* Correspondence: safeer@qu.edu.qa; safeerhussain5@yahoo.com 1 Department of Mathematics, Statistics and Physics, Qatar University, Doha, 2713, Qatar Full list of author information is a
Trang 1R E S E A R C H Open Access
Approximating fixed points of multivalued
ρ-nonexpansive mappings in modular
function spaces
Safeer Hussain Khan1*and Mujahid Abbas2
Dedicated to Professor Wataru Takahashi on his 70th birthday
* Correspondence:
safeer@qu.edu.qa;
safeerhussain5@yahoo.com
1 Department 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 multivaluedρ-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; multivaluedρ-nonexpansive mapping; iterative process; modular function space
1 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 Khamsi et
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
©2014 Khan and Abbas; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction
Trang 2therein) 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
Dhompongsa et al [] have proved that every ρ-contraction T : C → F ρ (C) has a fixed
point whereρ is a convex function modular satisfying the so-called -type condition,
C is a nonempty ρ-bounded ρ-closed subset of L ρ and F ρ (C) a family of ρ-closed
sub-sets of C 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
Let be a nonempty set and a nontrivial σ -algebra of subsets of Let P be a δ-ring of
subsets of, such that E ∩ A ∈ P for any E ∈ P and A ∈ Let us assume that there exists
an increasing sequence of sets K n∈P such that =K n(for instance,P can be the class
of sets of finite measure in aσ -finite measure space) By A, we denote the characteristic
function of the set A in By E we denote the linear space of all simple functions with
supports fromP By M∞we will denote the space of all extended measurable functions,
i.e., all functions f : → [–∞, ∞] such that there exists a sequence {g n} ⊂E, |g n | ≤ |f |
and g n(ω) → f (ω) for all ω ∈ .
Definition Let ρ : M∞→ [, ∞] be a nontrivial, convex and even function We say that
ρ is a regular convex function pseudomodular if
() ρ() = ;
() ρ is monotone, i.e., |f (ω)| ≤ |g(ω)| for any ω ∈ implies ρ(f ) ≤ ρ(g), where
f , g∈M∞; () ρ is orthogonally subadditive, i.e., ρ(f A ∪B)≤ ρ(f A) +ρ(f B ) for any A, B ∈ such that A ∩ B = φ, f ∈ M∞;
() ρ has Fatou property, i.e., |f n(ω)| ↑ |f (ω)| for all ω ∈ implies ρ(f n)↑ ρ(f ), where
f∈M∞; () ρ is order continuous in E, i.e., g n∈E, and |g n(ω)| ↓ implies ρ(g n)↓
A set A ∈ is said to be ρ-null if ρ(g A ) = for every g∈E A property p(ω) is said
to holdρ-almost everywhere (ρ-a.e.) if the set {ω ∈ : p(ω) does not hold} is ρ-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.
,
where f ∈M(, , P, ρ) is actually an equivalence class of functions equal ρ-a.e rather
than an individual function Where no confusion exists we will write M instead of
M(, , P, ρ).
Trang 3Definition Let ρ be a regular function pseudomodular We say that ρ is a regular convex
function modular ifρ(f ) = implies f = ρ-a.e.
It is known (see []) thatρ satisfies the following properties:
() ρ() = iff f = ρ-a.e.
() ρ(αf ) = ρ(f ) for every scalar α with |α| = and f ∈ M.
() ρ(αf + βg) ≤ ρ(f ) + ρ(g) if α + β = , α, β ≥ and f , g ∈ M.
ρ is called a convex modular if, in addition, the following property is satisfied:
() ρ(αf + βg) ≤ αρ(f ) + βρ(g) if α + β = , α, β ≥ and f , g ∈ M.
Definition The convex function modular ρ defines the modular function space L ρas
L ρ=
f ∈M; ρ(λf ) → as λ → Generally, the modularρ is not subadditive and therefore does not behave as a norm or
a distance However, the modular space L ρ can be equipped with an F-norm defined by
f ρ= inf
α > : ρ
f α
≤ α
In the caseρ is convex modular,
f ρ= inf
α > : ρ
f α
≤
defines a norm on the modular space L ρ, and it is called the Luxemburg norm
The following uniform convexity type properties ofρ can be found in [].
