JFSJournal of Function Spaces2314-88882314-8896Hindawi10.1155/2020/53935215393521Research ArticleNew Results for Set-Valued Mappings on Ordered Metric SpacesSabetghadamF.1MasihaH. P.1https://orcid.org/0000-0003-4606-7211AydiH.234PerssonLars E.1Faculty of MathematicsK. N. Toosi University of TechnologyP.O. Box 16315-1618TehranIrankntu.ac.ir2Nonlinear Analysis Research GroupTon Duc Thang UniversityHo Chi Minh CityVietnamtdtu.edu.vn3Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnamtdtu.edu.vn4China Medical University HospitalChina Medical UniversityTaichung 40402Taiwancmu.edu.cn202024920202020622020228202024920202020Copyright © 2020 F. Sabetghadam et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We establish some new fixed point results for order-closed multivalued mappings in complete metric spaces endowed with a partial order.

1. Introduction

In 1976, an order relation was defined in metric spaces . Later, many researchers proved various fixed point results in this setting (see ), while the authors in [4, 5, 9] considered coupled questions for so-called monotone conditions. Zhang  proved some (coupled) fixed point theorems for multivalued mappings with monotone conditions in metric spaces with a partial order. Since then, Agarwal and Khamsi  extended Caristi’s fixed point to vector-valued metric spaces. Also, Chung  considered nonlinear contraction mappings. Nadler  defined multivalued contractions, and Assad and Kirk  proved fixed point theorems for set-valued mappings (see also the related works ).

In this paper, we present some fixed point results in ordered metric spaces for order-closed multivalued operators. First, we need some facts.

Lemma 1.

 Let E,d be a metric space and ϕ:E,+ be a functional. Consider γ:0,+0,+ a nondecreasing, continuous, and subadditive function so that γ10=0. Take the relation “” on E given as (1)uviffγdu,vϕuϕv,forallu,vE.

Then, “” is a partial order relation on E. Apparently, uv then ϕuϕv.

Here, we state some definitions. Let E be a topological space. Denote by NE the family of nonempty subsets of E. Let be a partial order on E.

Definition 2.

 Given two nonempty subsets UandV of E. (2)r1IfforeachaU,thereisbVsothatab,thenU1Vr2IfforeachbV,thereisaUsothatab,thenU2Vr3IfU1VandU2V,thenUV.

Note that 1 and 2 are different relations between U and V (see Remark 114 of ). Also, 1, 2, and are not partial orders on NE (see Remark 2 of ).

Definition 3.

If unE satisfies u1u2un or u1u2un, then un is called a monotone sequence.

Definition 4.

A multivalued operator G:ENE is said to be order-closed if, for monotone sequences un and vnE, we have unu0, vnv0, and vnGun imply v0Gu0.

Definition 5.

A function g:E,+ is said to be order upper (lower) semicontinuous, if, for a monotone sequence unE with u0E, we have (3)unu0limn¯gungu0,gu0limn¯gun.

Note that an upper (lower) semicontinuous function is an order upper (lower) semicontinuous. But the converse is not true (see Remark 3 of ).

2. Main Results

Let E,d be a metric space. For ϕ:E, define the partial order “” on E induced by ϕ and γ in Lemma 1.

Theorem 6.

Let E,d, be a complete ordered metric space and ϕ:E,+ be a bounded below function. Suppose that G:ENE is order-closed with respect to “” so that (4)HforanyuE,thereisvGusothatuv

Then, there is a monotone sequence unn=0E, so that un+1Gun for n=0,1,2,, and un converges to u, which is a fixed point of G. If in addition, ϕ is order lower semicontinuous on E, then unu for each n.

Proof.

By the condition H, take u0E. From H, there is u1Gu0 so that u0u1. Again, from H, there is u2Gu1 with u1u2. Continuing this procedure, we have an increasing sequence un so that un+1Gun. Therefore, (5)ϕun1ϕunforalln.

That is, the real sequence ϕun is decreasing, so it is a convergent sequence because ϕ is bounded from below on E.

Using Remark 3 of  and Remark 2 of , we find that (6)lim0+γ=supγ:>0,provided that γ is subadditive. Thus, (7)liminf0+γ>0.

By (7), there are δ>0 and c>0 so that (8)γc,forall0,δ.

Since γ is nondecreasing, we have γγδ for each δ,+.

Let 0<ε<γδ. Then (9)γ>ε,foreachδ,+,i.e., (10)ifγε,then0,δ.

Therefore, we have (11)0:γε0,δ,which together with (8) implies that (12)γc,forall0:γε.

Note that ϕun is convergent; then, there is n0 so that for all mnn0, (13)γdun,umϕunϕum<ε.

Moreover, by (12), we get (14)cdun,umγdun,umϕunϕum,forallmnn0.

The convergence of ϕun implies that un is a Cauchy sequence. By the completeness of E, there is uE so that unu as n. Since G is order-closed, un is monotone and un+1Gun. Consequently, (15)uGu,i.e., u is a fixed point of G.

If ϕ is order lower semicontinuous on E, by definition of “,” then we have for each m(16)γdum,u=limnγdum,unlimn¯ϕumϕun=ϕumlimn¯ϕunϕumϕu.

It yields that umu. The proof is completed.

The proof of the following theorem carries over in the same manner as for Theorem 6.

Theorem 7.

Let E,d, be a complete ordered metric space and ϕ:E,+ be a bounded below function. Suppose that G:ENE is order-closed with respect to “.” Assume that (17)HforeachuE,thereisvGusothatvu

Then, there is a monotone sequence unn=0E, un+1Gun, n=0,1,2,, such that un converges to u, which is a fixed point of G. If, in addition, ϕ is order upper semicontinuous on E, then uun for all n.

