On Fixed Point Theory of Monotone Mappings with Respect to a Partial Order Introduced by a Vector Functional in Cone Metric Spaces

Abstract

We presented some maximal and minimal fixed point theorems of set-valued monotone mappings with respect to a partial order introduced by a vector functional in cone metric spaces. In addition, we proved not only the existence of maximal and minimal fixed points but also the existence of the largest and the least fixed points of single-valued increasing mappings. It is worth mentioning that the results on single-valued mappings in this paper are still new even in the case of metric spaces and hence they indeed improve the recent results.

1. Introductions

Throughout this paper, let (𝑋,𝑑) be a complete cone metric space over a total minihedral and continuous cone 𝑃 of a normed vector space 𝐸. A vector functional 𝜑∶𝑋→𝐸 introduces a partial order ≺ on 𝑋 as follows: 𝑥≺𝑦⟺𝑑𝑥,𝑦⪯𝜑(𝑥)−𝜑𝑦, (1) for all 𝑥,𝑦∈𝑋, where ⪯ is the partial order on 𝐸 determined by the cone 𝑃. Using the partial order introduced by the vector functional 𝜑, Agarwal and Khamsi [1] extended Caristi's fixed point theorem [2] to the case of cone metric space and proved that all mapping 𝑇∶𝑋→𝑋 (resp., 𝑇∶𝑋→2^𝑋) such that ∀𝑥∈𝑋,𝑥≺𝑇𝑥resp.,∀𝑥∈𝑋,∃𝑦∈𝑇𝑥,𝑥≺𝑦 (2) has a fixed point provided that 𝜑 is lower semicontinuous and bounded below on 𝑋. In [1, 3], the authors studied Kirk's problem [4, 5] in the case of cone metric spaces and obtained some generalized Caristi's fixed point theorems in cone metric spaces. For the researches on the generalization of primitive Caristi's result in the case of metric spaces, we refer the readers to [6–12]. For other references concerned with various fixed point results for one, two, three, or four self-mappings in the setting of metric, ordered metric, partial metric, Prešić-type mappings, cone metric, G-metric spaces, and so forth, we refer the readers to [13–24].

In particular, when 𝐸=ℝ, the partial order defined by (1) is reduced to the one defined by Caristi [2] who denote it by ≺₁. Zhang [25, 26] and Li [27] considered the existence of fixed points of a mapping 𝑇∶𝑋→𝑋 (resp., 𝑇∶𝑋→2^𝑋) such that 𝑥₀≺₁𝑇𝑥₀resp.,∃𝑦∈𝑇𝑥₀,𝑥₀≺₁𝑦, (3) for some 𝑥₀∈𝑋, and proved some maximal and minimal fixed point theorems at the expense that 𝑇 is monotone with respect to the partial order ≺₁.

In this paper, we shall extend the results of Zhang [25, 26] and Li [27] to the case of cone metric spaces. Some maximal and minimal fixed point theorems of set-valued monotone mappings with respect to the partial order ≺ are established in cone metric spaces. In addition, not only the existence of maximal and minimal fixed points but also the existence of largest and least fixed points is proved for single-valued increasing mappings. It is worth mentioning that the results on single-valued mappings in this paper are still new even in the case of metric spaces and hence they indeed improve the results of Zhang [25] and Li [27].

2. Preliminaries

First, we recall some definitions and properties of cones and cone metric spaces; these can be found in [1, 3, 17–24, 28–30].

Let 𝐸 be a topological vector space. A cone 𝑃 of 𝐸 is a nonempty closed subset of 𝐸 such that 𝑎𝑥+𝑏𝑦∈𝑃 for all 𝑥,𝑦∈𝑃 and all 𝑎,𝑏≥0, and 𝑃∩(−𝑃)={𝜃}, where 𝜃 is the zero element of 𝐸. A cone 𝑃 of 𝐸 determines a partial order ⪯ on 𝐸 by 𝑥⪯𝑦⇔𝑦−𝑥∈𝑃 for all 𝑥,𝑦∈𝑋. For all 𝑥,𝑦∈𝐸 with 𝑦−𝑥∈int𝑃, we write 𝑥≪𝑦, where int𝑃 is the interior of 𝑃.

Let 𝑃 be a cone of a topological vector space. 𝑃 is total order minihedral [29] if, for all upper bounded nonempty total ordered subset 𝐴 of 𝐸, sup𝐴 exists in 𝐸. Equivalently, 𝑃 is total order minihedral if, for all lower bounded nonempty total ordered subset 𝐴 of 𝐸, inf𝐴 exists in 𝐸.

