Volume 2009, Article ID 762478,17pages doi:10.1155/2009/762478
Research Article
Global Attractivity Results for Mixed-Monotone Mappings in Partially Ordered Complete
Metric Spaces
D ˇz. Burgi´c,1 S. Kalabuˇsi ´c,2 and M. R. S. Kulenovi ´c3
1Department of Mathematics, University of Tuzla, 75000 Tuzla, Bosnia and Herzegovina
2Department of Mathematics, University of Sarajevo, 71000 Sarajevo, Bosnia and Herzegovina
3Department of Mathematics, University of Rhode Island, Kingston, R I 02881-0816, USA
Correspondence should be addressed to M. R. S. Kulenovi´c,[email protected] Received 28 October 2008; Revised 17 January 2009; Accepted 9 February 2009 Recommended by Juan J. Nieto
We prove fixed point theorems for mixed-monotone mappings in partially ordered complete metric spaces which satisfy a weaker contraction condition than the classical Banach contraction condition for all points that are related by given ordering. We also give a global attractivity result for all solutions of the difference equationzn1Fzn, zn−1,n2,3, . . . ,whereFsatisfies mixed- monotone conditions with respect to the given ordering.
Copyrightq2009 Dˇz. Burgi´c 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.
1. Introduction and Preliminaries
The following results were obtained first in 1 and were extended to the case of higher- order difference equations and systems in2–6. For the sake of completeness and the readers convenience, we are including short proofs.
Theorem 1.1. Leta, bbe a compact interval of real numbers, and assume that
f :a, b×a, b−→a, b 1.1
is a continuous function satisfying the following properties:
afx, yis nondecreasing inx∈a, bfor eachy∈a, b, andfx, yis nonincreasing in y∈a, bfor eachx∈a, b;
bIfm, M∈a, b×a, bis a solution of the system
fm, M m, fM, m M, 1.2
thenmM.
Then
xn1f
xn, xn−1
, n0,1, . . . 1.3
has a unique equilibriumx∈a, band every solution of1.3converges tox.
Proof. Set
m0 a, M0b, 1.4
and fori1,2, . . .set
Mif
Mi−1, mi−1
, mif
mi−1, Mi−1
. 1.5
Now observe that for eachi≥0,
m0≤m1≤ · · · ≤mi≤ · · · ≤Mi≤ · · · ≤M1≤M0, mi≤xk≤Mi, fork≥2i1.
1.6
Set
m lim
i→ ∞mi, M lim
i→ ∞Mi. 1.7
Then
M≥lim sup
i→ ∞ xi≥lim inf
i→ ∞ xi ≥m 1.8
and by the continuity off,
mfm, M, MfM, m. 1.9
Therefore in view ofb,
mM 1.10
from which the result follows.
Theorem 1.2. Leta, bbe an interval of real numbers and assume that
f :a, b×a, b−→a, b 1.11
is a continuous function satisfying the following properties:
afx, yis nonincreasing inx∈a, bfor eachy∈a, b, andfx, yis nondecreasing in y∈a, bfor eachx∈a, b;
bthe difference equation1.3has no solutions of minimal period two ina, b. Then1.3 has a unique equilibriumx∈a, band every solution of 1.3converges tox.
Proof. Set
m0a, M0b 1.12
and fori1,2, . . .set
Mif
mi−1, Mi−1
, mif
Mi−1, mi−1
. 1.13
Now observe that for eachi≥0,
m0≤m1≤ · · · ≤mi≤ · · · ≤Mi≤ · · · ≤M1≤M0, mi≤xk≤Mi, fork≥2i1.
1.14
Set
m lim
i→ ∞mi, M lim
i→ ∞Mi. 1.15
Then clearly1.8holds and by the continuity off,
mfM, m, Mfm, M. 1.16
In view ofb,
mM 1.17
from which the result follows.
These results have been very useful in proving attractivity results for equilibrium or periodic solutions of 1.3 as well as for higher-order difference equations and systems of difference equations; see2,7–12. Theorems1.1and1.2have attracted considerable attention of the leading specialists in difference equations and discrete dynamical systems and have been generalized and extended to the case of maps inRn, see3, and maps in Banach space
with the cone see4–6. In this paper, we will extend Theorems 1.1and 1.2to the case of monotone mappings in partially ordered complete metric spaces.
