Volume 2011, Article ID 186237,12pages doi:10.1155/2011/186237
Research Article
Some Fixed-Point Theorems for Multivalued
Monotone Mappings in Ordered Uniform Space
Duran Turkoglu and Demet Binbasioglu
Department of Mathematics, Faculty of Science, University of Gazi, Teknikokullar 06500, Ankara, Turkey
Correspondence should be addressed to Duran Turkoglu,[email protected]
Received 22 September 2010; Accepted 8 March 2011
Academic Editor: Jong Kim
Copyrightq2011 D. Turkoglu and D. Binbasioglu. 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 use the order relation on uniform spaces defined by Altun and Imdad2009to prove some new fixed-point and coupled fixed-point theorems for multivalued monotone mappings in ordered uniform spaces.
1. Introduction
There exists considerable literature of fixed-point theory dealing with results on fixed or common fixed-points in uniform spacee.g., between1–14. But the majority of these results are proved for contractive or contractive type mapping notice from the cited references. Also some fixed-point and coupled fixed-point theorems in partially ordered metric spaces are given in15–20. Recently, Aamri and El Moutawakil2have introduced the concept of
E-distance function on uniform spaces and utilize it to improve some well-known results of the existing literature involving both E-contractive or E-expansive mappings. Lately, Altun and Imdad 21 have introduced a partial ordering on uniform spaces utilizing E -distance function and have used the same to prove a fixed-point theorem for single-valued nondecreasing mappings on ordered uniform spaces. In this paper, we use the partial ordering on uniform spaces which is defined by21, so we prove some fixed-point theorems of multivalued monotone mappings and some coupled fixed-point theorems of multivalued mappings which are given for ordered metric spaces in22on ordered uniform spaces.
Now, we recall some relevant definitions and properties from the foundation of uniform spaces. We call a pairX, ϑ to be a uniform space which consists of a nonempty setXtogether with an uniformityϑwherein the latter begins with a special kind of filter on
a Cauchy sequence with regard to uniformityϑif for anyV ∈ϑ, there existsN≥1 such that
xnandxmareV-close form, n≥ N. An uniformityϑdefines a unique topologyτϑonX
for which the neighborhoods ofx∈Xare the setsVx {y∈X :x, y∈V}whenV runs overϑ.
A uniform space X, ϑis said to be Hausdorff if and only if the intersection of all theV ∈ ϑ reduces to diagonalΔofX, that is,x, y ∈ V forV ∈ ϑ impliesx y. Notice that Hausdorffness of the topology induced by the uniformity guarantees the uniqueness of limit of a sequence in uniform spaces. An element of uniformityϑis said to be symmetrical ifV V−1 {y, x : x, y ∈ V}. Since eachV ∈ ϑ contains a symmetricalW ∈ ϑand if x, y∈Wthenxandyare bothWandV-close and then one may assume that eachV ∈ϑ is symmetrical. When topological concepts are mentioned in the context of a uniform space X, ϑ, they are naturally interpreted with respect to the topological spaceX, τϑ.
2. Preliminaries
We will require the following definitions and lemmas in the sequel.
Definition 2.1see2. LetX, ϑbe a uniform space. A functionp :X×X → R is said to be anE-distance if
p1for any V ∈ ϑ, there existsδ > 0, such thatpz, x ≤ δand pz, y ≤ δ for some
z∈Ximplyx, y∈V,
p2px, y≤px, z pz, y, for allx, y, z∈X.
The following lemma embodies some useful properties ofE-distance.
Lemma 2.2see1,2. LetX, ϑbe a Hausdorffuniform space andpbe anE-distance onX. Let {xn}and{yn}be arbitrary sequences inXand{αn},{βn}be sequences inR converging to 0. Then,
forx, y, z∈X, the following holds:
aifpxn, y≤αnandpxn, z≤βnfor alln∈N, thenyz. In particular, ifpx, y 0
andpx, z 0, thenyz,
bifpxn, yn≤αnandpxn, z≤βnfor alln∈N, then{yn}converges toz,
cifpxn, xm≤αnfor allm > n, then{xn}is a Cauchy sequence inX, ϑ.
