The Existence of Cone Critical Point and Common Fixed Point with Applications

Abstract

We first establish some new critical point theorems for nonlinear dynamical systems in cone metric spaces or usual metric spaces, and then we present some applications to generalizations of Dancš-Hegedüs-Medvegyev's principle and the existence theorem related with Ekeland's variational principle, Caristi's common fixed point theorem for multivalued maps, Takahashi's nonconvex minimization theorem, and common fuzzy fixed point theorem. We also obtain some fixed point theorems for weakly contractive maps in the setting of cone metric spaces and focus our research on the equivalence between scalar versions and vectorial versions of some results of fixed point and others.

1. Introduction

In 1983, Dancš et al. [1] proved the following interesting existence theorem of critical point (or stationary point) for a nonlinear dynamical system.

Dancš-Hegedüs-Medvegyev's Principle [1]
Let (𝑋,𝑑) be a complete metric space. Let Γ∶𝑋→2^𝑋 be a multivalued map with nonempty values. Suppose that the following conditions are satisfied:(i)for each 𝑥∈𝑋, we have 𝑥∈Γ(𝑥), and Γ(𝑥) is closed;(ii)𝑥, 𝑦∈𝑋 with 𝑦∈Γ(𝑥) implies Γ(𝑦)⊆Γ(𝑥);(iii)for each 𝑛∈ℕ and each 𝑥_𝑛+1∈Γ(𝑥_𝑛), we have lim_𝑛→∞𝑑(𝑥_𝑛,𝑥_𝑛+1)=0. Then there exists 𝑣∈𝑋 such that Γ(𝑣)={𝑣}.

Dancš-Hegedüs-Medvegyev's Principle has been popularly investigated and applied in various fields of applied mathematical analysis and nonlinear analysis, see, for example, [2, 3] and references therein. It is well known that the celebrated Ekeland's variational principle can be deduced by the detour of using Dancš-Hegedüs-Medvegyev's principle, and it is equivalent to the Caristi's fixed point theorem, to the Takahashi's nonconvex minimization theorem, to the drop theorem, and to the petal theorem. Many generalizations in various different directions of these results in metric (or quasi-metric) spaces and more general in topological vector spaces have been studied by several authors in the past; for detail, one can refer to [2–12].

Let 𝐸 be a topological vector space (t.v.s. for short) with its zero vector 𝜃_𝐸. A nonempty subset 𝐾 of 𝐸 is called a convex cone if 𝐾+𝐾⊆𝐾 and 𝜆𝐾⊆𝐾 for 𝜆≥0. A convex cone 𝐾 is said to be pointed if 𝐾∩(−𝐾)={𝜃_𝐸}. For a given proper, pointed, and convex cone 𝐾 in 𝐸, we can define a partial ordering ⪷_𝐾 with respect to 𝐾 by 𝑥⪷_𝐾𝑦⟺𝑦−𝑥∈𝐾. (1.1)𝑥≺_𝐾𝑦 will stand for 𝑥⪷_𝐾𝑦 and 𝑥≠𝑦, while 𝑥≪_𝐾𝑦 will stand for 𝑦−𝑥∈int𝐾, where int𝐾 denotes the interior of 𝐾.

In the following, unless otherwise specified, we always assume that 𝑌 is a locally convex Hausdorff t.v.s. with its zero vector 𝜃, 𝐾 a proper, closed, convex, and pointed cone in 𝑌 with int𝐾≠∅, 𝑒∈int𝐾 and ⪷_𝐾 a partial ordering with respect to 𝐾. Denote by ℝ and ℕ the set of real numbers and the set of positive integers, respectively.

Fixed point theory in 𝐾-metric and 𝐾-normed spaces was studied and developed by Perov [13], Kvedaras et al. [14], Perov and Kibenko [15], Mukhamadiev and Stetsenko [16], Vandergraft [17], Zabrejko [18], and references therein. In 2007, Huang and Zhang [19] reintroduced such spaces under the name of cone metric spaces and investigated fixed point theorems in such spaces in the same work. Since then, the cone metric fixed point theory is prompted to study by many authors; for detail, see [20–29] and references therein.

Very recently, in order to improve and extend the concept of cone metric space in the sense of Huang and Zhang, Du [23] first introduced the concepts of TVS-cone metric and TVS-cone metric space as follows.

Definition 1.1 (see [23]). Let 𝑋 be a nonempty set. A vector-valued function 𝑝∶𝑋×𝑋→𝑌 is said to be a TVS-cone metric, if the following conditions hold: (𝐶1)𝜃⪷_𝐾𝑝(𝑥,𝑦) for all 𝑥, 𝑦∈𝑋 and 𝑝(𝑥,𝑦)=𝜃 if and only if 𝑥=𝑦;(𝐶2)𝑝(𝑥,𝑦)=𝑝(𝑦,𝑥) for all 𝑥, 𝑦∈𝑋;(𝐶3)𝑝(𝑥,𝑧)⪷_𝐾𝑝(𝑥,𝑦)+𝑝(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋. The pair (𝑋,𝑝) is then called a TVS-cone metric space.

Definition 1.2 (see [23]). Let (𝑋,𝑝) be a TVS-cone metric space, 𝑥∈𝑋, and, {𝑥_𝑛}_𝑛∈ℕ let be a sequence in 𝑋. (i){𝑥_𝑛} is said to TVS-cone converge to 𝑥 if, for every 𝑐∈𝑌 with 𝜃≪_𝐾𝑐, there exists a natural number ℕ₀ such that 𝑝(𝑥_𝑛,𝑥)≪_𝐾𝑐 for all 𝑛≥ℕ₀. We denote this by cone-lim_𝑛→∞𝑥_𝑛=𝑥 or 𝑥−−−−→_𝑛^cone𝑥 as 𝑛→∞ and call 𝑥 the TVS-cone limit of {𝑥_𝑛}.(ii){𝑥_𝑛} is said to be a TVS-cone Cauchy sequence if, for every 𝑐∈𝑌 with 𝜃≪_𝐾𝑐, there is a natural number ℕ₀ such that 𝑝(𝑥_𝑛,𝑥_𝑚)≪_𝐾𝑐 for all 𝑛, 𝑚≥ℕ₀.(iii)(𝑋,𝑝) is said to be TVS-cone complete if every TVS-cone Cauchy sequence in 𝑋 is TVS-cone convergent. In [23], the author proved the following important results.

Theorem 1.3 (see [23]). Let (𝑋,𝑝) be a TVS-cone metric spaces. Then 𝑑_𝑝∶𝑋×𝑋→[0,∞) defined by 𝑑_𝑝∶=𝜉_𝑒∘𝑝 is a metric, where the nonlinear scalarization function 𝜉_𝑒∶𝑌→ℝ is defined by 𝜉_𝑒𝑦=inf𝑟∈ℝ∶𝑦∈𝑟𝑒−𝐾,∀𝑦∈𝑌. (1.2)

Example 1.4. Let 𝑋=[0,1], 𝑌=ℝ², 𝐾=ℝ²₊={(𝑥,𝑦)∈ℝ²∶𝑥,𝑦≥0}, 𝑒=(1,1), and 𝜃=(0,0). Define 𝑝∶𝑋×𝑋→𝑌 by 𝑝𝑥,𝑦=𝑥−𝑦,5𝑥−𝑦. (1.3) Then (𝑋,𝑝) is a TVS-cone complete metric space. It is easy to verify that 𝑑_𝑝𝑥,𝑦=𝜉_𝑒𝑝𝑥,𝑦=inf𝑟∈ℝ∶𝑝𝑥,𝑦∈𝑟𝑒−𝐾=5𝑥−𝑦, (1.4) so 𝑑_𝑝 is a metric on 𝑋, and (𝑋,𝑑_𝑝) is a complete metric space.

Theorem 1.5 (see [23]). Let (𝑋,𝑝) be a TVS-cone metric space, let 𝑥∈𝑋, and let {𝑥_𝑛}_𝑛∈ℕ be a sequence in 𝑋. Then the following statements hold. (a)If {𝑥_𝑛}TVS-cone converges to (i.e.,𝑥−−−−→_𝑛^cone𝑥as𝑛→∞), then 𝑑_𝑝(𝑥_𝑛,𝑥)→0 as 𝑛→∞(i.e.,𝑥−−→_𝑛^𝑑𝑝𝑥as𝑛→∞).(b)If {𝑥_𝑛} is a TVS-cone Cauchy sequence in (𝑋,𝑝), then {𝑥_𝑛} is a Cauchy sequence (in usual sense) in (𝑋,𝑑_𝑝).

The paper is organized as follows. In Section 2, we first establish some new critical point theorems for nonlinear dynamical systems in cone metric spaces or usual metric spaces, and then we present some applications to generalizations of Dancš-Hegedüs-Medvegyev's principle and the existence theorem related with Ekeland's variational principle, Caristi's common fixed point theorem for multivalued maps, Takahashi's nonconvex minimization theorem, and the common fuzzy fixed point theorem. Section 3 is dedicated to the study of fixed point theorems for weakly contractive maps in the setting of cone metric spaces. In Section 4, we focus our research on the equivalence between scalar versions and vectorial vesions of some results of fixed point and others.

2. Critical Point Theorems in Cone Metric Spaces

Let 𝑋 be a nonempty set. A fuzzy set in 𝑋 is a function of 𝑋 into [0,1]. Let ℱ(𝑋) be the family of all fuzzy sets in 𝑋. A fuzzy map on 𝑋 is a map from 𝑋 into ℱ(𝑋). This enables us to regard each fuzzy map as a two-variable function of 𝑋×𝑋 into [0,1]. Let 𝐹 be a fuzzy map on 𝑋. An element 𝑥 of 𝑋 is said to be a fuzzy fixed point of 𝐹 if 𝐹(𝑥,𝑥)=1 (see, e.g., [4, 5, 30–32]). Let Γ∶𝑋→2^𝑋 be a multivalued map. A point 𝑥∈𝑋 is called to be a critical point (or stationary point) [1–3, 7, 32] of Γ if Γ(𝑣)={𝑣}.

