Volume 2011, Article ID 979586,10pages doi:10.1155/2011/979586
Research Article
Fixed-Point Results for Generalized Contractions on Ordered Gauge Spaces with Applications
Cristian Chifu and Gabriela Petrus¸el
Faculty of Business, Babes¸-Bolyai University, Horia Street no. 7, 400174 Cluj-Napoca, Romania
Correspondence should be addressed to Cristian Chifu,[email protected] Received 6 December 2010; Accepted 31 December 2010
Academic Editor: Jen Chih Yao
Copyrightq2011 C. Chifu and G. Petrus¸el. 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.
The purpose of this paper is to present some fixed-point results for single-valuedϕ-contractions on ordered and complete gauge space. Our theorems generalize and extend some recent results in the literature. As an application, existence results for some integral equations on the positive real axis are given.
1. Introduction
Throughout this paper will denote a nonempty setEendowed with a separating gauge structureD{dα}α∈Λ, whereΛis a directed setsee1for definitions. Let :{0,1,2, . . .}
and∗ :\ {0}. We also denote bythe set of all real numbers and by : 0,∞.
A sequencexnof elements inEis said to be Cauchy if for everyε > 0 andα ∈ Λ, there is anN withdαxn, xnp ≤ εfor alln ≥ N andp ∈ ∗. The sequencexnis called convergent if there exists anx0 ∈Xsuch that for everyε > 0 andα∈Λ, there is anN ∈∗ withdαx0, xn≤ε, for alln≥N.
A gauge space is called complete if any Cauchy sequence is convergent. A subset of Xis said to be closed if it contains the limit of any convergent sequence of its elements. See also Dugundji1for other definitions and details.
Iff:E → Eis an operator, thenx∈Eis called fixed point forfif and only ifxfx.
The setFf :{x∈E|xfx}denotes the fixed-point set off.
On the other hand, Ran and Reurings2proved the following Banach-Caccioppoli type principle in ordered metric spaces.
Theorem 1.1Ran and Reurings2. LetXbe a partially ordered set such that every pairx, y∈X has a lower and an upper bound. Letdbe a metric onXsuch that the metric spaceX, dis complete.
Letf:X → Xbe a continuous and monotone (i.e., either decreasing or increasing) operator. Suppose that the following two assertions hold:
1there existsa∈0,1such thatdfx, fy≤a·dx, y, for eachx, y∈Xwithx≥y;
2there existsx0∈Xsuch thatx0≤fx0orx0≥fx0.
Thenfhas an unique fixed pointx∗∈X, that is,fx∗ x∗, and for eachx∈Xthe sequence fnxn∈
of successive approximations offstarting fromxconverges tox∗∈X.
Since then, several authors considered the problem of existenceand uniquenessof a fixed point for contraction-type operators on partially ordered sets.
In 2005, Nieto and Rodrguez-L ´opez proved a modified variant of Theorem 1.1, by removing the continuity of f. The case of decreasing operators is treated in Nieto and Rodrguez-L ´opez3, where some interesting applications to ordinary differential equations with periodic boundary conditions are also given. Nieto, Pouso, and Rodrguez-L ´opez, in a very recent paper, improve some results given by Petrus¸el and Rus in4in the setting of abstractL-spaces in the sense of Fr´echet, see, for example,5, Theorems 3.3 and 3.5. Another fixed-point result of this type was given by O’Regan and Petrus¸el in6for the case of ϕ- contractions in ordered complete metric spaces.
The aim of this paper is to present some fixed-point theorems forϕ-contractions on ordered and complete gauge space. As an application, existence results for some integral equations on the positive real axis are given. Our theorems generalize the above-mentioned theorems as well as some other ones in the recent literaturesee; Ran and Reurings2, Nieto and Rodrguez-L ´opez3,7, Nieto et al.5, Petrus¸el and Rus4, Agarwal et al.8, O’Regan and Petrus¸el6, etc..
2. Preliminaries
LetX be a nonempty set andf :X → Xbe an operator. Then,f0 : 1X,f1 :f, . . . , fn1 f ◦fn,n ∈ denote the iterate operators off. LetX be a nonempty set and letsX : {xnn∈N | xn ∈ X, n ∈ N}. Let cX ⊂ sX a subset of sX and Lim : cX → X an operator. By definition the tripleX, cX,Limis called anL-spaceFr´echet9if the following conditions are satisfied.
iIfxnx, for alln∈N, thenxnn∈N∈cXand Limxnn∈Nx.
iiIf xnn∈N ∈ cX and Limxnn∈N x, then for all subsequences, xnii∈N, of xnn∈Nwe have thatxnii∈N ∈cXand Limxnii∈Nx.