Let 𝐸 be a normed vector space. A cone 𝑃 of 𝐸 is continuous [1, 3] if, for all subset 𝐴 of 𝐸, inf𝐴 exists implies inf_𝑥∈𝐴‖𝑥−inf𝐴‖=0, and sup𝐴 exists implies sup_𝑥∈𝐴‖𝑥−sup𝐴‖=0. A cone 𝑃 of 𝐸 is normal [30] if there exists 𝑁>0 such that for all 𝑥,𝑦∈𝑃, 𝑥⪯𝑦 implies ‖𝑥‖≤𝑁‖𝑦‖, and the minimal 𝑁 is called a normal constant of 𝑃. Equivalently, A cone 𝑃 of 𝐸 is normal provided that for all {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛}⊆𝐸 with 𝑥_𝑛⪯𝑦_𝑛⪯𝑧_𝑛 for all 𝑛, 𝑥_𝑛→𝑥 and 𝑧_𝑛→𝑥 imply 𝑦_𝑛→𝑥 for some 𝑥∈𝑋.

Remark 1. A total order minihedral cone 𝑃 of a normed space 𝐸 is certainly normal see [29].

Let 𝑋 be a nonempty set and 𝑃 a cone of a topological vector space 𝐸. A cone metric [28] is a mapping 𝑑∶𝑋×𝑋→𝑃 such that for all 𝑥,𝑦,𝑥∈𝑋,(𝑑1) 𝑑(𝑥,𝑦)=𝜃 if and only if 𝑥=𝑦,(𝑑2) 𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥),(𝑑3) 𝑑(𝑥,𝑦)⪯𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦).

A pair (𝑋,𝑑) is called a cone metric space over 𝑃 if 𝑑∶𝑋×𝑋→𝑃 is a cone metric. Let (𝑋,𝑑) be a cone metric space over a cone 𝑃 of a topological vector space 𝐸. A sequence {𝑥_𝑛} in (𝑋,𝑑) converges [28] to 𝑥∈𝑋 (denote 𝑥→−_𝑛^𝑑𝑥) if, for all 𝜀∈𝑃 with 𝜃≪𝜀, there exists a positive integer 𝑛₀ such that 𝑑(𝑥_𝑛,𝑥)≪𝜀 for all 𝑛≥𝑛₀. A sequence {𝑥_𝑛} in (𝑋,𝑑) is Cauchy [28] if, for all 𝜀∈𝑃 with 𝜃≪𝜀, there exists a positive integer 𝑛₀ such that 𝑑(𝑥_𝑛,𝑥_𝑚)≪𝜀 for all 𝑚,𝑛≥𝑛₀. A cone metric space (𝑋,𝑑) is complete [28] if all Cauchy sequence {𝑥_𝑛} in (𝑋,𝑑) converges to a point 𝑥∈𝑋. A vector functional 𝜑∶𝑋→𝐸 is sequentially continuous at some 𝑥∈𝑋 if lim_𝑛→∞𝜑(𝑥_𝑛)=𝜑(𝑥) for all {𝑥_𝑛}⊆𝑋 such that 𝑥→−_𝑛^𝑑𝑥. If, for all 𝑥∈𝑋, 𝜑 is sequentially continuous at 𝑥, then 𝜑∶𝑋→𝐸 is sequentially continuous.

Remark 2. Let (𝑋,𝑑) be a cone metric space over a normal cone 𝑃 of a normed vector space 𝐸 and {𝑥_𝑛} a sequence in (𝑋,𝑑). Then 𝑥→−_𝑛^𝑑𝑥 if and only if lim_𝑛→∞𝑑(𝑥_𝑛,𝑥)=𝜃, and {𝑥_𝑛} is Cauchy if and only if lim_{𝑚,𝑛→∞}𝑑(𝑥_𝑛,𝑥_𝑚)=𝜃 see [28].

Let 𝑋 be a nonempty set and ≺ a partial order on 𝑋. For all 𝑥,𝑦∈𝑋 with 𝑥≺𝑦, set [𝑥,+∞)={𝑧∈𝑋∶𝑥≺𝑧}, (−∞,𝑥]={𝑧∈𝑋∶𝑧≺𝑥}, and [𝑥,𝑦]={𝑧∈𝑋∶𝑥≺𝑧≺𝑦}. Let 𝐴 be a nonempty subset of 𝑋. A set-valued mapping 𝑇∶𝑋→2^𝑋 is increasing on 𝐴 if, for all 𝑥,𝑦∈𝐴 with 𝑥≺𝑦 and all 𝑢∈𝑇𝑥, there exists 𝑣∈𝑇𝑦 such that 𝑢≺𝑣. A set-valued mapping 𝑇∶𝑋→2^𝑋 is quasi-increasing if, for all 𝑥,𝑦∈𝐴 with 𝑥≺𝑦 and all 𝑣∈𝑇𝑦, there exists 𝑢∈𝑇𝑥 such that 𝑢≺𝑣. In particular, a single-valued mapping 𝑇∶𝑋→𝑋 is increasing on 𝐴 if, for all 𝑥,𝑦∈𝐴 with 𝑥≺𝑦, 𝑇𝑥≺𝑇𝑦.

