Fixed Point Theorems for Asymptotically Contractive Multimappings

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Volume 2012, Article ID 862791,12pages doi:10.1155/2012/862791

Research Article

Fixed Point Theorems for Asymptotically

Contractive Multimappings

M. Djedidi

1

_{and K. Nachi}

2

1_{Department of Mathematics, El Oued University, Alger, Algeria}

2_{Department of Mathematics, Oran University, Oran, Algeria}

Correspondence should be addressed to K. Nachi,[email protected]

Received 19 February 2012; Accepted 4 April 2012

Academic Editor: Yonghong Yao

Copyrightq2012 M. Djedidi and K. Nachi. 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 present fixed point theorems for a nonexpansive set-valued mapping from a closed convex subset of a reflexive Banach space into itself under some asymptotic contraction assumptions. Some existence results of coincidence points and eigenvalues for multimappings are given.

1. Introduction

In this paper, we investigate fixed point theorems for nonexpansive multifunctionsrelations, multimaps, set-valued mappings, or correspondencessatisfying some asymptotic condition. This study has been the subject of numerous works1–6for an asymptotically contractive mapping. Our aim here is to obtain some generalization by using the notion of semi- asymptotically contractive multimappings which is introduced below. For doing so, we need to fix some notations and conventions. Given a normed vector spacen.v.s. X,·, the open ball with centerxand radiusrinXis denoted byBx, r; the closed, unit ball is denoted by BX. For any subsetsC,D⊂X, we set

dx, D inf

y∈D

_x₋_y_{with the convention inf}

∅ ∞,

eC, D sup

x∈C

dx, D if C /∅, e∅, D 0,

dC, D maxeC, D, eD, C.

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Recall that a multifunctionF : C → 2X _{is a contractive}_{resp., nonexpansive}_{multifunction}

onC⊂Xif there existsθ∈0,1such that for anyx, x∈C, one has

Fx⊂Fxθx−xBX,

resp., Fx⊂Fxx−xBX

. 1.2

Note that whenFx:{fx}, wheref :C → Xis a mapping,Fis a contraction with rateθ

resp., nonexpansiveonCif and only iff is a contraction with rateθresp., nonexpansive mapping onC: for anyx, x∈C

fx−fx≤θx−x, resp.,fx−fx≤x−x. 1.3

The existence theorem of fixed points for contractive multifunction is well known

see7. More generally, a generalization of Picard-Banach theorem to pseudo-contractive multifunction is given in8,9,10, Lemma 1, page 31and11, Proposition 2.5. Let us recall that result for the sake of clarity.

Proposition 1.1see8,10,11. LetX, dbe a complete metric space, and letF :X → 2X_{be a}

multifunction with closed, nonempty values. Suppose thatFis pseudo-θ-contractive with respect to

some ballBx0, r0for someθ∈0,1(i.e.,eFx∩Bx0, r0, Fx≤θdx, xforx, x∈Bx0, r0

andr : 1−θ−1dx0, Fx0< r0. Then the fixed point set FixF : {x∈X :x∈ Fx}ofF is nonempty and

dx0,FixF∩Bx0, r0≤r. 1.4

In this work, the reflexivity of Banach spaces and the property of demiclosedness of multifunctions play an important role to have fixed points results. Let us recall thatF:C → 2X_{is said to be demiclosed if its graph Gr}_F_{is sequentially closed in the product of the weak}

topology onCwith the norm topology on a Banach spaceX, that is,

xn, yn

n⊂GrF, xn x,

yn

−→y⇒x∈C, y∈Fx, 1.5

where GrF:{x, y∈C×X :y∈Fx}.

It is well known that iff :C → Xis nonexpansive onC, a closed convex subset of a uniformly convex Banach spaceX, thenI−fis demi-closed6,12, Proposition 10.9, page 476, where a Banach spaceX,·is uniformly convex if and only if for anyε∈0,2, there existsδε∈0,1such that for anyx, y∈X, r >0, one has

x ≤r,y≤r,x−y≥εr⇒xy 2

≤1−δεr. 1.6

As examples, every Hilbert space is uniformly convex, the spaces lp and LpΩare

uniformly convex for 1< p <∞Ωis a domain inRn_{, which is not the case for}_p_{∈ {}₁_,_∞}_.

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2. Fixed Point Theorem under Asymptotical Conditions

The following definition generalizes the notion of asymptotically contractive mapping to set-valued mappings. Note that the meaning of the word “asymptotic” is not related to the iterations of the multimapping as in13but bears on the behavior of the set-valued mapping at infinity. This behavior can be studied using concepts of asymptotic cones and asymptotic compactness as in14–19.

