R E S E A R C H
Open Access
Approximating fixed points of multivalued
ρ
-nonexpansive mappings in modular
function spaces
Safeer Hussain Khan
1*and Mujahid Abbas
2Dedicated to Professor Wataru Takahashi on his 70th birthday.
*Correspondence: [email protected]; [email protected] 1Department of Mathematics, Statistics and Physics, Qatar University, Doha, 2713, Qatar Full list of author information is available at the end of the article
Abstract
The existence of fixed points of single-valued mappings in modular function spaces has been studied by many authors. The approximation of fixed points in such spaces via convergence of an iterative process for single-valued mappings has also been attempted very recently by Dehaish and Kozlowski (Fixed Point Theory Appl. 2012:118, 2012). In this paper, we initiate the study of approximating fixed points by the convergence of a Mann iterative process applied on multivalued
ρ-nonexpansive
mappings in modular function spaces. Our results also generalize the corresponding results of (Dehaish and Kozlowski in Fixed Point Theory Appl. 2012:118, 2012) to the case of multivalued mappings.MSC: 47H09; 47H10; 54C60
Keywords: fixed point; multivalued
ρ-nonexpansive mapping; iterative process;
modular function space1 Introduction and preliminaries
The theory of modular spaces was initiated by Nakano [] in connection with the theory of ordered spaces, which was further generalized by Musielak and Orlicz []. The fixed point theory for nonlinear mappings is an important subject of nonlinear functional anal-ysis and is widely applied to nonlinear integral equations and differential equations. The study of this theory in the context of modular function spaces was initiated by Khamsiet al.[] (see also [–]). Kumam [] obtained some fixed point theorems for nonexpansive mappings in arbitrary modular spaces. Kozlowski [] has contributed a lot towards the study of modular function spaces both on his own and with his collaborators. Of course, most of the work done on fixed points in these spaces was of existential nature. No results were obtained for the approximation of fixed points in modular function spaces until re-cently Dehaish and Kozlowski [] tried to fill this gap using a Mann iterative process for asymptotically pointwise nonexpansive mappings.
All above work has been done for single-valued mappings. On the other hand, the study of fixed points for multivalued contractions and nonexpansive mappings using the Hausdorff metric was initiated by Markin [] (see also []). Later, an interesting and rich fixed point theory for such maps was developed which has applications in control theory, convex optimization, differential inclusion, and economics (see [] and references cited
therein). Moreover, the existence of fixed points for multivalued nonexpansive mappings in uniformly convex Banach spaces was proved by Lim []. The theory of multivalued nonexpansive mappings is harder than the corresponding theory of single-valued nonex-pansive mappings. Different iterative processes have been used to approximate the fixed points of multivalued nonexpansive mappings in Banach spaces.
Dhompongsaet al.[] have proved that everyρ-contractionT:C→Fρ(C) has a fixed
point whereρ is a convex function modular satisfying the so-called-type condition,
Cis a nonemptyρ-boundedρ-closed subset ofLρandFρ(C) a family ofρ-closed
sub-sets ofC. By using this result, they asserted the existence of fixed points for multivalued ρ-nonexpansive mappings. Again their results are existential in nature. See also Kutbi and Latif [].
In this paper, we approximate fixed points ofρ-nonexpansive multivalued mappings in modular function spaces using a Mann iterative process. We make the first ever effort to fill the gap between the existence and the approximation of fixed points ofρ-nonexpansive multivalued mappings in modular function spaces. In a way, the corresponding results of Dehaish and Kozlowski [] are also generalized to the case of multivalued mappings.
Some basic facts and notation needed in this paper are recalled as follows.
Letbe a nonempty set anda nontrivialσ-algebra of subsets of. LetPbe aδ-ring of subsets of, such thatE∩A∈Pfor anyE∈PandA∈. Let us assume that there exists an increasing sequence of setsKn∈Psuch that=
Kn(for instance,Pcan be the class
of sets of finite measure in aσ-finite measure space). By A, we denote the characteristic
function of the setAin. ByE we denote the linear space of all simple functions with supports fromP. ByM∞we will denote the space of all extended measurable functions, i.e., all functionsf :→[–∞,∞] such that there exists a sequence{gn} ⊂E,|gn| ≤ |f|
andgn(ω)→f(ω) for allω∈.
