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

Authors: Zhilong Li; Shujun Jiang
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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:
(1)x≺y⟺d(x,y)⪯φ(x)-φ(y),
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→2X) such that
(2)∀x∈X,x≺Tx(resp.,∀x∈X,∃y∈Tx,x≺y)
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, whenE=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→2X) such that
(3)x0≺1Tx0(resp.,∃y∈Tx0,x0≺1y),
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].LetE 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.LetP 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.LetE be a normed vector space. A cone P of E is continuous [1, 3] if, for all subset A of E, infA exists implies infx∈A∥x-infA∥=0, and supA exists implies supx∈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 {xn},{yn},{zn}⊆E with xn⪯yn⪯zn for all n, xn→x and zn→x imply yn→x for some x∈X.Remark 1.
A total order minihedral coneP of a normed space E is certainly normal see [29].LetX 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 {xn} in (X,d) converges [28] to x∈X (denote xn→dx) if, for all ɛ∈P with θ≪ɛ, there exists a positive integer n0 such that d(xn,x)≪ɛ for all n≥n0. A sequence {xn} in (X,d) is Cauchy [28] if, for all ɛ∈P with θ≪ɛ, there exists a positive integer n0 such that d(xn,xm)≪ɛ for all m,n≥n0. A cone metric space (X,d) is complete [28] if all Cauchy sequence {xn} in (X,d) converges to a point x∈X. A vector functional φ:X→E is sequentially continuous at some x∈X if limn→∞φ(xn)=φ(x) for all {xn}⊆X such that xn→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 {xn} a sequence in (X,d). Then xn→dx if and only if limn→∞d(xn,x)=θ, and {xn} is Cauchy if and only if limm,n→∞d(xn,xm)=θ see [28].LetX 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→2X 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→2X 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 pointx*∈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→2X 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.
SinceP is a total order minihedral cone and E is a normed space, then P is a normal cone by Remark 1. Set
(4)Q1={x∈[x0,+∞):Tx∩[x,+∞)≠∅}.
Clearly, Q1 is nonempty since x0∈Q1. Let {xα}α∈Γ⊆Q1 be an increasing chain, where Γ is a directed set. Then by (1) we have
(5)d(xα,xβ)⪯φ(xα)-φ(xβ),
for all α,β∈Γ with α⪯β. This implies that {φ(xα)} is a decreasing chain in E. Since P is total order minihedral and φ is bounded below on [x0,+∞), then infα∈Γφ(xα) exists in E. Moreover, infα∈Γ∥φ(xα)-infα∈Γφ(xα)∥=0 since P is continuous. Therefore there exists an increasing sequence {xαn}⊆{xα} such that limn→∞∥φ(xαn)-infα∈Γφ(xα)∥=0, that is,
(6)limn→∞φ(xαn)=infα∈Γφ(xα).
By (1) we have for all m∈ℕ such that m≥n,
(7)d(xαn,xαm)⪯φ(xαn)-φ(xαm).
Let n→∞, by (6) we have limm,n→∞[φ(xαn)-φ(xαm)]=θ and hence limm,n→∞d(xαn,xαm)=θ by the normality of P. Moreover by Remark 2, {xαn} is a Cauchy sequence in X. Therefore by the completeness of X, there exists some x¯∈X such that
(8)xαn→dx¯.
Note that {xαn} is an increasing sequence of Q1, then by (1), we have for all n,
(9)d(x0,xαn)⪯φ(x0)-φ(xαn),
And, for all n≥n0,
(10)d(xαn0,xαn)⪯φ(xαn0)-φ(xαn),
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(xαn0,x¯)⪯φ(xαn0)-φ(x¯), that is, x¯∈[x0,+∞) and xαn0≺x¯. Moreover the arbitrary property of n0 forces that
(11)xαn≺x¯,
for all n. By xαn∈Q1, there exists yn∈Txαn such that
(12)xαn≺yn,
for all n. Since T is increasing on [x0,+∞), then by (11) and x¯∈[x0,+∞), there exists zn∈Tx¯ such that
(13)yn≺zn,
for all n. This together with (12) implies that
(14)xαn≺zn,
for all n. Note that Tx¯ is compact, and there exists a subsequence {znk}⊆{zn} and z∈Tx¯ such that
(15)znk→z.
From (14) we have xαnk≺znk for all nk and hence by (1),
(16)d(xαnk,znk)⪯φ(xαnk)-φ(znk),
for all nk. Let nk→∞, 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 allα∈Γ, if there exists some n0 such that xα≺xαn0, then x¯ is an upper bound of {xα} by (11). Otherwise, there exists some β∈Γ such that xαn≺xβ for all n. Thus by (1) we have φ(xαn)-φ(xβ)∈P for all n. Let n→∞, by (6) we have infα∈Γφ(xα)-φ(xβ)∈P; that is, φ(xβ)⪯infα∈Γφ(xα). So we have φ(xβ)=infα∈Γφ(xα) and hence φ(xβ)⪯φ(xα) for all α∈Γ. Note that {φ(xα)}α∈Γ is a decreasing chain, then β≥α for all α∈Γ. Moreover xα≺xβ for all α∈Γ since {xα}α∈Γ is an increasing chain. Hence {xα}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→2X 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(17)Q2={x∈(-∞,y0]:Tx∩(-∞,x]≠∅}.