Recall that the nonlinear scalarization function 𝜉_𝑒∶𝑌→ℝ is defined by 𝜉_𝑒𝑦=inf𝑟∈ℝ∶𝑦∈𝑟𝑒−𝐾,∀𝑦∈𝑌. (2.1)

Lemma 2.1 (see [6, 23, 29, 33]). For each 𝑟∈ℝ and 𝑦∈𝑌, the following statements are satisfied: (i)𝜉_𝑒(𝑦)≤𝑟⇔𝑦∈𝑟𝑒−𝐾,(ii)𝜉_𝑒(𝑦)>𝑟⇔𝑦∉𝑟𝑒−𝐾,(iii)𝜉_𝑒(𝑦)≥𝑟⇔𝑦∉𝑟𝑒−int𝐾,(iv)𝜉_𝑒(𝑦)<𝑟⇔𝑦∈𝑟𝑒−int𝐾,(v)𝜉_𝑒(⋅) is positively homogeneous and continuous on 𝑌,(vi)if 𝑦₁∈𝑦₂+𝐾 (i.e., 𝑦₂⪷_𝐾𝑦₁), then 𝜉_𝑒(𝑦₂)≤𝜉_𝑒(𝑦₁),(vii)𝜉_𝑒(𝑦₁+𝑦₂)≤𝜉_𝑒(𝑦₁)+𝜉_𝑒(𝑦₂) for all 𝑦₁,𝑦₂∈𝑌.

Remark 2.2. Notice that the reverse statement of (vi) in Lemma 2.1 (i.e., 𝜉_𝑒(𝑦₂)≤𝜉_𝑒(𝑦₁)⇒𝑦₁∈𝑦₂+𝐾 or 𝑦₂⪷_𝐾𝑦₁) does not hold in general. For example, let 𝑌=ℝ², let 𝐾=ℝ²₊={(𝑥,𝑦)∈ℝ²∶𝑥,𝑦≥0}, and let 𝑒=(1,1). Then 𝐾 is a proper, closed, convex, and pointed cone in 𝑌 with int𝐾={(𝑥,𝑦)∈ℝ²∶𝑥,𝑦>0}≠∅ and 𝑒∈int𝐾. For 𝑟=1, it is easy to see that 𝑦₁=(5,−6)∉𝑟𝑒−int𝐾, and 𝑦₂=(0,0)∈𝑟𝑒−int𝐾. By applying (iii) and (iv) of Lemma 2.1, we have 𝜉_𝑒(𝑦₂)<1≤𝜉_𝑒(𝑦₁) while 𝑦₁∉𝑦₂+𝐾.

Definition 2.3. Let 𝐴 be a nonempty subset of a TVS-cone metric space (𝑋,𝑝). (i)The TVS-cone closure of 𝐴, denoted tvsc-cl(𝐴), is defined by tvsc-cl(𝐴)=𝑥∈𝑋∶∃𝑥_𝑛⊂𝐴suchthat𝑥⟶_𝑛^cone𝑥as𝑛⟶∞. (2.2) Obviously,𝐴⊆tvsc-cl(𝐴).(ii)𝐴 is said to be TVS-cone closed if 𝐴=tvsc-cl(𝐴).(iii)𝐴 is said to be TVS-cone open if the complement 𝑋⧵𝐴 of A is TVS-cone closed.
If 𝑌=ℝ, 𝐾=[0,∞)⊂ℝ and 𝑒=1, then 𝑝≡𝑑 is a metric in usual sense, and the closure of 𝐴 is denoted by cl_𝑑(𝐴).

Theorem 2.4. Let (𝑋,𝑝) be a TVS-cone metric space and let 𝒯_𝑝=𝑈⊆𝑋∶𝑈is𝑇𝑉𝑆-𝑐𝑜𝑛𝑒𝑜𝑝𝑒𝑛in𝑋,𝑝. (2.3) Then 𝒯_𝑝 is a topology on (𝑋,𝑝) induced by 𝑝.

Proof. Clearly, ∅ and 𝑋 are TVS-cone closed in (𝑋,𝑝). Thus, 𝑋 and ∅ are TVS-cone open in (𝑋,𝑝), and hence ∅, 𝑋∈𝒯_𝑝. Let 𝑈₁, 𝑈₂∈𝒯_𝑝. Then 𝑉₁=𝑋⧵𝑈₁ and 𝑉₂=𝑋⧵𝑈₂ are TVS-cone closed in (𝑋,𝑝). We claim that 𝑈₁∩𝑈₂∈𝒯_𝑝. Let 𝑣∈tvsc-cl(𝑉₁∪𝑉₂). Then ∃{𝑣_𝑛}⊂𝑉₁∪𝑉₂ such that 𝑣−−−−→_𝑛^cone𝑣 as 𝑛→∞. Without loss of generality, we may assume that there exists a subsequence {𝑣_𝑛𝑘} of {𝑣_𝑛}∩𝑉₁. Since 𝑣−−−−→_𝑛𝑘^cone𝑣 as 𝑘→∞, we have 𝑣∈tvsc-cl(𝑉₁)=𝑉₁⊆𝑉₁∪𝑉₂. So tvsc-cl(𝑉₁∪𝑉₂)⊆𝑉₁∪𝑉₂, and; hence, 𝑉₁∪𝑉₂ is TVS-cone closed in (𝑋,𝑝). From 𝑈₁∩𝑈₂=𝑋⧵𝑉₁∪𝑉₂, (2.4) we see that 𝑈₁∩𝑈₂ is TVS-cone open in (𝑋,𝑝) and 𝑈₁∩𝑈₂∈𝒯_𝑝.
Let 𝐼 be any index set, and let {𝑈_𝑖}_𝑖∈𝐼⊂𝒯_𝑝. We show that ⋃_𝑖∈𝐼𝑈_𝑖∈𝒯_𝑝. For each 𝑖∈𝐼, set 𝑉_𝑖=𝑋⧵𝑈_𝑖. Thus, 𝑉_𝑖 is TVS-cone closed in 𝑋 for all 𝑖∈𝐼. Let 𝑤∈tvsc-cl(⋂_𝑖∈𝐼𝑉_𝑖). Then ∃{𝑤_𝑛}⊂⋂_𝑖∈𝐼𝑉_𝑖 such that 𝑤−−−−→_𝑛^cone𝑤 as 𝑛→∞. For each 𝑖∈𝐼, since {𝑤_𝑛}⊂𝒱_𝑖 and 𝑤−−−−→_𝑛^cone𝑤, 𝑤∈tvsc-cl(𝑉_𝑖)=𝑉_𝑖. Hence, 𝑤∈⋂_𝑖∈𝐼𝑉_𝑖. So tvsc-cl(⋂_𝑖∈𝐼𝑉_𝑖)⊆⋂_𝑖∈𝐼𝑉_𝑖, and then ⋂_𝑖∈𝐼𝑉_𝑖 is TVS-cone closed in (𝑋,𝑝). Since _𝑖∈𝐼𝑈_𝑖=𝑋⧵_𝑖∈𝐼𝑉_𝑖, (2.5)⋃_𝑖∈𝐼𝑈_𝑖 is TVS-cone open in (𝑋,𝑝), and ⋃_𝑖∈𝐼𝑈_𝑖∈𝒯_𝑝.
Therefore, by above, we prove that 𝒯_𝑝 is a topology on (𝑋,𝑝).

The following result is simple, but it is very useful in this paper.

Lemma 2.5. Let 𝐸 be a t.v.s., 𝐾 a convex cone with int𝐾≠∅ in 𝐸, and let 𝑎, 𝑏, 𝑐∈𝐸. Then the following statements hold. (i)int𝐾+𝐾⊆int𝐾.(ii)If 𝑎⪷_𝐾𝑏 and 𝑏⪷_𝐾𝑐, then 𝑎⪷_𝐾𝑐.(iii)If 𝑎⪷_𝐾𝑏 and 𝑏≪_𝐾𝑐, then 𝑎≪_𝐾𝑐.(iv)If 𝑎≪_𝐾𝑏 and 𝑏⪷_𝐾𝑐, then 𝑎≪_𝐾𝑐.(v)If 𝑎≪_𝐾𝑏 and 𝑏≪_𝐾𝑐, then 𝑎≪_𝐾𝑐.(vi)If 𝑎⪷_𝐾𝑐 and 𝑏⪷_𝐾𝑐, then 𝑎+𝑏⪷_𝐾2𝑐.(vii)If 𝑎⪷_𝐾𝑐 and 𝑏≪_𝐾𝑐, then 𝑎+𝑏≪_𝐾2𝑐.(viii)If 𝑎≪_𝐾𝑐 and 𝑏≪_𝐾𝑐, then 𝑎+𝑏≪_𝐾2𝑐.

Proof. The conclusion (i) follows from the facts that the set int𝐾+𝐾 is open in 𝐸, and 𝐾 is a convex cone. By the transitivity of partial ordering ⪷_𝐾, we have the conclusion (ii). To see (iii), since 𝑎⪷_𝐾𝑏⇔𝑏−𝑎∈𝐾 and 𝑏≪_𝑘𝑐⇔𝑐−𝑏∈int𝐾, it follows from (i) that 𝑐−𝑎=(𝑐−𝑏)+(𝑏−𝑎)∈int𝐾+𝐾⊆int𝐾, (2.6) which means that 𝑎≪_𝑘𝑐. The proofs of conclusions (iv)–(viii) are similar to (iii).

Definition 2.6. Let (𝑋,𝑝) be a TVS-cone metric space. A nonempty subset 𝐶 of 𝑋 is said to be TVS-cone compact if every sequence in 𝐶 has a TVS-cone convergent subsequence whose TVS-cone limit is an element of 𝐶.
If 𝑋 is TVS-cone compact, then we say that (𝑋,𝑝) is a TVS-cone compact metric space.

Theorem 2.7. Let 𝐶 be a nonempty subset of a TVS-cone metric space (𝑋,𝑝). Then the following statements hold. (a)If 𝐶 is a closed set in the metric space (𝑋,𝑑_𝑝), then 𝐶 is TVS-cone closed in (𝑋,𝑝) and tvsc-cl(𝐶)=cl_𝑑𝑝(𝐶), where 𝑑_𝑝∶=𝜉_𝑒∘𝑝.(b)If 𝐶 is TVS-cone compact, then it is TVS-cone closed.(c)If 𝐶 is TVS-cone closed and (𝑋,𝑝) is TVS-cone complete, then (𝐶,𝑝) is also TVS -cone complete.(d)If 𝐶 is TVS-cone compact, then (𝐶,𝑝) is TVS-cone complete.(e)If 𝐶 is TVS-cone compact, then 𝐶 is (sequentially) compact in the metric space (𝑋,𝑑_𝑝).