Definition 2.1. LetCbe a subset of a Banach spaceX, and letF :C → 2X_{be a multimapping with}

nonempty values. We say thatFis asymptotically contractive onCif there existsx0∈Csuch that

lim sup

x∈C,x → ∞

eFx, Fx0

x−x0

<1. 2.1

Let us note that whenFx:{fx}, wheref :C → Xis a mapping, we get the definition of the asymptotically contractive mapping onCgiven in6as a variant of the notion introduced in17.

If eFx, Fx < ∞ for any x, x ∈ C particularly, if F is a multimapping with bounded values, then the condition2.1 is independent of the choice ofx0 ∈ C: indeed,

letx1∈Cx1/x0. SinceeFx, Fx1≤eFx, Fx0 eFx0, Fx1, we have

eFx, Fx1

x−x1 ≤

eFx, Fx0

x−x0

eFx0, Fx1

x−x0

x−x1,

lim sup

x∈C,x → ∞

eFx, Fx1

x−x1 ≤xlim sup∈C,x → ∞

eFx, Fx0

x−x0 <1.

2.2

Proposition2.2is a multivalued version of the main result of6.

Proposition 2.2. LetXbe a reflexive Banach space andCa (nonempty) closed convex subset ofX.

LetF :C → 2X be a multifunction with closed and nonempty values such thatF is nonexpansive

onC. Assume thatFis asymptotically contractive onCatx0withFx0bounded. IfFC⊂Cand

I−Fis demi-closed, thenFadmits a fixed point.

Proof. Letθnbe a sequence in0,1such thatθn → 1. For anyn ∈ N, we define a

multi-functionFn:C⇒Xby setting

Fnx:θnFx 1−θnx0. 2.3

It is clear thatFnx⊂Cfor anynandx∈C. On the other hand, forx, x∈Candvn∈Fnx,

from 2.3, there exists un ∈ Fxsuch that vn θnun 1−θnx0. Applying 1.2 since

F is nonexpansive, there existsu_n ∈ Fxsatisfyingun −un ≤ x−x. Thus, for vn

θnun 1−θnx0∈Fnx, one has

_v_n₋_v

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Then Fn is a contraction with rate θn on C. The Nadler’s theorem 7 ensures that each

multivaluedFnadmits a fixed pointxninC. So, from2.3and for someyn∈Fxn, one has

yn−x0θ−1n xn−x0, 2.5

1−θn

x0−yn

xn−yn∈I−Fxn. 2.6

Observe that if the sequencexnhas a bounded subsequence, the proof is finished. Indeed,

taking a subsequence if necessary,xnadmits a weak limitx∈CCis closed, convex in the

reflexive spaceX. Asynis boundedby equality2.5, the sequencexn−ynconverges

to 0. We conclude that 0∈I−Fx, that is,xis a fixed point ofF.

Thus, to complete the proof of the proposition, let us show that the sequence xn

is bounded. If this is not the case, taking a subsequence if necessary, we may assume that

xn → ∞. As condition2.1is satisfied, there existc∈0,1andρ >0 such that

∀x∈C, x ≥ρ:eFx, Fx0< cx−x0. 2.7

For largen, we haveθn> candxn ≥ρ, so that

dyn, Fx0

< cxn−x0. 2.8

There exists then a sequencezninFx0such that

_y_n₋_z_n_≤_c_x_n₋_x₀_. _2.9

On the other hand, from equalities2.5and2.6, we get

xn ≤xn−ynyn−znzn, ≤1−θnx0−yncxn−x0zn

≤1−θnθn−1c

xn−x0zn.

2.10

Dividing byxn, we obtain

1≤θ−1_n −1c 1 x0

xn

zn

xn. 2.11

Passing to the limit and using the fact thatznis bounded,xn → ∞andθn → 1, a

contradiction follows. So the sequencexnhas a bounded subsequence and the proposition

is proved.

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Definition 2.3. LetX be a set, let Y be a linear space, and letF, G : X → 2Y _{be two}

multi-mappings. We say thatFandGpresent a coincidence onXif there existsu∈Xsuch that

0∈F−Gu. 2.12

The pointuis called a coincidence point ofFandG.

Note that ifY XandGx : {x}for allx∈ X, we obtain the definition of a fixed point of the multifunctionF. Also observe that the relation 0 ∈ F −Gu can be written Fu∩Gu/∅, so that two mappingsf, g :X → Y present a coincidence onXif and only if there existsu∈Xsuch thatfu gu.

The following corollary is an immediate consequence giving the existence of a fixed point of a sumresp., a coincidence point of two multifunctions.