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.
IfT: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 whenT 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 Theorem3 and Remark 5, T has a maximal fixed point x*∈[x0,+∞) and hence F={x∈[x0,+∞):x=Tx}≠∅. Set
(18)S={I=[x,+∞):x∈[x0,+∞),x≺Tx,F⊆I}.
Clearly, [x0,+∞)∈S and hence S≠∅. Define a relation ⊑ on S by
(19)I1⊑I2⟺I1⊆I2,
for all I1,I2∈S, then it is easy to check that ⊑ is a partial order on S.
Let{Iα}α∈Γ be a decreasing chain of S, where Iα=[xα,+∞). From (1), (18), and (19) we find that {xα}α∈Γ is an increasing chain of M, where
(20)M={x∈[x0,+∞):x≺Tx,F⊆[x,+∞)}.
Set Q¯1={x∈[x0,+∞):x≺Tx}. Clearly, M⊆Q¯1. Following the proof of Theorem 3, there exists x¯∈Q¯1 and an increasing sequence {xαn}⊆{xα} satisfying (6) such that (8) and (11) are satisfied. From xαn∈M we have that xαn≺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,
(21)xαn≺Txαn≺Tx=x,
and hence by (1),
(22)d(xαn,x)⪯φ(xαn)-φ(x),
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,
(23)x¯≺x,
for all x∈F. This together with x¯∈Q¯1 implies x¯∈M. Then in analogy to the proof of Theorem 3, by (6), (8), and x¯∈M we can prove {xα}α∈Γ has an upper bound x^∈M. By (18), we have [x^,+∞)∈S. Note that x^ is an upper bound of {xα}α∈Γ in M, then [x^,+∞)⊆Iα for all α∈Γ and hence by (19),
(24)[x^,+∞)⊑Iα,
for all α∈Γ. This means [x^,+∞) is a lower bound of {Iα}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
(25)x*≺x,
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 Theorem4 and Remark 5, T has a minimal fixed point in x*∈(-∞,y0]. Set
(26)S¯={J=(-∞,x]:x∈(-∞,y0],Tx≺x,F⊆J}.
Define a relation ⊑S¯ on S¯ as follows:
(27)J1⊑S¯J2⟺J1⊆J2,
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 allx∈[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
(28)S1={I=[x,y0]:x∈[x0,y0],x≺Tx,G⊆I},S2={J=[x0,x]:x∈[x0,y0],Tx≺x,G⊆J},
where G={x∈[x0,y0]:Tx=x} is nonempty. Define ⊑1 on S1 and ⊑2 on S2, respectively, by
(29)I1⊑1I2⟺I1⊆I2,∀I1,I2∈S1,J1⊑2J2⟺J1⊆J2,∀J1,J2∈S2,
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],
(30)x*≺x≺y*,(31)x*≺Tx*≺Ty*≺y*.
Moreover by (30) and the increasing property of T on [x0,y0], for all x∈G, we have
(32)x0≺Tx0≺Tx*≺x≺Ty*≺Ty0≺y0,
and so by (31),
(33)x*≺Tx*≺T(Tx*)≺T(Ty*)≺Ty*≺y*.
From (32) and (33) we have that [Tx*,y0]∈S1, [x0,Ty*]∈S2, and
(34)[Tx*,y0]⊑1I*,[x0,Ty*]⊑2J*,
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.
Theorems3–8 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 Theorem3.Example 10.
LetX={1,2,3,4}, E=ℝ2 with the norm ∥u∥=u12+u22 for all u=(u1,u2)∈ℝ2 and P=ℝ+2. Clearly, P is a strongly minihedral and continuous cone of E. Define a mapping d:ℝ×ℝ→P by
(35)d(x,y)=(|x-y|,|x-y|1/2),∀x,y∈ℝ,
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
(36)φ(x)=(6x,32+23x),
for all x∈[1,+∞). For arbitrary x∈[1,+∞), let {xn}⊆[1,+∞) be a sequence such that xn→dx, then xn→|·|x and hence ∥φ(xn)-φ(x)∥→0, that is, limn→∞φ(xn)=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
(37)1≺1,1≺2,1≺3,1≺4,2≺2,2≺3,2≺4,3≺3,3⊀4,4≺4,4⊀3,
where ≺ is the partial order defined by (1). Let T:X→2X be a set-valued mapping such that
(38)T1={3,4},T2={1,3},T3={1,2,3,4},T4={1,2,3}.
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→2X is increasing on [x0,+∞). Therefore all the conditions of Theorem 3 are satisfied and hence T has a fixed point 3∈[x0,+∞).
Fixx=4; for all y∈T4, we have x=4⊀y 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].



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