LetX, ϑbe a uniform space equipped withE-distancep. A sequence inX isp-Cauchy if it satisfies the usual metric condition. There are several concepts of completeness in this setting.
Definition 2.3see1,2. LetX, ϑbe a uniform space andpbe anE-distance onX. Then iXsaid to beS-complete if for everyp-Cauchy sequence{xn}there existsx∈Xwith
limn→ ∞pxn, x 0,
iiXis said to bep-Cauchy complete if for everyp-Cauchy sequence{xn}there exists
x∈Xwith limn→ ∞xnxwith respect toτϑ,
iiif :X → Xisp-continuous if limn→ ∞pxn, x 0 implies
lim
n→ ∞p
ivf : X → X is τϑ-continuous if limn→ ∞xn x with respect to τϑ implies
limn→ ∞fxnfxwith respect toτϑ.
Remark 2.4see2. LetX, ϑbe a Hausdorffuniform space and let {xn} be a p-Cauchy
sequence. Suppose thatXisS-complete, then there existsx∈Xsuch that limn→ ∞pxn, x 0.
ThenLemma 2.2bgives that limn→ ∞xnxwith respect to the topologyτϑwhich shows thatS-completeness impliesp-Cauchy completeness.
Lemma 2.5see15. LetX, ϑbe a Hausdorffuniform space,pbeE-distance onXandϕ:X →
R. Define the relation “” onXas follows:
xy⇐⇒xy or px, y≤ϕx−ϕy. 2.2
Then “” is a (partial) order onXinduced byϕ.
3. The Fixed-Point Theorems of Multivalued Mappings
Theorem 3.1. Let X, ϑ a Hausdorff uniform space andp is an E-distance on X,ϕ : X → R
be a function which is bounded below and “” the order introduced byϕ. LetXbe also ap-Cauchy complete space,T :X → 2Xbe a multivalued mapping,x, ∞ {y∈X:xy}andM{x∈
X|Tx∩x, ∞/∅}. Suppose that:
iT is upper semicontinuous, that is,xn∈Xandyn∈Txnwithxn → x0andyn → y0, impliesy0∈Tx0,
iiM /∅,
iiifor eachx∈M,Tx∩M∩x, ∞/∅.
ThenT has a fixed-pointx∗and there exists a sequence{xn}with
xn−1xn∈Txn−1, n1,2,3, . . . 3.1
such thatxn → x∗. Moreover ifϕis lower semicontinuous, thenxnx∗for alln.
Proof. By the conditionii, takex0 ∈M. Fromiii, there existx1 ∈Tx0∩Mandx0x1.
Again fromiii, there existx2 ∈Tx1∩M. Thusx1x2.
Continuing this procedure we get a sequence{xn}satisfying
xn−1xn∈Txn−1, n1,2,3, . . . . 3.2
So by the definition of “”, we have· · ·ϕx2≤ ϕx1≤ϕx0, that is, the sequence{ϕxn}
hence it is Cauchy, that is, for allε >0, there existsn0∈Nsuch that for allm > n > n0we have |ϕxm−ϕxn|< ε. Sincexnxm, we havexnxmorpxn, xm≤ϕxn−ϕxm. Therefore,
pxn, xm≤ϕxn−ϕxm
ϕxn−ϕxm
< ε,
3.3
which shows thatin view ofLemma 2.2cthat{xn}isp-Cauchy sequence. By thep-Cauchy completeness ofX,{xn}converges tox∗. SinceTis upper semicontinuous,x∗∈Tx∗.
Moreover, whenϕis lower semicontinuous, for eachn
pxn, x∗ mlim→ ∞pxn, xm
≤ lim
m→ ∞sup
ϕxn−ϕxm
ϕxn−mlim→ ∞infϕxm
≤ϕxn−ϕx∗.
