,𝑑) be a complete quasimetric space with one of seven completeness defined in Definition 11(vii) and 𝑞∶𝑋×𝑋→[0,+∞) a 𝑄-function on 𝑋. Let 𝑓∶𝑋→(−∞,+∞] be a proper, bounded from below, sequentially lower monotone function and 𝜑∶(−∞,+∞)→(0,+∞) a nondecreasing function. Let 𝐷 be a nonempty subset of 𝑋, 𝑔∶𝐷→𝑋 a surjective function, 𝐼 an index set and for each 𝑖∈𝐼, 𝑇_𝑖∶𝐷→2^𝑋 a multivalued map. Then the following conclusions hold and are equivalent. (1)Suppose that for each 𝑥∈𝐷 with 𝑔(𝑥)∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists 𝑦∈𝑋⧵{𝑔(𝑥)}, such that 𝑞𝑔(𝑥),𝑦≤𝜑𝑓𝑔(𝑥)𝑓𝑔(𝑥)−𝑓𝑦. (56) Then for any 𝑥₀∈𝑋, there exists a coincidence point 𝑢∈𝐷 of 𝑔 and {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑔(𝑢)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that 𝑓𝑔(𝑢)≤𝑓𝑥₀. (57)(2)Suppose that for each 𝑥∈𝐷 with 𝑔(𝑥)∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists an 𝑖₀∈𝐼 and 𝑦∈𝑇_𝑖0(𝑥)⧵{𝑔(𝑥)}, such that (56) holds. Then for any 𝑥₀∈𝑋, there exists a coincidence point 𝑢∈𝐷 of 𝑔 and {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑔(𝑢)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that (57) holds.(3)Let 𝐼 be an index set, and for each 𝑖∈𝐼, let 𝑇_𝑖∶𝑋→2^𝑋 be a multivalued map. Suppose further that for each 𝑥∈𝑋 with 𝑥∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists 𝑦∈𝑋⧵{𝑥}, such that 𝑞𝑥,𝑦≤𝜑𝑓(𝑥)𝑓(𝑥)−𝑓𝑦. (58) Then for any 𝑥₀∈𝑋, there exists a common fixed point 𝑢∈𝑋 of {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑢∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that 𝑓(𝑢)≤𝑓𝑥₀. (59)(4) (Ekeland type variational principle in quasimetric spaces) For any 𝑥₀∈𝑋, there exists 𝑣∈𝑋, such that 𝑓(𝑣)≤𝑓(𝑥₀) and 𝑞(𝑣,𝑥)>𝜑𝑓(𝑣)𝑓(𝑣)−𝑓(𝑥),∀𝑥∈𝑋with𝑥≠𝑣. (60)

Proof. The equivalence of the conclusions (1)–(4) is clear. We only prove (4). It follows from (4) of Theorem 15 that for any 𝑥₀∈𝑋, there exists 𝑣∈𝑋, such that 𝑓(𝑣)≤𝑓(𝑥₀) and 𝑞()𝑓(𝑣)𝑓(𝑣)(𝑥),()𝑞()𝑓(𝑣)𝑓(𝑣)(𝑥),()𝑣,𝑥=𝜑−𝑓∀𝑥∈𝑋with𝑞𝑣,𝑥=0,𝑣,𝑥>𝜑−𝑓∀𝑥∈𝑋with𝑞𝑣,𝑥>0. (61) If 𝑣 satisfies (60), then conclusion (4) is proved. Otherwise, there exists 𝑥∈𝑋⧵{𝑣}, such that 0=𝑞(𝑣,𝑥)=𝜑𝑓(𝑣)𝑓(𝑣)−𝑓(𝑥). (62) If 𝑥 and 𝑦 satisfy (62), then 𝑞(𝑣,𝑥)=0 and 𝑞(𝑣,𝑦)=0. By using (Q3) in Definition 13, we get that 𝑑(𝑥,𝑦)=0. It follows from (QM1) that 𝑑(𝑥,𝑥)=0, and then (QM3) implies that 𝑥=𝑦. That is, there is only one point 𝑥 which satisfies (62). Let 𝑆(𝑥)=𝑦∈𝑋∶𝑞𝑥,𝑦≤𝜑𝑓(𝑥)𝑓(𝑥)−𝑓𝑦. (63) Then 𝑆(𝑣)={𝑥}. Since 𝑥∈𝑆(𝑣), we can imply that 𝑆(𝑥)⊂𝑆(𝑣)={𝑥}. This shows that for any 𝑥≠𝑥, 𝑥∉𝑆(𝑥); that is, 𝑥 satisfies (60). The proof is completed.

Definition 17 (see, e.g., [37]). Let 𝑋 be a real vector space; a collection ℬ of subsets of 𝑋 is called a vector bornology on 𝑋, if it satisfies the following conditions: (B1)𝑥∈𝑋 implies that {𝑥}∈ℬ; (B2)𝐵₁⊂𝐵₂ and 𝐵₂∈ℬ imply that 𝐵₁∈ℬ; (B3)𝐵₁,𝐵₂∈ℬ implies that 𝐵₁∪𝐵₂∈ℬ; (B4)𝐵₁,𝐵₂∈ℬ implies that 𝐵₁+𝐵₂=𝑥+𝑦∶𝑥∈𝐵₁,𝑦∈𝐵₂∈ℬ; (64) (B5) for any bounded interval 𝐼⊂(−∞,+∞), 𝐵∈ℬ implies that 𝐼⋅𝐵={𝛼𝑥∶𝛼∈𝐼,𝑥∈𝐵}∈ℬ. (65) In view of (B5), if 𝐵∈ℬ, so is its balanced hull 𝐵_𝑏 which is defined by 𝐵_𝑏=[−1,1]⋅𝐵.