By definition, an element ofcX is a convergent sequence, x : Limxnn∈N is the limit of this sequence and we also writexn → xasn → ∞.
In what follow we denote anL-space byX,→.
In this setting, if U ⊂ X ×X, then an operatorf : X → X is called orbitally U- continuoussee5ifx ∈ Xandfnix → a∈ X, asi → ∞andfnix, a ∈ Ufor anyi∈implyfni1x → fa, asi → ∞. In particular, ifUX×X, thenfis called orbitally continuous.
LetX,≤be a partially ordered set, that is,X is a nonempty set and≤is a reflexive, transitive, and antisymmetric relation onX. Denote
X≤ : x, y
∈X×X|x≤y ory≤x
. 2.1
Also, ifx, y ∈ X, withx ≤ y then byx, y≤we will denote the ordered segment joining x andy, that is,x, y≤ :{z∈X|x≤z≤y}. In the same setting, considerf :X → X. Then, LFf :{x∈X|x≤fx}is the lower fixed-point set off, whileUFf :{x∈X|x≥fx}
is the upper fixed-point set off. Also, iff:X → Xandg :Y → Y, then the cartesian product offandgis denoted byf×g, and it is defined in the following way:f×g:X×Y → X×Y, f×gx, y: fx, gy.
Definition 2.1. LetXbe a nonempty set. By definitionX,→,≤is an orderedL-space if and only if
i X,→is anL-space;
ii X,≤is a partially ordered set;
iii xnn∈ → x,ynn∈ → yandxn≤yn, for eachn∈ ⇒x≤y.
If : E,Dis a gauge space, then the convergence structure is given by the family of gaugesD{dα}α∈Λ. Hence,E,D,≤is an orderedL-space, and it will be called an ordered gauge space, see also10,11.
Recall thatϕ : → is said to be a comparison function if it is increasing and ϕkt → 0, ask → ∞. As a consequence, we also haveϕt < t, for eacht > 0,ϕ0 0 andϕis right continuous at 0. For example,ϕt atwherea∈0,1 ,ϕt t/1tand ϕt ln1t,t∈ are comparison functions.
Recall now the following important abstract concept.
Definition 2.2Rus12. LetX,→be anL-space. An operatorf:X → Xis, by definition, a Picard operator if
iFf {x∗};
ii fnxn∈ → x∗asn → ∞, for allx∈X.
Several classical results in fixed-point theory can be easily transcribed in terms of the Picard operators, see4,13,14. In Rus12the basic theory of Picard operators is presented.
3. Fixed-Point Results
Our first main result is the following existence, uniqueness, and approximation fixed-point theorem.
Theorem 3.1. LetE,D,≤ be an ordered complete gauge space andf : E → Ebe an operator.
Suppose that
ifor eachx, y ∈ Ewithx, y∈/ X≤there existscx, y ∈ Esuch thatx, cx, y ∈X≤ andy, cx, y∈X≤;
iiX≤∈If×f;
iiiifx, y∈X≤andy, z∈X≤, thenx, z∈X≤; ivthere existsx0∈X≤such thatx0, fx0∈X≤;
vfis orbitally continuous;
vithere exists a comparison functionϕ: → such that, for eachα∈Λone has dα
fx, f y
≤ϕ dα
x, y
, for each x, y
∈X≤. 3.1
Then,fis a Picard operator.
Proof. Letx0 ∈ Ebe such thatx0, fx0∈ X≤. Suppose first thatx0/fx0. Then, fromii we obtain
fx0, f2x0 ,
f2x0, f3x0 , . . . ,
fnx0, fn1x0
, . . . ,∈X≤. 3.2
Fromvi, by induction, we get, for eachα∈Λ, that dα
fnx0, fn1x0
≤ϕn dα
x0, fx0
, for eachn∈. 3.3
Sinceϕndαx0, fx0 → 0 asn → ∞, for an arbitraryε >0 we can chooseN ∈∗ such thatdαfnx0, fn1x0< ε−ϕε, for eachn≥N. Sincefnx0, fn1x0∈X≤for alln∈, we have for alln≥Nthat
dα
fnx0, fn2x0
≤dα
fnx0, fn1x0 dα
fn1x0, fn2x0
< ε−ϕε ϕ dα
fnx0, fn1x0
≤ε.
3.4
Now sincefnx0, fn2x0∈X≤seeiiiwe have for anyn≥Nthat dα
fnx0, fn3x0
≤dα
fnx0, fn1x0 d
fn1x0, fn3x0
< ε−ϕε ϕ dα
fnx0, fn2x0
≤ε.
3.5
By induction, for eachα∈Λ, we have dα
fnx0, fnkx0
< ε, for anyk∈∗, n≥N. 3.6
Hencefnx0n∈is a Cauchy sequence in . From the completeness of the gauge space we havefnx0n∈ → x∗, asn → ∞.
