Volume 2009, Article ID 786357,12pages doi:10.1155/2009/786357
Research Article
Fixed Points of Multivalued Maps in
Modular Function Spaces
Marwan A. Kutbi and Abdul Latif
Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
Correspondence should be addressed to Abdul Latif,[email protected]
Received 7 February 2009; Accepted 14 April 2009
Recommended by Jerzy Jezierski
The purpose of this paper is to study the existence of fixed points for contractive-type and nonexpansive-type multivalued maps in the setting of modular function spaces. We also discuss the concept ofw-modular function and prove fixed point results forweakly-modular contractive maps in modular function spaces. These results extend several similar results proved in metric and Banach spaces settings.
Copyrightq2009 M. A. Kutbi and A. Latif. 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 and Preliminaries
The well-known Banach fixed point theorem on complete metric spaces specifically, each contraction self-map of a complete metric space has a unique fixed pointhas been extended and generalized in different directions. For example, see Edelstein 1, 2, Kasahara 3, Rhoades4, Siddiq and Ansari5, and others. One of its generalizations is for nonexpansive single-valued maps on certain subsets of a Banach space. Indeed, these fixed points are not necessarily unique. See, for example, Browder 6–8 and Kirk 9. Fixed point theorems for contractive and nonexpansive multivalued maps have also been established by several authors. LetHdenote the Hausdorffmetric on the space of all bounded nonempty subsets of a metric spaceX, d. A multivalued mapJ:X → 2Xwhere 2Xdenotes the collection of all
nonempty subsets ofXwith bounded subsets as values is called contractive10if
HJx, Jy≤hdx, y 1.1
metric and Banach spaces. Note that an elementx∈Xis called a fixed point of a multivalued mapJ : X → 2X ifx ∈ Jx. Among others, without using the concept of the Hausdorff
metric, Husain and Tarafdar13introduced the notion of a nonexpansive-type multivalued map and proved a fixed point theorem on compact intervals of the real line. Using such type of notions Husain and Latif14extended their result to general Banach space setting.
The fixed point results in modular function spaces were given by Khamsi et al.15. Even though a metric is not defined, many problems in metric fixed point theory can be reformulated in modular spaces. For instance, fixed point theorems are proved in15,16for nonexpansive maps.
In this paper, we define nonexpansive-type and contractive-type multivalued maps in modular function spaces, investigate the existence of fixed points of such mappings, and prove similar results found in17.
Now, we recall some basic notions and facts about modular spaces as formulated by Kozlowski18. For more details the reader may consult15,16.
LetΩbe a nonempty set and letΣbe a nontrivialσ-algebra of subsets ofΩ. LetPbe a
δ-ring of subsets ofΣ, such thatE∩A∈ Pfor anyE∈ PandA∈Σ.
Let us assume that there exists an increasing sequence of setsKn ∈ Psuch thatΩ
Kn. ByEwe denote the linear space of all simple functions with supports fromP. ByM
we will denote the space of all measurable functions, that is, all functionsf :Ω → Rsuch that there exists a sequence{gn} ∈ E,|gn| ≤ |f|andgnω → fωfor allω ∈Ω. By 1A we
denote the characteristic function of the setA.
Definition 1.1. A functionalρ:E ×Σ → 0,∞is called a function modular if
P1ρ0, E 0 for anyE∈Σ,
P2ρf, E≤ρg, Ewhenever|fω| ≤ |gω|for anyω∈Ω,f, g∈ EandE∈Σ,
P3ρf,·:Σ → 0,∞is aσ-subadditive measure for everyf∈ E,
P4ρα, A → 0 asαdecreases to 0 for everyA∈ P, whereρα, A ρα1A, A,
P5if there existsα >0 such thatρα, A 0, thenρβ, A 0 for everyβ >0, and
P6for anyα >0, ρα, .is order continuous onP, that is,ρα, An → 0 if{An} ∈ P
and decreases to∅.
The definition ofρis then extended tof∈ Mby
ρf, Esupρg, E;g∈ε,gω≤fω,for everyω∈Ω. 1.2
For the sake of simplicity we writeρfinstead ofρf,Ω.
Definition 1.2. A setEis said to beρ-null ifρα, E 0 for everyα >0.A propertypwis said to holdρ-almost everywhereρ-a.e.if the set{w∈Ω:pwdoes not hold}isρ-null.
ρ:M → 0,∞is a modular and satisfies the following properties:
iρf 0 if and only iff0 ρ-a.e.,
iiραf ρffor every scalarαwith|α|1 andf∈ M, and
iiiραfβg≤ρf ρgifαβ1,α≥0, β≥0 andf, g∈ M.
In addition, if the following property is satisfied,
iii’ραfβg≤αρf βρgifαβ1 ,α≥0, β≥0 and,f, g∈ M,
we say thatρis a convex modular.