Corollary 2.4. Let C be a nonempty closed convex cone of a reflexive Banach space X. Let θ ∈

0,1, F : C → 2C _{be a}_θ_{-contraction (resp.,}_G _: _C _→ ₂C _{be a}₁₋_θ_{-contraction) set-valued}

mapping onCwith closed and nonempty values. Assume thatI−FGis demi-closed and there

existsx0 ∈Csuch thatFx0,Gx0are bounded and one has

lim sup

x∈C,x → ∞

eFx, Fx0

x−x0

eGx, Gx0

x−x0

<1. 2.13

Then the multifunctionH:FGadmits a fixed point onC, which is a coincidence point ofI−F

andG.

Proof. Since for any subsetsA,A,B,BofXone has

eAB, AB≤eA, AeB, B, 2.14

the multimappingHis nonexpansive and

lim sup

x∈C,x → ∞

eHx, Hx0

x−x0

<1. 2.15

SinceCis a convex cone,HCis contained inC, the result is a consequence of Proposition2.2.

Observe that ifxis a fixed point ofHsuch thatFx Gxand ifFxis a convex cone, thenxis a common fixed point ofFandG.

Corollary 2.5. LetCbe a nonempty closed convex cone of a Banach uniformly convex spaceX. Let

θ ∈ 0,1,f :C → Cbe aθ-contraction (resp.,g :C → Cbe a1−θ-contraction) set-valued

mapping onC. Assume that

lim sup

x∈C,x → ∞

_f_x₋_f_x₀

x−x0

_g_x₋_g_x₀

x−x0

<1. 2.16

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The notion of eigenvalue is very important in nonlinear analysis. It has many appli-cations as the notion of fixed point. We present now some results related to eigenvalues. We obtain in particular an existence result for eigenvalues of nonexpansive mappings.

Let us recall that a real numberλis said to be an eigenvalue for a set-valued mapping F : C → 2X if there exists an element x ∈ C,x /0 such that λx ∈ Fx. WhenFx :

{fx}, wheref:C → Xis a mapping, we obtain the usual definition of an eigenvalue for a mapping.

The next proposition gives an existence result.

Proposition 2.6. Let Cbe a closed convex cone of a reflexive Banach space X. Letλ > 1 and let

F : C → 2C _{be a nonexpansive set-valued mapping on}_C_{whose values are nonempty, closed and}

0/∈F0. Assume thatI−λ−1Fis demi-closed and that there existsx0∈Csuch thatFx0is bounded

and one has

lim sup

x∈C,x → ∞

eFx, Fx0

x−x0

<1. 2.17

Thenλis an eigenvalue forFassociated to an eigenvectorx∈C. And ifFxis a cone, thenxis a

fixed point ofF.

Proof. By takingH:θIλ−11−θFwithθ∈0,1, we haveHC⊂CCa convex cone,

Hx0bounded, andI−H 1−θI−λ−1Fso thatI−His demi-closed. Moreover, using

the inequality2.14, we get

eHx, Hx0

x−x0 ≤

θλ−11−θeFx, Fx0

x−x0

,

lim sup

x∈C,x → ∞

eHx, Hx0

x−x0 ≤

θλ−11−θ lim sup

x∈C,x → ∞

eFx, Fx0

x−x0

< θλ−11−θ< θ 1−θ 1.

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Therefore, there exists a fixed pointxofθIλ−11−θF, that is, we have

x∈θxλ−11−θFx,

1−θx∈λ−11−θFx,

x∈λ−1Fx,

2.19

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Corollary 2.7. LetCbe a closed convex cone of an uniformly convex Banach spaceX. Letλ >1, and

letf:C → Cbe a nonexpansive mapping onCsuch thatf0/0. Assume that there existsx0 ∈C

such that

lim sup

x∈C,x → ∞

_f_x₋_f_x₀

x−x0

<1. 2.20

Thenλis an eigenvalue forFassociated to an eigenvectorx∈C.

2.1. Asymptotic Contraction Condition with Respect to Semi-Inner Product

In this section, we will present some fixed points results for multimappings under another asymptotic condition. This study is inspired by the work20. For this aim, let us introduce some definitions. Recall that a semi-inner product on a vector spaceX is a function·,· : X×X → Rsatisfying the following properties for anyx, y, z∈Xandλ∈R:

xy, z x, z y, z,

λx, yλx, y,

x, x>0 for x /0,

x, y2≤x, xy, y.