A point 𝑥^∗∈𝑋 is called a fixed point of a set-valued (resp., single-valued) mapping 𝑇 if 𝑥^∗∈𝑇𝑥^∗(resp. 𝑥^∗=𝑇𝑥^∗). Let 𝐴 be a nonempty subset of 𝑋 and let 𝑥^∗∈𝐴 be a fixed point of a mapping 𝑇. 𝑥^∗ is called a maximal (resp. minimal) fixed point of 𝑇 in 𝐴 if for all fixed point 𝑥∈𝐴 of 𝑇, 𝑥^∗≺𝑥 (resp., 𝑥≺𝑥^∗) implies 𝑥^∗=𝑥. 𝑥^∗∈𝐴 is called a largest (resp., least) fixed point of 𝑇 in 𝐴 if, for all fixed point 𝑥∈𝐴 of 𝑇, 𝑥≺𝑥^∗ (resp., 𝑥^∗≺𝑥). A largest (resp., least) fixed point of 𝑇 in 𝐴 is naturally a maximal (resp., minimal) fixed point in 𝐴, but the converse may not be true.

3. Fixed Point Theorems

In this section, we always assume that the partial order ≺ is defined by (1).

Theorem 3. Let (𝑋,𝑑,≺) be a complete partially ordered cone metric space over a total order minihedral and continuous cone 𝑃 of a normed vector space 𝐸. Let 𝜑∶𝑋→𝐸 be a sequentially continuous vector functional and let 𝑇∶𝑋→2^𝑋 be a set-valued mapping such that 𝑇𝑥 is compact for all 𝑥∈𝑋. Assume that there exists 𝑥₀∈𝑋 such that 𝜑 is bounded below on [𝑥₀,+∞), 𝑇 is increasing on [𝑥₀,+∞), and 𝑇𝑥₀∩[𝑥₀,+∞)≠∅. Then 𝑇 has a maximal fixed point 𝑥^∗∈[𝑥₀,+∞).