On the other hand, there has been recent interest in establishing fixed point theorems in partially ordered complete metric spaces with a contractivity condition which holds for all points that are related by partial ordering; see 13–20. These fixed point results have been applied mainly to the existence of solutions of boundary value problems for differential equations and one of them, namely20, has been applied to the problem of solving matrix equations. See also21, where the application to the boundary value problems for integro- differential equations is given and 22 for application to some classes of nonexpansive mappings and 23 for the application of the Leray-Schauder theory to the problems of an impulsive boundary value problem under the condition of non-well-ordered upper and lower solutions. None of these results is global result, but they are rather existence results. In this paper, we combine the existence results with the results of the type of Theorems1.1and 1.2to obtain global attractivity results.
2. Main Results: Mixed Monotone Case I
LetXbe a partially ordered set and letdbe a metric onXsuch thatX, dis a complete metric space. ConsiderX×X.We will use the following partial ordering.
Forx, y,u, v∈X×X, we have
x, yu, v⇐⇒ {x≤u, y≥v}. 2.1
This partial ordering is well known as “south-east ordering” in competitive systems in the plane; see5,6,12,24,25.
Letd1be a metric onX×Xdefined as follows:
d1x, y,u, v dx, u dy, v. 2.2
Clearly
d1x, y,u, v d1y, x,v, u. 2.3 We prove the following theorem.
Theorem 2.1. LetF :X×X → Xbe a map such thatFx, yis nonincreasing inxfor ally∈X, and nondecreasing inyfor allx∈X.Suppose that the following conditions hold.
iThere existsk∈0,1with dFx, y, Fu, v≤ k
2d1x, y,u, v ∀x, yu, v. 2.4 iiThere existsx0, y0∈Xsuch that the following condition holds:
x0≤F y0, x0
, y0≥F
x0, y0
. 2.5
iiiIf {xn} ∈ X is a nondecreasing convergent sequence such that limn→ ∞xn x, then xn ≤ x, for all n ∈ N and if {yn} ∈ Y is a nonincreasing convergent sequence such that limn→ ∞yn y, then yn ≥ y, for all n ∈ N; if xn ≤ yn for every n, then limn→ ∞xn≤limn→ ∞ yn.
Then we have the following.
aFor every initial point x0, y0 ∈ X ×X such that condition2.5holds,Fnx0, y0 → x, Fny0, x0 → y, n → ∞, wherex, ysatisfy
xFy, x, yFx, y. 2.6
Ifx0≤y0in condition2.5, thenx≤y.If in additionxy, then{xn},{yn}converge to the equilibrium of the equation
xn1F yn, xn
, yn1F
xn, yn
, n1,2, . . . . 2.7
bIn particular, every solution{zn}of
zn1F
zn, zn−1
, n2,3, . . . 2.8
such thatx0≤z0, z1≤y0converges to the equilibrium of 2.8.
cThe following estimates hold:
d Fn
y0, x0
, x
≤ 1 2
kn 1−k
d F
x0, y0
, y0
d
F y0, x0
, x0
, 2.9
d Fn
x0, y0
, y
≤ 1 2
kn 1−k
d F
y0, x0
, x0
d
F x0, y0
, y0
. 2.10
Proof. Letx1Fy0, x0andy1Fx0, y0.Sincex0≤Fy0, x0 x1andy0≥Fx0, y0 y1, forx2 Fy1, x1, y2Fx1, y1,we have
F2 y0, x0
:F
F x0, y0
, F y0, x0
F
y1, x1
x2,
F2 x0, y0
:F
F y0, x0
, F x0, y0
F
x1, y1
y2.
2.11
Now, we have
x2F2 y0, x0
F
y1, x1
≥F
y0, x0
x1, y2F2
x0, y0
F
x1, y1
≤F
x0, y0
y1.
2.12
Forn1,2, . . . ,we let
xn1Fn1 y0, x0
F
Fn x0, y0
, Fn y0, x0
, yn1Fn1
x0, y0
F
Fn y0, x0
, Fn x0, y0
.