Definition Let ρ be a nonzero regular convex function modular defined on Let t ∈
(, ), r > , ε > Define
D(r,ε) =(f , g) : f , g ∈ L ρ,ρ(f ) ≤ r, ρ(g) ≤ r, ρ(f – g) ≥ εr Let
δ t
(r, ε) = inf
–
r ρ tf + ( – t)g
: (f , g) ∈ D(r,ε)
if D(r,ε) = φ,
andδ(r, ε) = if D(r,ε) = φ.
As a conventional notation,δ=δ
Definition A nonzero regular convex function modular ρ is said to satisfy (UC) if for
every r > , ε > , δ(r, ε) > Note that for every r > , D(r, ε) = φ for ε > small enough.
ρ is said to satisfy (UUC) if for every s ≥ , ε > , there exists η(s, ε) > depending only
upon s and ε such that δ(r, ε) > η(s, ε) > for any r > s.
Definition Let L ρbe a modular space The sequence{f n } ⊂ L ρis called:
Trang 4• ρ-convergent to f ∈ L ρifρ(f n – f ) → as n → ∞;
• ρ-Cauchy, if ρ(f n – f m)→ as n and m → ∞.
Consistent with [], theρ-distance from an f ∈ L ρ to a set D ⊂ L ρis given as follows:
distρ (f , D) = inf
ρ(f – h) : h ∈ D
Definition A subset D ⊂ L ρis called:
• ρ-closed if the ρ-limit of a ρ-convergent sequence of D always belongs to D;
• ρ-a.e closed if the ρ-a.e limit of a ρ-a.e convergent sequence of D always belongs
to D;
• ρ-compact if every sequence in D has a ρ-convergent subsequence in D;
• ρ-a.e compact if every sequence in D has a ρ-a.e convergent subsequence in D;
• ρ-bounded if
diamρ (D) = sup
ρ(f – g) : f , g ∈ D<∞
A set D ⊂ L ρ is called ρ-proximinal if for each f ∈ L ρ there exists an element g ∈ D
such that ρ(f – g) = dist ρ (f , D) We shall denote the family of nonempty ρ-bounded
ρ-proximinal subsets of D by P ρ (D), the family of nonempty ρ-closed ρ-bounded
sub-sets of D by C ρ (D) and the family of ρ-compact subsets of D by K ρ (D) Let H ρ(·, ·) be the
ρ-Hausdorff distance on C ρ (L ρ), that is,
H ρ (A, B) = max
sup
f ∈A
distρ (f , B), sup
g ∈Bdistρ (g, A) , A, B ∈ C ρ (L ρ)
A multivalued mapping T : D → C ρ (L ρ) is said to beρ-nonexpansive if
H ρ (Tf , Tg) ≤ ρ(f – g), f , g ∈ D.
A sequence{t n } ⊂ (, ) is called bounded away from if there exists a > such that t n ≥ a
for every n ∈ N Similarly, {t n } ⊂ (, ) is called bounded away from if there exists b <
such that t n ≤ b for every n ∈ N.
Lemma (Lemma . []) Let ρ satisfy (UUC) and let {t k } ⊂ (, ) be bounded away
from and If there exists R > such that
lim sup
n→∞ ρ(f n)≤ R, lim sup
n→∞ ρ(g n)≤ R
and
lim
n→∞ρ t n f n + ( – t n )g n
= R,
then lim n→∞ρ(f n – g n) =
The above lemma is an analogue of a famous lemma due to Schu [] in Banach spaces
A function f ∈ L ρ is called a fixed point of T : L ρ → P ρ (D) if f ∈ Tf The set of all fixed points of T will be denoted by F ρ (T).
Trang 5Lemma Let T : D → P ρ (D) be a multivalued mapping and
P T ρ (f ) =
g ∈ Tf : ρ(f – g) = dist ρ (f , Tf )
Then the following are equivalent:
() f ∈ F ρ (T), that is, f ∈ Tf () P T
ρ (f ) = {f }, that is, f = g for each g ∈ P T
ρ (f ).
() f ∈ F(P T
ρ (f )), that is, f ∈ P T
ρ (f ) Further F ρ (T) = F(P T
ρ (f )) where F(P T
ρ (f )) denotes the
set of fixed points of P T ρ (f ).
Proof () ⇒ () Since f ∈ F ρ (T) ⇒ f ∈ Tf , so dist ρ (f , Tf ) = Therefore, for any g ∈
P T
ρ (f ), ρ(f – g) = dist ρ (f , Tf ) = implies that ρ(f – g) = Hence f = g That is, P T
ρ (f ) = {f }.