Proof. Since 𝑃 is a total order minihedral cone and 𝐸 is a normed space, then 𝑃 is a normal cone by Remark 1. Set 𝑄₁=𝑥∈𝑥₀,+∞∶𝑇𝑥∩[𝑥,+∞)≠∅. (4) Clearly, 𝑄₁ is nonempty since 𝑥₀∈𝑄₁. Let {𝑥_𝛼}_𝛼∈Γ⊆𝑄₁ be an increasing chain, where Γ is a directed set. Then by (1) we have 𝑑𝑥_𝛼,𝑥_𝛽⪯𝜑𝑥_𝛼−𝜑𝑥_𝛽, (5) for all 𝛼,𝛽∈Γ with 𝛼⪯𝛽. This implies that {𝜑(𝑥_𝛼)} is a decreasing chain in 𝐸. Since 𝑃 is total order minihedral and 𝜑 is bounded below on [𝑥₀,+∞), then inf_𝛼∈Γ𝜑(𝑥_𝛼) exists in 𝐸. Moreover, inf_𝛼∈Γ‖𝜑(𝑥_𝛼)−inf_𝛼∈Γ𝜑(𝑥_𝛼)‖=0 since 𝑃 is continuous. Therefore there exists an increasing sequence {𝑥_𝛼𝑛}⊆{𝑥_𝛼} such that lim_𝑛→∞‖𝜑(𝑥_𝛼𝑛)−inf_𝛼∈Γ𝜑(𝑥_𝛼)‖=0, that is, lim_𝑛→∞𝜑𝑥_𝛼𝑛=inf_𝛼∈Γ𝜑𝑥_𝛼. (6) By (1) we have for all 𝑚∈ℕ such that 𝑚≥𝑛, 𝑑𝑥_𝛼𝑛,𝑥_𝛼𝑚⪯𝜑𝑥_𝛼𝑛−𝜑𝑥_𝛼𝑚. (7) Let 𝑛→∞, by (6) we have lim_{𝑚,𝑛→∞}[𝜑(𝑥_𝛼𝑛)−𝜑(𝑥_𝛼𝑚)]=𝜃 and hence lim_{𝑚,𝑛→∞}𝑑(𝑥_𝛼𝑛,𝑥_𝛼𝑚)=𝜃 by the normality of 𝑃. Moreover by Remark 2, {𝑥_𝛼𝑛} is a Cauchy sequence in 𝑋. Therefore by the completeness of 𝑋, there exists some 𝑥∈𝑋 such that 𝑥−−−_𝛼𝑛^𝑑→𝑥. (8) Note that {𝑥_𝛼𝑛} is an increasing sequence of 𝑄₁, then by (1), we have for all 𝑛, 𝑑𝑥₀,𝑥_𝛼𝑛⪯𝜑𝑥₀−𝜑𝑥_𝛼𝑛, (9) And, for all 𝑛≥𝑛₀, 𝑑𝑥_𝛼𝑛0,𝑥_𝛼𝑛⪯𝜑𝑥_𝛼𝑛0−𝜑𝑥_𝛼𝑛, (10) where 𝑛₀ is an arbitrary integer. Let 𝑛→∞, then by (8) and the continuity of 𝜑 we have 𝑑(𝑥₀,𝑥)⪯𝜑(𝑥₀)−𝜑(𝑥) and 𝑑(𝑥_𝛼𝑛0,𝑥)⪯𝜑(𝑥_𝛼𝑛0)−𝜑(𝑥), that is, 𝑥∈[𝑥₀,+∞) and 𝑥_𝛼𝑛0≺𝑥. Moreover the arbitrary property of 𝑛₀ forces that 𝑥_𝛼𝑛≺𝑥, (11) for all 𝑛. By 𝑥_𝛼𝑛∈𝑄₁, there exists 𝑦_𝑛∈𝑇𝑥_𝛼𝑛 such that 𝑥_𝛼𝑛≺𝑦_𝑛, (12) for all 𝑛. Since 𝑇 is increasing on [𝑥₀,+∞), then by (11) and 𝑥∈[𝑥₀,+∞), there exists 𝑧_𝑛∈𝑇𝑥 such that 𝑦_𝑛≺𝑧_𝑛, (13) for all 𝑛. This together with (12) implies that 𝑥_𝛼𝑛≺𝑧_𝑛, (14) for all 𝑛. Note that 𝑇𝑥 is compact, and there exists a subsequence {𝑧_𝑛𝑘}⊆{𝑧_𝑛} and 𝑧∈𝑇𝑥 such that 𝑧_𝑛𝑘→𝑧. (15) From (14) we have 𝑥_𝛼𝑛𝑘≺𝑧_𝑛𝑘 for all 𝑛_𝑘 and hence by (1), 𝑑𝑥_𝛼𝑛𝑘,𝑧_𝑛𝑘⪯𝜑𝑥_𝛼𝑛𝑘−𝜑𝑧_𝑛𝑘, (16) for all 𝑛_𝑘. Let 𝑛_𝑘→∞, then by (8), (15), and the continuity of 𝜑 we have 𝑑(𝑥,𝑧)⪯𝜑(𝑥)−𝜑(𝑧), that is, 𝑥≺𝑧. This implies that 𝑇𝑥∩[𝑥,+∞)≠∅ and hence 𝑥∈𝑄₁ by 𝑥∈[𝑥₀,+∞).
For all 𝛼∈Γ, if there exists some 𝑛₀ such that 𝑥_𝛼≺𝑥_𝛼𝑛0, then 𝑥 is an upper bound of {𝑥_𝛼} by (11). Otherwise, there exists some 𝛽∈Γ such that 𝑥_𝛼𝑛≺𝑥_𝛽 for all 𝑛. Thus by (1) we have 𝜑(𝑥_𝛼𝑛)−𝜑(𝑥_𝛽)∈𝑃 for all 𝑛. Let 𝑛→∞, by (6) we have inf_𝛼∈Γ𝜑(𝑥_𝛼)−𝜑(𝑥_𝛽)∈𝑃; that is, 𝜑(𝑥_𝛽)⪯inf_𝛼∈Γ𝜑(𝑥_𝛼). So we have 𝜑(𝑥_𝛽)=inf_𝛼∈Γ𝜑(𝑥_𝛼) and hence 𝜑(𝑥_𝛽)⪯𝜑(𝑥_𝛼) for all 𝛼∈Γ. Note that {𝜑(𝑥_𝛼)}_𝛼∈Γ is a decreasing chain, then 𝛽≥𝛼 for all 𝛼∈Γ. Moreover 𝑥_𝛼≺𝑥_𝛽 for all 𝛼∈Γ since {𝑥_𝛼}_𝛼∈Γ is an increasing chain. Hence {𝑥_𝛼}_𝛼∈Γ has an upper bound in 𝑄₁. By Zorn's lemma, (𝑄₁,≺) has a maximal element 𝑥^∗; that is, for all 𝑥∈𝑄₁, 𝑥^∗≺𝑥 implies 𝑥=𝑥^∗. By 𝑥^∗∈𝑄₁, there exists 𝑦^∗∈𝑇𝑥^∗ such that 𝑥^∗≺𝑦^∗. Moreover by the increasing property of 𝑇 on [𝑥₀,+∞), there exists 𝑧^∗∈𝑇𝑦^∗ such that 𝑦^∗≺𝑧^∗. Thus we have 𝑥^∗≺𝑧^∗ by 𝑥^∗≺𝑦^∗. This indicates 𝑧^∗∈𝑇𝑥^∗∩[𝑥^∗,+∞) and hence 𝑧^∗∈𝑄₁. Finally the maximality of 𝑥^∗ in 𝑄₁ forces that 𝑥^∗=𝑧^∗∈𝑇𝑥^∗; that is, 𝑥^∗ is a maximal fixed point of 𝑇 in [𝑥₀,+∞). The proof is complete.