3.4
Soxn x∗, for alln.
Similarly, we can prove the following.
Theorem 3.2. LetX, ϑa Hausdorff uniform space andp anE-distance onX,ϕ : X → Rbe a function which is bounded above and “” the order introduced byϕ. Let X be also ap-Cauchy complete space,T :X → 2Xbe a multivalued mapping,−∞, x {y∈X:yx}andM{x∈
X|Tx∩−∞, x/∅}. Suppose that
iT is upper semicontinuous, that is,xn∈Xandyn∈Txnwithxn → x0andyn → y0, impliesy0∈Tx0,
iiM /∅,
iiifor eachx∈M,Tx∩M∩−∞, x/∅.
ThenT has a fixed-pointx∗and there exists a sequence{xn}with
xn−1xn∈Txn−1, n1,2,3, . . . 3.5
such thatxn → x∗. Moreover, ifϕis upper semicontinuous, thenx∗xnfor alln.
Corollary 3.3. Let X, ϑa Hausdorffuniform space andp is an E-distance onX,ϕ : X → R
be a function which is bounded below and “” the order introduced byϕ. LetXbe also ap-Cauchy complete space,T :X → 2X be a multivalued mapping andx, ∞ {y ∈X : xy}. Suppose
that:
iiT satisfies the monotonic condition: for anyx,y∈Xwithxyand anyu∈Tx, there existsv∈Tysuch thatuv,
iiithere exists anx0∈Xsuch thatTx0∩x0, ∞/∅.
ThenT has a fixed-pointx∗and there exists a sequence{xn}with
xn−1xn∈Txn−1, n1,2,3, . . ., 3.6
such thatxn → x∗. Moreover ifϕis lower semicontinuous, thenxnx∗for alln.
Proof. By iii,x0 ∈ M {x ∈ X : Tx∩x, ∞/∅}. Forx ∈ M, takey ∈ Txand
x y. By the monotonicity ofT, there exists z ∈ Ty such that y z. So y ∈ M, and
Tx∩M∩x, ∞/∅. The conclusion follows fromTheorem 3.1.
Corollary 3.4. Let X, ϑa Hausdorffuniform space andp is an E-distance onX,ϕ : X → R
be a function which is bounded above and “” the order introduced byϕ. LetX be also ap-Cauchy complete space,T :X → 2X be a multivalued mapping and−∞, x {y ∈X : y x}. Suppose
that:
iT is upper semicontinuous,
iiT satisfies the monotonic condition; for anyx, y∈Xwithxyand anyv∈Ty, there existsu∈Txsuch thatuv,
iiithere exists anx0∈Xsuch thatTx0∩−∞, x0/∅.
ThenT has a fixed-pointx∗and there exists a sequence{xn}with
xn−1 xn ∈Txn−1, n1,2, . . . , 3.7
such thatxn → x∗. Moreover ifϕis upper semicontinuous, thenxnx∗for alln.
Corollary 3.5. Let X, ϑa Hausdorffuniform space andp is an E-distance onX,ϕ : X → R
be a function which is bounded below and “” the order introduced byϕ. LetXbe also ap-Cauchy complete space,f:X → Xbe a map andM{x∈X:xfx}. Suppose that:
ifisτϑ-continuous,
iiM /∅,
iiifor eachx∈M,fx∈M.
Thenfhas a fixed-pointx∗and the sequence
xn−1xnfxn−1, n1,2,3, . . . 3.8
converges tox∗. Moreover ifϕis lower semicontinuous, thenxnx∗for alln.
complete space,f:X → Xbe a map andM{x∈X:xfx}. Suppose that: ifisτϑ-continuous,
iiM /∅,
iiifor eachx∈M,fx∈M.