Definition Letρ:M∞→[,∞] be a nontrivial, convex and even function. We say that
ρis a regular convex function pseudomodular if () ρ() = ;
() ρis monotone,i.e.,|f(ω)| ≤ |g(ω)|for anyω∈impliesρ(f)≤ρ(g), where
f,g∈M∞;
() ρis orthogonally subadditive,i.e.,ρ(fA∪B)≤ρ(fA) +ρ(fB)for anyA,B∈such
thatA∩B=φ,f ∈M∞;
() ρhas Fatou property,i.e.,|fn(ω)| ↑ |f(ω)|for allω∈impliesρ(fn)↑ρ(f), where f∈M∞;
() ρis order continuous inE,i.e.,gn∈E, and|gn(ω)| ↓impliesρ(gn)↓.
A setA∈ is said to beρ-null ifρ(gA) = for everyg∈E. A propertyp(ω) is said
to holdρ-almost everywhere (ρ-a.e.) if the set{ω∈:p(ω) does not hold}isρ-null. As usual, we identify any pair of measurable sets whose symmetric difference isρ-null as well as any pair of measurable functions differing only on aρ-null set. With this in mind we define
M(,,P,ρ) =f ∈M∞:f(ω)<∞ρ-a.e.,
Definition Letρbe a regular function pseudomodular. We say thatρis a regular convex function modular ifρ(f) = impliesf= ρ-a.e.
It is known (see []) thatρsatisfies the following properties:
() ρ() = ifff = ρ-a.e.
() ρ(αf) =ρ(f)for every scalarαwith|α|= andf ∈M.
() ρ(αf +βg)≤ρ(f) +ρ(g)ifα+β= ,α,β≥andf,g∈M.
ρis called a convex modular if, in addition, the following property is satisfied:
() ρ(αf +βg)≤αρ(f) +βρ(g)ifα+β= ,α,β≥andf,g∈M.
Definition The convex function modularρdefines the modular function spaceLρas
Lρ=
f ∈M;ρ(λf)→ asλ→.
Generally, the modularρis not subadditive and therefore does not behave as a norm or a distance. However, the modular spaceLρcan be equipped with anF-norm defined by
fρ=inf
α> :ρ
f
α
≤α .
In the caseρis convex modular,
fρ=inf
α> :ρ
f
α
≤
defines a norm on the modular spaceLρ, and it is called the Luxemburg norm.
The following uniform convexity type properties ofρcan be found in [].
Definition Letρbe a nonzero regular convex function modular defined on. Lett∈
(, ),r> ,ε> . Define
D(r,ε) =
(f,g) :f,g∈Lρ,ρ(f)≤r,ρ(g)≤r,ρ(f –g)≥εr
.
Let
δt(r,ε) =inf
–
rρ
tf+ ( –t)g: (f,g)∈D(r,ε) ifD(r,ε)=φ,
andδ(r,ε) = ifD(r,ε) =φ.
As a conventional notation,δ=δ .
Definition A nonzero regular convex function modularρis said to satisfy (UC) if for everyr> ,ε> ,δ(r,ε) > . Note that for everyr> ,D(r,ε)=φforε> small enough. ρis said to satisfy (UUC) if for everys≥,ε> , there existsη(s,ε) > depending only uponsandεsuch thatδ(r,ε) >η(s,ε) > for anyr>s.
• ρ-convergent tof ∈Lρifρ(fn–f)→asn→ ∞;
• ρ-Cauchy, ifρ(fn–fm)→asnandm→ ∞.
Consistent with [], theρ-distance from anf ∈Lρto a setD⊂Lρis given as follows:
distρ(f,D) =inf
ρ(f –h) :h∈D.