Definition 18. The ordered pair (𝑋,ℬ) is called a bornological vector space (in short: BVS), and every element of ℬ is called a bounded subset (with respect to ℬ).

Definition 19 (see, e.g., [28, 37]). Let (𝑋,ℬ) be a bornological vector space. (i)A sequence {𝑥_𝑛} in 𝑋 is said to be Mackey-convergent (or 𝑀-convergent) to a point 𝑥, denoted by lim^𝑏_𝑛→∞𝑥_𝑛=𝑥, if there is a balanced 𝐵∈ℬ and a sequence of positive real numbers {𝜆_𝑛} such that lim_𝑛→∞𝜆_𝑛=0 and 𝑥_𝑛−𝑥∈𝜆_𝑛𝐵 for any 𝑛∈𝑁. Also, we say that 𝑥 is a bornological limit of {𝑥_𝑛}.(ii)A sequence {𝑥_𝑛} in 𝑋 is said to be Mackey-Cauchy (or 𝑀-Cauchy) if there is a balanced 𝐵∈ℬ and a double sequence of positive real numbers {𝜆_𝑚𝑛} such that lim_{𝑚,𝑛→∞}𝜆_𝑚𝑛=0 and 𝑥_𝑚−𝑥_𝑛∈𝜆_𝑚𝑛𝐵 for any 𝑚,𝑛∈𝑁.(iii)𝐴⊂𝑋 is said to be Mackey-closed (or 𝑀-closed) if it contains all bornological limits of any sequences in 𝐴.(iv)𝐴⊂𝑋 is said to be Mackey-complete (or 𝑀-complete) if every 𝑀-Cauchy sequence in 𝐴 will be 𝑀-convergent to some element in 𝐴.(v)A BVS (𝑋,ℬ) is said to be separated if every 𝑀-convergent sequence is 𝑀-convergent to exactly one bornological limit.

Remark 20. From Lemma 2.13 in [28] we know that if 𝐴⊂𝑋 is a 𝑀-complete subset, then 𝐴 is 𝑀-closed. On the other hand, if (𝑋,ℬ) is 𝑀-complete and 𝐴⊂𝑋 is 𝑀-closed, then 𝐴 is 𝑀-complete. For the details about BVS, one can refer to [17, 28, 37].

The collection of all (complements of) 𝑀-closed subsets of 𝑋 defines a topology on 𝑋, and we called it bornological topology. Therefore, (𝑋,ℬ) endowed this topology is a topological space (but, from Remark 2.4 in [28], we can see that it is rarely a vector topology with respect to the algebraic structure of 𝑋). In the following, we will assume that (𝑋,ℬ) is separated; that is (v) in Definition 19 holds.

Let 𝑋 be a separated bornological vector space and 𝑃∶𝑋→(−∞,+∞] a positively homogeneous subadditive function. By Lemma 4.4 in [28], 𝑃(𝑥)>0 for any nonzero 𝑥 if 𝑃 satisfies the following condition: (P1) the set 𝐶={𝑥∈𝑋∶𝑃(𝑥)≤1} is 𝑀-complete and bounded.

Lemma 21. Let 𝑝∶𝑋×𝑋→(−∞,+∞) be defined by 𝑝(𝑥,𝑦)=𝑃(𝑥−𝑦). If (P1) holds, then (𝑋,ℬ) is sequentially lower complete with respect to 𝑝.

