Volume 2009, Article ID 809315,8pages doi:10.1155/2009/809315
Research Article
A Continuation Method for Weakly Contractive Mappings under the Interior Condition
David Ariza-Ruiz and Antonio Jim ´enez-Melado
Departamento de An´alisis Matem´atico, Facultad de Ciencias, Universidad de M´alaga, 29071 M´alaga, Spain
Correspondence should be addressed to Antonio Jim´enez-Melado,[email protected] Received 29 July 2009; Accepted 8 October 2009
Recommended by Marlene Frigon
Recently, Frigon proved that, for weakly contractive maps, the property of having a fixed point is invariant by a certain class of homotopies, obtaining as a consequence a Leray-Schauder alternative for this class of maps in a Banach space. We prove here that the Leray-Schauder condition in the aforementioned result can be replaced by a modification of it, the interior condition. We also show that our arguments work for a certain class of generalized contractions, thus complementing a result of Agarwal and O’Regan.
Copyrightq2009 D. Ariza-Ruiz and A. Jim´enez-Melado. 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
Suppose thatX is a Banach space, thatU ⊂ X is an open bounded subset ofX, containing the origin, and thatf :U → X is a mapping. It is well known that iff satisfies the Leray- Schauder condition defined as
fx/λx, forx∈∂U, λ >1 L-S
andf is a strict set-contraction or, more generally, condensing, thenfhas a fixed point inU see, e.g.,1or2. The first continuation method in the setting of a complete metric space for contractive maps comes from the hands of Granas3, in 1994, who gave a homotopy result for contractive mapsfor more information on this topic see, e.g.,4,5or6.
On the other hand, it has been recently shown in7that, for condensing mappings, the conditionL-Scan be replaced by a modification of it which we call the interior condition,
and is defined as follows: a mappingf :U → Xsatisfies the Interior ConditionI-C, if there existsδ >0 such that
fx/λx, forx∈Uδ, λ >1, fx/∈U, I-C whereUδ {x ∈ U: distx, ∂U< δ}some generalizations of this result can be found in 8,9.
We remark that the conditionI-Cby itself cannot be a substitute for the condition L-S, and an additional assumption on the domain of f needs to be made in order to guarantee the existence of a fixed point for f. The class of sets that we need is defined as follows: suppose that U ⊂ X is an open neighborhood of the origin. We say that Uis strictly star shaped if for any x ∈ ∂U we have that {λx : λ > 0} ∩∂U {x}. It was shown in7that ifUis bounded and strictly star shaped andf : U → Xis a condensing mapping satisfying the conditionI-C, thenfhas a fixed point. Of course, this result includes the case of a contractive map i.e., a map f for which there exists k ∈ 0,1 such that dfx, fy ≤ kdx, yfor allx, y ∈ U, but our aim in this note is, following the pattern of Granas3 and Frigon et al.10, to give a continuation method for weakly contractive mappings, in the setting of a complete metric space, under some conditions on the homotopy which are the counterpart of the conditionI-Cand the notion of a strictly star shaped set in a space without a vector structure. Finally, in the last section we show that our arguments also work for a class of generalized contractions, thus complementing a result of Agarwal and O’Regan11.
2. Weakly Contractive Maps
In this chapter we deal with the concept of weakly contractive maps, as it was introduced by Dugundji and Granas in12.
Definition 2.1. LetX, dbe a complete metric space andUan open subset ofX. A function f : U → X is said to be weakly contractive if there existsψ : X×X → 0,∞compactly positivei.e., inf{ψx, y:a≤dx, y≤b}θa, b>0 for every 0< a≤bsuch that
d
fx, f y
≤d x, y
−ψ x, y
. 2.1
Ifψis a compactly positive function, we define for 0< a≤b
γa, b min{a, θa, b}. 2.2
It was shown in 12 that any weakly contractive map f : X → X defined on a complete metric spaceXhas a unique fixed point. Some years later, Frigon5proved that, for weakly contractive maps, the property of having a fixed point is invariant by a certain class of homotopies, obtaining as a consequence a Leray-Schauder alternative for weakly contractive maps in the setting of a Banach space. We prove here that the Leray-Schauder condition in the aforementioned result can be replaced by the conditionI-C, and it will also be obtained as a consequence of a continuation method. The definition of homotopy that we need for our purposes is the following.