Letx∈Ebe arbitrarily chosen. Then;
1If x, x0 ∈ X≤ then fnx, fnx0 ∈ X≤ and thus, for each α ∈ Λ, we have dαfnx, fnx0≤ ϕndαx, x0, for eachn∈. Lettingn → ∞we obtain that fnxn∈ → x∗.
2If x, x0 /∈ X≤ then, byi, there existscx, x0 ∈ E such thatx, cx, x0 ∈ X≤ and x0, cx, x0 ∈ X≤. From the second relation, as before, we get, for each α ∈ Λ, that dαfnx0, fncx, x0 ≤ ϕndαx0, cx, x0, for each n ∈ and hence fncx, x0n∈ → x∗, asn → ∞. Then, using the first relation we infer that, for eachα ∈ Λ, we havedαfnx, fncx, x0 ≤ ϕndαx, cx, x0, for eachn ∈ . Letting againn → ∞, we concludefnxn∈ → x∗.
By the orbital continuity offwe get thatx∗ ∈Ff. Thusx∗ fx∗. If we havefy y for somey∈E, then from above, we must havefny → x∗, soyx∗.
Iffx0 x0, thenx0plays the role ofx∗.
Remark 3.2. Equivalent representation of conditionivare as follows.
iv’ There existsx0∈Esuch thatx0≤fx0orx0≥fx0 iv”LFf∪UFf/∅.
Remark 3.3. Conditioniican be replaced by each of the following assertions:
ii’f :E,≤ → E,≤is increasing, ii”f :E,≤ → E,≤is decreasing.
However, it is easy to see that assertioniiinTheorem 3.1. is more general.
As a consequence of Theorem 3.1, we have the following result very useful for applications.
Theorem 3.4. LetE,D,≤be an ordered complete gauge space andf:E → Ebe an operator. One supposes that
ifor eachx, y ∈ Ewithx, y∈/ X≤there existscx, y ∈ Esuch thatx, cx, y ∈X≤ andy, cx, y∈X≤;
iif :E,≤ → E,≤is increasing;
iiithere existsx0∈Esuch thatx0≤fx0; iv
afis orbitally continuous or
bif an increasing sequencexnn∈converges toxinE, thenxn≤xfor alln∈; vthere exists a comparison functionϕ: → such that
dα
fx, f y
≤ϕ dα
x, y
, for each x, y
∈X≤, α∈Λ; 3.7
viifx, y∈X≤andy, z∈X≤, thenx, z∈X≤. Thenfis a Picard operator.
Proof. Since f : E,≤ → E,≤ is increasing and x0 ≤ fx0 we immediately have x0 ≤ fx0 ≤ f2x0 ≤ · · ·fnx0 ≤ · · ·. Hence from v we obtaindαfnx0, fn1x0 ≤ ϕndαx0, fx0, for each n ∈ . By a similar approach as in the proof ofTheorem 3.1we obtain
dα
fnx0, fnkx0
< ε, for anyk∈∗, n≥N, 3.8 Hencefnx0n∈is a Cauchy sequence in . From the completeness of the gauge space we have thatfnx0n∈ → x∗asn → ∞.
By the orbital continuity of the operatorf we get thatx∗ ∈Ff. Ifivbtakes place, then, sincefnx0n∈ → x∗, given any >0 there existsN∈∗ such that for eachn≥N
we havedαfnx0, x∗< . On the other hand, for eachn≥N, sincefnx0≤x∗, we have, for eachα∈Λthat
dα
x∗, fx∗
≤dα
x∗, fn1x0 dα
f
fnx0 , fx∗
≤dα
x∗, fn1x0 ϕ
dα
fnx0, x∗
<2.
3.9
Thusx∗∈Ff.
The uniqueness of the fixed point follows by contradiction. Suppose there existsy∗ ∈ Ff, withx∗/y∗. There are two possible cases.
aIf x∗, y∗ ∈ X≤, then we have 0 < dαy∗, x∗ dαfny∗, fnx∗ ≤ ϕndαy∗, x∗ → 0 asn → ∞, which is a contradiction. Hencex∗y∗.
bIfx∗, y∗ ∈/ X≤ then there existsc∗ ∈ Esuch thatx∗, c∗ ∈ X≤andy∗, c∗ ∈X≤. The monotonicity condition implies that fnx∗ and fnc∗ are comparable as well as fnc∗ and fny∗. Hence 0 < dαy∗, x∗ dαfny∗, fnx∗ ≤ dαfny∗, fnc∗ dαfnc∗, fnx∗ ≤ ϕndαy∗, c∗ ϕndαc∗, x∗ → 0 as n → ∞, which is again a contradiction. Thusx∗y∗.