Proof. Let {𝑥_𝑛} be a sequence in 𝑋 and ∑^∞_𝑛=1𝑝(𝑥_𝑛,𝑥_𝑛+1)=∑^∞_𝑛=1𝑃(𝑥_𝑛−𝑥_𝑛+1)<+∞. Then for any 0<𝛿<1, there exists a positive integer 𝑛₀, such that ∑^∞_𝑛=𝑛0𝑃(𝑥_𝑛−𝑥_𝑛+1)<𝛿. Since 𝑃 is subadditive, we get that for any 𝑚>𝑛₀, 𝑃𝑥_𝑛0−𝑥_𝑚≤^𝑚−1_𝑛=𝑛0𝑃𝑥_𝑛−𝑥_𝑛+1<𝛿. (66) From this we have 𝑃𝑥_𝑛0−𝑥_𝑚≤𝛿<1. (67) So we have 𝑥_𝑛0−𝑥_𝑚∈𝐶. For any 𝑚>𝑛, let 𝜆_𝑚𝑛=∑^𝑚−1_𝑘=𝑛𝑃(𝑥_𝑘−𝑥_𝑘+1), and then we have 𝑃(𝑥_𝑛−𝑥_𝑚)≤𝜆_𝑚𝑛, and lim_{𝑚,𝑛→∞}𝜆_𝑚𝑛=0. It follows from 𝑃𝑥_𝑛0−𝑥_𝑚−𝑥_𝑛0−𝑥_𝑛=𝑃𝑥_𝑛−𝑥_𝑚≤𝜆_𝑚𝑛 (68) that (𝑥_𝑛0−𝑥_𝑚)−(𝑥_𝑛0−𝑥_𝑛)∈𝜆_𝑚𝑛𝐶⊂𝜆_𝑚𝑛𝐶_𝑏 where 𝐶_𝑏, is the (bounded) balanced hull of 𝐶. That is, {𝑥_𝑛0−𝑥_𝑛} is 𝑀-Cauchy. Since 𝐶 is 𝑀-complete, {𝑥_𝑛0−𝑥_𝑛} is 𝑀-convergent. Thus, {𝑥_𝑛} is also 𝑀-convergent. Assume that {𝑥_𝑛} is 𝑀-convergent to a point 𝑥. If we set 𝜆_𝑛=^∞_𝑘=𝑛𝑃𝑥_𝑘−𝑥_𝑘+1, (69) then 𝜆_𝑚𝑛≤𝜆_𝑛, and hence 𝜆_𝑚𝑛𝐶⊂𝜆_𝑛𝐶 whenever 𝑚>𝑛. It follows from (68) that 𝑥_𝑛−𝑥_𝑚∈𝐶 for 𝑚>𝑛. Since 𝜆_𝑛𝐶 is 𝑀-closed by (P1), we have lim^𝑏_𝑚→∞𝑥_𝑛−𝑥_𝑚=𝑥_𝑛−𝑥∈𝜆_𝑛𝐶. (70) Consequently, 𝑃(𝑥_𝑛−𝑥)≤𝜆_𝑛, and hence lim_𝑛→∞𝑃(𝑥_𝑛−𝑥)=0. Since 𝑃(𝑥_𝑛−𝑦)≤𝑃(𝑥_𝑛−𝑥)+𝑃(𝑥−𝑦), we have liminf_𝑛→∞𝑃𝑥_𝑛−𝑦≤lim_𝑛→∞𝑃𝑥_𝑛.−𝑥+𝑃𝑥−𝑦=𝑃𝑥−𝑦 (71) That is, liminf_𝑛→∞𝑝𝑥_𝑛,𝑦≤𝑝𝑥,𝑦. (72) Thus, (𝑋,ℬ) is sequentially lower complete with respect to 𝑝. The proof is completed.

From Lemma 21 and Theorem 9, we can get the following Ekeland type variational principle in bornological vector space, which is also proved in [17, 28].

Theorem 22 (Ekeland type variational principle in bornological vector space). Let (𝑋,ℬ) be a separated bornological vector space and 𝑃∶𝑋→(−∞,+∞] a positively homogeneous subadditive function satisfying the condition (P1). Let 𝑓∶𝑋→(−∞,+∞] be a proper, bounded from below, sequentially lower monotone function and 𝜑∶(−∞,+∞)→(0,+∞) a nondecreasing function. Then for any 𝑥₀∈𝑋, there exists 𝑣∈𝑋, such that 𝑓(𝑣)≤𝑓(𝑥₀) and 𝑃(𝑣−𝑥)>𝜑𝑓(𝑣)𝑓(𝑣)−𝑓(𝑥),∀𝑥≠𝑣. (73) Moreover, the corresponding equivalent principles in Theorem 9 hold.

Definition 23 (see [25]). Let 𝑋 be a uniform space. An extended real-valued function 𝑝∶𝑋×𝑋→[0,+∞] is called a 𝑞-distance on 𝑋 if the following conditions are satisfied: (q1) for any 𝑥,𝑦,𝑧∈𝑋, 𝑝(𝑥,𝑧)≤𝑝(𝑥,𝑦)+𝑝(𝑦,𝑧); (q2) every sequence {𝑦_𝑛}⊂𝑋 with 𝑝(𝑦_𝑛,𝑦_𝑚)→0(𝑚>𝑛→∞) is a Cauchy sequence and in the case 𝑝(𝑦_𝑛,𝑦)→0 implies that 𝑦_𝑛→𝑦 in 𝑋; (q3) for 𝑥,𝑦,𝑧∈𝑋, 𝑝(𝑧,𝑥)=0 and 𝑝(𝑧,𝑦)=0 imply 𝑥=𝑦.

Here 𝑝(𝑦_𝑛,𝑦_𝑚)→0(𝑚>𝑛→∞) means that for any 𝜀>0, there exists 𝑛₀∈𝑁 such that 𝑝(𝑦_𝑛,𝑦_𝑚)<𝜀 for all 𝑚>𝑛≥𝑛₀.

Definition 24 (see [25]). Let (𝑋,𝒰) be a uniform space and 𝑝 a 𝑞-distance on 𝑋. A proper function 𝑓∶𝑋→(−∞,+∞] is said to be sequentially lower monotone with respect to 𝑝 (in short, sequentially lower monotone with respect to 𝑝) if for any sequence {𝑥_𝑛} in 𝑋 satisfying 𝑝(𝑥_𝑛,𝑥_𝑚)→0(𝑚>𝑛→∞), 𝑝(𝑥_𝑛,𝑥)→0(𝑛→∞) and 𝑓(𝑥_𝑛+1)≤𝑓(𝑥_𝑛), we have 𝑓(𝑥)≤𝑓(𝑥_𝑛) for each 𝑛∈𝑁.

Definition 25 (see [25]). Let (𝑋,𝒰) be a uniform space, 𝑝 a 𝑞-distance on 𝑋, and 𝑓∶𝑋→(−∞,+∞] a proper function. (𝑋,𝒰) is said to be sequentially complete with respect to (𝑝,𝑓↓) if for any sequence {𝑥_𝑛} in 𝑋 satisfying 𝑝(𝑥_𝑛,𝑥_𝑚)→0(𝑚>𝑛→∞) and 𝑓(𝑥_𝑛+1)≤𝑓(𝑥_𝑛) for each 𝑛∈𝑁, there exists 𝑥∈𝑋 such that 𝑝(𝑥_𝑛,𝑥)→0(𝑛→∞).