Theorem 4. Let (𝑋,𝑑,≺) be a complete partially ordered cone metric space over a total order minihedral and continuous cone 𝑃 of a normed vector space 𝐸. Let 𝜑∶𝑋→𝐸 be a sequentially continuous vector functional and 𝑇∶𝑋→2^𝑋 be a set-valued mapping such that 𝑇𝑥 is compact for all 𝑥∈𝑋. Assume that there exists 𝑦₀∈𝑋 such that 𝜑 is bounded above on (−∞,𝑦₀], 𝑇 is quasi-increasing on (−∞,𝑦₀], and 𝑇𝑦₀∩(−∞,𝑦₀]≠∅. Then 𝑇 has a minimal fixed point 𝑥_∗∈(−∞,𝑦₀].

Proof. Set 𝑄₂=𝑥∈−∞,𝑦₀∶𝑇𝑥∩(−∞,𝑥]≠∅. (17) Clearly, 𝑄₂≠∅. By the same method used in the proof of Theorem 3, we can prove that (𝑄₂,≺) has a minimal element 𝑥_∗ which is also a minimal fixed point of 𝑇 in (−∞,𝑦₀]. The proof is complete.

Remark 5. If 𝑇∶𝑋→𝑋 is a single-valued mapping, then 𝑇𝑥 is naturally compact for all 𝑥∈𝑋. Hence both of Theorems 3 and 4 are still valid for a single-valued mapping.

In particular when 𝑇 is a single-valued mapping, we have the following further results.

Theorem 6. Let (𝑋,𝑑,≺) be a complete partiallly ordered cone metric space over a total order minihedral and continuous cone 𝑃 of a normed vector space 𝐸. Let 𝜑∶𝑋→𝐸 be a sequentially continuous vector functional and let 𝑇∶𝑋→𝑋 be a single-valued mapping. Assume that there exists 𝑥₀∈𝑋 such that 𝜑 is bounded below on [𝑥₀,+∞), 𝑇 is increasing on [𝑥₀,+∞), and 𝑥₀≺𝑇𝑥₀. Then 𝑇 has a maximal fixed point 𝑥^∗ and a least fixed point 𝑥_∗ in [𝑥₀,+∞) such that 𝑥_∗≺𝑥^∗.