4. Applications
We will apply the above result to nonlinear integral equations on the real axis
xt t
0
Kt, s, xsdsgt, t∈. 4.1
Theorem 4.1. Consider4.1. Suppose that
iK:× ×n → n andg: → n are continuous;
iiKt, s,·:n → n is increasing for eacht, s∈;
iiithere exists a comparison functionϕ: → , withϕλt≤λϕtfor eacht∈ and anyλ≥1, such that
|Kt, s, u−Kt, s, v| ≤ϕ|u−v|, for eacht, s∈, u, v∈n, u≤v; 4.2
ivthere existsx0∈C,nsuch that
x0t≤ t
0
Kt, s, x0sdsgt, for anyt∈. 4.3
Then the integral equation4.1has a unique solutionx∗inC0,∞,n. Proof. LetE:C0,∞,nand the family of pseudonorms
xn: max
t∈0,n|xt|e−τt, whereτ >0. 4.4
Define nowdnx, y:x−ynforx, y∈E.
ThenD: dnn∈∗is family of gauges onE. Consider onEthe partial order defined by
x≤y if and only ifxt≤yt for any t∈. 4.5
ThenE,D,≤is an ordered and complete gauge space. Moreover, for any increasing sequencexnn∈inEconverging to somex∗ ∈Ewe havexnt≤x∗t, for anyt∈0,∞.
Also, for everyx, y∈Ethere existscx, y∈Ewhich is comparable toxandy.
DefineA:C0,∞,n → C0,∞,n, by the formula
Axt: t
0
Kt, s, xsdsgt, t∈. 4.6
First observe that fromiiAis increasing. Also, for eachx, y∈Ewithx≤yand for t∈0, n, we have
Axt−Ayt ≤ t
0
Kt, s, xs−K
t, s, ys ds≤ t
0
ϕ xs−ys ds
t
0
ϕ xs−ys e−τseτs ds≤
t
0
eτsϕ xs−ys e−τs ds
≤ϕ dn
x, y
t 0
eτsds≤ 1 τϕ
dn
x, y eτt.
4.7
Hence, forτ ≥1 we obtain
dn
Ax, Ay
≤ϕ dn
x, y
, for eachx, y∈X, x≤y. 4.8
Fromivwe have thatx0≤Ax0. The conclusion follows now fromTheorem 3.4.
Consider now the following equation:
xt t
−tKt, s, xsdsgt, t∈. 4.9
Theorem 4.2. Consider4.9. Suppose that
iK:××n → n andg: → nare continuous;
iiKt, s,·:n → n is increasing for eacht, s∈;
iiithere exists a comparison functionϕ: → , withϕλt≤λϕtfor eacht∈ and anyλ≥1, such that
|Kt, s, u−Kt, s, v| ≤ϕ|u−v|, for eacht, s∈, u, v∈n, u≤v; 4.10
ivthere existsx0∈C,nsuch that
x0t≤ t
−tKt, s, x0sdsgt, for anyt∈. 4.11 Then the integral equation4.9has a unique solutionx∗inC,n.
Proof. We consider the gauge spaceE: C,n,D: dnn∈where
dn
x, y max
−n≤t≤n xt−yt ·e−τ|t|
, τ >0, 4.12
and the operatorB:E → Edefined by
Bxt t
−tKt, s, xsdsgt. 4.13
As before, consider onEthe partial order defined by
x≤y iffxt≤yt for anyt∈. 4.14
Then E,D,≤ is an ordered and complete gauge space. Moreover, for any increasing sequencexnn∈inEconverging to a certainx∗ ∈ Ewe havexnt ≤ x∗t, for anyt ∈ . Also, for everyx, y ∈Ethere existscx, y∈Ewhich is comparable toxandy. Notice that iiimplies thatBis increasing.
From conditioniii, forx, y∈Ewithx≤y, we have Bxt−Byt ≤
t
−tϕ xs−ys e−τ|s|eτ|s|
ds
≤ t
−teτ|s|ϕ xs−ys e−τ|s|
ds≤ϕ dn
x, y t
−teτ|s|ds
≤ϕ dn
x, y
|t|
−|t|eτ|s|ds≤ 2 τϕ
dn
x, y
eτ|t|, t∈−n;n.
4.15
Thus, for anyτ≥2, we obtain
dn
Bx, B y
≤ϕ dn
x, y
, ∀ x, y∈E, x≤y, forn∈. 4.16
As before, fromivwe have thatx0≤Bx0. The conclusion follows again byTheorem 3.4.
Remark 4.3. It is worth mentioning that it could be of interest to extend the above technique for other metrical fixed-point theorems, see15,16, and so forth.
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