Thenfhas a fixed-pointx∗. And the sequence
xn−1xnfxn−1, n1,2,3, . . . 3.9
converges tox∗. Moreover, ifϕis upper semicontinuous, thenxnx∗for alln.
Corollary 3.7. LetX, ϑbe a Hausdorffuniform space,pis anE-distance onX,ϕ : X → Rbe a function which is bounded below, and “” the order introduced byϕ. LetX be also ap-Cauchy complete space,f:X → Xbe a map andM{x∈X:xfx}. Suppose that:
ifisτϑ-continuous,
iifis monotone increasing, that is, forxywe havefxfy,
iiithere exists anx0, withx0fx0.
Thenfhas a fixed-pointx∗and the sequence
xn−1xnfxn−1, n1,2,3, . . . 3.10
converges tox∗. Moreover ifϕis lower semicontinuous, thenxnx∗for alln.
Example 3.8. LetX {k, l, m}andϑ {V ⊂ X×X : Δ ⊂ V}. Definep : X×X → R as
px, x 0 for allx∈ X,pk, l pl, k 2,pk, m pm, k 1 vepl, m pm, l 3. Since definition ofϑ,V∈ϑV Δand this show that the uniform spaceX, ϑis a Hausdorff uniform space. On the other hand,pk, l≤ pk, m pm, l,pk, m≤ pk, l pl, mand
pl, m ≤ pl, k pk, m for k, l, m ∈ X and thus p is anE-distance as it is a metric on
X. Next defineϕ : X → Rϕk 3, ϕl 2,ϕm 1. Sincepk, m pm, k 1 ≤
ϕk−ϕm, thereforek m. But aspl, k pk, l 2 |ϕk−ϕl|thereforekland
lk. Again similarlylmandmlwhich show that this ordering is partial and henceXis a partially ordered uniform space. Definef :X → Xasfk k,fl landfm m, then by a routine calculation one can verify that all the conditions ofCorollary 3.7are satisfied andf has a fixed-point. Notice thatpfk, fl pk, lwhich shows thatf is neitherE -contractive norEexpansive, therefore the results of 2are not applicable in the context of this example. Thus, this example demonstrates the utility of our result.
Corollary 3.9. LetX, ϑbe a Hausdorffuniform space,pis anE-distance onX,ϕ:X → Rbe a function which is bounded above and “” the order introduced byϕ. LetXbe also ap-Cauchy complete space andf:X → Xbe a map. Suppose that
ifisτϑ-continuous,
iifis monotone increasing, that is, forxywe havefxfy,
Thenfhas a fixed-pointx∗. And the sequence
xn−1xnfxn−1, n1,2,3, . . . 3.11
converges tox∗. Moreover ifϕis upper semicontinuous, thenxnx∗for alln.
Theorem 3.10. LetX, ϑbe a Hausdorffuniform space,pis anE-distance onX,ϕ : X → Rbe a continuous function bounded below and “” the order introduced byϕ. LetX be also ap-Cauchy complete space,T :X → 2X be a multivalued mapping andx, ∞ {y ∈X : xy}. Suppose
that
iTsatisfies the monotonic condition: for eachxyand eachu∈Txthere existsv∈Ty such thatuv,
iiTxis compact for eachx∈X,
iiiM{x∈X:Tx∩x, ∞/∅}/∅.
ThenT has a fixed-pointx0.
Proof. We will prove thatMhas a maximum element. Let{xv}v∈Λbe a totally ordered subset inM, where Λis a directed set. Forv, μ ∈ Λ andv ≤ μ, one hasxv xμ, which implies
thatϕxv≥ ϕxμforv ≤μ. Sinceϕis bounded below,{ϕxv}is a convergence net inR. Frompxv, xμ ≤ ϕxv−ϕxμ, we get that{xv}is a p-cauchy net inX. By the p-Cauchy
completeness ofX, letxvconverge tozinX.