Proof. By Theorem 3 and Remark 5, 𝑇 has a maximal fixed point 𝑥^∗∈[𝑥₀,+∞) and hence 𝐹={𝑥∈[𝑥₀,+∞)∶𝑥=𝑇𝑥}≠∅. Set 𝑆=𝐼=[𝑥,+∞)∶𝑥∈𝑥₀,+∞,𝑥≺𝑇𝑥,𝐹⊆𝐼. (18) Clearly, [𝑥₀,+∞)∈𝑆 and hence 𝑆≠∅. Define a relation ⊑ on 𝑆 by 𝐼₁⊑𝐼₂⟺𝐼₁⊆𝐼₂, (19) for all 𝐼₁,𝐼₂∈𝑆, then it is easy to check that ⊑ is a partial order on 𝑆.
Let {𝐼_𝛼}_𝛼∈Γ be a decreasing chain of 𝑆, where 𝐼_𝛼=[𝑥_𝛼,+∞). From (1), (18), and (19) we find that {𝑥_𝛼}_𝛼∈Γ is an increasing chain of 𝑀, where 𝑀=𝑥∈𝑥₀,+∞∶𝑥≺𝑇𝑥,𝐹⊆[𝑥,+∞). (20) Set 𝑄₁={𝑥∈[𝑥₀,+∞)∶𝑥≺𝑇𝑥}. Clearly, 𝑀⊆𝑄₁. Following the proof of Theorem 3, there exists 𝑥∈𝑄₁ and an increasing sequence {𝑥_𝛼𝑛}⊆{𝑥_𝛼} satisfying (6) such that (8) and (11) are satisfied. From 𝑥_𝛼𝑛∈𝑀 we have that 𝑥_𝛼𝑛≺𝑥 for all 𝑥∈𝐹 and all 𝑛. Thus the increasing property of 𝑇 on [𝑥₀,+∞) implies that, for all 𝑥∈𝐹 and all 𝑛, 𝑥_𝛼𝑛≺𝑇𝑥_𝛼𝑛≺𝑇𝑥=𝑥, (21) and hence by (1), 𝑑𝑥_𝛼𝑛,𝑥⪯𝜑𝑥_𝛼𝑛−𝜑(𝑥), (22) for all 𝑥∈𝐹 and all 𝑛. Let 𝑛→∞, then by (8) and the continuity of 𝜑 we have 𝑑(𝑥,𝑥)⪯𝜑(𝑥)−𝜑(𝑥); that is, 𝑥≺𝑥, (23) for all 𝑥∈𝐹. This together with 𝑥∈𝑄₁ implies 𝑥∈𝑀. Then in analogy to the proof of Theorem 3, by (6), (8), and 𝑥∈𝑀 we can prove {𝑥_𝛼}_𝛼∈Γ has an upper bound 𝑥∈𝑀. By (18), we have [𝑥,+∞)∈𝑆. Note that 𝑥 is an upper bound of {𝑥_𝛼}_𝛼∈Γ in 𝑀, then [𝑥,+∞)⊆𝐼_𝛼 for all 𝛼∈Γ and hence by (19), [𝑥,+∞)⊑𝐼_𝛼, (24) for all 𝛼∈Γ. This means [𝑥,+∞) is a lower bound of {𝐼_𝛼}_𝛼∈Γ in 𝑆. By Zorn's lemma, (𝑆,⊑) has a minimal element; denote it by 𝐼^∗=[𝑥_∗,+∞). By (18) we have 𝑥₀≺𝑥_∗≺𝑇𝑥_∗ and 𝑥_∗≺𝑥, (25) for all 𝑥∈𝐹. By the increasing property of 𝑇, we have 𝑥₀≺𝑥_∗≺𝑇𝑥_∗≺𝑇(𝑇𝑥_∗) and 𝑇𝑥_∗≺𝑇𝑥=𝑥 for all 𝑥∈𝐹, which implies [𝑇𝑥_∗,+∞)∈𝑆 and [𝑇𝑥_∗,+∞)⊆𝐼^∗. Moreover by (19), [𝑇𝑥_∗,+∞)⊑𝐼^∗. The minimality of 𝐼^∗ in 𝑆 forces that [𝑇𝑥_∗,+∞)=𝐼^∗ and so we have 𝑥_∗=𝑇𝑥_∗. Finally by (25), 𝑥_∗ is a least fixed point of 𝑇 in [𝑥₀,+∞) and 𝑥_∗≺𝑥^∗. The proof is complete.

Theorem 7. Let (𝑋,𝑑,≺) be a complete partially ordered cone metric space over a total order minihedral and continuous cone 𝑃 of a normed vector space 𝐸. Let 𝜑∶𝑋→𝐸 be a sequentially continuous vector functional and let 𝑇∶𝑋→𝑋 be a single-valued mapping. Assume that there exists 𝑥₀∈𝑋 such that 𝜑 is bounded above on (−∞,𝑦₀], 𝑇 is increasing on (−∞,𝑦₀], and 𝑇𝑦₀≺𝑦₀. Then 𝑇 has a minimal fixed point 𝑥_∗ and a largest fixed point in 𝑥^∗ in (−∞,𝑥₀] such that 𝑥_∗≺𝑥^∗.

Proof. By Theorem 4 and Remark 5, 𝑇 has a minimal fixed point in 𝑥_∗∈(−∞,𝑦₀]. Set 𝑆=𝐽=(−∞,𝑥]∶𝑥∈−∞,𝑦₀,𝑇𝑥≺𝑥,𝐹⊆𝐽. (26) Define a relation ⊑_𝑆 on 𝑆 as follows: 𝐽₁⊑_𝑆𝐽₂⟺𝐽₁⊆𝐽₂, (27) for all 𝐽₁,𝐽₂∈𝑆, then ⊑_𝑆 is a partial order on 𝑆. In an analogy to the proof of Theorem 4, we can prove (𝑆,⊑_𝑆) has a minimal element (−∞,𝑥^∗] and 𝑥^∗ is a largest fixed point of 𝑇 in (−∞,𝑦₀]. The proof is complete.