Proof. Applying Theorem 1.3, 𝑑_𝑝 is a metric on 𝑋. Let 𝐶 be a closed set in the metric space (𝑋,𝑑_𝑝). By Theorem 1.5, we have tvsc-cl(𝐶)=𝑥∈𝑋∶∃𝑥_𝑛⊂𝐶suchthat𝑥_𝑛^cone⊆𝑥⟶𝑥as𝑛⟶∞𝑥∈𝑋∶∃_𝑛⊂𝐶suchthat𝑥⟶_𝑛^𝑑𝑝𝑥as𝑛⟶∞=cl_𝑑𝑝(𝐶)=𝐶, (2.7) which implies that 𝐶 is TVS-cone closed in (𝑋,𝑝) and tvsc-cl(𝐶)=cl_𝑑𝑝(𝐶). Hence, the conclusion (a) holds.
Next, assume that 𝐶 is TVS-cone compact in (𝑋,𝑝). Let 𝑥∈tvsc-cl(𝐶). Then there exists {𝑥_𝑛}⊂𝐶 such that 𝑥−−−−→_𝑛^cone𝑥 as 𝑛→∞. By the TVS-cone compactness of 𝐶, there exist {𝑥_𝑛𝑗}⊂{𝑥_𝑛} and 𝑤∈𝐶 such that 𝑥−−−−→_𝑛𝑗^cone𝑤 as 𝑗→∞. Applying Theorem 1.5, 𝑑_𝑝(𝑥_𝑛,𝑥)→0 as 𝑛→∞ and 𝑑_𝑝(𝑥_𝑛𝑗,𝑤)→0 as 𝑗→∞. By the uniqueness of limit, 𝑥=𝑤∈𝐶, and; hence, tvsc-cl(𝐶)⊆𝐶. So 𝐶 is TVS-cone closed in (𝑋,𝑝), and (b) is proved.
To see (c), let {𝑥_𝑛} be a TVS-cone Cauchy sequence in (𝐶,𝑝)⊆(𝑋,𝑝). Since (𝑋,𝑝) is TVS-cone complete, there exists 𝑣∈𝑋 such that 𝑥−−−−→_𝑛^cone𝑣 as 𝑛→∞. Hence, 𝑣∈tvsc-cl(𝐶)=𝐶, which show that (𝐶,𝑝) is TVS-cone complete.
Let us verify (d). Given 𝑐∈𝑌 with 𝜃≪_𝐾𝑐, and let {𝑧_𝑛} be a TVS-cone Cauchy sequence in (𝐶,𝑝). Then there exists 𝜈₁∈ℕ such that 𝑝(𝑧_𝑛,𝑧_𝑚)≪_𝐾(1/2)𝑐 for all 𝑚, 𝑛≥𝜈₁. Since 𝐶 is TVS-cone compact, there exists a subsequence {𝑧_𝑛𝑡} of {𝑧_𝑛}, and 𝑧∈𝐶 such that 𝑧−−−−→_𝑛𝑡^cone𝑧 as 𝑡→∞. For 𝑐, there exists 𝜈₂∈ℕ such that 𝑝(𝑧_𝑛𝑡,𝑧)≪_𝐾(1/2)𝑐 for all 𝑡≥𝜈₂. Let 𝜈=max{𝜈₁,𝜈₂}. For any 𝑛≥𝜈, since 𝑝𝑧_𝑛,𝑧⪷_𝐾𝑝𝑧_𝑛,𝑧_𝑛𝜈+𝑝𝑧_𝑛𝜈,𝑧,𝑝𝑧_𝑛,𝑧_𝑛𝜈+𝑝𝑧_𝑛𝜈,𝑧≪_𝐾𝑐, (2.8) by (iii) of Lemma 2.5, we have 𝑝(𝑧_𝑛,𝑧)≪_𝐾𝑐. So {𝑥_𝑛} is TVS-cone convergent to 𝑥. Therefore, (𝐶,𝑝) is TVS-cone complete.
The conclusion (e) is obvious. The proof is completed.

Let 𝐶 be a subset of a TVS-cone metric space (𝑋,𝑝). We denote diam_𝑑𝑝(𝐶)𝑑∶=0,if𝐶=∅,sup_𝑝𝑥,𝑦∶𝑥,𝑦∈𝐶,if𝐶≠∅. (2.9)

It is obvious that 𝐴⊆𝐵 in (𝑋,𝑝) implies diam_𝑑𝑝(𝐴)≤diam_𝑑𝑝(𝐵).

Now, we first introduce the concepts of fitting nest.

Definition 2.8. A sequence {𝐴_𝑛}_𝑛∈ℕ of subsets of a TVS-cone metric space (𝑋,𝑝) is said to be a fitting nest if it satisfies the following properties: (FN1)𝐴_𝑛+1⊆𝐴_𝑛 for each 𝑛∈ℕ,(FN2)for any 𝑐∈𝑌 with 𝜃≪_𝐾𝑐, there exists 𝑛_𝑐∈ℕ such that 𝑝(𝑥,𝑦)≪_𝐾𝑐 for all 𝑥,𝑦∈𝐴_𝑛𝑐.

Remark 2.9. (a) It is easy to observe that if 𝑌=ℝ, 𝐾=[0,∞)⊂ℝ, and 𝑒=1, then 𝑝 is a metric, and Assumption (FN2) is equivalent to lim_𝑛→∞diam(𝐴_𝑛)=0 if Assumption (FN1) holds, where diam(𝐴_𝑛) is the diameter of 𝐴_𝑛. Indeed, “lim_𝑛→∞diam(𝐴_𝑛)=0⇒ (FN2)” is obvious. Conversely, if (FN2) holds, then, by (FN1), for any 𝜀>0, there exists 𝑛_𝜀∈ℕ such that diam(𝐴_𝑛)≤𝜀 for all 𝑛≥𝑛_𝜀. This show that lim_𝑛→∞diam(𝐴_𝑛)=0.
(b) Let (𝑋,𝑑) be a metric space. Then a sequence {𝐴_𝑛}_𝑛∈ℕ in (𝑋,𝑑) is a fitting nest ⇔𝐴_𝑛+1⊆𝐴_𝑛 for each 𝑛∈ℕ and lim_𝑛→∞diam(𝐴_𝑛)=0.

The following intersection theorem in TVS-cone metric spaces is one of the main results of this paper.

Theorem 2.10. Let {𝐴_𝑛} be a fitting nest in a TVS-cone metric space (𝑋,𝑝). Then the following statements hold. (a)lim_𝑛→∞diam_𝑑𝑝(𝐴_𝑛)=0.(b)If 𝑋 is TVS-cone complete and 𝐴_𝑛 is TVS-cone closed in (𝑋,𝑝) for all 𝑛∈ℕ, then ⋂^∞_𝑛=1𝐴_𝑛 contains precisely one point.

Proof. (a) Let 𝜀>0 be given. Then 𝜃≪_𝐾𝜀𝑒. By (FN2) and (iv) of Lemma 2.1, there exists 𝑛₀∈ℕ such that 𝑝𝑥,𝑦≪_𝐾𝜀𝑒⟺𝑑_𝑝𝑥,𝑦=𝜉_𝑒∘𝑝𝑥,𝑦<𝜀, (2.10) for all 𝑥,𝑦∈𝐴_𝑛0, which implies diam_𝑑𝑝(𝐴_𝑛0)≤𝜀. By (FN1), we obtain diam_𝑑𝑝𝐴_𝑛≤diam_𝑑𝑝𝐴_𝑛0≤𝜀,∀𝑛≥𝑛₀. (2.11) Hence, lim_𝑛→∞diam_𝑑𝑝(𝐴_𝑛)=0.
(b) Given 𝑐∈𝑌 with 𝜃≪_𝐾𝑐. By (FN2), there exists 𝑛_𝑐∈ℕ such that 𝑝(𝑎,𝑏)≪_𝐾𝑐 for all 𝑎,𝑏∈𝐴_𝑛𝑐. For each 𝑛∈ℕ, choose 𝑥_𝑛∈𝐴_𝑛. Then, for 𝑚,𝑛∈ℕ with 𝑚≥𝑛≥𝑛_𝑐; since 𝑥_𝑚∈𝐴_𝑚⊆𝐴_𝑛⊆𝐴_𝑛𝑐 from (FN1), we have 𝑝𝑥_𝑚,𝑥_𝑛≪_𝐾𝑐. (2.12) Hence, {𝑥_𝑛} is a TVS-cone Cauchy sequence in (𝑋,𝑝). By the TVS-cone completeness of (𝑋,𝑝), there exists 𝑤∈𝑋, such that {𝑥_𝑛}TVS-cone converges to 𝑤. For any 𝑛∈ℕ, from the TVS-cone closedness of 𝐴_𝑛 and 𝑥−−−−→_𝑚^cone𝑤 as 𝑚→∞, we have 𝑤∈tvsc-cl𝐴_𝑛=𝐴_𝑛. (2.13) So 𝑤∈⋂^∞_𝑛=1𝐴_𝑛, and hence ⋂^∞_𝑛=1𝐴_𝑛≠∅. Finally, we claim ⋂^∞_𝑛=1𝐴_𝑛={𝑤}. For each 𝑧∈⋂^∞_𝑛=1𝐴_𝑛, applying (a), we have 𝑑_𝑝(𝑧,𝑤)≤diam_𝑑𝑝𝐴_𝑛⟶0as𝑛⟶∞. (2.14) Hence, 𝑑_𝑝(𝑧,𝑤)=0 or, equivalency, 𝑧=𝑤, which gives the required result (b).

Theorem 2.11. Let (𝑋,𝑝) be a TVS-cone complete metric space. Then there exists a nonempty proper subset ℳ of 𝑋, such that ℳ contains infinite points of 𝑋, and (ℳ,𝑑_𝑝) is a complete metric space, where 𝑑_𝑝∶=𝜉_𝑒∘𝑝.

Proof. Let {𝐴_𝑛} be a fitting nest in (𝑋,𝑝). Following the same argument as in the proof of (b), we can obtain a sequence {𝑥_𝑛} satisfying (1)𝑥_𝑛∈𝐴_𝑛,(2){𝑥_𝑛} is a TVS-cone Cauchy sequence in (𝑋,𝑝),(3){𝑥_𝑛}TVS-cone converges to some point 𝑤 in 𝑋.
Applying Theorem 1.5 with (2) and (3), we know that {𝑥_𝑛} is a Cauchy sequence in (𝑋,𝑑_𝑝), and 𝑑_𝑝(𝑥_𝑛,𝑤)→0 or 𝑥−−→_𝑛^𝑑𝑝𝑤 as 𝑛→∞. Let ℳ={𝑥_𝑛}_𝑛∈ℕ∪{𝑤}. Therefore, (ℳ,𝑑_𝑝) is a complete metric space.