2.13
By using the monotonicity ofF, we obtain
x0≤F y0, x0
x1≤F2 y0, x0
x2≤ · · · ≤Fn1 y0, x0
≤ · · ·, y0≥F
x0, y0
y1 ≥F2 x0, y0
y2≥ · · · ≥Fn1 x0, y0
≥ · · · 2.14
that is
x0≤x1≤x2≤ · · · y0≥y1≥y2≥ · · · .
2.15
We claim that for alln∈Nthe following inequalities hold:
d xn1, xn
d
Fn1 y0, x0
, Fn y0, x0
≤ kn 2 d1
x1, y1
, x0, y0
, 2.16
d yn1, yn
d
Fn1 x0, y0
, Fn x0, y0
≤ kn 2 d1
x1, y1
, x0, y0
. 2.17
Indeed, forn1,usingx0≤Fy0, x0,y0 ≥Fx0, y0, and2.3, we obtain
d x2, x1
d
F y1, x1
, F y0, x0
≤ k
2d1
y1, x1
, y0, x0
k
2d1
x1, y1
, x0, y0
,
d y2, y1
d
Fx1, y1
, F x0, y0
≤ k
2d1
x1, y1
, x0, y0
.
2.18
Assume that2.16holds. Using the inequalities
Fn1 y0, x0
≥Fn y0, x0
, Fn1
x0, y0
≤Fn x0, y0
,
2.19
and the contraction condition2.4, we have d
xn2, xn1 d
Fn2 y0, x0
, Fn1 y0, x0
d
F Fn1
x0, y0
, Fn1 y0, x0
, F Fn
x0, y0
, Fn y0, x0
≤ k 2
d Fn1
x0, y0
, Fn x0, y0
d
Fn1 y0, x0
, Fn y0, x0
≤ k 2
kn 2
d F
x0, y0
, y0
d
F y0, x0
, x0
d
F y0, x0
, x0
d
F x0, y0
, y0
kn1 2 d1
x1, y1
, x0, y0
.
2.20
Similarly,
d
yn2, yn1 d
Fn2 x0, y0
, Fn1 x0, y0
≤ kn1
2 d1
x1, y1
, x0, y0
. 2.21
This implies that{xn}{Fny0, x0}and{yn}{Fnx0, y0}are Cauchy sequences inX.
Indeed, d
Fn y0, x0
, Fnp y0, x0
≤d
Fn y0, x0
, Fn1 y0, x0
· · · d
Fnp−1 y0, x0
, Fnp y0, x0
≤ kn 2 d
F x0, y0
, y0
d
F y0, x0
, x0
· · · knp−1
2 d
F x0y0
, y0
d
F y0, x0
, x0
kn
2
1kk2· · ·kp−1 d
F x0, y0
, y0
dF
y0, x0
, x0
kn
2 1−kp
1−k d
F x0, y0
, y0
d
F y0, x0
, x0
.
2.22
Sincek∈0,1,we have
d
xn, xnp d
Fn y0, x0
, Fnp y0, x0
≤ kn
21−kd1
x1, y1
, x0, y0
. 2.23
Using 2.23, we conclude that {xn} {Fny0, x0} is a Cauchy sequence. Similarly, we conclude that{yn}{Fnx0, y0}is a Cauchy sequence. SinceX is a complete metric space, then there existx, y∈Xsuch that
nlim→ ∞xn lim
n→ ∞Fn y0, x0
x , lim
n→ ∞yn lim
m→ ∞Fm x0, y0
y. 2.24
Using the continuity ofF,which follows from contraction condition2.4, the equations xn1F
yn, xn
, yn1F
xn, yn
2.25
imply2.6.
Assume thatx0≤y0.Then, in view of the monotonicity ofF x1F
y0, x0
≤F
x0, y0
y1, x2F
y1, x1
≤F
x1, y1
y2, x3F
y2, x2
≤F
x2, y2
y3.
2.26
By using induction, we can show thatxn≤ynfor alln.Assume thatx0 ≤z0, z1 ≤y0.Then, in view of the monotonicity ofF, we have
x1F y0, x0
≤F
z1, z0
z2≤F x0, y0
y1, x1F
y0, x0
≤F
z2, z1
z3≤F x0, y0
y1.