2.21

It is proved in21,22that a semi-inner-product space is a normed linear space with the norm x_s : x, x1/2 and every Banach space can be endowed with diﬀerent semi-inner-products unless for Hilbert spaces where ·,· is the inner product. We say that the semi-inner-product on an n.v.s.X, · is compatible with the norm · ifx, x x2.

Let us introduce the following definition of asymptotically contractive multimappings with respect tow.r.ta semi-inner product·,·on a BanachX.

Definition 2.8. LetCbe a subset of a Banach spaceX, and letF :C → 2X _{a multimapping}

with nonempty values. We say thatFis asymptotically contractive onCwith respect to·,· if there existsx0, y0∈GrFsuch that

lim sup

x∈C,x → ∞

sup

y∈Fx

y−y0, x−x0

x−x02

<1. 2.22

Note that whenFx: {fx}, wheref :C → X is a mapping, we get a definition of the asymptotically contractive mapping onCas a variant of the notion introduced in20. Indeed, condition2.22becomes in this case as follows: there existsx0, y0∈GrFso that

lim sup

x∈C,x → ∞

fx−y0, x−x0

x−x02

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Observe that if X is a Hilbert space endowed with the scalar product noted by· | ·_X, the above inequality is then written

lim sup

x∈C,x → ∞

fx−y0 |x−x0

X x−x02

<1, 2.24

and a mapf :C → Xis said to be scalarly asymptotically contractive onCif2.24is satisfied for somex0, y0∈GrF.

In the sequel, we consider only semi-inner products on Banach spacesX, · which are compatible with the norm · . The next theorem is a multivalued version of the main result of20, Theorem 3.2for correspondances.

Theorem 2.9. LetXbe a reflexive Banach space andCa (nonempty) closed convex subset ofX. Let

F:C → 2X_{be a nonexpansive multifunction on}_C_{with closed and nonempty values. Assume that}_F

is asymptotically contractive onCwith respect to·,·. IfFC⊂CandI−Fis demi-closed, thenF

admits a fixed point onC.

Proof. Letx0, y0∈GrFsuch that2.22is satisfied, and letθnbe a sequence in0,1such

thatθn → 1. For anyn∈N, we define a multifunctionFn:C⇒Xby setting

Fnx:θnFx 1−θny0. 2.25

It is clear thatFnx⊂Cfor anynandx∈C. On the other hand, forx, x∈Candvn∈Fnx,

from2.25, there existsun ∈ Fxsuch thatvn θnun 1−θny0. Applying1.2sinceF

is nonexpansive andFxis closed, convex in the reflexive spaceX, there existsu_n ∈Fx satisfyingun−un ≤ x−x. Thus, forvn θnun 1−θny0∈Fnx, one has

_v_n₋_v

n≤θnx−x. 2.26

Then Fn is a contraction with rate θn on C. The Nadler’s theorem 7 ensures that each

multivaluedFnadmits a fixed pointxninC. So, from2.25and for someyn∈Fxn, one has

yn−y0θn−1

xn−y0

,

1−θn

y0−yn

xn−yn∈I−Fxn.

2.27

As in the proof of Proposition2.2, it suﬃces to show thatxnis bounded. Suppose on the

contrary, by taking a subsequence if necessary, that xn → ∞. As condition 2.24 is

satisfied, there existc∈0,1andρ >0 such that

∀x∈C, x ≥ρ, ∀y∈Fx:y−y0, x−x0

< cx−x02. 2.28

For largen, we haveθn> c,xn ≥ρandxnθnyn 1−θny0 ∈Fnxnwithyn∈Fxnso

that

yn−y0, xn−x0

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From the properties of the semi-inner product, we get

xn−x02 xn−x0, xn−x0

xn−y0, xn−x0

y0−x0, xn−x0

≤θn

yn−y0, xn−x0

y0−x0xn−x0

< θncxn−x02y0−x0xn−x0.

2.30

Dividing byxn−x02and taking the limit, we obtainc≥ 1, which leads to a contradiction

and the conclusion of the proposition follows.

Let us remark that whenFx : {fx}, wheref :C → X is a mapping, we get the following corollary, which is a variant of6, Proposition 1.

Corollary 2.10. LetXbe a reflexive Banach space andCa (nonempty) closed convex subset ofX. Let

f:C → Xbe a nonexpansive function onC. Assume that for somex0∈C, one has

lim sup

x∈C,x → ∞

fx−fx0, x−x0

x−x02

<1. 2.31

IffC⊂CandI−fis demi-closed, thenfadmits a fixed point.

We introduce now the following concept, which generalizes the definition ofϕ-asymptotically

bounded maps to multimaps.