Definition A subsetD⊂Lρis called:
• ρ-closed if theρ-limit of aρ-convergent sequence ofDalways belongs toD; • ρ-a.e. closed if theρ-a.e. limit of aρ-a.e. convergent sequence ofDalways belongs
toD;
• ρ-compact if every sequence inDhas aρ-convergent subsequence inD; • ρ-a.e. compact if every sequence inDhas aρ-a.e. convergent subsequence inD; • ρ-bounded if
diamρ(D) =sup
ρ(f –g) :f,g∈D<∞.
A setD⊂Lρ is called ρ-proximinal if for eachf ∈Lρ there exists an element g∈D
such that ρ(f –g) = distρ(f,D). We shall denote the family of nonempty ρ-bounded
ρ-proximinal subsets ofDby Pρ(D), the family of nonemptyρ-closedρ-bounded
sub-sets ofDbyCρ(D) and the family ofρ-compact subsets ofDbyKρ(D). LetHρ(·,·) be the
ρ-Hausdorff distance onCρ(Lρ), that is,
Hρ(A,B) =max
sup
f∈A
distρ(f,B),sup g∈B
distρ(g,A)
, A,B∈Cρ(Lρ).
A multivalued mappingT:D→Cρ(Lρ) is said to beρ-nonexpansive if
Hρ(Tf,Tg)≤ρ(f–g), f,g∈D.
A sequence{tn} ⊂(, ) is called bounded away from if there existsa> such thattn≥a
for everyn∈N. Similarly,{tn} ⊂(, ) is called bounded away from if there existsb<
such thattn≤bfor everyn∈N.
Lemma (Lemma . []) Letρ satisfy(UUC)and let{tk} ⊂(, )be bounded away fromand.If there exists R> such that
lim sup
n→∞ ρ(fn)≤R, lim supn→∞ ρ(gn)≤R
and
lim
n→∞ρ
tnfn+ ( –tn)gn
=R,
thenlimn→∞ρ(fn–gn) = .
The above lemma is an analogue of a famous lemma due to Schu [] in Banach spaces. A functionf ∈Lρis called a fixed point ofT:Lρ→Pρ(D) iff ∈Tf. The set of all fixed
Lemma Let T:D→Pρ(D)be a multivalued mapping and
PTρ(f) =
g∈Tf :ρ(f –g) =distρ(f,Tf)
.
Then the following are equivalent: () f ∈Fρ(T),that is,f ∈Tf.
() PTρ(f) ={f},that is,f=gfor eachg∈PTρ(f).
() f ∈F(PTρ(f)),that is,f∈PTρ(f).FurtherFρ(T) =F(PTρ(f))whereF(PρT(f))denotes the set of fixed points ofPT
ρ(f).
Proof ()⇒(). Sincef ∈Fρ(T)⇒f ∈Tf, sodistρ(f,Tf) = . Therefore, for any g∈ PT
ρ(f),ρ(f–g) =distρ(f,Tf) = implies thatρ(f–g) = . Hencef=g. That is,PρT(f) ={f}.
()⇒(). Obvious.
()⇒(). Sincef ∈F(PTρ(f)), so by definition ofPρT(f) we havedistρ(f,Tf) =ρ(f–f) = .
Thusf ∈Tf byρ-closedness ofTf.
Definition A multivalued mappingT:D→Cρ(D) is said to satisfy condition (I) if there
exists a nondecreasing functionl: [,∞)→[,∞) withl() = ,l(r) > for allr∈(,∞) such thatdistρ(f,Tf)≥l(distρ(f,Fρ(T))) for allf ∈D.
It is a multivalued version of condition (I) of Senter and Dotson [] in the framework of modular function spaces.
2 Main results
We prove a key result giving a major support to ourρ-convergence result for approxi-mating fixed points of multivaluedρ-nonexpansive mappings in modular function spaces using a Mann iterative process.