The celebrated Cantor intersection theorem [2] in metric spaces can be proved by Theorem 2.10 and Remark 2.9.

Corollary 2.12 (Cantor). Let (𝑋,𝑑) be a metric space, and let {𝐴_𝑛}_𝑛∈ℕ be a sequence of closed subsets of 𝑋 satisfying 𝐴_𝑛+1⊆𝐴_𝑛 for each 𝑛∈ℕ and lim_𝑛→∞diam(𝐴_𝑛)=0. Then ⋂^∞_𝑛=1𝐴_𝑛 contains precisely one point.

The following existence theorems relate with critical point and common fuzzy fixed point for a nonlinear dynamical system in TVS-cone complete metric spaces or complete metric spaces.

Theorem 2.13. Let (𝑋,𝑝) be a TVS-cone complete metric space, let 𝑔∶𝑋→𝑋 a map, and let Γ∶𝑋→2^𝑋 be a multivalued map with nonempty values. Let 𝐼 be any index set. For each 𝑖∈𝐼, let 𝐹_𝑖 be a fuzzy map on 𝑋. Suppose that the following conditions are satisfied. (H1)For each 𝑥∈𝑋, Γ(𝑥) is TVS-cone closed in (𝑋,𝑝).(H2)𝑥,𝑦∈𝑋 with 𝑦∈Γ(𝑥) implies 𝑔(𝑦)∈Γ(𝑥) and Γ(𝑦)⊆Γ(𝑥).(H3)For any {𝑥_𝑛}⊂𝑋 with 𝑔(𝑥_𝑛+1)∈Γ(𝑥_𝑛) for each 𝑛∈ℕ, it satisfies the following:For any 𝑐∈𝑌 with 𝜃≪_𝐾𝑐, there exists 𝑛_𝑐∈ℕ such that 𝑝(𝑠,𝑡)≪_𝐾𝑐 for all 𝑠,𝑡∈Γ(𝑥_𝑛𝑐).(H4)For any (𝑖,𝑥)∈𝐼×𝑋, there exists 𝑦_(𝑖,𝑥)∈Γ(𝑥) such that 𝐹_𝑖(𝑥,𝑦_(𝑖,𝑥))=1. Then there exists 𝑣∈𝑋 such that(a)𝐹_𝑖(𝑣,𝑣)=1 for all 𝑖∈𝐼.(b)Γ(𝑣)={𝑔(𝑣)}={𝑣}.

Proof. Let 𝑢∈𝑋 be given. Define a sequence {𝑥_𝑛} by 𝑥₁=𝑢 and 𝑥_𝑛+1∈Γ(𝑥_𝑛) for 𝑛∈ℕ. Hence, 𝑔(𝑥_𝑛+1)∈Γ(𝑥_𝑛) for all 𝑛∈ℕ from (H2). For each 𝑛∈ℕ, let 𝐴_𝑛=Γ(𝑥_𝑛). By (H2) and (H3), we know that {𝐴_𝑛} is a fitting nest in (𝑋,𝑝). By (H1), 𝐴_𝑛 is TVS-cone closed in (𝑋,𝑝) for all 𝑛∈ℕ. Applying Theorem 2.10, there exists 𝑣∈𝑋 such that ⋂^∞_𝑛=1Γ(𝑥_𝑛)=⋂^∞_𝑛=1𝐴_𝑛={𝑣}. Since 𝑣∈Γ(𝑥_𝑛) for all 𝑛∈ℕ, by (H2), we obtain ∅≠Γ(𝑣)⊆^∞_𝑛=1Γ𝑥_𝑛={𝑣}, (2.15) which implies Γ(𝑣)={𝑣}. For each 𝑖∈𝐼, by (H4), 𝐹_𝑖(𝑣,𝑣)=1. By (H2) again, we have 𝑔(𝑣)∈Γ(𝑣)={𝑣}. Therefore, Γ(𝑣)={𝑔(𝑣)}={𝑣}. The proof is completed.

Remark 2.14. Let 𝑌=ℝ, let 𝐾=[0,∞), and let 𝑒=1, then 𝑝 is a metric, and Assumption (H3) in Theorem 2.13 is equivalent to
(H3)_ℝ for any {𝑥_𝑛}⊂𝑋 with 𝑔(𝑥_𝑛+1)∈Γ(𝑥_𝑛) for each 𝑛∈ℕ, we have lim_𝑛→∞diam(Γ(𝑥_𝑛))=0.

The following critical point theorems are immediate from Theorem 2.13.

Theorem 2.15. Let (𝑋,𝑑) be a complete metric space, let 𝑔∶𝑋→𝑋 be a map and let Γ∶𝑋→2^𝑋 be a multivalued map with nonempty values. Let 𝐼 be any index set. For each 𝑖∈𝐼, let 𝐹_𝑖 be a fuzzy map on 𝑋. Suppose that the following conditions are satisfied. (H1)_ℝFor each 𝑥∈𝑋, Γ(𝑥) is closed.(H2)𝑥,𝑦∈𝑋 with 𝑦∈Γ(𝑥) implies 𝑔(𝑦)∈Γ(𝑥) and Γ(𝑦)⊆Γ(𝑥).(H3)_ℝFor any {𝑥_𝑛}⊂𝑋 with 𝑔(𝑥_𝑛+1)∈Γ(𝑥_𝑛) for each 𝑛∈ℕ, we have lim_𝑛→∞diam(Γ(𝑥_𝑛))=0.(H4)For any (𝑖,𝑥)∈𝐼×𝑋, there exists 𝑦_(𝑖,𝑥)∈Γ(𝑥) such that 𝐹_𝑖(𝑥,𝑦_(𝑖,𝑥))=1. Then there exists 𝑣∈𝑋 such that (a)𝐹_𝑖(𝑣,𝑣)=1 for all 𝑖∈𝐼.(b)Γ(𝑣)={𝑔(𝑣)}={𝑣}.

Corollary 2.16. Let (𝑋,𝑝) be a TVS- cone complete metric space, and let Γ∶𝑋→2^𝑋 be a multivalued map with nonempty values. Suppose that the following conditions are satisfied. (i)For each 𝑥∈𝑋, Γ(𝑥) is TVS-cone closed in (𝑋,𝑝).(ii)𝑥,𝑦∈𝑋 with 𝑦∈Γ(𝑥) implies Γ(𝑦)⊆Γ(𝑥).(iii)For any {𝑥_𝑛}⊂𝑋 with 𝑥_𝑛+1∈Γ(𝑥_𝑛), for each 𝑛∈ℕ, it satisfies the following:For any 𝑐∈𝑌 with 𝜃≪_𝐾𝑐, there exists 𝑛_𝑐∈ℕ such that 𝑝(𝑠,𝑡)≪_𝐾𝑐 for all 𝑠,𝑡∈Γ(𝑥_𝑛𝑐).
Then there exists 𝑣∈𝑋 such that Γ(𝑣)={𝑣}.

Proof. Let 𝐹 be a fuzzy map on 𝑋 defined by 𝐹(𝑥,𝑦)=1 for all 𝑥, 𝑦∈𝑋, and let 𝑔≡id be an identity map. Therefore, the conclusion follows from Theorem 2.13.

Corollary 2.17. Let (𝑋,𝑑) be a complete metric space, and let Γ∶𝑋→2^𝑋 be a multivalued map with nonempty values. Suppose that the following conditions are satisfied. (i)For each 𝑥∈𝑋, Γ(𝑥) is closed.(ii)𝑥,𝑦∈𝑋 with 𝑦∈Γ(𝑥) implies Γ(𝑦)⊆Γ(𝑥).(iii)For any {𝑥_𝑛}⊂𝑋 with 𝑥_𝑛+1∈Γ(𝑥_𝑛), for each 𝑛∈ℕ, we have lim_𝑛→∞diam(Γ(𝑥_𝑛))=0.
Then there exists 𝑣∈𝑋 such that Γ(𝑣)={𝑣}.

Theorem 2.18. Let (𝑋,𝑑) be a complete metric space, and let Γ∶𝑋→2^𝑋 be a multivalued map with nonempty values such that 𝑥,𝑦∈𝑋 with 𝑦∈Γ(𝑥) implies Γ(𝑦)⊆Γ(𝑥). Then the following statements holds. (1)If a sequence {𝑥_𝑛} in 𝑋 satisfies 𝑥_𝑛+1∈Γ(𝑥_𝑛) for each 𝑛∈ℕ and lim_𝑛→∞diam(Γ(𝑥_𝑛))=0, then lim_𝑛→∞𝑑(𝑥_𝑛,𝑥_𝑛+1)=0.(2)If (𝑋,𝑑) has the following property (𝒫): forany𝑥_𝑛⊂𝑋with𝑥_𝑛+1∈Γ𝑥_𝑛foreach𝑛∈ℕ,wehavelim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=0,( (𝒫)) then there exists a sequence {𝑧_𝑛} in 𝑋 satisfying 𝑧_𝑛+1∈Γ(𝑧_𝑛) for each 𝑛∈ℕ and lim_𝑛→∞diam(Γ(𝑧_𝑛))=0.