Theorem 8.

Let E,d, be a complete ordered metric space and ϕ:E,+ be bounded below function. Suppose that G:ENE is order-closed with respect to “.” Assume that

for each u,vE with uvGu1Gv

there exists u0E such that, u01Gu0.

Then, G has a fixed point u, and there is a sequence un with un1unGun1 for n1, such that unx. Moreover, if ϕ is order lower semicontinuous, then unu for all n.

Proof.

Since Gu, by ii, we can choose u1Gu0 so that x0u1. This implies that Gu01Gu1, by definition of 1, there is u2Gu1 so that u1u2. Continuing this procedure, we can find an increasing sequence un such that unGun1. The rest of the proof is similar as in Theorem 6.

The following supports Theorem 8.

Example 9.

Let E=0,1 and dξ,σ=ξσ, for ξ,σE. The metric space E,d is complete. Consider the multivalued mapping G:ENE given as (18)Gξ=ξ5,ξ4forξEξ5=3,ξ2forξE,where Q is the set of rational numbers. Let γs=ss0 and ϕξ=ξξE. Note that ϕ is bounded from below. Take the order induced by ϕ, that is, given as follows: (19)ξσγdξ,σϕξϕσ.

Mention that G verifies the following assertions:

for each u,vE with uv, we have Gu1Gv

01G0

G is order-closed on E.

Hence, G has a fixed point on E, which is u=0.

Theorem 10.

Let E,d, be a complete ordered metric space and ϕ:E,+ be a bounded below function. Suppose G:ENE is order-closed with respect to “.” If the following conditions hold:

for each u,vE with uvGu2Gv

there exists u0E such that, Gu02u0

then G has a fixed point u. Also, there is a sequence un with un1unGun1 for any n1 such that unu. Moreover, if ϕ is order upper semicontinuous, then unu for all n.

The following example illustrates Theorem 10.

Example 11.

Let E=0,1, dξ,σ=ξσ, ϕξ=ξ for ξ,σE and γt=t for each t0. Here, E is a complete metric space, and ϕ is a bounded below function. Consider the order induced by ϕ: (20)ξσdξ,σϕξϕσ.

Clearly, this partial order is the usual order on E. Define G:ENE by Gξ=ξ/3,ξ/2. It is obvious that G satisfies the following:

for each u,vE with uvGu2Gv

G121

G is order-closed on E.

Hence, G has a fixed point on E, which is u=0.

Now, a multivalued version of Theorem 2 of  may be obtained. Here, the considered multivalued mapping is not necessarily continuous.

Theorem 12.

Let E,d, be a complete ordered metric space and ϕ:E,+ be a continuous function bounded below. Let G:ENE be a multivalued mapping. Suppose that (21)H1foranyuEthereisvGusothatuvH2GuiscompactforeachuE.

Then, G has a fixed point.

Proof.

Set (22)MuEvGu:vu.

We claim that M has a maximum element. For a directed set I, let uiiI be a totally ordered subset in M. For i,jI with ij, the fact that uiuj yields ϕuiϕuj. Due to the fact that ϕ is bounded below, ϕui is a convergent set in . Consider (23)γdui,ujϕuiϕuj.

As in proof of Theorem 6, ui is Cauchy in E, which is complete, so ui converges to u in E. For jI, (24)γduj,u=limiγduj,uilimiϕujϕui=ϕujϕu.

Hence, uju for each jI. By H1, for each vjGuj, there is wjGu so that vjwj. By the compactness of Gu, there is a convergence subset vj of vj. Assume that vj converges to vGu. Take I such that ij implies vjwjwj. One writes (25)γdvj,v=limjγdvj,wjlimjϕvjϕwj=ϕvjϕv.

So vjv for all j. Also, (26)γdu,v=limjγdvj,vlimjϕvjϕv=ϕuϕv.

So uv and uM. Thus, uj has an upper bound in M.

By Zorn’s Lemma, there is a maximum element uM. also, there is vGu so that uv. Using H1, there is zGv so that vz. Hence, vM. The element u is maximum in M, so v=u and uGu. That is, u is a fixed point of G.

Theorem 13.

Let E,d, be a complete ordered metric space and ϕ:E,+ be a continuous function bounded below. Let G:ENE be a multivalued mapping. Assume that (27)H1ForanyuE,thereisvGusothatvuH2GuiscompactforanyuE.

Then, G has a fixed point.

Remark 14.

If G is a continuous single-valued mapping in Theorem 6 (resp., Theorem 7), we can replace the condition H by (28)H: foreachuE,uGuresp.,Guu,and we can obtain the same result.

Remark 15.

If in Theorem 8 (resp., Theorem 10), G is assumed to be a continuous single-valued mapping, then we get the same result when replacing the conditions i and ii by

G is monotone increasing (resp., decreasing), that is, for uv, we have GuGvresp.,GvGu

there exists u0 with u0Gu0 (resp., Gu0u0).

If in addition ϕ is order upper (resp., lower) semicontinuous on E, then u is the smallest (resp., largest) fixed point of G in K=uEssu0u (resp., K=uEuu0).

Proof.

Let z be a fixed point of G in K, i.e., z=Gz. Since u0z, we get Gu0Gz. Hence, u1=Gu0Gz=z, i.e., u1z. Suppose unz, then GunGz, i.e., un+1=GunGz=z. By a mathematical induction, we have unz for all n, then (29)γdu,z=limndun,zlimn¯ϕunϕz=limn¯ϕunϕzϕuϕz.

That is, uz.

Note that if we omit the continuity of G and add above condition on ϕ, the results remain true.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no competing interests regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.