()⇒ () Obvious
()⇒ () Since f ∈ F(P T
ρ (f )), so by definition of P T
ρ (f ) we have dist ρ (f , Tf ) = ρ(f –f ) = .
Definition A multivalued mapping T : D → C ρ (D) is said to satisfy condition (I) if there
exists a nondecreasing function l : [, ∞) → [, ∞) with l() = , l(r) > for all r ∈ (, ∞)
such that distρ (f , Tf ) ≥ l(dist ρ (f , F ρ (T))) for all f ∈ 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 nonempty ρ-closed, ρ-bounded and convex
subset of L ρ Let T : D → P ρ (D) be a multivalued mapping such that P T
ρ is a ρ-nonexpansive mapping Suppose that F ρ (T) = φ Let {f n } ⊂ D be defined by the Mann iterative process:
f n+= ( –α n )f n+α n u n,
where u n ∈ P T
ρ (f n ) and {α n } ⊂ (, ) is bounded away from both and Then
lim
n→∞ρ(f n – c) exists for all c ∈ F ρ (T)
and
lim
n→∞ρ f n – P ρ T (f n)
=
Proof Let c ∈ F ρ (T) By Lemma , P T
ρ (c) = {c} Moreover, by the same lemma, F ρ (T) =
F(P T
ρ) To prove that limn→∞ρ(f n – c) exists for all c ∈ F ρ (T), consider
ρ(f n+ – c) = ρ( –α n )f n+α n u n – c
=ρ( –α )(f – c) + α (u – c)
Trang 6
By convexity ofρ, we have
ρ(f n+ – c) ≤ ( – α n)ρ(f n – c) + α n ρ(u n – c)
≤ ( – α n )H ρ P T ρ (f n ), P T ρ (c)
+α n H ρ P ρ T (f n ), P ρ T (c)
≤ ( – α n)ρ(f n – c) + α n ρ(f n – c)
=ρ(f n – c).
Hence limn→∞ρ(f n – c) exists for each c ∈ F ρ (T).
Suppose that
lim
where L≥
We now prove that
lim
n→∞ρ f n – P T
ρ (f n)
=
As distρ (f n , P T
ρ (f n))≤ ρ(f n – u n), it suffices to prove that lim
n→∞ρ(f n – u n) =
Since
ρ(u n – c) ≤ H ρ P T ρ (f n ), P T ρ (c)
≤ ρ(f n – c),
therefore
lim sup
n→∞ ρ(u n – c)≤ lim sup
n→∞ ρ(f n – c)
and so in view of (.), we have
lim sup
As
lim
n→∞ρ(f n+ – c) = lim
n→∞ρ( –α n )f n+α n u n – c
(.)
= lim
n→∞ρ( –α n )(f n – c) + α n (u n – c)
(.)
from (.), (.), (.), and Lemma , we have
lim
n→∞ρ(f n – u n) =
Hence
lim
→∞distρ f n , P T ρ (f n)
Trang 7Now 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 Let ρ satisfy (UUC) and D a nonempty ρ-compact, ρ-bounded and convex
subset of L ρ Let T : D → P ρ (D) be a multivalued mapping such that P T
ρ is ρ-nonexpansive mapping Suppose that F ρ (T) = φ Let {f n } be as defined in Theorem Then {f n } ρ-converges
to a fixed point of T
Proof From ρ-compactness of D, there exists a subsequence {f nk } of {f n} such that
limk→∞(f nk – q) = for some q ∈ D To prove that q is a fixed point of T, let g be an
arbitrary point in P T
ρ (q) and f in P T
ρ (f nk) Note that
ρ
q – g
=ρ
q – f nk
f nk – f
f – g
ρ(q – f nk) +
ρ(f nk – f ) +
ρ(f – g)
≤ ρ(q – f n k) + distρ f n k , P ρ T (f n k)
+ distρ P ρ T (f n k ), g
≤ ρ(q – f nk) + distρ f nk , P ρ T (f nk)
+ H ρ P T ρ (f nk ), P ρ T (q)
≤ ρ(q – f nk) + distρ f nk , P ρ T (f nk)
+ρ(q – f nk)
By Theorem , we have limn→∞distρ (f n , P T
ρ (f n)) = This givesρ( q–g
) = Hence q is a fixed point of P T
ρ Since the set of fixed points of P T
ρ is the same as that of T by Lemma ,
Theorem Let ρ satisfy (UUC) and D a nonempty ρ-closed, ρ-bounded and convex
sub-set of L ρ Let T : D → P ρ (D) be a multivalued mapping with and F ρ (T) = φ and satisfying
condition (I) such that P T
ρ is ρ-nonexpansive mapping Let {f n } be as defined in Theorem .