Proof. (1) Let {𝑥_𝑛} in 𝑋 with 𝑥_𝑛+1∈Γ(𝑥_𝑛) for each 𝑛∈ℕ, and let lim_𝑛→∞diam(Γ(𝑥_𝑛))=0. For any 𝑛∈ℕ, by our hypothesis, Γ(𝑥_𝑛+1)⊆Γ(𝑥_𝑛), and; hence, 𝑥_𝑚+1∈Γ𝑥_𝑚⊆Γ𝑥_𝑛,∀𝑚∈ℕwith𝑚≥𝑛. (2.16) So 𝑑(𝑥_𝑛+1,𝑥_𝑛+2)≤diam(Γ(𝑥_𝑛)) for each 𝑛∈ℕ. Since lim_𝑛→∞diam(Γ(𝑥_𝑛))=0, we have lim_𝑛→∞𝑑𝑥_𝑛+1,𝑥_𝑛+2=lim_𝑛→∞diamΓ𝑥_𝑛=0. (2.17) That is, lim_𝑛→∞𝑑(𝑥_𝑛,𝑥_𝑛+1)=0.
(2) Suppose that (𝑋,𝑑) has the property (𝒫). Define a function 𝜏∶𝑋→ℝ by 𝜏(𝑥)=sup_{𝑦∈Γ(𝑥)}𝑑𝑥,𝑦. (2.18) We first note that 𝜏(𝑤)<∞ for some 𝑤∈𝑋. Indeed, on the contrary, assume that 𝜏(𝑥)=∞ for all 𝑥∈𝑋. Let 𝑢∈𝑋 be given. Set 𝑢₁=𝑢. Since 𝜏(𝑢₁)=∞, 𝜏(𝑢₁)=sup_{𝑦∈Γ(𝑢1)}𝑑(𝑢₁,𝑦)>1, and then there exists 𝑢₂∈Γ(𝑢₁) such that 𝑑(𝑢₁,𝑢₂)>1. Since 𝜏(𝑢₂)>2, there exists 𝑢₃∈Γ(𝑢₂) such that 𝑑(𝑢₂,𝑢₃)>2. Continuing in the process, we can obtain a sequence {𝑢_𝑛}⊂𝑋, such that, for each 𝑛∈ℕ,
(𝒦₁)𝑢_𝑛+1∈Γ(𝑢_𝑛),(𝒦₂)𝑑(𝑢_𝑛,𝑢_𝑛+1)>𝑛. By (𝒫), we have lim_𝑛→∞𝑑(𝑢_𝑛,𝑢_𝑛+1)=0. On the other hand, by (𝒦₂), we also obtain lim_𝑛→∞𝑑(𝑢_𝑛,𝑢_𝑛+1)=∞, a contradiction. Therefore, there exists 𝑤∈𝑋 such that 𝜏(𝑤)<∞. Let 𝑧₁=𝑤. Since ∞>𝜏𝑧₁=sup_{𝑦∈Γ(𝑧1)}𝑑𝑧₁≥1,𝑦2diamΓ𝑧₁, (2.19) we have 0≤diam(Γ(𝑧₁))<∞, and there exists 𝑧₂∈Γ(𝑧₁) such that 𝑑𝑧₁,𝑧₂>12diamΓ𝑧₁−12. (2.20) Since Γ(𝑧₂)⊆Γ(𝑧₁), we have 𝜏(𝑧₂)≤𝜏(𝑧₁)<∞ and 0≤diam(Γ(𝑧₂))<∞. So there exists 𝑧₃∈Γ(𝑧₂) such that 𝑑𝑧₂,𝑧₃>12diamΓ𝑧₂−12². (2.21) Continuing in this way, we can construct a sequence {𝑧_𝑛} in 𝑋 satisfying, for each 𝑛∈ℕ, (𝒦₃)𝑧_𝑛+1∈Γ(𝑧_𝑛),(𝒦₄)𝑑(𝑧_𝑛,𝑧_𝑛+1)>(1/2)diam(Γ(𝑧_𝑛))−(1/2^𝑛). From (𝒫), we have lim_𝑛→∞𝑑(𝑧_𝑛,𝑧_𝑛+1)=0. By (𝒦₄), it follows that lim_𝑛→∞diam(Γ(𝑧_𝑛))=0.

Remark 2.19. In general, under the same assumptions of Theorem 2.18, lim_𝑛→∞𝑑(𝑥_𝑛,𝑥_𝑛+1)=0 does not always imply lim_𝑛→∞diam(Γ(𝑥_𝑛))=0. For example, let 𝑋=[0,5] with the metric 𝑑(𝑥,𝑦)=|𝑥−𝑦|. Then (𝑋,𝑑) is a complete metric space. For each 𝑥∈𝑋, define Γ∶𝑋→2^𝑋 by Γ(𝑥)=[𝑥,5]. Thus, 𝑥, 𝑦∈𝑋 with 𝑦∈Γ(𝑥) implies Γ(𝑦)⊆Γ(𝑥). Choose {𝑥_𝑛} in 𝑋 with 𝑥_𝑛=2; for all 𝑛∈ℕ, we have lim_𝑛→∞diam(Γ(𝑥_𝑛))=3≠0 while lim_𝑛→∞𝑑(𝑥_𝑛,𝑥_𝑛+1)=0.

The following result is also a generalized Dancš-Hegedüs-Medvegyev's principle with common fuzzy fixed point. Notice that we do not assume 𝑥∈Γ(𝑥) for all 𝑥∈𝑋.

Theorem 2.20. Let (𝑋,𝑑) be a complete metric space. Let Γ∶𝑋→2^𝑋 be a multivalued map with nonempty values. Let 𝐼 be any index set. For each 𝑖∈𝐼, let 𝐹_𝑖 be a fuzzy map on 𝑋. Suppose that the following conditions are satisfied. (D1)For each 𝑥∈𝑋, Γ(𝑥) is closed.(D2)𝑥, 𝑦∈𝑋 with 𝑦∈Γ(𝑥) implies Γ(𝑦)⊆Γ(𝑥).(D3)(Property  (𝒫)) for any {𝑥_𝑛}⊂𝑋 with 𝑥_𝑛+1∈Γ(𝑥_𝑛) for each 𝑛∈ℕ, we have lim_𝑛→∞𝑑(𝑥_𝑛,𝑥_𝑛+1)=0.(D4)For any (𝑖,𝑥)∈𝐼×𝑋, there exists 𝑦_(𝑖,𝑥)∈Γ(𝑥) such that 𝐹_𝑖(𝑥,𝑦_(𝑖,𝑥))=1. Then there exists 𝑣∈𝑋 such that (a)𝐹_𝑖(𝑣,𝑣)=1 for all 𝑖∈𝐼,(b)Γ(𝑣)={𝑣}.

Proof. By conclusion (2) of Theorem 2.18, there exists a sequence {𝑧_𝑛} in 𝑋 satisfying 𝑧_𝑛+1∈Γ(𝑧_𝑛) for each 𝑛∈ℕ and lim_𝑛→∞diam(Γ(𝑧_𝑛))=0. For each 𝑛∈ℕ, let 𝐴_𝑛=Γ(𝑧_𝑛). By (D2) and lim_𝑛→∞diam(Γ(𝑧_𝑛))=0, we see that {𝐴_𝑛} is a fitting nest in (𝑋,𝑑). Applying Theorem 2.10, there exists 𝑣∈𝑋 such that ⋂^∞_𝑛=1Γ(𝑧_𝑛)=⋂^∞_𝑛=1𝐴_𝑛={𝑣}. Since 𝑣∈Γ(𝑧_𝑛) for all 𝑛∈ℕ, by (D2) again, we obtain ∅≠Γ(𝑣)⊆^∞_𝑛=1Γ𝑧_𝑛={𝑣}, (2.22) which implies Γ(𝑣)={𝑣}. For each 𝑖∈𝐼, by (D4), 𝐹_𝑖(𝑣,𝑣)=1. The proof is completed.

The following existence theorem relate with common fixed point for multivalued maps and critical point for a nonlinear dynamical system in TVS-cone complete metric spaces.

Theorem 2.21. Let (𝑋,𝑝), 𝑔, and Γ be the same as in Theorem 2.13. Assume that conditions (H1), (H2), and (H3) in Theorem 2.13 hold. Let 𝐼 be any index set. For each 𝑖∈𝐼, let 𝑇_𝑖∶𝑋→2^𝑋 be a multivalued map with nonempty values. Suppose that, for each (𝑖,𝑥)∈𝐼×𝑋, there exists 𝑦_(𝑖,𝑥)∈𝑇_𝑖(𝑥)∩Γ(𝑥). Then there exists 𝑣∈𝑋 such that (a)𝑣 is a common fixed point for the family {𝑇_𝑖}_𝑖∈𝐼 (i.e., 𝑣∈𝑇_𝑖(𝑣) for all 𝑖∈𝐼),(b)Γ(𝑣)={𝑔(𝑣)}={𝑣}.

Proof. For each 𝑖∈𝐼, define a fuzzy map 𝐹_𝑖 on 𝑋 by 𝐹_𝑖𝑥,𝑦=𝜒_{𝑇𝑖(𝑥)}𝑦, (2.23) where 𝜒_𝐴 is the characteristic function for an arbitrary set 𝐴⊂𝑋. Note that 𝑦∈𝑇_𝑖(𝑥)⇔𝐹_𝑖(𝑥,𝑦)=1 for 𝑖∈𝐼. Then, for any (𝑖,𝑥)∈𝐼×𝑋, there exists 𝑦_𝑥∈Γ(𝑥) such that 𝐹_𝑖(𝑥,𝑦_𝑥)=1. So (H4) in Theorem 2.13 holds. Therefore, the result follows from Theorem 2.13.

Remark 2.22. (a) Theorems 2.20 and 2.21 all generalize and improve the primitive Dancš-Hegedüs-Medvegyev's principle.
(b) Corollary 2.17 is a special case of Theorem 2.20 or Theorem 2.21.

The following result is a special case of [32, Theorem  4.1], but it can also be proved by applying Theorem 2.20 (please follow a similar argument as in the proof of [32, Theorem  4.1]).

Theorem 2.23. Let (𝑋,𝑑) be a complete metric space, let 𝑓∶𝑋→(−∞,∞] be a proper l.s.c. and bounded from below function, and let 𝜑∶(−∞,∞]→(0,∞) be a nondecreasing function. Let 𝐼 be any index set. For each 𝑖∈𝐼, let 𝐹_𝑖 be a fuzzy map on 𝑋. Suppose that, for each (𝑖,𝑥)∈𝐼×𝑋, there exists 𝑦_(𝑖,𝑥)∈𝑋 such that 𝐹_𝑖(𝑥,𝑦_(𝑖,𝑥))=1 and 𝑑(𝑥,𝑦_(𝑖,𝑥))≤𝜑(𝑓(𝑥))(𝑓(𝑥)−𝑓(𝑦_(𝑖,𝑥))). Then, for each 𝑢∈𝑋 with 𝑓(𝑢)<∞, there exists 𝑣∈𝑋 such that (a)𝑑(𝑢,𝑣)≤𝜑(𝑓(𝑢))(𝑓(𝑢)−𝑓(𝑣)),(b)𝑑(𝑣,𝑥)>𝜑(𝑓(𝑣))(𝑓(𝑣)−𝑓(𝑥)) for all 𝑥∈𝑋 with 𝑥≠𝑣,(c)𝐹_𝑖(𝑣,𝑣)=1 for all 𝑖∈𝐼. Moreover, if further assume that
(H) for any 𝑥∈𝑋 with 𝑓(𝑥)>inf_𝑧∈𝑋𝑓(𝑧), there exists 𝑦∈𝑋 with 𝑦≠𝑥 such that𝑑(𝑥,𝑦)≤𝜑(𝑓(𝑥))(𝑓(𝑥)−𝑓(𝑦)),then 𝑓(𝑣)=inf_𝑧∈𝑋𝑓(𝑧).