2.27
Continuing in a similar way we can prove thatxi≤zk≤yifor allk≥2i1.By using condition iiiwe conclude that whenever limn→ ∞zkexists we must have
x≤ lim
k→ ∞zk≤y 2.28
which in the case whenxyimplies limk→ ∞zkx.
By lettingp → ∞in2.23, we obtain the estimate2.9.
Remark 2.2. Propertyiiiis usually called closedness of the partial ordering, see6, and is an important ingredient of the definition of an orderedL-space; see17,19.
Theorem 2.3. Assume that along with conditions (i) and (ii) ofTheorem 2.1, the following condition is satisfied:
ivevery pair of elements has either a lower or an upper bound.
Then, the fixed pointx, yis unique andxy.
Proof. First, we prove that the fixed pointx, yis unique. Conditionivis equivalent to the following. For everyx, y,x∗, y∗∈X×X,there existsz1, z2∈X×Xthat is comparable tox, y,x∗, y∗.See16.
Letx, yandx∗, y∗be two fixed points of the mapF.
We consider two cases.
Case 1. If x, y is comparable to x∗, y∗, then for all n 0,1,2, . . .Fny, x, Fnx, y is comparable toFny∗, x∗, Fnx∗, y∗ x∗, y∗.We have to prove that
d1
x, y,
x∗, y∗
0. 2.29
Indeed, using2.2, we obtain d1
x, y,
x∗, y∗ d
x, x∗ d
y, y∗ d
Fny, x, Fn
y∗, x∗ d
Fnx, y, Fn
x∗, y∗
. 2.30
We estimatedFny, x, Fny∗, x∗, anddFnx, y, Fnx∗, y∗. First, by using contraction condition2.4, we have
d
Fy, x, F
y∗, x∗
≤ k 2
d y, y∗
d x, x∗
k 2d1
x, y,
x∗, y∗ , d
Fx, y, F
x∗, y∗
≤ k 2
d x, x∗
d y, y∗
k 2d1
x, y,
x∗, y∗ .
2.31
Now, by using2.31and2.30, we have d1
x, y,
x∗, y∗
≤kd1
x, y,
x∗, y∗
< d1
x, y,
x∗, y∗
, 2.32
which implies that
d1
x, y,
x∗, y∗
0. 2.33
Case 2. If x, y is not comparable to x∗, y∗, then there exists an upper bound or a lower bound z1, z2 of x, y and x∗, y∗. Then, Fnz2, z1, Fnz1, z2 is comparable to Fny, x, Fnx, yandFny∗, x∗, Fnx∗, y∗.
Therefore, we have d1
x, y,
x∗, y∗ d1
Fny, x, Fnx, y ,
Fn y∗, x∗
, Fn
x∗, y∗
≤d1
Fny, x, Fnx, y ,
Fn z2, z1
, Fn z1, z2
d1
Fn z2, z1
, Fn z1, z2
,
Fn y∗, x∗
, Fn
x∗, y∗ d
Fny, x, Fn z2, z1
d
Fn z2, z1
, Fn
y∗, x∗ d
Fnz1, z2
, Fn
x∗, y∗ d
Fn
z2, z1, Fn
y∗, x∗ .
2.34
Now, we obtain d1
x, y,
x∗, y∗ d
Fny, x, Fn z2, z1
d
Fn z2, z1
, Fn
y∗, x∗ d
Fn z1, z2
, Fn
x∗, y∗ d
Fn z2, z1
, Fn
y∗, x∗ .
2.35 We now estimate the right-hand side of2.35.
First, by using
d
Fy, x, F z2, z1
≤ k
2 d
y, z2
d
x, z1
, 2.36
we have d
F2y, x, F2 z2, z1
d
FFx, y, Fy, x, F F
z1, z2
, F z2, z1
≤ k 2d
Fx, y, F z1, z2
d
Fy, x, F z2, z1
≤ k 2
k 2
d x, z1
d
y, z2
k
2 d
y, z2
d
x, z1
k2 2
d x, z1
dy, z2
.
2.37
Similarly, d
F2x, y, F2 z1, z2
d
FFy, x, Fx, y, F F
z2, z1
, F z1, z2
≤ k 2
d
Fy, x, F z2, z1
d
Fx, y , F
z1, z2
≤ k 2
k 2
d y, z2
d
x, z1
k
2 d
y, z2
d
x, z1
k2 2
d x, z1
d
y, z2
.