Lemma 26. Let (𝑋,𝒰) be a uniform space, 𝑝 a 𝑞-distance on 𝑋, and 𝑓∶𝑋→(−∞,+∞] a sequentially lower monotone with respect to 𝑝, proper function, bounded from below. If (𝑋,𝒰) is sequentially lower complete with respect to (𝑝,𝑓↓), then (𝑋,𝒰) is sequentially lower complete with respect to 𝑝 and 𝑓.

Proof. Assume that {𝑥_𝑛} is a sequence in 𝑋 with 𝑓(𝑥_𝑛+1)≤𝑓(𝑥_𝑛) for each 𝑛∈𝑁 and ∑^∞_𝑛=1𝑝(𝑥_𝑛,𝑥_𝑛+1)<+∞. Let 𝜆_𝑛=∑^∞_𝑘=𝑛+1𝑝(𝑥_𝑛,𝑥_𝑛+1), then we have that lim_𝑛→∞𝜆_𝑛=0, and for any 𝑚>𝑛, 𝑝𝑥_𝑛,𝑥_𝑚≤𝜆_𝑛. (74) This shows that 𝑝(𝑥_𝑛,𝑥_𝑚)→0(𝑚>𝑛→∞). By using Definition 23 and Definition 25, we get that there exists 𝑥∈𝑋 such that 𝑝(𝑥_𝑛,𝑥)→0(𝑛→∞) and {𝑥_𝑛} converges to 𝑥. Then for any 𝑦∈𝑋, by 𝑝(𝑥_𝑛,𝑦)≤𝑝(𝑥_𝑛,𝑥)+𝑝(𝑥,𝑦) we have that liminf_𝑛→∞𝑝𝑥_𝑛,𝑦≤lim_𝑛→∞𝑝𝑥_𝑛,𝑥+𝑝𝑥,𝑦=𝑝𝑥,𝑦. (75) Thus, (𝑋,𝒰) is sequentially lower complete w.r.t. 𝑝 and 𝑓. The proof is completed.

From Theorems 5, 7, and 9, we can get the following Caristi type coincidence point theorem and Ekeland type variational principle in uniform space equipped with 𝑞-distance. From Lemma 26, we also see that this is a transformation of the results appeared in [25].

Theorem 27. Let (𝑋,𝒰) be a uniform space, 𝑝 a 𝑞-distance on 𝑋, and 𝑓∶𝑋→(−∞,+∞] a sequentially lower monotone with respect to 𝑝, proper function, bounded from below. Let 𝐷 be a nonempty subset of 𝑋, 𝑔∶𝐷→𝑋 a surjective function, 𝐼 an index set, and, for each 𝑖∈𝐼, 𝑇_𝑖∶𝐷→2^𝑋 a multivalued map. If (𝑋,𝒰) is sequentially complete with repect to 𝑝 and 𝑓 and 𝜑∶(−∞,+∞)→(0,+∞) is a nondecreasing function, then the following conclusions hold and are equivalent.(1)Suppose that for each 𝑥∈𝐷 with 𝑔(𝑥)∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists 𝑦∈𝑋⧵{𝑔(𝑥)}, such that 𝑝𝑔(𝑥),𝑦≤𝜑𝑓𝑔(𝑥)𝑓𝑔(𝑥)−𝑓𝑦. (76) Then for any 𝑥₀∈𝑋, there exists a coincidence point 𝑢∈𝐷 of 𝑔 and {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑔(𝑢)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that 𝑓𝑔(𝑢)≤𝑓𝑥₀. (77)(2)Suppose that for each 𝑥∈𝐷 with 𝑔(𝑥)∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists an 𝑖₀∈𝐼 and 𝑦∈𝑇_𝑖0(𝑥)⧵{𝑔(𝑥)}, such that (76) holds. Then for any 𝑥₀∈𝑋, there exists a coincidence point 𝑢∈𝐷 of 𝑔 and {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑔(𝑢)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that (77) holds.(3)Let 𝐼 be an index set, and, for each 𝑖∈𝐼, let 𝑇_𝑖∶𝑋→2^𝑋 be a multivalued map.Suppose further that for each 𝑥∈𝑋 with 𝑥∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists 𝑦∈𝑋⧵{𝑥}, such that 𝑝𝑥,𝑦≤𝜑𝑓(𝑥)𝑓(𝑥)−𝑓𝑦. (78) Then for any 𝑥₀∈𝑋, there exists a common fixed point 𝑢∈𝑋 of {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑢∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that 𝑓(𝑢)≤𝑓𝑥₀. (79)(4) (Ekeland type variational principle in uniform spaces) For any 𝑥₀∈𝑋, there exists 𝑣∈𝑋, such that 𝑓(𝑣)≤𝑓(𝑥₀) and 𝑝(𝑣,𝑥)>𝜑𝑓(𝑣)𝑓(𝑣)−𝑓(𝑥),∀𝑥∈𝑋with𝑥≠𝑣. (80) Moreover, the rest of corresponding equivalent principles in Theorem 9 hold.