For givenμ∈Λ
pxμ, z limvpxμ, xv≤limvϕxμ−ϕxv ϕxμ−ϕxz. Soxμzfor allμ∈Λ.
Forμ∈Λ, by the conditioni, for eachuμ∈Txμ, there exists avμ ∈Tzsuch that
uμvμ. By the compactness ofTz, there exists a convergence subnet{vμ|}of{vμ}. Suppose
that{vμ|}converges tow∈Tz. TakeΛ|such thatμ|≥Λ|impliesuμvμvμ|.
We have
puμ, wlim
μ| p
uμ, vμ|
≤lim
μ|
ϕuμ−ϕ
vμ|
ϕuμ−ϕw. 3.12
Souμ wfor allμand
pz, w lim
μ p
uμ, w≤limμ ϕuμ−ϕwϕz−ϕw. 3.13
Soz w and this gives thatz ∈ M. Hence we have proven that{xμ}has an upper
bound inM.
By Zorn’s Lemma, there exists a maximum elementx0 inM. By the definition ofM,
there exists ay0 ∈Tx0such thatx0y0. By the conditioni, there exists az0 ∈Ty0such
thaty0 z0. Hencey0 ∈M. Sincex0is the maximum element inM, it follows thaty0 x0
andx0∈Tx0. Sox0is a fixed-point ofT.
that
iTsatisfies the following condition; for eachxyandv∈Tx, there existsu∈Tysuch thatuv,
iiTxis compact for eachx∈X,
iiiM{x∈X:Tx∩−∞, x/∅}/∅.
ThenT has a fixed-point.
Corollary 3.12. LetX, ϑbe a Hausdorffuniform space,pis anE-distance onX,ϕ :X → Rbe a continuous function bounded below and “” the order introduced byϕ. LetX be also ap-Cauchy complete space andf:X → Xbe a map. Suppose that;
ifis monotone increasing, that is forxy,fxfy,
iithere is anx0∈Xsuch thatx0 fx0. Thenfhas a fixed-point.
Corollary 3.13. LetX, ϑbe a Hausdorffuniform space,pis anE-distance onX,ϕ :X → Rbe a continuous function bounded above and “” the order introduced byϕ. LetX be also ap-Cauchy complete space andf:X → Xbe a map. Suppose that;
ifis monotone increasing, that is, forxy,fxfy;
iithere is anx0∈Xsuch thatx0 fx0. Thenfhas a fixed-point.
4. The Coupled Fixed-Point Theorems of Multivalued Mappings
Definition 4.1. An elementx, y∈ X×Xis called a coupled fixed-point of the multivalued mappingT :X×X → 2Xifx∈Tx, y,y∈Ty, x.
Theorem 4.2. LetX, ϑbe a Hausdorff uniform space,pis anE-distance onX,ϕ : X → Rbe a function bounded below and “” be the order in X introduced byϕ. LetX be also ap-Cauchy complete space, T : X ×X → 2X be a multivalued mapping, x, ∞ {y ∈ X : x y}, −∞, y {x ∈ X : x y}, andM {x, y ∈ X ×X : x y,Tx, y∩x, ∞/∅ and
Ty, x∩−∞, y/∅}. Suppose that:
iT is upper semicontinuous, that is,xn ∈X,yn ∈X andzn ∈Txn, yn, withxn → x0,
yn → y0andzn → z0impliesz0 ∈Tx0, y0, iiM /∅,
iiifor each x, y ∈ M, there isu, v ∈ Msuch that u ∈ Tx, y∩x, ∞ and v ∈
Ty, x∩−∞, y.
ThenT has a coupled fixed-pointx∗, y∗, that is,x∗ ∈ Tx∗, y∗andy∗ ∈ Ty∗, x∗. And there exist two sequences{xn}and{yn}with
xn−1xn∈Txn−1, yn−1
, yn−1yn∈Tyn−1, xn−1
, n1,2,3, . . . 4.1
Proof. By the conditionii, takex0, y0∈ M. Fromiii, there existx1, y1 ∈Msuch that
x1 ∈Tx0, y0,x0 x1 andy1 ∈Ty0, x0,y1 y0. Again fromiii, there existx2, y2 ∈M
such thatx2∈Tx1, y1,x1x2andy2∈Ty1, x1,y2y1.