Definition 2.2. LetX, dbe a complete metric space, andUan open subset ofX. Letf, g : U → Xbe two weakly contractive maps. We say thatfisI-C-homotopic togif there exists H:U×0,1 → Xwith the following properties:
P1Hx,1 fxandHx,0 gxfor everyx∈U;
P2there existsδ >0 such thatx /Hx, tfor everyx∈Uδ, withfx/∈U, andt∈0,1, whereUδ{x∈U: distx, ∂U< δ};
P3there exists a compactly positive function ψ : X × X → 0,∞ such that dHx, t, Hy, t≤dx, y−ψx, yfor everyx, y∈U, andt∈0,1;
P4there exists a continuous functionφ : 0,1 → Rsuch that, for everyx ∈ Uand t, s∈0,1,dHx, t, Hx, s≤ |φt−φs|;
P5ifx∈∂Uand 0≤λ <1, withHx, λ∈∂U, thenHx,1/∈U.
In the proof of the main result of this chapter we shall make use of the following lemma see Frigon5.
Lemma 2.3. Letx0 ∈ X, r > 0, and h : Bx0, r → X weakly contractive. If dx0, hx0 <
γr/2, r, thenhhas a fixed point.
Theorem 2.4. Letf, g:U → Xbe two weakly contractive maps. Suppose thatfis homotopic tog andgUis bounded. Ifghas a fixed point inU, thenfhas a fixed point inU.
Proof. We argue by contradiction. Suppose thatfdoes not have any fixed point inU, and let Hbe a homotopy betweenfandg, in the sense ofDefinition 2.1. Consider the set
A{λ∈0,1:xHx, λfor somex∈U}, 2.3 and notice thatAis nonempty sinceghas a fixed point inU, that is, 0∈A. We will show that Ais both open and closed in0,1, and hence, by connectedness, we will have thatA 0,1.
As a result,fwill have a fixed point inU, which establishes a contradiction.
To show thatAis closed, suppose that{λn}is a sequence inAconverging toλ∈0,1 and let us show that λ ∈ A. Sinceλn ∈ A, there existsxn ∈ Uwith xn Hxn, λn. Fix ε > 0. Using thatgUis bounded and thatφis continuous on the compact interval0,1, it is easy to show that there existsM > ε such that diamHU×0,1 ≤ M, and hence dxn, xm ≤ M for all n, m ∈ N. Define μ θε, Mand let n0 ∈ N be such that for all n, m ≥ n0,|φλn−φλm| < μ. Thendxn, xm < εfor alln, m ≥ n0 because, otherwise, we would havedxn, xm≥εfor somen, m≥n0, and then
dxn, xm dHxn, λn, Hxm, λm
≤dHxn, λn, Hxn, λm dHxn, λm, Hxm, λm
≤φλn−φλmdxn, xm−ψxn, xm
< μdxn, xm−ψxn, xm
≤dxn, xm,
2.4
which is a contradiction. Then{xn}is a Cauchy sequence and, sinceX, dis complete, there existsx0 ∈Usuch thatxn → x0asn → ∞. In addition,x0 Hx0, λsince for alln∈Nwe have that
dxn, Hx0, λ dHxn, λn, Hx0, λ
≤dHxn, λn, Hxn, λ dHxn, λ, Hx0, λ
≤φλn−φλdxn, x0−ψxn, x0
≤φλn−φλdxn, x0.
2.5
Observe that 0≤λ <1, because ifλ1,thenx0Hx0,1 fx0, which contradicts the fact thatfdoes not have any fixed point inU. Notice thatx0∈U, because, otherwise, we would havex0 ∈∂U, that is,Hx0, λ∈∂U,and since 0≤λ <1, byP5, we have thatHx0,1/∈U.