Proof. The equivalence of the conclusions (1)–(4) is clear. We only need to prove (4). The proof of (4) is similar to the proof of Theorem 16, by using (q3). So we delete the detail of the proof.

Remark 28. If 𝜑∶(−∞,+∞)→(0,+∞) is upper semicontinuous and (76) is replaced by 𝑝𝑔(𝑥)𝜑𝑓𝑔(𝑥)𝑓𝑦⋅𝑓𝑔(𝑥)𝑦,,𝑦≤max,𝜑−𝑓 (81) then the conclusions of Theorem 27 hold. In this case, the proof is similar to the proof of Theorem 2.1 in [11].

As noted in Remark 5.1 in [25], from our Theorem 27, we can deduce [13, Theorems 3.1 and 3.2, and Corollary 3.3] and [15, Theorems 4–6]. Furthermore, we will show that Theorem 27 improves some coincidence point theorems and their equivalences in 𝐹-type separated topological space (or equivalently, generating spaces of quasimetric family) and fuzzy metric spaces, which were proved in [8–11].

In the following, we will assume that (𝑋,ℑ) is an 𝐹-type separated topological space (or equivalently, a uniform space, see [13, 15]) whose topology is generated by a separated family {𝑞_𝜆}_𝜆∈Λ of quasimetrics, where (Λ,≺) is a directed set. Moreover, let 𝛼∶Λ→(0,+∞) be a nondecreasing function; that is, 𝜆,𝜇∈Λ, 𝜆≺𝜇 implies 𝛼(𝜆)≤𝛼(𝜇). An extended real-valued function 𝑝∶𝑋×𝑋→[0,+∞] is defined as follows: 𝑝𝑥,𝑦=sup_𝜆∈Λ𝛼(𝜆)𝑞_𝜆𝑥,𝑦,∀𝑥,𝑦∈𝑋×𝑋. (82)

Lemma 29. Let (𝑋,ℑ) be an 𝐹-type separated topological space (or equivalently, a uniform space) whose topology is generated by a family {𝑞_𝜆}_𝜆∈Λ of quasimetrics, and let 𝑝∶𝑋×𝑋→[0,+∞] be defined by (82). If (𝑋,ℑ) is sequentially complete, then (𝑋,ℑ) is sequentially lower complete w.r.t. 𝑝.

Proof. Assume that a sequence {𝑥_𝑛} in 𝑋 satisfies ∑^∞_𝑛=1𝑝(𝑥_𝑛,𝑥_𝑛+1)<+∞. This implies that the sequence {𝑥_𝑛} satisfies 𝑝(𝑥_𝑛,𝑥_𝑚)→0(𝑚>𝑛→∞). It follows from Examples 2.3 and 3.1 in [25] that 𝑝 is a 𝑞-distance on 𝑋, and there exists an 𝑥∈𝑋, such that {𝑥_𝑛} converges to 𝑥 and 𝑝(𝑥_𝑛,𝑥)→0. Then for any 𝑦∈𝑋, liminf_𝑛→∞𝑝𝑥_𝑛,𝑦≤liminf_𝑛→∞𝑝𝑥_𝑛,𝑥+𝑝𝑥,𝑦=𝑝𝑥,𝑦. (83) Thus, (𝑋,ℑ) is sequentially lower complete w.r.t. 𝑝. The proof is completed.

By using Lemma 29 and Theorem 27, we have the following results.

Theorem 30. Let (𝑋,ℑ) be a sequentially complete and separated 𝐹-type topological space (or equivalently, a uniform space) whose topology is generated by a family {𝑞_𝜆}_𝜆∈Λ of quasi-metrics, 𝜑∶(−∞,+∞)→(0,+∞) a nondecreasing function, and 𝑓∶𝑋→(−∞,+∞] a proper, bounded from below, sequentially lower monotone function. Let 𝐷 be a nonempty subset of 𝑋, 𝑔∶𝐷→𝑋 a surjective function, and 𝐼 an index set, and, for each 𝑖∈𝐼, let 𝑇_𝑖∶𝐷→2^𝑋 be a multivalued map. Then the following conclusions hold and are equivalent.(1)Suppose that for each 𝑥∈𝐷 with 𝑔(𝑥)∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists 𝑦∈𝑋⧵{𝑔(𝑥)}, such that 𝛼(𝜆)𝑞_𝜆𝑔(𝑥),𝑦≤𝜑𝑓𝑔(𝑥)𝑓𝑔(𝑥)−𝑓𝑦,∀𝜆∈Λ. (84) Then for any 𝑥₀∈𝑋, there exists a coincidence point 𝑢∈𝐷 of 𝑔 and {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑔(𝑢)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that 𝑓𝑔(𝑢)≤𝑓𝑥₀. (85)(2)Suppose that for each 𝑥∈𝐷 with 𝑔(𝑥)∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists an 𝑖₀∈𝐼 and 𝑦∈𝑇_𝑖0(𝑥)⧵{𝑔(𝑥)}, such that (82) holds. Then for any 𝑥₀∈𝑋, there exists a coincidence point 𝑢∈𝐷 of 𝑔 and {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑔(𝑢)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that (84) holds.(3)Let 𝐼 be an index set and, for each 𝑖∈𝐼, 𝑇_𝑖∶𝑋→2^𝑋 a multivalued map.Suppose further that for each 𝑥∈𝑋 with 𝑥∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists 𝑦∈𝑋⧵{𝑥}, such that 𝛼(𝜆)𝑞_𝜆𝑥,𝑦≤𝜑𝑓(𝑥)𝑓(𝑥)−𝑓𝑦,∀𝜆∈Λ. (86) Then for any 𝑥₀∈𝑋, there exists a common fixed point 𝑢∈𝑋 of {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑢∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that 𝑓(𝑢)≤𝑓𝑥₀. (87)(4) (Ekeland type variational principle in 𝐹-type topological spaces) For any 𝑥₀∈𝑋, there exists 𝑣∈𝑋, such that 𝑓(𝑣)≤𝑓(𝑥₀) and for any 𝑥∈𝑋 with 𝑣≠𝑥, there exists 𝜆₀∈Λ, such that 𝛼𝜆₀𝑞_𝜆0(𝑣,𝑥)>𝜑𝑓(𝑣)𝑓(𝑣)−𝑓(𝑥). (88)