By using Theorem 2.23, one can immediately obtain the following existence theorem related to generalized Ekeland's variational principle, generalized Takahashi's nonconvex minimization theorem, and generalized Caristi's common fixed point theorem for multivalued maps.

Theorem 2.24. Let (𝑋,𝑑), 𝑓, and 𝜑 be be the same as in Theorem 2.23. Let 𝐼 be any index set. For each 𝑖∈𝐼, let 𝑇_𝑖∶𝑋→2^𝑋 be a multivalued map with nonempty values such that, for each (𝑖,𝑥)∈𝐼×𝑋, there exists 𝑦_(𝑖,𝑥)∈𝑇_𝑖(𝑥) such that 𝑑(𝑥,𝑦_(𝑖,𝑥))≤𝜑(𝑓(𝑥))(𝑓(𝑥)−𝑓(𝑦_(𝑖,𝑥))). Then, for each 𝑢∈𝑋 with 𝑓(𝑢)<∞, there exists 𝑣∈𝑋 such that (a)𝑑(𝑢,𝑣)≤𝜑(𝑓(𝑢))(𝑓(𝑢)−𝑓(𝑣)).(b)𝑑(𝑣,𝑥)>𝜑(𝑓(𝑣))(𝑓(𝑣)−𝑓(𝑥)) for all 𝑥∈𝑋 with 𝑥≠𝑣.(c)𝑣 is a common fixed point for the family {𝑇_𝑖}_𝑖∈𝐼. Moreover, if further assume that
(H) for any 𝑥∈𝑋 with 𝑓(𝑥)>inf_𝑧∈𝑋𝑓(𝑧), there exists 𝑦∈𝑋 with 𝑦≠𝑥 such that 𝑑(𝑥,𝑦)≤𝜑(𝑓(𝑥))(𝑓(𝑥)−𝑓(𝑦)), then 𝑓(𝑣)=inf_𝑧∈𝑋𝑓(𝑧).

3. Fixed Point Theorems in Cone Metric Spaces

In this section, motivated by the recent results of Abbas and Rhoades [21], we will present some generalizations of those in TVS-cone complete metric spaces.

Theorem 3.1. Let (𝑋,𝑝) be a TVS-cone complete metric space. Suppose that 𝑇,𝑆∶𝑋→𝑋 are two self-maps of 𝑋 satisfying 𝑝𝑇𝑥,𝑆𝑦⪷_𝐾𝛼𝑝𝑥,𝑦+𝛽𝑝(𝑥,𝑇𝑥)+𝑝𝑦,𝑆𝑦+𝛾𝑝𝑥,𝑆𝑦+𝑝𝑦,𝑇𝑥, (3.1) for all 𝑥, 𝑦∈𝑋, where 𝛼, 𝛽, 𝛾≥0 and 𝛼+2𝛽+2𝛾<1. Then the following statements hold:(a)There exists a nonempty proper subset 𝑊 of 𝑋, such that 𝑊 contains infinite points of 𝑋 and (𝑊,𝑑_𝑝) is a complete metric space, where 𝑑_𝑝∶=𝜉_𝑒∘𝑝.(b)𝑇 and 𝑆 have a unique common fixed point in 𝑋 (in fact, the unique common fixed point of 𝑇 and 𝑆 belongs to 𝑊). Moreover, for each 𝑥₀∈𝑋, the mixed iterative sequence {𝑥_𝑛}_{𝑛∈ℕ∪{0}}, defined by 𝑥_2𝑛+1=𝑇𝑥_2𝑛 and 𝑥_2𝑛+2=𝑆𝑥_2𝑛+1 for 𝑛∈ℕ∪{0}, TVS-cone converges to the common fixed point.(c)Any fixed point of 𝑇 is a fixed point of 𝑆, and conversely.