However, sincex0 ∈∂U,{xn} → x0andxn ∈Ufor alln∈N, there existsn0 ∈Nsuch that xn∈Uδfor alln≥n0. Hence, sincexnHxn, λnfor alln≥n0, applyingP2, we have that fxn∈ Ufor alln≥ n0, that is,Hxn,1 ∈Ufor alln≥ n0. Taking limits, we arrive to the contradictionHx0,1∈U.
Therefore,x0∈Uand, consequently,λ∈A.
Next we show thatAis open in0,1. Letλ0 ∈A. Then there existsx0 ∈Uwithx0 Hx0, λ0. Letr >0 be such thatBx0, r⊂U, and letδ >0 such that|φλ−φλ0|< γr/2, r for everyλ∈0,1with|λ0−λ|< δ. Then, ifλ∈λ0−δ, λ0δ∩0,1,
dx0, Hx0, λ dHx0, λ0, Hx0, λ
≤φλ0−φλ
< γr 2, r
.
2.6
UsingLemma 2.3, we obtain thatH·, λhas a fixed point inUfor everyλ∈0,1such that
|λ0−λ|< δ. Thusλ∈Afor anyλ∈λ0−δ, λ0δ∩0,1, and thereforeAis open in0,1.
As an immediate consequence of the previous theorem, we obtain the following fixed point result of the Leray-Schauder type for weakly contractive maps under the condition I-C.
Theorem 2.5. Suppose thatUis an open and strictly star shaped subset of a Banach spaceX, · , with 0∈U, and thatf:U → Xis a weakly contractive map withfUbeing bounded. Iffsatisfies the conditionI-C, thenfhas a fixed point inU.
Proof. Since f satisfies the conditionI-C, there existsδ > 0 such thatfx/λx forλ > 1 andx∈Uδwithfx/∈U. We may assume thatx /fxfor everyx∈Uδ, because otherwise we are finished. DefineH :U×0,1 → X asHx, t tfx, and letg be the zero map.
Notice thatghas a fixed point inU, that is, 0 g0and also thatf andg are two weakly contractive mappings. So, the result will follow from Theorem 2.4once we prove that f is I-C-homotopic tog. Let us check it.
P1For allx∈U,Hx,0 0·fx 0gxandHx,1 1·fx fx.
P2Sincefsatisfies the conditionI-C, we have thatfx/λxforx∈Uδwithfx/∈U andλ >1. Hence,x /Hx, tfor everyx∈Uδ, withfx/∈U, andt∈0,1.
P3Sincefis weakly contractive, there exists a compactly positive functionψ:X×X → 0,∞such thatdfx, fy≤dx, y−ψx, yfor everyx, y∈U. Then, ifx, y∈U andt∈0,1,
d
Hx, t, H y, t
tfx−f y
≤d
fx, f y
≤d x, y
−ψ x, y
.
2.7
P4SincefUis bounded, there existsM ≥ 0 such that fx ≤ Mfor all x ∈ U.
Hence,
dHx, t, Hx, s fx|t−s|
≤M|t−s|
φt−φs, 2.8
whereφ:0,1 → Ris the continuous function defined asφt Mt.
P5Suppose that for somex∈∂Uandλ <1 we have thatHx, λ∈∂U. Then,fx/0 sinceHx, λ λfx, 0∈ UandUis open. Let us see thatHx,1/∈U: suppose, on the contrary, thatHx,1∈U, that is,fx∈Uand define
λ:sup t≥1 :tfx∈U
. 2.9
Then, it is easy to see that λfx ∈ ∂U, which contradicts that U is strictly star shaped, since we also have thatλfx∈∂U.
3. A Class of Generalized Contractions
A multitude of generalizations and variants of Banach’s contractive condition have been given after Banach’s theoremsee, e.g., Rhoades13and, recently, Agarwal and O’Regan 11have given a homotopy resultthus generalizing a fixed point theorem of Hardy and Rogers14 under the following generalized contractive condition: there existsa ∈ 0,1 such that for allx, y∈X
d
fx, f y
≤amax
d x, y
, d
x, fx , d
y, f y
,1 2
d x, f
y d
y, fx . 3.1
In this section we give a homotopy result for this class of mappings under the conditionI-C. In the proof of our theorem we shall use the following result11.