The following version of coincidence point theorem is an improvement for the coincidence point theorems proved in [8, 9].

Theorem 31. Let (𝑋,𝑑_𝜆∶𝜆∈Λ) and (𝑌,𝛿_𝜆∶𝜆∈Λ) be two sequentially complete and separated generating spaces of quasimetric family, 𝛼∶Λ→(0,+∞) a nondecreasing function, 𝐷 a nonempty subset of 𝑋, 𝑔∶𝐷→𝑋 a surjective function, ℎ∶𝑋→𝑌 a closed mapping, 𝑓∶ℎ(𝑋)→(−∞,+∞] a proper, bounded from below, sequentially lower monotone function, and 𝜑∶(−∞,+∞)→(0,+∞) a non-decreasing function. Let 𝐼 be an index set, and, for each 𝑖∈𝐼, let 𝑇_𝑖∶𝐷→2^𝑋 be a multivalued map. Suppose further that for each 𝑥∈𝐷 with 𝑔(𝑥)∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists 𝑦∈𝑋⧵{𝑔(𝑥)}, such that 𝛼(𝜆)max𝑑_𝜆𝑔(𝑥),𝑦,𝑐𝛿_𝜆ℎ𝑔(𝑥)𝑦𝑓ℎ𝑔(𝑥)𝑓ℎ𝑔(𝑥)ℎ𝑦,ℎ≤𝜑−𝑓,∀𝜆∈Λ, (89) where 𝑐>0 is a given constant. Then for any 𝑥₀∈𝑋, there exists a coincidence point 𝑢∈𝐷 of 𝑔 and {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑔(𝑢)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that 𝑓ℎ𝑔(𝑢)≤𝑓ℎ𝑥₀. (90)

Proof. For each 𝜆∈Λ, we define 𝑞_𝜆∶𝑋×𝑋→𝑅 by 𝑞_𝜆𝑥,𝑦=max𝑑_𝜆𝑥,𝑦,𝑐𝛿_𝜆ℎ(𝑥),ℎ𝑦, (91) and then, by Definition 2 in [15], we can verify that the collection {𝑞_𝜆}_𝜆∈Λ defined by (91) is a family of quasimetrics on 𝑋. Since (𝑋,𝑑_𝜆∶𝜆∈Λ) and (𝑌,𝛿_𝜆∶𝜆∈Λ) are sequentially complete generating spaces of quasimetric family and ℎ is a closed mapping, we can deduce that (𝑋,𝑞_𝜆∶𝜆∈Λ) is also a sequentially complete generating space of quasimetric family. Next, we assume that {𝑥_𝑛}⊂𝑋 is a sequence which converges to 𝑥 in (𝑋,𝑞_𝜆∶𝜆∈Λ) and satisfies 𝑓ℎ𝑥₁≥𝑓ℎ𝑥₂≥⋯≥𝑓ℎ𝑥_𝑛≥⋯. (92) By (91) we know that {𝑥_𝑛} converges to 𝑥 in (𝑋,𝑑_𝜆∶𝜆∈Λ) and {ℎ(𝑥_𝑛)} converges to ℎ(𝑥) in (𝑌,𝛿_𝜆∶𝜆∈Λ). Since 𝑓∶ℎ(𝑋)→(−∞,+∞] is a sequentially lower monotone function, we have 𝑓(ℎ(𝑥))≤𝑓(ℎ(𝑥_𝑛)), for each 𝑛, that is, 𝑓∘ℎ is a sequentially lower monotone function on (𝑋,𝑞_𝜆∶𝜆∈Λ). Then by using Theorem 27 (1) for (𝑋,𝑞_𝜆∶𝜆∈Λ) and 𝑓∘ℎ, we can get the conclusion of Theorem 31. The proof is completed.

The following version of coincidence point theorem is an improvement for the coincidence point theorems proved in [11].