Continuing this procedure we get two sequences{xn}and{yn}satisfyingxn, yn∈M
and
xn−1xn∈Txn−1, yn−1
, n1,2, . . . ,
yn−1yn∈Tyn−1, xn−1
, n1,2, . . . . 4.2
So
x0x1 · · · xn · · · yn · · · y2y1. 4.3
Hence,
ϕx0≥ϕx1≥ · · · ≥ϕxn≥ · · · ≥ϕyn≥ · · · ≥ϕy1
≥ϕy0
. 4.4
From this we get thatϕxnandϕynare convergent sequences. By the definition of “” as in
the proof ofTheorem 3.1, it is easy to prove that{xn}and{yn}arep-Cauchy sequences. Since
X isp-Cauchy complete, let{xn}converge tox∗ and{yn}converge toy∗. SinceT is upper
semicontinuous,x∗ ∈Tx∗, y∗andy∗ ∈Ty∗, x∗. Hencex∗, y∗is a coupled fixed-point of
T.
Corollary 4.3. LetX, ϑbe a Hausdorffuniform space,pis anE-distance onX,ϕ : X → Rbe a function bounded below, and “” be the order in X introduced byϕ. Let X be also ap-Cauchy complete space,f :X×X → Xbe a mapping andM{x, y∈X×X :xyandxfx, y andfx, yy}. Suppose that;
ifisτϑ-continuous,
iiM /∅,
iiifor eachx, y∈M,xfx, yandfy, xy.
Thenf has a coupled fixed-pointx∗, y∗, that is,x∗ fx∗, y∗andy∗ fy∗, x∗. And there exist two sequences{xn}and{yn}withxn−1xnfxn−1, yn−1,yn−1 ynfyn−1, xn−1,
n1,2, . . .such thatxn → x∗andyn → y∗.
Corollary 4.4. LetX, ϑbe a Hausdorffuniform space,pis anE-distance on X,ϕ : X → Rbe a function bounded below, and “” be the order in X introduced byϕ. Let X be also ap-Cauchy complete space,f :X×X → Xbe a mapping andM{x, y∈X×X :xyandxfx, y andfx, yy}. Suppose that;
ifisτϑ-continuous,
iiM /∅,
iiifis mixed monotone, that is for eachx1x2andy1y2,fx1, y1fx2, y2.
Thenfhas a coupled fixed-pointx∗, y∗. And there exist two sequences{xn}and{yn}with
xn−1 xn fxn−1, yn−1, yn−1 yn fyn−1, xn−1,n 1,2, . . . such thatxn → x∗ and
Theorem 4.5. Let X, ϑbe a Hausdorffuniform space, p is anE-distance on X, ϕ : X → R
be a continuous function, and “” be the order in X introduced by ϕ. Let X be also ap-Cauchy complete space, T : X ×X → 2X be a multivalued mapping, x, ∞ {y ∈ X : x y}, −∞, y {x ∈ X : x y}, andM {x, y ∈ X ×X : x y,Tx, y∩x, ∞/∅ and
Ty, x∩−∞, y/∅}. Suppose that;
iT is mixed monotone, that is, forx1 y1,x2 y2andu∈Tx1, y1,v∈Ty1, x1, there existw∈Tx2, y2,z∈Ty2, x2such thatuw,vz,
iiM /∅,
iiiTx, yis compact for eachx, y∈X×X.
ThenT has a coupled fixed-point.
Proof. Byii, there existsx0, y0 ∈Mwithx0 y0,Tx0, y0∩x0, ∞/∅andTy0, x0∩ −∞, y0/∅. LetC{x, y:x0 x,yy0,Tx, y∩x, ∞/∅andTy, x∩−∞, y/∅}.