The modularρ defines a corresponding modular space, that is, the vector spaceLρ
given by
Lρ
f ∈ M;ρλf−→0 as λ−→0. 1.3
Whenρis convex, the formula
fpinf α >0;ρ
f
α
≤1
1.4
defines a norm in the modular spaceLρwhich is frequently called the Luxemburg norm. We
can also consider the space
Eρ
f∈M;ραf, An
→0 asn→ ∞for everyAn∈Σthat decreases to∅ andα >0
. 1.5
Definition 1.4. A function modular is said to satisfy theΔ2-condition if supn≥1ρ2fn, Dk →
0 ask → ∞whenever{fn}n≥1 ⊂ M, Dk ∈Σ decreases to ∅and supn≥1ρfn, Dk → 0 as k → ∞.
We know from18thatEρLρwhenρsatisfies theΔ2-condition.
Definition 1.5. A function modular is said to satisfy theΔ2-type condition if there existsK >0
such that for anyf∈Lρwe haveρ2f≤Kρf.
In general,Δ2-type condition andΔ2-condition are not equivalent, even though it is
obvious thatΔ2-type condition impliesΔ2-condition on the modular spaceLρ.
Definition 1.6. LetŁρbe a modular space.
1The sequence{fn} ⊂ Lρis said to beρ-convergent tof ∈ Lρ ifρfn−f → 0 as n → ∞.
2The sequence{fn} ⊂ Lρ is said to beρ-a.e. convergent tof ∈ Lρ if the set{ω ∈ Ω;fnωfω}isρ-null.
3The sequence{fn} ⊂Lρis said to beρ-Cauchy ifρfn−fm → 0 asnandmgo to
∞.
4A subsetCofLρ is calledρ-closed if theρ-limit of aρ-convergent sequence of C
5A subset C of Lρ is called ρ-a.e. closed if the ρ-a.e. limit of a ρ-a.e. convergent
sequence ofCalways belongs toC.
6A subset C of Lρ is called ρ-a.e. compact if every sequence in C has a ρ-a.e.
convergent subsequence inC.
7A subsetCofLρis calledρ-bounded if
δρC sup
ρf−g;f, g∈C<∞. 1.6
We recall two basic resultssee15in the theory of modular spaces.
iIf there exists a number α > 0 such that ραfn −f → 0, then there exists a
subsequence{gn}of{fn}such thatgn → fρ-a.e.
ii Lebesgue’s TheoremIffn, f ∈ M,fn → fρ-a.e. and there exists a functiong∈Eρ
such that|fn| ≤ |g|ρ-a.e. for alln,thenfn−fp → 0.
We know, by 15, 16 that under Δ2-condition the norm convergence and modular
convergence are equivalent, which implies that the norm and modular convergence are also the same when we deal with theΔ2-type condition. In the sequel we will assume that the
modular functionρis convex and satisfies theΔ2-type condition.
Definition 1.7. Letρbe as aforementioned. We define a growth functionωby
ωt sup
ρtf
ρf, f∈Lρ\ {0}
∀0≤t <∞. 1.7
We have the following:
Lemma 1.8see19. Letρbe as aforementioned. Then the growth functionωhas the following properties:
1ωt<∞,∀t∈0,∞,
2ω:0,∞ → 0,∞is a convex, strictly increasing function. So, it is continuous,
3ωαβ≤ωαωβ;∀α, β∈0,∞,
4ω−1αω−1β≤ω−1αβ;∀α, β∈0,∞,whereω−1is the function inverse ofω.
The following lemma shows that the growth function can be used to give an upper bound for the norm of a function.
Lemma 1.9see19. Letρbe a convex function modular satisfying theΔ2-type condition. Then
f
p≤
1
ω−11/ρf whenever f ∈Lρ. 1.8
Lemma 1.10see16. Letρbe a function modular satisfying theΔ2-condition and let{fn}be a sequence inLρsuch thatfn
ρ−a.e
→ f ∈Lρ, and there existsk > 1such thatsupnρkfn−f< ∞. Then,
lim inf
n→ ∞ ρ
fn−g
lim inf
n→ ∞ ρ
fn−f
ρf−g ∀g∈Lρ. 1.9
Moreover, one has
ρf≤lim inf
n→ ∞ ρ
fn
. 1.10
2. Fixed Points of Contractive-Type and Nonexpansive-Type Maps
In the sequel we assume thatρis a convex,σ-finite modular function satisfying theΔ2-type
condition, and C is a nonemptyρ-bounded subset of the modular function spaceLρ. We
denote thatCCis a collection of all nonemptyρ-closed subsets ofC, andKCis a collection of all nonemptyρ-compact subsets ofC.
We say that a multivalued mapT : C → 2C isρ-contractive-type if there existsk ∈ 0,1such that for anyf, g∈Cand for anyF∈Tf, there existsG∈Tgsuch that
ρF−G≤kρf−g, 2.1
andρ-nonexpansive-type if for anyf, g∈Cand for anyF ∈Tf, there existsG∈Tgsuch that
ρF−G≤ρf−g. 2.2
We have the following fixed point theoremfor which a similar result may be found in17.