Proof. Since 𝑌 is a locally convex Hausdorff’s t.v.s. with its zero vector 𝜃, let 𝜏 denote the topology of 𝑌, and let 𝒰_𝜏 be the base at 𝜃 consisting of all absolutely convex neighborhood of 𝜃. Let ℒ=ℓ∶ℓbeaMinkowskifunctionalof𝑈for𝑈∈𝒰_𝜏. (3.2) Then ℒ is a family of seminorms on 𝑌. For each ℓ∈ℒ, let 𝑉(ℓ)=𝑦∈𝑌∶ℓ𝑦<1, (3.3) and let 𝒰_ℒ=𝑈∶𝑈=𝑟₁𝑉ℓ₁∩𝑟₂𝑉ℓ₂∩⋯∩𝑟_𝑛𝑉ℓ_𝑛,𝑟_𝑘>0,ℓ_𝑘∈ℒ,1≤𝑘≤𝑛,𝑛∈ℕ. (3.4) Then 𝒰_ℒ is a base at 𝜃, and the topology Γ_ℒ generated by 𝒰_ℒ is the weakest topology for 𝑌 such that all seminorms in ℒ are continuous and 𝜏=Γ_ℒ. Moreover, given any neighborhood 𝒪_𝜃 of 𝜃, there exists 𝑈∈𝒰_ℒ such that 𝜃∈𝑈⊂𝒪_𝜃 (see, e.g., [34, Theorem  12.4 in II.12, Page 113] or the proofs of [28, Theorem  3.1] and [29, Theorem  2.1]).
Let 𝑥₀∈𝑋 be given. First, from our hypothesis, we have 0≤𝛼+𝛽+𝛾<1−𝛽−𝛾. (3.5) If 𝑥₀ is a fixed point of 𝑇, then, by using (3.1), 𝑝𝑥₀,𝑆𝑥₀=𝑝𝑇𝑥₀,𝑆𝑥₀⪷_𝐾𝛼𝑝𝑥₀,𝑥₀+𝛽𝑝𝑥₀,𝑇𝑥₀+𝑝𝑥₀,𝑆𝑥₀+𝛾𝑝𝑥₀,𝑆𝑥₀+𝑝𝑥₀,𝑇𝑥₀, (3.6) implies that 𝑝𝑥₀,𝑆𝑥₀⪷_𝐾𝛽+𝛾𝑝𝑥₀,𝑆𝑥₀, (3.7) or 1−𝛽−𝛾𝑝𝑥₀,𝑆𝑥₀∈−𝐾. (3.8) Since 1−𝛽−𝛾>0 and 𝐾 is pointed, we have 𝑝(𝑥₀,𝑆𝑥₀)∈𝐾∩(−𝐾)={𝜃}. So 𝑝(𝑥₀,𝑆𝑥₀)=𝜃, and; hence, 𝑆𝑥₀=𝑥₀=𝑇𝑥₀; that is, 𝑥₀ is a common fixed point of 𝑇 and 𝑆. Otherwise, if 𝑇𝑥₀≠𝑥₀, we will define the mixed iterative sequence {𝑥_𝑛} by 𝑥_2𝑛+1=𝑇𝑥_2𝑛 and 𝑥_2𝑛+2=𝑆𝑥_2𝑛+1 for 𝑛∈ℕ∪{0}. Then 𝑝(𝑥₁,𝑥₀)≠𝜃. We claim that {𝑥_𝑛} is a TVS-cone Cauchy’s sequence in (𝑋,𝑝). Let 𝜆=(𝛼+𝛽+𝛾)/(1−𝛽−𝛾).
By (3.5), we know 𝜆∈[0,1). For each 𝑛∈ℕ∪{0}, we have 𝑝𝑥_2𝑛+1,𝑥_2𝑛+2=𝑝𝑇𝑥_2𝑛,𝑆𝑥_2𝑛+1⪷_𝐾𝛼𝑝𝑥_2𝑛,𝑥_2𝑛+1+𝛽𝑝𝑥_2𝑛,𝑇𝑥_2𝑛+𝑝𝑥_2𝑛+1,𝑆𝑥_2𝑛+1+𝛾𝑑𝑥_2𝑛,𝑆𝑥_2𝑛+1+𝑑𝑥_2𝑛+1,𝑇𝑥_2𝑛⪷_𝐾𝛼+𝛽+𝛾𝑝𝑥_2𝑛,𝑥_2𝑛+1+𝛽+𝛾𝑝𝑥_2𝑛+1,𝑥_2𝑛+2, (3.9) which implies that 𝑝𝑥_2𝑛+1,𝑥_2𝑛+2⪷_𝐾𝜆𝑝𝑥_2𝑛,𝑥_2𝑛+1. (3.10) Similarly, we also obtain 𝑝𝑥_2𝑛+2,𝑥_2𝑛+3⪷_𝐾𝜆𝑝𝑥_2𝑛+1,𝑥_2𝑛+2. (3.11) Hence, for each 𝑛∈ℕ, 𝑝𝑥_𝑛+1,𝑥_𝑛⪷_𝐾𝜆𝑝𝑥_𝑛,𝑥_𝑛−1⪷_𝐾⋯⪷_𝐾𝜆^𝑛𝑝𝑥₁,𝑥₀. (3.12) Therefore, for 𝑚,𝑛∈ℕ with 𝑚>𝑛, 𝑝𝑥_𝑚,𝑥_𝑛⪷_𝐾^𝑚−1_𝑗=𝑛𝑝𝑥_𝑗+1,𝑥_𝑗⪷_𝐾𝜆^𝑛𝑝𝑥1−𝜆¹,𝑥₀. (3.13) Given 𝑐∈𝑌 with 𝜃≪_𝐾𝑐 (i.e., 𝑐∈int𝐾=int(int𝐾)), there exists a neighborhood 𝑁_𝜃 of 𝜃 such that 𝑐+𝑁_𝜃⊆int𝐾. Therefore, there exists 𝑈_𝑐∈𝒰_ℒ with 𝑈_𝑐⊆𝑁_𝜃 such that 𝑐+𝑈_𝑐⊆𝑐+𝑁_𝜃⊆int𝐾, where 𝑈_𝑐=𝑟₁𝑉ℓ₁∩𝑟₂𝑉ℓ₂∩⋯∩𝑟_𝑠𝑉ℓ_𝑠, (3.14) for some 𝑟_𝑖>0, ℓ_𝑖∈ℒ and 1≤𝑖≤𝑠. Let 𝛿=min{𝑟_𝑖∶1≤𝑖≤𝑠}>0, and let 𝜌=max{ℓ_𝑖(𝑝(𝑥₁,𝑥₀))∶1≤𝑖≤𝑠}. If 𝜌=0, since each ℓ_𝑖 is a seminorm, we have ℓ_𝑖(𝑝(𝑥₁,𝑥₀))=0 and ℓ_𝑖−𝜆^𝑛𝑝𝑥1−𝜆¹,𝑥₀=𝜆^𝑛ℓ1−𝜆^𝑖𝑝𝑥₁,𝑥₀=0<𝑟_𝑖, (3.15) for all 1≤𝑖≤𝑠 and all 𝑛∈ℕ. If 𝜌>0, since 𝜆∈[0,1), lim_𝑛→∞(𝜆^𝑛/(1−𝜆))=0, and hence there exists 𝑛₀∈ℕ such that 𝜆^𝑛/(1−𝜆)<𝛿/𝜌 for all 𝑛≥𝑛₀. So, for each 𝑖∈{1,2,…,𝑠} and any 𝑛≥𝑛₀, we obtain ℓ_𝑖−𝜆^𝑛𝑝𝑥1−𝜆¹,𝑥₀=𝜆^𝑛ℓ1−𝜆^𝑖𝑝𝑥₁,𝑥₀<𝛿𝜌ℓ_𝑖𝑝𝑥₁,𝑥₀≤𝛿≤𝑟_𝑖. (3.16) Therefore, for any 𝑛≥𝑛₀, (−𝜆^𝑛/(1−𝜆))𝑝(𝑥₁,𝑥₀)∈𝑟_𝑖𝑉(ℓ_𝑖) for all 1≤𝑖≤𝑠, and hence (−𝜆^𝑛/(1−𝜆))𝑝(𝑥₁,𝑥₀)∈𝑈_𝑐. So we obtain 𝜆𝑐−^𝑛𝑝𝑥1−𝜆¹,𝑥₀∈𝑐+𝑈_𝑐⊆int𝐾, (3.17) or 𝜆^𝑛𝑝𝑥1−𝜆¹,𝑥₀≪_𝐾𝑐, (3.18) for all 𝑛≥𝑛₀. For 𝑚, 𝑛∈ℕ with 𝑚>𝑛≥𝑛₀, by (3.13), (3.18), and Lemma 2.5, it follows that 𝑝𝑥_𝑚,𝑥_𝑛≪_𝐾𝑐for𝑚,𝑛∈ℕwith𝑚>𝑛≥𝑛₀. (3.19) Hence, {𝑥_𝑛} is a TVS-cone Cauchy sequence in (𝑋,𝑝). By the TVS-cone completeness of (𝑋,𝑝), there exists 𝑣∈𝑋, such that {𝑥_𝑛}TVS-cone converges to 𝑣. On the other hand, applying Theorem 1.5, {𝑥_𝑛} is a Cauchy sequence in (𝑋,𝑑_𝑝), and 𝑑_𝑝(𝑥_𝑛,𝑣)→0 or 𝑥−−→_𝑛^𝑑𝑝𝑣 as 𝑛→∞. Let 𝑊={𝑥_𝑛}_𝑛∈ℕ∪{𝑣}. Then (𝑊,𝑑_𝑝) is a complete metric space, and the conclusion (a) holds.
To see (b), it suffices to show that 𝑣 is the unique common fixed point of 𝑇 and 𝑆. By (v) and (vi) of Lemma 2.1, the assumption (3.1) implies 𝑑_𝑝𝑇𝑥,𝑆𝑦≤𝛼𝑑_𝑝𝑥,𝑦+𝛽𝑑_𝑝(𝑥,𝑇𝑥)+𝑑_𝑝𝑦,𝑆𝑦+𝛾𝑑_𝑝𝑥,𝑆𝑦+𝑑_𝑝𝑦,𝑇𝑥,∀𝑥,𝑦∈𝑋. (3.20) For any 𝑛∈ℕ∪{0}, by (3.20), 𝑑_𝑝(𝑣,𝑆𝑣)≤𝑑_𝑝𝑣,𝑥_2𝑛+1+𝑑_𝑝𝑇𝑥_2𝑛,𝑆𝑣≤𝑑_𝑝𝑣,𝑥_2𝑛+1+𝛼𝑑_𝑝𝑥_2𝑛,𝑣+𝛽𝑑_𝑝𝑥_2𝑛,𝑥_2𝑛+1+𝑑_𝑝(𝑑𝑣,𝑆𝑣)+𝛾_𝑝𝑥_2𝑛,𝑆𝑣+𝑑_𝑝𝑣,𝑥_2𝑛+1≤𝑑_𝑝𝑣,𝑥_2𝑛+1+𝛼𝑑_𝑝𝑥_2𝑛,𝑣+𝛽𝑑_𝑝𝑥_2𝑛,𝑥_2𝑛+1+𝑑_𝑝(𝑑𝑣,𝑆𝑣)+𝛾_𝑝𝑥_2𝑛,𝑣+𝑑_𝑝(𝑣,𝑆𝑣)+𝑑_𝑝𝑣,𝑥_2𝑛+1, (3.21) which implies that 𝑑_𝑝()≤1𝑣,𝑆𝑣𝑑1−𝛽−𝛾^𝑝𝑣,𝑥_2𝑛+1+𝛼𝑑_𝑝𝑥_2𝑛,𝑣+𝛽𝑑_𝑝𝑥_2𝑛,𝑥_2𝑛+1+𝛾𝑑_𝑝𝑥_2𝑛,𝑣+𝑑_𝑝𝑣,𝑥_2𝑛+1. (3.22) Since 𝑑_𝑝 is a metric and 𝑥−−→_𝑛^𝑑𝑝𝑣 as 𝑛→∞, the right-hand side of (3.22) approaches zero as 𝑛→∞. Hence, 𝑑_𝑝(𝑣,𝑆𝑣)=0 or 𝑆𝑣=𝑣. Also, since 𝑑_𝑝(𝑇𝑣,𝑣)=𝑑_𝑝(𝑇𝑣,𝑆𝑣)≤𝛼𝑑_𝑝(𝑣,𝑣)+𝛽𝑑_𝑝(𝑣,𝑇𝑣)+𝑑_𝑝(𝑣,𝑆𝑣)+𝛾𝑑_𝑝(𝑣,𝑆𝑣)+𝑑_𝑝(=𝑑𝑣,𝑇𝑣)𝛽+𝛾_𝑝(𝑇𝑣,𝑣), (3.23) this implies that 1−𝛽−𝛾𝑑_𝑝(𝑇𝑣,𝑣)≤0. (3.24) Since 1−𝛽−𝛾>0, we have 𝑑_𝑝(𝑇𝑣,𝑣)=0 or 𝑇𝑣=𝑣. Therefore, 𝑣 is a common fixed point of 𝑇 and 𝑆. Suppose that there exists 𝑤∈𝑋 such that 𝑇𝑤=𝑆𝑤=𝑤. Since 𝑑_𝑝(𝑣,𝑤)=𝑑_𝑝(𝑇𝑣,𝑆𝑤)≤𝛼𝑑_𝑝(𝑣,𝑤)+𝛽𝑑_𝑝(𝑣,𝑇𝑣)+𝑑_𝑝(𝑤,𝑆𝑤)+𝛾𝑑_𝑝(𝑣,𝑆𝑤)+𝑑_𝑝(=𝑑𝑤,𝑇𝑣)𝛼+2𝛾_𝑝(𝑣,𝑤), (3.25) and 1−𝛼−2𝛾>2𝛽≥0, it follows that 𝑑_𝑝(𝑣,𝑤)=0, and hence 𝑤=𝑣. So the uniqueness of common fixed point in 𝑋 of 𝑇, and 𝑆 is proved. Following a similar argument as above, one can verify conclusion (c). The proof is completed.

The following result is immediate from Theorem 3.1.

Corollary 3.2. Let (𝑋,𝑑) be a complete metric space. Suppose that 𝑇,𝑆∶𝑋→𝑋 are two self-maps of 𝑋 satisfying 𝑑𝑇𝑥,𝑆𝑦≤𝛼𝑑𝑥,𝑦+𝛽𝑑(𝑥,𝑇𝑥)+𝑑𝑦,𝑆𝑦+𝛾𝑑𝑥,𝑆𝑦+𝑑𝑦,𝑇𝑥, (3.26) for all 𝑥, 𝑦∈𝑋, where 𝛼, 𝛽, 𝛾≥0 and 𝛼+2𝛽+2𝛾<1. Then the following statements hold. (a)𝑇 and 𝑆 have a unique common fixed point in 𝑋. Moreover, for each 𝑥₀∈𝑋, the mixed iterative sequence {𝑥_𝑛}_{𝑛∈ℕ∪{0}}, defined by 𝑥_2𝑛+1=𝑇𝑥_2𝑛 and 𝑥_2𝑛+2=𝑆𝑥_2𝑛+1 for 𝑛∈ℕ∪{0}, converges to the common fixed point;(b)any fixed point of 𝑇 is a fixed point of 𝑆, and conversely.

Remark 3.3. [21, Theorem  2.1] is a special case of Theorem 3.1.

The following result improves and extends [23, Theorem  2.3], and it is immediate from Theorem 3.1.

Theorem 3.4. Let (𝑋,𝑝) be a TVS-cone complete metric space, and let the map 𝑇∶𝑋→𝑋 be a cone-contraction; that is, 𝑇 satisfies the contractive condition 𝑝𝑇𝑥,𝑇𝑦⪷_𝐾𝛼𝑝𝑥,𝑦 (3.27) for all 𝑥,𝑦∈𝑋, where 𝛼∈[0,1) is a constant. Then the following statements hold:(a)there exists a nonempty proper subset 𝑀 of 𝑋, such that 𝑀 contains infinite points of 𝑋, and (𝑀,𝑑_𝑝) is a complete metric space, where 𝑑_𝑝∶=𝜉_𝑒∘𝑝;(b)𝑇 has a unique fixed point in 𝑋 (in fact, the unique fixed point of 𝑇 belongs to 𝑀). Moreover, for each 𝑥∈𝑋, the iterative sequence {𝑇^𝑛𝑥}_𝑛∈ℕTVS -cone converges to the fixed point.

Proof. Set 𝑆=𝑇 and 𝛽=𝛾=0 in Theorem 3.1.