Theorem 8. Let (𝑋,𝑑,≺) be a complete partially ordered cone metric space over a total order minihedral and continuous cone 𝑃 of a normed vector space 𝐸. Let 𝜑∶𝑋→𝐸 be a sequentially continuous mapping and let 𝑇∶𝑋→𝑋 be a single-valued mapping. Assume that there exists 𝑥₀,𝑦₀∈𝑋 with 𝑥₀≺𝑦₀ such that 𝑇 is increasing on [𝑥₀,𝑦₀] and 𝑥₀≺𝑇𝑥₀,𝑇𝑦₀≺𝑦₀. Then 𝑇 has a largest fixed point 𝑥^∗ and a least fixed point 𝑥_∗ in [𝑥₀,𝑦₀] such that 𝑥_∗≺𝑥^∗.

Proof. For all 𝑥∈[𝑥₀,𝑦₀], by (1) we have 𝜑(𝑦₀)⪯𝜑(𝑥)≺𝜑(𝑥₀); that is, 𝜑 is bounded on [𝑥₀,𝑦₀]. In an analogy to the proof of Theorem 3, we can prove 𝑇 has a maximal fixed point and a minimal fixed point in [𝑥₀,𝑦₀] by investigating the existence of maximal element and minimal element, respectively, in 𝐷₁={𝑥∈[𝑥₀,𝑦₀]∶𝑥≺𝑇𝑥} and 𝐷₂={𝑥∈[𝑥₀,𝑦₀]∶𝑇𝑥≺𝑥}. Let 𝑆₁=𝐼=𝑥,𝑦₀∶𝑥∈𝑥₀,𝑦₀,𝑆,𝑥≺𝑇𝑥,𝐺⊆𝐼₂=𝐽=𝑥₀,𝑥∶𝑥∈𝑥₀,𝑦₀,𝑇𝑥≺𝑥,𝐺⊆𝐽, (28) where 𝐺={𝑥∈[𝑥₀,𝑦₀]∶𝑇𝑥=𝑥} is nonempty. Define ⊑₁ on 𝑆₁ and ⊑₂ on 𝑆₂, respectively, by 𝐼₁⊑₁𝐼₂⟺𝐼₁⊆𝐼₂,∀𝐼₁,𝐼₂∈𝑆₁,𝐽₁⊑₂𝐽₂⟺𝐽₁⊆𝐽₂,∀𝐽₁,𝐽₂∈𝑆₂, (29) then it is easy to check that ⊑₁ and ⊑₂ are partial orders on 𝑆₁ and 𝑆₂, respectively. In an analogy to the proof of Theorem 4, we can prove (𝑆₁,⊑₁) has a minimal element 𝐼_∗=[𝑥_∗,𝑦₀] and (𝑆₂,⊑₂) has a minimal element 𝐽_∗=[𝑥₀,𝑦^∗]. By the definitions of 𝑆₁ and 𝑆₂, we have 𝑥_∗,𝑦^∗∈[𝑥₀,𝑦₀], 𝑥_∗≺𝑥≺𝑦^∗,𝑥_∗≺𝑇𝑥_∗≺𝑇𝑦^∗≺𝑦^∗. (30)(31) Moreover by (30) and the increasing property of 𝑇 on [𝑥₀,𝑦₀], for all 𝑥∈𝐺, we have 𝑥₀≺𝑇𝑥₀≺𝑇𝑥_∗≺𝑥≺𝑇𝑦^∗≺𝑇𝑦₀≺𝑦₀, (32) and so by (31), 𝑥_∗≺𝑇𝑥_∗≺𝑇𝑇𝑥_∗≺𝑇𝑇𝑦^∗≺𝑇𝑦^∗≺𝑦^∗. (33) From (32) and (33) we have that [𝑇𝑥_∗,𝑦₀]∈𝑆₁, [𝑥₀,𝑇𝑦^∗]∈𝑆₂, and 𝑇𝑥_∗,𝑦₀⊑₁𝐼_∗,𝑥₀,𝑇𝑦^∗⊑₂𝐽_∗, (34) which implies [𝑇𝑥_∗,𝑦₀]=𝐼_∗ and [𝑥₀,𝑇𝑦^∗]=𝐽_∗ by the minimality of 𝐼_∗ and 𝐽_∗. This means that 𝑇𝑥_∗=𝑥_∗ and 𝑇𝑦^∗=𝑦^∗. Hence 𝑥_∗ is the least fixed point and 𝑦^∗ is the largest fixed point of 𝑇 in [𝑥₀,𝑦₀] by (31). The proof is complete.