Then {f n } ρ-converges to a fixed point of T.
Proof From Theorem , lim n→∞ρ(f n – c) exists for all c ∈ F(P T
ρ ) = F ρ (T) If lim n→∞ρ(f n–
c) = , there is nothing to prove We assume lim n→∞ρ(f n – c) = L > Again from
Theo-rem ,ρ(f n+ – c) ≤ ρ(f n – c) so that
distρ f n+ , F ρ (T)
≤ distρ f n , F ρ (T)
Hence limn→∞distρ (f n , F ρ (T)) exists We now prove that lim n→∞distρ (f n , F ρ (T)) = By
using condition (I) and Theorem , we have
lim
n→∞l distρ f n , F ρ (T)
≤ lim
n→∞distρ (f n , Tf n) =
That is,
lim
n→∞l distρ f n , F ρ (T)
=
Since l is a nondecreasing function and l() = , it follows that lim n→∞distρ (f n , F ρ (T)) = .
Trang 8Next, we show that{f n } is a ρ-Cauchy sequence in D Let ε > be arbitrarily chosen.
Since limn→∞distρ (f n , F ρ (T)) = , there exists a constant nsuch that for all n ≥ n, we
have
distρ f n , F ρ (T)
<ε
.
In particular, inf{ρ(fn– c) : c ∈ F ρ (T)} < ε
There must exist a c∗∈ F ρ (T) such that
ρ f n– c∗
<ε.
Now for m, n ≥ n, we have
ρ
f n+m – f n
ρ f n+m – c∗
+
ρ f n – c∗
≤ ρ f n– c∗
<ε.
Hence{f n } is a ρ-Cauchy sequence in a ρ-closed subset D of L ρ, and so it must converge
in D Let lim n→∞f n = q That q is a fixed point of T now 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 Let L ρ = M[, ] (the collection of all real valued measurable functions on
[, ]) Note that M[, ] is a modular function space with respect to
ρ(f ) =
|f |.
Let D = {f ∈ L ρ:≤ f (x) ≤ } Obviously D is a nonempty closed and convex subset of L ρ
Define T : D → C ρ (L ρ) as
Tf =
g ∈ L ρ:
≤ g(x) ≤ + f (x)
Define a continuous and nondecreasing function l : [, ∞) → [, ∞) by l(r) = r
It is obvi-ous that distρ (f , Tf ) ≥ l(dist ρ (f , F T )) for all f ∈ D Hence T satisfies the condition (I).
Example The real number system R is a space modulared by ρ(f ) = |f | Let D = [, ].
Obviously D is a nonempty closed and convex subset of R Define T : D → P ρ (D) as
Tf =
, +f
Define a continuous and nondecreasing function l : [, ∞) → [, ∞) by l(r) = r
It is obvi-ous that distρ (f , Tf ) ≥ l(dist ρ (f , F T )) for all f ∈ D.
Trang 9Note that P ρ T (f ) = {f } when f ∈ D Hence P T
ρ is nonexpansive Moreover, by Lemma ,
P T
ρ (f ) = {f } ⇒ f ∈ Tf for all f ∈ D Thus {f n } ⊂ D defined by f n+= ( –α n )f n+α n u nwhere
u n ∈ P T
ρ (f n)ρ-converges to a fixed point of T.
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
1 Department of Mathematics, Statistics and Physics, Qatar University, Doha, 2713, Qatar 2 Department 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
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10.1186/1687-1812-2014-34
Cite this article as: Khan and Abbas: Approximating fixed points of multivaluedρ-nonexpansive mappings in
modular function spaces Fixed Point Theory and Applications 2014, 2014:34
... class="page_container" data-page="7">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... and Abbas: Approximating fixed points of multivalued< /small>ρ -nonexpansive mappings in< /small>
modular function spaces Fixed Point Theory and Applications... 1123-1126 (1974)
16 Kutbi, MA, Latif, A: Fixed points of multivalued mappings in modular function spaces Fixed Point Theory Appl 2009,
Article ID 786357