2.38
So,
d
F2y, x, F2 z2, z1
≤ k2 2
d x, z1
d
y, z2
,
d
F2x, y, F2 z1, z2
≤ k2 2
d x, z1
d
y, z2
.
2.39
Using induction, we obtain
d
Fny, x, Fn z2, z1
≤ kn 2
d x, z1
d
y, z2
,
d
Fnx, y, Fn z1, z2
≤ kn 2
d x, z1
dy, z2
,
d Fn
z2, z1
, Fn
y∗, x∗
≤ kn 2
d z1, x∗
, d z2, y∗
, d
Fn z1, z2
, Fn
x∗, y∗
≤ kn 2
d z1, x∗
, d z2, y∗
.
2.40
Using2.40, relation2.35becomes
d1
x, y,
x∗, y∗
≤ kn 2
d x, z1
d
y, z2
kn
2 d
x, z1
d
y, z2
kn
2 d
z1, x∗ d
z2, y∗ kn
2 d
z1, x∗ d
z2, y∗ kn
d x, z1
d
y, z2
d
z1, x∗ d
z2, y∗
−→0, n−→ ∞.
2.41
So,
d1
x, y,
x∗, y∗
0. 2.42
Finally, we prove thatxy.We will consider two cases.
Case A. Ifxis comparable toy,thenFy, x xis comparable toFx, y y.Now, we obtain
dx, y dFy, x, Fx, y≤ k
2dx, y dy, x kdx, y, 2.43
sincek∈0,1,this implies
dx, y 0⇐⇒xy. 2.44
Case B. Ifxis not comparable toy,then there exists an upper bound or alower bound ofx andy, that is, there existsz∈Xsuch thatx≤z, y≤z.Then by using monotonicity character
ofF,we have
Fx, y≤Fx, z, Fy, x≤Fy, z,
Fx, y≥Fz, y, Fy, x≥Fz, x. 2.45
Now,
F2x, y FFy, x, Fx, y≤FFz, x, Fx, z F2x, z, 2.46
that is
F2x, y≤F2x, z. 2.47
Furthermore,
F2x, y FFy, x, Fx, y≥FFy, z, Fz, y F2y, z, 2.48
that is
F2x, y≥F2y, z. 2.49
Similarly,
F2y, x FFx, y, Fy, x≤FFz, y, Fy, z F2y, z, 2.50
that is
F2y, x≤F2y, z, 2.51
and
F2y, x FFx, y, Fy, x≥FFx, z, Fz, x F2z, x. 2.52
By using induction, we have
Fn1x, y≤Fn1x, z, Fn1x, y≥Fn1y, z, Fn1y, x≤Fn1y, z, Fn1y, x≥Fn1z, x.
2.53
Sincex, yis a fixed point, we obtain
dx, y d
Fn1y, x, Fn1x, y d
F
Fnx, y, Fny, x , F
Fny, x, Fnx, y
≤d F
Fnx, y, Fny, x , F
Fnx, z, Fnz, x d
F
Fnx, z, Fnz, x , F
Fny, x, Fnx, y
≤d F
Fnx, y, Fny, x , F
Fnx, z, Fnz, x d
F
Fnz, x, Fnx, z , F
Fnx, z, Fnz, x d
F
Fny, x, Fnx, y , F
Fnz, x, Fnx, z .
2.54
Using the contractivity condition2.4onF,we have
dx, y≤ k 2
d
Fnx, y, Fnx, z d
Fny, x, Fnz, x k
2 d
Fnz, x, Fnx, z d
Fnx, z, Fnz, x k
2 d
Fny, x, Fnz, x d
Fny, x, Fnx, y k
2 2d
Fnx, y, Fnx, z 2d
Fny, x, Fnz, x 2d
Fnx, z, Fnz, x k
d
Fnx, y, Fnx, z d
Fny, x, Fnz, x d
Fnx, z, Fnz, x .