Theorem Let ρ satisfy(UUC)and D a nonemptyρ-closed, ρ-bounded and convex subset of Lρ.Let T:D→Pρ(D)be a multivalued mapping such that PTρis aρ-nonexpansive mapping.Suppose that Fρ(T)=φ.Let{fn} ⊂D be defined by the Mann iterative process:
fn+= ( –αn)fn+αnun,
where un∈PTρ(fn)and{αn} ⊂(, )is bounded away from bothand.Then
lim
n→∞ρ(fn–c)exists for all c∈Fρ(T)
and
lim
n→∞ρ
fn–PρT(fn)
= .
Proof Letc∈Fρ(T). By Lemma ,PTρ(c) ={c}. Moreover, by the same lemma,Fρ(T) = F(PT
ρ). To prove thatlimn→∞ρ(fn–c) exists for allc∈Fρ(T), consider
ρ(fn+–c) =ρ
( –αn)fn+αnun–c
=ρ( –αn)(fn–c) +αn(un–c)
By convexity ofρ, we have
ρ(fn+–c)≤( –αn)ρ(fn–c) +αnρ(un–c) ≤( –αn)Hρ
PTρ(fn),PTρ(c)
+αnHρ
PρT(fn),PρT(c)
≤( –αn)ρ(fn–c) +αnρ(fn–c)
=ρ(fn–c).
Hencelimn→∞ρ(fn–c) exists for eachc∈Fρ(T).
Suppose that
lim
n→∞ρ(fn–c) =L, (.)
whereL≥. We now prove that
lim
n→∞ρ
fn–PρT(fn)
= .
Asdistρ(fn,PTρ(fn))≤ρ(fn–un), it suffices to prove that
lim
n→∞ρ(fn–un) = .
Since
ρ(un–c)≤Hρ
PTρ(fn),PTρ(c)
≤ρ(fn–c),
therefore
lim sup
n→∞ ρ(un–c)≤lim supn→∞ ρ(fn–c)
and so in view of (.), we have
lim sup
n→∞
ρ(un–c)≤L. (.)
As
lim
n→∞ρ(fn+–c) =nlim→∞ρ
( –αn)fn+αnun–c
(.)
= lim
n→∞ρ
( –αn)(fn–c) +αn(un–c)
(.)
=L, (.)
from (.), (.), (.), and Lemma , we have
lim
n→∞ρ(fn–un) = .
Hence
lim
n→∞distρ
fn,PTρ(fn)
Now we are all set for our convergence result for approximating fixed points of mul-tivaluedρ-nonexpansive mappings in modular function spaces using the Mann iterative process as follows.
Theorem Letρsatisfy(UUC)and D a nonemptyρ-compact,ρ-bounded and convex subset of Lρ.Let T:D→Pρ(D)be a multivalued mapping such that PρTisρ-nonexpansive mapping.Suppose that Fρ(T)=φ.Let{fn}be as defined in Theorem.Then{fn}ρ-converges to a fixed point of T.
Proof From ρ-compactness of D, there exists a subsequence {fnk} of {fn} such that
limk→∞(fnk –q) = for someq∈D. To prove thatq is a fixed point ofT, letg be an
arbitrary point inPT
ρ(q) andf inPρT(fnk). Note that
ρ
q–g
=ρ
q–fnk
+
fnk–f
+
f –g
≤
ρ(q–fnk) +
ρ(fnk–f) +
ρ(f –g)
≤ρ(q–fnk) +distρ
fnk,P T ρ(fnk)
+distρ
PρT(fnk),g
≤ρ(q–fnk) +distρ
fnk,P T ρ(fnk)
+Hρ
PTρ(fnk),P T ρ(q)
≤ρ(q–fnk) +distρ
fnk,P T ρ(fnk)
+ρ(q–fnk).
By Theorem , we havelimn→∞distρ(fn,PρT(fn)) = . This givesρ(q–g) = . Henceqis a
fixed point ofPT
ρ. Since the set of fixed points ofPρTis the same as that ofTby Lemma , {fn}ρ-converges to a fixed point ofT.