Thenx0, y0∈C. Define the order relation “” inCby
x1, y1
x2, y2
⇐⇒x1x2, y2y1. 4.5
It is easy to prove thatC,becomes an ordered space.
We will prove thatC has a maximum element. Let{xv, yv}v∈Λ be a totally ordered subset inC, whereΛis a directed set. Forv, μ∈Λandv≤μ, one hasxv, yvxμ, yμ. So
xvxμandyμyv, which implies that
ϕx0≥ϕxv≥ϕxμ≥ϕy0
, ϕy0
≤ϕyμ≤ϕyv≤ϕx0
4.6
forv≤μ.
Since{ϕxv}and{ϕyv}are convergence nets inR. From
pxv, xμ≤ϕxv−ϕxμ, pyμ, yv≤ϕyμ−ϕyv, 4.7
we get that{xv}and{yv}arep-Cauchy nets inX. By thep-Cauchy completeness ofX, letxv
convergence tox∗andyvconvergence toy∗inX. For givenμ∈Λ,
pxμ, x∗limv pxμ, xv≤limv ϕxμ−ϕxvϕxμ−ϕx∗,
pyμ, y∗limv pyμ, yv≤limv ϕyv−ϕyμϕyv−ϕy∗.
4.8
Sox0xμx∗andyμy∗y0for allμ∈Λ.
Forμ∈Λ, by the conditioni, for eachuμ∈Txμ, yμwithxμuμandvμ∈Tyμ, xμ
withvμ yμ, there existwμ ∈Tx∗, y∗andzμ ∈Ty∗, x∗such thatuμ wμ andvμ zμ.
and {zμ|} of {zμ}. Suppose that {wμ|} converges to w ∈ Tx∗, y∗and {zμ|} converges to
z∈Ty∗, x∗. TakeΛ|, such thatμ|≥Λ|impliesu
μvμvμ|. We have
puμ, wlim
μ| p
uμ, uμ|
≤lim
μ|
ϕuμ−ϕ
uμ|
ϕuμ−ϕw,
pz, vμlim
μ| p
vμ|, vμ
≤lim
μ|
ϕvμ|
−ϕvμϕz−ϕvμ.
4.9
Soxμ uμwandzvμ yμfor allμ. And
px∗, w lim
μ| p
xμ|, uμ|
≤lim
μ|
ϕxμ|
−ϕuμ|
ϕx∗−ϕw,
pz, y∗lim
μ| p
vμ|, yμ|
≤lim
μ|
ϕvμ|
−ϕyμ|
ϕz−ϕy∗. 4.10
Sox∗wandzy∗, this gives thatx∗, y∗∈C. Hence we have proven that{xμ, yμ}μ∈Λhas an upper bound inC.
By Zorn’s lemma, there exists a maximum elementx, yinC. By the definition ofC, there existu ∈ Tx, y,v ∈ Ty, x, such thatx0 u,v y0 andx u,v y. By the
conditionithere existw ∈ Tu, v,z−∈ Tv, u such thatx0 u w and z v y0.
Henceu, v ∈ Candx, y u, v. Sincex, yis maximum element inC, it follows that x, y u, v, and it follows thatx u ∈Tx, uandy v ∈Ty, x. Sox, yis a coupled fixed-point ofT.
Corollary 4.6. LetX, ϑbe a Hausdorffuniform space,pis anE-distance onX,ϕ:X → Rbe a continuous function, and “” be the order inXintroduced byϕ. LetXbe also ap-Cauchy complete space andf:X×X → Xbe a mapping. Suppose that;
ifis mixed monotone, that is forx1y1,x2 y2andfx1, y1fy2, x2, iithere existx0, y0∈Xsuch thatx0fx0, y0andfy0, x0y0.
Thenfhas a coupled fixed-point.
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