Theorem 32. Let (𝑋,𝑑_𝜆∶𝜆∈Λ) and (𝑌,𝛿_𝜆∶𝜆∈Λ) be two sequentially complete generating spaces of quasimetric family, 𝛼∶Λ→(0,+∞) a nondecreasing function, 𝐷 a non-empty subset of 𝑋, 𝑔∶𝐷→𝑋 a surjective function, ℎ∶𝑋→𝑌 a closed mapping, 𝑓∶ℎ(𝑋)→(−∞,+∞] a proper, bounded from below, sequentially lower monotone function, and 𝜑∶(−∞,+∞)→(0,+∞) an upper semicontinuous function. Let 𝐼 be an index set, and, for each 𝑖∈𝐼, let 𝑇_𝑖∶𝐷→2^𝑋 be a multivalued map. Suppose further that for each 𝑥∈𝐷 with 𝑔(𝑥)∉⋂_𝑖∈𝐼𝑇_𝑖(𝑥), there exists 𝑦∈𝑋⧵{𝑔(𝑥)}, such that 𝛼(𝜆)max𝑑_𝜆𝑔(𝑥),𝑦,𝑐𝛿_𝜆ℎ𝑔(𝑥)𝑦𝜑𝑓ℎ𝑔(𝑥)𝑓ℎ𝑦×𝑓ℎ𝑔(𝑥)ℎ𝑦,ℎ≤max,𝜑−𝑓,∀𝜆∈Λ, (93) where 𝑐>0 is a given constant. Then for any 𝑥₀∈𝑋, there exists a coincidence point 𝑢∈𝐷 of 𝑔 and {𝑇_𝑖}_𝑖∈𝐼; that is, 𝑔(𝑢)∈⋂_𝑖∈𝐼𝑇_𝑖(𝑢), such that 𝑓ℎ𝑔(𝑢)≤𝑓ℎ𝑥₀. (94)

Proof. It follows from the proof of Theorem 16 that 𝑓∘ℎ is a sequentially lower monotone function on (𝑋,𝑞_𝜆∶𝜆∈Λ), where {𝑞_𝜆}_𝜆∈Λ is defined by (91). Then by using Remark 28 for (𝑋,𝑞_𝜆∶𝜆∈Λ) and 𝑓∘ℎ, we can get the conclusion of Theorem 32. The proof is completed.

Remark 33. Similarly, under the conditions of Theorem 31 or Theorem 32, we can get corresponding common fixed point theorems, Ekeland's variational principle, and other equivalences of Theorems 31 and 32. On the other hand, a fuzzy metric space in the sense of Kaleva and Seikkala [39] is a generating space of quasimetric family (see, e.g., [8, 9, 11]); thus, by using our results in this section we can get coincidence point theorems and its equivalences in fuzzy metric spaces. Therefore, our results are also unified improvements of the results in [8–11].

Let (𝑋,𝒰) be a uniform space, 𝑞∶𝑋×𝑋→[0,+∞) a 𝑞-distance on 𝑋, and ℎ∶[0,+∞)→[0,+∞) a nondecreasing function, such that ₀^+∞𝑑𝑟1+ℎ(𝑟)=+∞, (95) and 𝑥₀ is a given point in 𝑋. Let 𝑝∶𝑋×𝑋→[0,+∞) be defined by 𝑝𝑥,𝑦=^{𝑞(𝑥0,𝑥)+𝑞(𝑥,𝑦)}_{𝑞(𝑥0,𝑥)}𝑑𝑟(𝑟).1+ℎ (96)

Lemma 34. Let (𝑋,𝒰) be a uniform space, 𝑞 a 𝑞-distance on 𝑋, ℎ∶[0,+∞)→[0,+∞) a nondecreasing function satisfying (95), and 𝑓∶𝑋→(−∞,+∞] a sequentially lower monotone with respect to 𝑞, proper function, bounded from below. If (𝑋,𝒰) is sequentially lower complete with respect to (𝑞,𝑓↓), then (𝑋,𝒰) is sequentially lower complete with respect to 𝑝 and 𝑓, where 𝑝 is defined by (96).