Theorem Letρsatisfy(UUC)and D a nonemptyρ-closed,ρ-bounded and convex sub-set of Lρ.Let T:D→Pρ(D)be a multivalued mapping with and Fρ(T)=φand satisfying condition(I)such that PT
ρ isρ-nonexpansive mapping.Let{fn}be as defined in Theorem. Then{fn}ρ-converges to a fixed point of T.
Proof From Theorem ,limn→∞ρ(fn–c) exists for allc∈F(PTρ) =Fρ(T). Iflimn→∞ρ(fn– c) = , there is nothing to prove. We assumelimn→∞ρ(fn–c) =L> . Again from
Theo-rem ,ρ(fn+–c)≤ρ(fn–c) so that
distρ
fn+,Fρ(T)
≤distρ
fn,Fρ(T)
.
Hencelimn→∞distρ(fn,Fρ(T)) exists. We now prove thatlimn→∞distρ(fn,Fρ(T)) = . By
using condition (I) and Theorem , we have
lim
n→∞l
distρ
fn,Fρ(T)
≤ lim
n→∞distρ(fn,Tfn) = .
That is,
lim
n→∞l
distρ
fn,Fρ(T)
= .
Next, we show that{fn}is aρ-Cauchy sequence inD. Letε> be arbitrarily chosen.
Sincelimn→∞distρ(fn,Fρ(T)) = , there exists a constantnsuch that for alln≥n, we have
distρ
fn,Fρ(T)
<ε
.
In particular,inf{ρ(fn–c) :c∈Fρ(T)}<
ε
. There must exist ac∗∈Fρ(T) such that
ρfn–c∗
<ε.
Now form,n≥n, we have
ρ
fn+m–fn
≤
ρ
fn+m–c∗
+
ρ
fn–c∗
≤ρfn–c∗
<ε.
Hence{fn}is aρ-Cauchy sequence in aρ-closed subsetDofLρ, and so it must converge
inD. Letlimn→∞fn=q. Thatqis a fixed point ofTnow follows from Theorem .
We now give some examples. The first one shows the existence of a mapping satisfying the condition (I) whereas the second one shows the existence of a mapping satisfying all the conditions of Theorem .
Example LetLρ=M[, ] (the collection of all real valued measurable functions on
[, ]). Note thatM[, ] is a modular function space with respect to
ρ(f) =
|f|.
LetD={f ∈Lρ:≤f(x)≤}. ObviouslyDis a nonempty closed and convex subset ofLρ.
DefineT:D→Cρ(Lρ) as
Tf =
g∈Lρ:
≤g(x)≤ +
f(x)
.
Define a continuous and nondecreasing functionl: [,∞)→[,∞) byl(r) =r. It is obvi-ous thatdistρ(f,Tf)≥l(distρ(f,FT)) for allf ∈D. HenceT satisfies the condition (I).
Example The real number systemRis a space modulared byρ(f) =|f|. LetD= [, ]. ObviouslyDis a nonempty closed and convex subset ofR. DefineT:D→Pρ(D) as
Tf =
, +f
.
Define a continuous and nondecreasing functionl: [,∞)→[,∞) byl(r) = r
Note thatPT
ρ(f) ={f}whenf ∈D. HencePTρ is nonexpansive. Moreover, by Lemma , PT
ρ(f) ={f} ⇒f ∈Tf for allf ∈D. Thus{fn} ⊂Ddefined byfn+= ( –αn)fn+αnunwhere un∈PTρ(fn)ρ-converges to a fixed point ofT.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors worked on the manuscript. Both read and approved the final manuscript.
Author details
1Department of Mathematics, Statistics and Physics, Qatar University, Doha, 2713, Qatar.2Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood road, Pretoria, 0002, South Africa.
Acknowledgements
The first author owes a lot to Professor Wataru Takahashi from whom he started learning the very alphabets of Fixed Point Theory during his doctorate at Tokyo Institute of Technology, Tokyo, Japan. He is extremely indebted to Professor Takahashi and wishes him a long healthy active life. The authors are thankful to the anonymous referees for giving valuable comments.
Received: 3 October 2013 Accepted: 24 January 2014 Published:11 Feb 2014 References
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