Definition 2.11. Let X be a Banach space, Ca nonemptyclosed convex subset of X, F :

C → 2X a multifunction with nonempty values, and letϕ : R → R. We say thatF isϕ

-symptotically bounded onCif for somex0, y0∈GrF, there existρ, c >0 such that for any

x∈C\B0, ρandy∈Fxone has

_y₋_y₀_≤_cϕ_x₋_x₀_. _2.32

We want to give a fixed point result for a nonexpansive multimappingFwhenF−Gsatisfies some asymptotic contraction condition under the assumption that G is ϕ-asymptotically bounded multifunction. More precisely, we have the following proposition.

Proposition 2.12. LetXbe a reflexive Banach space andCa (nonempty) closed convex subset ofX.

LetF:C → 2X_{be a nonexpansive multifunction with (nonempty) closed values, and let}_G_:_C _→ ₂X

be aϕ-asymptotically bounded multifunction on Catx0, z0 ∈ GrG with limt→ ∞ϕt/t 0.

Assume that there existc∈0,1,ρ >0 such that for somey0∈Fx0one has

∀x∈C, x ≥ρ, ∀y∈Fx, ∃z∈Gx:y−z−y0, x−x0

< cx−x02. 2.33

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Proof. It is enough to prove thatthe condition2.22is satisfied. Letx0, z0∈GrG,c∈0,1,

c, ρ > 0 such that2.32and2.33are hold. Considerx∈C\B0, ρandy∈Fx. We have then, for somez∈Gx,

y−y0, x−x0

x−x02

y−zz−z0z0−y0, x−x0

x−x02

y−z−y0, x−x0

x−x02

z−z0, x−x0

x−x02

z0, x−x0

x−x02

< c z−z0

x−x0

z0

x−x0

.

2.34

We conclude that

sup

y∈Fx

y−y0, x−x0

x−x02

≤ccϕx−x0

x−x0

z0

x−x0

,

lim sup

x∈C,x → ∞ysup∈Fx

y−y0, x−x0

x−x02

≤c lim

x → ∞ c

ϕx−x0

x−x0

z0

x−x0

,

≤c <1.

2.35

Hence by Theorem2.9,Fadmits a fixed point.

Corollary 2.13. Let X be a reflexive Banach space and C a (nonempty) closed convex cone ofX.

Let f : C → X be a θ-contraction mapping with θ ∈ 0,1and F : C → 2X a1 −θ

-con-traction multifunction with closed and nonempty values. Assume thatFisϕ-asymptotically bounded

multifunction atx0, y0∈GrFwith limt→ ∞ϕt/t 0.

IffC⊂C, FC⊂CandI−fFis demi-closed, thenfFadmits a fixed point onC.

Proof. Let us verify the assumptions of Proposition 2.12 with ·,· a semi-inner-product

compatible with the norm inX. It is clear thatH : fF is nonexpansive on Cand asC is a convex cone,HC fC FC⊂CC⊂ C. Consider nowz0 fx0 y0 ∈Hx0

andx, z∈GrH. There exists then somey∈Fxsuch thatzfx y. By the properties of·,·, we get the following inequalities:

z−y−z0, x−x0

fx−fx0−y0, x−x0

fx−fx0, x−x0

−y0, x−x0

≤fx−fx0x−x0y0x−x0

≤θx−x02y0x−x0.

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Thus for allx∈Csuch thatx−x0 ≥21−θ−1y0, we obtain

z−y−z0, x−x0

≤θx−x02

1

21−θx−x0

2

≤ 1

21θx−x0

2_.

2.37

Hence the property2.33 is satisfied with the constantc : 1/21θ ∈ 0,1and the corollary follows.

The following result is close to Corollary2.7giving the existence of an eigenvalue of a nonexpansive andϕ-asymptotically bounded multifunction.

Corollary 2.14. Let C be a nonempty closed convex cone of a real reflexive Banach space X. Let

λ ≥1,θ ∈0,1, and letF :C → 2C _{be a nonexpansive multifunction with closed and nonempty}

values onCsuch that 0 ∈/ F0andFC ⊂ C. Assume thatI−λ−1F is demiclosed and thatF is

ϕ-asymptotically bounded multifunction with limt→ ∞ϕt/t 0. Thenλis an eigenvalue of the

multifunctionFassociated to an eigenvectorx∈C.

Proof. Since all assumptions of the above corollary are satisfied, there existsx∈θxλ−11−

θFxor1−θx∈λ−11−θFx. Thusλx∈Fxand as 0/∈F0, x /0. And the conclusion follows.

Acknowledgment

This paper was supported by the National Research Program of the Ministry of Higher Education and Scientific ResearchAlgeria, under Grant no. 50/03.

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