2.55
Now, we estimate the terms on the right-hand side
d
Fnx, y, Fnx, z d
F
Fn−1y, x, Fn−1x, y , F
Fn−1z, x, Fn−1x, z
≤ k 2
d
Fn−1y, x, Fn−1z, x d
Fn−1x, y, Fn−1x, z , d
Fny, x, Fnz, x d
F
Fn−1x, y, Fn−1y, x , F
Fn−1x, z, Fn−1z, x
≤ k 2
d
Fn−1x, y, Fn−1x, z d
Fn−1y, x, Fn−1z, x , d
Fnx, z, Fnz, x d
F
Fn−1z, x, Fn−1x, z , F
Fn−1x, z, Fn−1z, x
≤ k 2
d
Fn−1z, x, Fn−1x, z d
Fn−1x, z, Fn−1z, x .
2.56
Now, we have dx, y≤k2
d
Fn−1y, x, Fn−1z, x d
Fn−1x, y, Fn−1x, z d
Fn−1z, x, Fn−1x, z . 2.57
Continuing this process, we obtain
dx, y≤kndFy, x, Fz, x dFx, y, Fx, z dFz, x, Fx, z. 2.58
Using the contractivity ofF,we have dx, y≤kn
k
2dx, x dy, z dy, z dx, x dx, z dz, x kn1dy, z dz, x.
2.59
That is
dx, y≤kn1dy, z dz, x−→0, n−→ ∞. 2.60
So,
dx, y 0⇐⇒xy. 2.61
3. Main Results: Mixed Monotone Case II
LetXbe a partially ordered set and letdbe a metric onXsuch thatX, dis a complete metric space. ConsiderX×X.We will use the following partial order.
Forx, y,u, v∈X×X, we have
x, yu, v⇐⇒ {x≥u, y≤v}. 3.1
Letd1be a metric onX×Xdefined as follows:
d1x, y,u, v dx, u dy, v. 3.2 The following two theorems have similar proofs to the proofs of Theorems 2.1 and 2.3, respectively, and so their proofs will be skipped. Significant parts of these results have been included in 14 and applied successfully to some boundary value problems in ordinary differential equations.
Theorem 3.1. LetF :X×X → Xbe a map such thatFx, yis nondecreasing inxfor ally∈X, and nonincreasing inyfor allx∈X.Suppose that the following conditions hold.
iThere existsk∈0,1with dFx, y, Fu, v≤ k
2d1x, y,u, v ∀x, yu, v. 3.3 iiThere existsx0, y0∈Xsuch that the following condition holds:
x0≤F x0, y0
, y0≥F
y0, x0
. 3.4
iiiIf {xn} ∈ X is a nondecreasing convergent sequence such that limn→ ∞xn x, then xn ≤ x, for all n ∈ N and if {yn} ∈ Y is a nonincreasing convergent sequence such that limn→ ∞yn y, then yn ≥ y, for all n ∈ N; if xn ≤ yn for every n, then limn→ ∞xn≤limn→ ∞yn.
Then we have the following.
aFor every initial pointx0, y0∈X×Xsuch that the condition3.2holds,Fnx0, y0 → x, Fny0, x0 → y, n → ∞, wherex, ysatisfy
xFx, y, yFy, x. 3.5
Ifx0≤y0in condition3.4, thenx≤y.If in additionxy, then{xn},{yn}converge to the equilibrium of the equation
xn1F xn, yn
, yn1F
yn, xn
, n1,2, . . . . 3.6
bIn particular, every solution{zn}of
zn1F
zn, zn−1
, n2,3, . . . 3.7
such thatx0≤z0, z1≤y0converges to the equilibrium of 3.7.
cThe following estimates hold:
d Fn
x0, y0
, x
≤ 1 2
kn 1−k
d F
x0, y0
, x0
d
F y0, x0
, y0
,
d Fn
y0, x0
, y
≤ 1 2
kn 1−k
d F
x0, y0
, x0
d
F y0, x0
, y0
.
3.8
Theorem 3.2. Assume that along with conditions (i) and (ii) ofTheorem 3.1, the following condition is satisfied:
ivevery pair of elements has either a lower or an upper bound.
Then, the fixed pointx, yis unique andxy.
Remark 3.3. Theorems3.1and3.2generalize and extend the results in14. The new feature of our results is global attractivity part that extends Theorems1.1and1.2. Most of presented ideas were presented for the first time in14.
Acknowledgment
The authors are grateful to the referees for pointing out few fine details that improved the presented results.
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