Proof. Let {𝑥_𝑛} be a sequence in 𝑋 with 𝑓(𝑥_𝑛+1)≤𝑓(𝑥_𝑛) for each 𝑛∈𝑁 and satisfy ^∞_𝑛=1𝑝𝑥_𝑛,𝑥_𝑛+1=^∞_𝑛=1^{𝑞(𝑥0,𝑥𝑛)+𝑞(𝑥𝑛,𝑥𝑛+1)}_{𝑞(𝑥0,𝑥𝑛)}𝑑𝑟1+ℎ(𝑟)<+∞. (97) For any 𝑛>1, by noting that ℎ∶[0,+∞)→[0,+∞) is nondecreasing, we have ^{∑𝑛−1𝑘=0𝑞(𝑥𝑘,𝑥𝑘+1)}_{𝑞(𝑥0,𝑥1)}𝑑𝑟(𝑟)=1+ℎ_{𝑞(𝑥0,𝑥1)+𝑞(𝑥1,𝑥2)𝑞(𝑥0,𝑥1)}𝑑𝑟(𝑟)+1+ℎ+⋯_{𝑞(𝑥0,𝑥1)+⋯+𝑞(𝑥𝑛−1,𝑥𝑛)𝑞(𝑥0,𝑥1)+⋯+𝑞(𝑥𝑛−2,𝑥𝑛−1)}𝑑𝑟1+ℎ(𝑟)≤^{𝑞(𝑥0,𝑥1)+𝑞(𝑥1,𝑥2)}_{𝑞(𝑥0,𝑥1)}𝑑𝑟(𝑟)+1+ℎ+⋯_{𝑞(𝑥0,𝑥𝑛)+𝑞(𝑥𝑛−1,𝑥𝑛)𝑞(𝑥0,𝑥𝑛)}𝑑𝑟(𝑟)≤1+ℎ_{∞𝑛=1𝑞(𝑥0,𝑥𝑛)+𝑞(𝑥𝑛,𝑥𝑛+1)𝑞(𝑥0,𝑥𝑛)}𝑑𝑟1+ℎ(𝑟)<+∞. (98) This shows that ^{∑∞𝑛=1𝑞(𝑥𝑛,𝑥𝑛+1)}_{𝑞(𝑥0,𝑥1)}𝑑𝑟(𝑟)≤1+ℎ_{∞𝑛=1𝑞(𝑥0,𝑥𝑛)+𝑞(𝑥𝑛,𝑥𝑛+1)𝑞(𝑥0,𝑥𝑛)}𝑑𝑟1+ℎ(𝑟)<+∞. (99) By (95) we obtain that ∑^∞_𝑛=1𝑞(𝑥_𝑛,𝑥_𝑛+1)<+∞. Since (𝑋,𝒰) is sequentially lower complete w.r.t. (𝑞,𝑓↓), by Lemma 26, we get that there exists 𝑥∈𝑋 such that 𝑞(𝑥_𝑛,𝑥)→0(𝑛→∞), {𝑥_𝑛} converges to 𝑥, and, for any 𝑦∈𝑋, liminf_𝑛→∞𝑞𝑥_𝑛,𝑦≤𝑝𝑥,𝑦. (100) Then there exists a subsequence {𝑥_𝑛𝑘} such that lim_𝑘→∞𝑞𝑥_𝑛𝑘,𝑦=liminf_𝑛→∞𝑞𝑥_𝑛,𝑦≤𝑞𝑥,𝑦. (101) Also, there exists a subsequence {𝑥_𝑛𝑘𝑖} of {𝑥_𝑛𝑘} such that lim_𝑖→∞𝑞(𝑥₀,𝑥_𝑛𝑘𝑖)=liminf_𝑘→∞𝑞(𝑥₀,𝑥_𝑛𝑘). It follows from 𝑞(𝑥_𝑛,𝑥)→0(𝑛→∞) and 𝑞(𝑥₀,𝑥)≤𝑞(𝑥₀,𝑥_𝑛𝑘𝑖)+𝑞(𝑥_𝑛𝑘𝑖,𝑥) that 𝑞(𝑥₀,𝑥)≤lim_𝑖→∞𝑞(𝑥₀,𝑥_𝑛𝑘𝑖). Then, by noting that ℎ is nondecreasing, we have liminf_𝑛→∞𝑝𝑥_𝑛,𝑦≤liminf_𝑘→∞𝑝𝑥_𝑛𝑘,𝑦≤liminf_𝑖→∞𝑝𝑥_𝑛𝑘𝑖,𝑦=liminf_𝑖→∞^{𝑞(𝑥0,𝑥𝑛𝑘𝑖)+𝑞(𝑥𝑛𝑘𝑖,𝑦)}_{𝑞(𝑥0,𝑥𝑛𝑘𝑖)}𝑑𝑟1+ℎ(𝑟)=^{lim𝑖→∞𝑞(𝑥0,𝑥𝑛𝑘𝑖)+lim𝑖→∞𝑞(𝑥𝑛𝑘𝑖,𝑦)}_{lim𝑖→∞𝑞(𝑥0,𝑥𝑛𝑘𝑖)}𝑑𝑟1+ℎ(𝑟)≤^{𝑞(𝑥0,𝑥)+lim𝑖→∞𝑞(𝑥𝑛𝑘𝑖}_𝑞𝑥^,𝑦)_0,𝑥𝑑𝑟1+ℎ(𝑟)≤^{𝑞(𝑥0,𝑥)+𝑞(𝑥,𝑦)}_{𝑞(𝑥0,𝑥)}𝑑𝑟1+ℎ(𝑟)=𝑝𝑥,𝑦. (102) Thus, (𝑋,𝒰) is sequentially lower complete with respect to 𝑝 and 𝑓. The proof is completed.

Remark 35. By using Lemma 34 and Theorem 27, we can get coincidence point theorems and its equivalences for (𝑋,𝒰) and 𝑝, which improve the results in [29, 30].

Remark 36. Let (𝑋,ℑ) be a sequentially complete 𝐹-type separated topological space (or equivalently, a uniform space) whose topology is generated by a family {𝑞_𝜆}_𝜆∈Λ of pseudometrics (see [15]), 𝛼∶Λ→(0,+∞) a nondecreasing function, and ℎ∶[0,+∞)→[0,+∞) a nondecreasing function satisfying (95). Let 𝑝_𝜆∶𝑋×𝑋→[0,+∞) be defined by 𝑝_𝜆𝑥,𝑦=^{𝑞𝜆(𝑥0,𝑥)+𝑞𝜆}_𝑞^(𝑥,𝑦)_{𝜆𝑥0,𝑥}𝑑𝑟1+ℎ(𝑟),𝜆∈Λ, (103) and let an extended real-valued function 𝑝∶𝑋×𝑋→[0,+∞] be defined by 𝑝𝑥,𝑦=sup_𝜆∈Λ𝛼(𝜆)𝑝_𝜆𝑥,𝑦,∀𝑥,𝑦∈𝑋×𝑋. (104) Similar to the proof of Lemma 34, one can prove that (𝑋,ℑ) is sequentially lower complete w.r.t. 𝑝. Then, by using Theorem 27, we can get coincidence point theorems and its equivalences for (𝑋,ℑ) and 𝑝, which improve the results in [32].

Acknowledgments

First, the authors are very grateful to the referees for their careful reading of the manuscript and a lot of valuable comments and suggestions, which greatly improve this paper. Next, this paper was supported by the National Natural Science Foundation of China (11171286, 11126344).