Remark 3.5. (a) It is obvious that the classical Banach’s contraction principle is a special case of Theorem 3.4;
(b) Theorem 3.4 generalizes and improves [19, Theorem 1] and [24, Theorem  2.3].
(c) In fact, following a very similar argument as in the proof of Theorem 3.1 under the assumptions of Theorem 3.4, we can obtain an important fact that there exists a nonempty subset 𝑀 of 𝑋, such that (𝑀,𝑑_𝑝) is a complete metric space and 𝑇𝑀⊆𝑀. Since 𝑝𝑇𝑥,𝑇𝑦⪷_𝐾𝛼𝑝𝑥,𝑦⟹𝑑_𝑝𝑇𝑥,𝑇𝑦≤𝛼𝑑_𝑝𝑥,𝑦,∀𝑥,𝑦∈𝑋, (3.28) one can apply the Banach’s contraction principle to prove that 𝑇 has a unique fixed point in 𝑋. So the classical Banach contraction principle and [23, Theorem  2.3] are equivalent if we are only asked to find the fixed point of 𝑇.
(d) Another proof of Theorem 2.11 is given hereunder. Let 𝜑∶𝑋→𝑋 be any cone-contraction. Take 𝑣₀∈𝑋 and let 𝑣_𝑛=𝜑𝑣_𝑛−1=𝜑^𝑛𝑣₀ for 𝑛∈ℕ. Following a similar argument as in the proof of Theorem 3.1, there exists 𝑣∈𝑋, such that 𝑣−−→_𝑛^𝑑𝑝𝑣 as 𝑛→∞. Let ℳ={𝑣_𝑛}_𝑛∈ℕ∪{𝑣}. Then (ℳ,𝑑_𝑝) is a complete metric space.

Theorem 3.6. Let (𝑋,𝑝) be a TVS-cone complete metric space, and let the map 𝑇∶𝑋→𝑋 satisfy 𝑝𝑇𝑥,𝑇𝑦⪷_𝐾𝛽𝑝(𝑥,𝑇𝑥)+𝑝𝑦,𝑇𝑦 (3.29) for all𝑥,𝑦∈𝑋, where 𝛽∈[0,1/2) is a constant. Then the following statements hold: (a)there exists a nonempty proper subset 𝑀 of 𝑋, such that 𝑀 contains infinite points of 𝑋, and (𝑀,𝑑_𝑝) is a complete metric space, where 𝑑_𝑝∶=𝜉_𝑒∘𝑝;(b)𝑇 has a unique fixed point in 𝑋 (in fact, the unique fixed point of 𝑇 belongs to 𝑀). Moreover, for each 𝑥∈𝑋, the iterative sequence {𝑇^𝑛𝑥}_𝑛∈ℕTVS-cone converges to the fixed point.

Proof. Set 𝑆=𝑇 and 𝛼=𝛾=0 in Theorem 3.1.

Remark 3.7. Theorem 3.6 generalizes the fixed point theorems of Kannan's type [21, 24, 35].

Theorem 3.8. Let (𝑋,𝑝) be a TVS-cone complete metric space, and let the map 𝑇∶𝑋→𝑋 satisfy 𝑝𝑇𝑥,𝑇𝑦⪷_𝐾𝛾𝑝𝑥,𝑇𝑦+𝑝𝑦,𝑇𝑥, (3.30) for all 𝑥, 𝑦∈𝑋, where 𝛾∈[0,1/2) is a constant. Then the following statements hold:(a)there exists a nonempty proper subset 𝑀 of 𝑋, such that 𝑀 contains infinite points of 𝑋 and (𝑀,𝑑_𝑝) is a complete metric space, where 𝑑_𝑝∶=𝜉_𝑒∘𝑝;(b)𝑇 has a unique fixed point in 𝑋 (in fact, the unique fixed point of 𝑇 belongs to 𝑀). Moreover, for each 𝑥∈𝑋, the iterative sequence {𝑇^𝑛𝑥}_𝑛∈ℕTVS -cone converges to the fixed point.

Proof. Set 𝑆=𝑇 and 𝛼=𝛽=0 in Theorem 3.1.

Remark 3.9. Theorem 3.8 improves the fixed point theorems of Chatterjea's type [21, 24, 36].

4. Some Equivalences

In this final section, we introduce the following new concepts.

Definition 4.1. Let 𝑋 be a nonempty set with a TVS-cone metric 𝑝,𝑥∈𝑋, and let {𝑥_𝑛}_𝑛∈ℕ be a sequence in 𝑋. (i){𝑥_𝑛}𝑤-cone converges to𝑥 if for any 𝜀>0, there is a natural number ℕ₀ such that 𝑝𝑥_𝑛,𝑥≪_𝐾𝜀𝑒,∀𝑛≥ℕ₀. (4.1) We denote this by 𝑤-cone lim_𝑛→∞𝑥_𝑛=𝑥 or 𝑥−−−−−−→_𝑛^𝑤-cone𝑥 as 𝑛→∞ and call 𝑥 the 𝑤-cone limit of {𝑥_𝑛};(ii){𝑥_𝑛} is a 𝑤-cone Cauchy’s sequence; if for any 𝜀>0, there is a natural number ℕ₀ such that 𝑝(𝑥_𝑛,𝑥_𝑚)≪_𝐾𝜀𝑒 for all 𝑛, 𝑚≥ℕ₀;(iii)(𝑋,𝑝) is 𝑤-cone complete if every 𝑤-cone Cauchy sequence in 𝑋 is 𝑤-cone convergent. We establish the following crucial and useful properties.

Theorem 4.2. Let 𝑋 be a nonempty set with a TVS-cone metric 𝑝, 𝑥∈𝑋, and let {𝑥_𝑛}_𝑛∈ℕ be a sequence in 𝑋. Then the following statements hold. (a){𝑥_𝑛}𝑤-cone converges to 𝑥if and only if 𝑑_𝑝(𝑥_𝑛,𝑥)→0 as 𝑛→∞.(b){𝑥_𝑛} is a 𝑤-cone Cauchy sequence in(𝑋,𝑝)if and only if {𝑥_𝑛} is a Cauchy sequence (in usual sense) in (𝑋,𝑑_𝑝).(c)(𝑋,𝑝) is 𝑤-cone complete if and only if (𝑋,𝑑_𝑝) is a complete metric space.

Proof. Let 𝜀>0 be given. If {𝑥_𝑛}𝑤-cone converges to 𝑥, then, from (iv) of Lemma 2.1, there exists 𝑛₀∈ℕ such that 𝑝𝑥_𝑛,𝑥≪_𝐾𝜀𝑒⟺𝑑_𝑝𝑥_𝑛,𝑥=𝜉_𝑒∘𝑝𝑥_𝑛,𝑥<𝜀 (4.2) for all 𝑛≥𝑛₀. Hence, 𝑑_𝑝(𝑥_𝑛,𝑥)→0 as 𝑛→∞. Conversely, if 𝑑_𝑝(𝑥_𝑛,𝑥)→0 as 𝑛→∞, then, by (4.2), we show that {𝑥_𝑛}𝑤-cone converges to 𝑥. Therefore, (a) holds.
To see (b), let {𝑥_𝑛} be a 𝑤-cone Cauchy sequence in (𝑋,𝑝). Then there exists 𝑛₁∈ℕ such that 𝑝𝑥_𝑛,𝑥_𝑚≪_𝐾𝜀𝑒⟺𝑑_𝑝𝑥_𝑛,𝑥_𝑚<𝜀, (4.3) for all 𝑛, 𝑚≥𝑛₁. So {𝑥_𝑛} is a Cauchy sequence in (𝑋,𝑑_𝑝). The converse holds obviously.
Conclusion (c) is immediate from conclusions (a) and (b).

Theorem 4.3. Let (𝑋,𝑝) be a 𝑤-cone complete metric space. Suppose that 𝑇,𝑆∶𝑋→𝑋 are two self-maps of 𝑋 satisfying 𝑝𝑇𝑥,𝑆𝑦⪷_𝐾𝛼𝑝𝑥,𝑦+𝛽𝑝(𝑥,𝑇𝑥)+𝑝𝑦,𝑆𝑦+𝛾𝑝𝑥,𝑆𝑦+𝑝𝑦,𝑇𝑥 (4.4) for all𝑥,𝑦∈𝑋, where 𝛼, 𝛽, 𝛾≥0 and 𝛼+2𝛽+2𝛾<1. Then the following statements hold: (a)𝑇 and 𝑆 have a unique common fixed point in 𝑋. Moreover, for each 𝑥₀∈𝑋, the mixed iterative sequence {𝑥_𝑛}_{𝑛∈ℕ∪{0}}, defined by 𝑥_2𝑛+1=𝑇𝑥_2𝑛 and 𝑥_2𝑛+2=𝑆𝑥_2𝑛+1 for 𝑛∈ℕ∪{0}, 𝑤-cone converges to the common fixed point;(b)any fixed point of 𝑇 is a fixed point of 𝑆, and conversely.

Proof. Applying (c) of Theorem 4.2, (𝑋,𝑑_𝑝) is a complete metric space. By Lemma 2.1, the assumption (4.4) implies 𝑑_𝑝𝑇𝑥,𝑆𝑦≤𝛼𝑑_𝑝𝑥,𝑦+𝛽𝑑_𝑝(𝑥,𝑇𝑥)+𝑑_𝑝𝑦,𝑆𝑦+𝛾𝑑_𝑝𝑥,𝑆𝑦+𝑑_𝑝𝑦,𝑇𝑥,∀𝑥,𝑦∈𝑋. (4.5) Therefore, the conclusion follows from Corollary 3.2 and Theorem 4.2.

It is obvious that Theorem 4.3 implies Corollary 3.2, so we obtain the following equivalence between Theorem 4.3 and Corollary 3.2.

Theorem 4.4. Theorem 4.3 and Corollary 3.2 are equivalent.

Remark 4.5. Using the same argument as above, we can prove the equivalences between scalar version and vectorial version of the Banach contraction principle, Kannan's fixed point theorem, Chatterjea's fixed point theorem, and others (e.g., [20, 21, 24, 25]) in 𝑤-cone complete metric spaces.

The following result tell us the relationship between the TVS-cone completeness and the 𝑤-cone completeness.

Theorem 4.6. If (𝑋,𝑝) is TVS-cone complete, then there exists a nonempty proper subset 𝑀 of 𝑋, such that 𝑀 contains infinite points of 𝑋, and (𝑀,𝑝) is a 𝑤-cone complete metric space.

Proof. Applying Theorem 2.11, there exists a nonempty proper subset 𝑀 of 𝑋, such that 𝑀 contains infinite points of 𝑋, and (𝑀,𝑑_𝑝) is a complete metric space. By using (c) of Theorem 4.2, (𝑀,𝑝) is a 𝑤-cone complete metric space.

Acknowledgment

This research was supported by the National Science Council of the Republic of China.