Remark 9. Theorems 3–8 are extensions of [4, Theorems 3 and 4] and [2, Theorems 3, 4, and 5] to the case of cone metric spaces. It is worth mentioning that in Theorems 4, 7, and 8, not only the existence of maximal and minimal fixed points but also the existence of largest and least fixed points is obtained. Therefore Theorems 4, 7, and 8 are still new even in the case of metric space and hence they indeed improve [2, Theorems 3, 4, and 6].

Now we give an example to demonstrate Theorem 3.

Example 10. Let 𝑋={1,2,3,4}, 𝐸=ℝ² with the norm ‖𝑢‖=𝑢²₁+𝑢²₂ for all 𝑢=(𝑢₁,𝑢₂)∈ℝ² and 𝑃=ℝ²₊. Clearly, 𝑃 is a strongly minihedral and continuous cone of 𝐸. Define a mapping 𝑑∶ℝ×ℝ→𝑃 by 𝑑𝑥,𝑦=𝑥−𝑦,𝑥−𝑦^1/2,∀𝑥,𝑦∈ℝ, (35) then (ℝ,𝑑) is a complete cone metric space over 𝑃 and hence (𝑋,𝑑) is a complete cone metric subspace of (ℝ,𝑑). Define a vector functional 𝜑∶[1,+∞)→𝐸 by 𝜑(𝑥)=6𝑥,3√2+2√3√𝑥, (36) for all 𝑥∈[1,+∞). For arbitrary 𝑥∈[1,+∞), let {𝑥_𝑛}⊆[1,+∞) be a sequence such that 𝑥→−_𝑛^𝑑𝑥, then 𝑥−→_𝑛^|⋅|𝑥 and hence ‖𝜑(𝑥_𝑛)−𝜑(𝑥)‖→0, that is, lim_𝑛→∞𝜑(𝑥_𝑛)=𝜑(𝑥). This means that 𝜑∶[1,+∞)→𝐸 is sequentially continuous; in particular, 𝜑∶𝑋→𝐸 is sequentially continuous. From (35) and (36) it is easy to check that ̸̸1≺1,1≺2,1≺3,1≺4,2≺2,2≺3,2≺4,3≺3,3≺4,4≺4,4≺3, (37) where ≺ is the partial order defined by (1). Let 𝑇∶𝑋→2^𝑋 be a set-valued mapping such that {}{},{}{}.𝑇1=3,4,𝑇2=1,3𝑇3=1,2,3,4,𝑇4=1,2,3 (38) Fix 𝑥₀=2, then [𝑥₀,+∞)={𝑥∈𝑋∶2≺𝑥}={2,3,4} by (37), and so 𝑇𝑥₀∩[𝑥₀,+∞)={3}≠∅. For 𝑥,𝑦∈[𝑥₀,+∞), if 𝑥≺𝑦 and 𝑥≠𝑦, then we have only two cases: 𝑥=2≺3=𝑦 and 𝑥=2≺4=𝑦 by (37). Fix 𝑥=2 and 𝑦=3, for all 𝑢∈𝑇𝑥, there exists 𝑣=3,4∈𝑇𝑦 such that 𝑢≺𝑣. Fix 𝑥=2 and 𝑦=4, for all 𝑢∈𝑇𝑥, there exists 𝑣=3∈𝑇𝑦 such that 𝑢≺𝑣. This means that 𝑇∶𝑋→2^𝑋 is increasing on [𝑥₀,+∞). Therefore all the conditions of Theorem 3 are satisfied and hence 𝑇 has a fixed point 3∈[𝑥₀,+∞).
Fix 𝑥=4; for all 𝑦∈𝑇4, we have 𝑥=4≺𝑦̸ by (37); that is, (2) is not satisfied. Therefore the existence of fixed points could not be obtained by generalized Caristi's fixed point theorems in cone metric spaces of [1, 3].

Acknowledgments

The work was supported by Natural Science Foundation of China (11161022), Natural Science Foundation of Jiangxi Province (20114BAB211006, 20122BAB201015), Educational Department of Jiangxi Province (GJJ12280), and Program for Excellent Youth Talents of JXUFE (201201). The authors are grateful to the editor and referees for their critical suggestions led to the improvement of the presentation of the work.