Lemma 3.1. LetX, dbe a complete metric space,x0∈X,r >0, andh:Bx0, r → X. Suppose that there existsa∈0,1such that forx, y∈Bx0, rone has
d
hx, h y
≤amax
d x, y
, dx, hx, d y, h
y ,1
2 d
x, h y
d
y, hx , dx0, hx0<1−ar.
3.2
Then there existsx∈Bx0, rwithxhx.
The proof of the following theorem is very similar to the proof ofTheorem 2.4, and we give a sketch of it.
Theorem 3.2. LetX, dbe a complete metric space, andUan open subset ofX. Letf, g:U → X be two maps such that there existsH:U×0,1 → Xwith the following properties:
P1Hx,1 fxandHx,0 gxfor everyx∈U;
P2there existsδ >0 such thatx /Hx, tfor everyx∈Uδ, withfx/∈U, andt ∈0,1, whereUδ{x∈U: distx, ∂U< δ};
P3there existsa∈0,1such that for allx, y∈Uandλ∈0,1one has d
Hx, λ, H y, λ
≤amax
d x, y
, dx, Hx, λ, d y, H
y, λ ,1
2 d
x, H y, λ
d
y, Hx, λ
; 3.3
P4there exists a continuos functionφ:0,1 → Rsuch that, for everyx∈Uandt, s∈0,1, dHx, t, Hx, s≤ |φt−φs|;
P5ifx∈∂Uand 0≤λ <1, withHx, λ∈∂U, thenHx,1/∈U.
Ifghas a fixed point inU, thenfhas a fixed point inU.
Proof. Suppose thatfdoes not have any fixed point inUand consider the nonempty set A{λ∈0,1:Hx, λ xfor somex∈U}. 3.4 We will arrive to a contradiction by showing thatA 0,1, and for this we only need prove thatAis closed and open in0,1.
To show thatAis closed in0,1, consider a sequence{λn}inA, withλn → λ∈0,1 asn → ∞, and show thatλ∈A; that is, that there existsx0 ∈UwithHx0, λ x0. To prove thatx0 exists, take any sequence{xn}inUwithxn Hxn, λn, prove that{xn}is Cauchy, and definex0as the limit of{xn}, asn → ∞.
That {xn} is a Cauchy sequence, as well asx0 Hx0, λ, follows from standard arguments which can be seen in 11, Theorem 3.1. It remains to show that x0 ∈ U.
To prove this, suppose that it is not true and arrive to a contradiction as follows: we have thatHx0, λ x0 ∈ U\U ∂U, and also that 0 ≤ λ < 1, because f does not have any fixed point inU. Then, byP5fx0/∈∂U. On the other hand,fx0 limfxn∈Ubecause fxn∈Ufornlarge enough. To be convinced of it, just applyP2: sincex0∈∂U,{xn} → x0
andxn∈Ufor alln∈N, there existsn0∈Nsuch thatxn∈Uδfor alln≥n0. Then,fxn∈U for alln≥n0sincexnHxn, λn.
To prove thatAis open argue as inTheorem 2.4, useLemma 3.1instead ofLemma 2.3.
As an immediate consequence, we obtain the following result, whose proof is omitted because it is analogous to the proof ofTheorem 2.5.
Theorem 3.3. Suppose thatUis an open and strictly star shaped subset of a Banach spaceX, · , with 0 ∈ U, and thatf : U → Xis map withfUbeing bounded. Assume also that there exists a∈0,1such that for allx, y∈Uandλ∈0,1one has
d
λfx, λf y
≤amax
d x, y
, d
x, λfx , d
y, λf y
,1 2
d x, λf
y d
y, λfx
. 3.5
Iffsatisfies the conditionI-C, thenfhas a fixed point inU.
Acknowledgment
This research is partially supported by the SpanishGrant MTM2007-60854 and regional AndalusianGrants FQM210, FQM1504Governments.
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