Theorem 2.1. Let C be a nonemptyρ-closed subset of the modular function spaceLρ. Then any T : C → CCρ-contractive-type map has a fixed point, that is, there exists f ∈ C such that
f∈Tf.
Proof. Letf0 ∈ C. Without loss of generality, assume thatf0 is not a fixed point ofT. Then
there exists f1 ∈ Tf0such thatf1/f0. Henceρf0, f1 > 0. SinceT isρ-contractive-type,
then there existsf2∈Tf1such that
ρf1−f2
≤kρf0−f1
. 2.3
By induction, one can easily construct a sequence{fn} ∈Csuch thatfn1∈Tfnand
ρfn1−fn
≤kρfn−fn−1
for anyn≥1. In particular we have
ρfn1−fn
≤knρf1−f0
. 2.5
Without loss of generality, we may assumeρfn1, fn/0, otherwisefnis a fixed point ofT.
Hence
1
knρf
1−f0
≤ 1
ρfn1−fn
2.6
UsingLemma 1.9, we get
fn1−fnρ≤
1
ω−11/ρfn1−fn. 2.7
Using the properties ofωt, we get
ω−1
1
knρf1−f0
≤ω−1
1
ρfn1−fn
. 2.8
So
ω−1
1
k
n ω−1
1
ρf1−f0
≤ω−1
1
ρfn1−fn
, 2.9
which implies
fn1−fnρ≤
1
ω−11/knω−11/ρf1−f0. 2.10
Sinceω1 1 andk <1, then 1< ω−11/k. This forces{fn}to be · ρ-Cauchy. Hence the
sequence{fn} · ρ-converges to somef ∈Lρ. Sinceρsatisfies theΔ2-condition, then{fn}ρ
-converges tof. SinceCisρ-closed, thenf ∈C. Let us prove thatfis indeed a fixed point of
T. SinceT is aρ-contractive-type mapping, then for anyn ≥ 1, there existsFn ∈Tfsuch
that
ρfn1−Fn
≤kρfn−f
. 2.11
Hence{ρfn1−Fn}converges to 0. Sinceρsatisfies theΔ2-condition, we have{fn1−Fnρ}
converges to 0. Since{fn} · ρ-converges tof, then{Fn} · ρ-converges tof. Hence{Fn}ρ
-converges tof. SinceTfisρ-closed and{Fn} ∈Tf, we getf ∈Tf.
Remark 2.2. Consider the multivalued map TAf A, where A is a nonempty ρ-closed
point ofTAis exactly the setA. In particular,ρ-contractive-type maps may not have a unique
fixed point.
As an application of the above theorem, we have the following result.
Proposition 2.3. LetCbe aρ-closed convex subset of the modular function spaceLρ. LetT :C →
CCbeρ-nonexpansive-type map. Then there exists an approximate fixed points sequence{fn}inC, that is, for anyn≥1there existsFn∈Tfnsuch that
lim
n→ ∞ρ
fn−Fn
0. 2.12
In particular one haslimn→ ∞distρfn, Tfn 0, where
distρ
fn, T
fn
infρfn−g
;g ∈Tfn
. 2.13
Proof. Letλ∈0,1and letf0be a fixed point inC. For eachf∈C, define a map
Tλ
fλf0 1−λT
fλf0 1−λg; g∈T
f. 2.14
Note that Tλf is nonempty and ρ-closed subset of C because Tfis ρ-closed and C is
convex. SinceT is aρ-nonexpansive-type map, for eachf, g ∈Cand for anyF ∈Tf, there existsG∈Tgsuch that
ρF−G≤ρf−g. 2.15
Sinceρis convex we get
ρλf0 1−λF
−λf0 1−λG
ρ1−λF−G≤1−λρF−G, 2.16
which implies
ρλf0 1−λF
−λf0 1−λG
≤1−λρf−g. 2.17
In other words, the map Tλ is aρ-contractive-type. Theorem 2.1implies the existence of a
fixed pointfλofTλ, thus there existsFλ∈Tfλsuch that
fλλf0 1−λFλ. 2.18
In particular, we have
ρfλ−Fλ
ρλf0−Fλ
≤λρf0−Fλ
whereδρC supf,g∈Cρf−gis theρ-diameter ofC. Note that sinceCisρ-bounded, then δρC<∞. If we chooseλ1/n, forn≥1 and writefn fλn andFnFλn, we get
ρfn−Fn
≤ δρC
n , 2.20
for anyn≥1, which implies limn→ ∞ρfn−Fn 0.
Using the above result, we are now ready to prove the main fixed point result for
ρ-nonexpansive-type multivalued maps.
Theorem 2.4. LetCbe a nonemptyρ-closed convex subset of the modular function spaceLρ. Assume thatCisρ-a.e. compact. Then eachρ-nonexpansive-type mapT :C → KChas a fixed point.
Proof. Proposition 2.3ensures the existence of a sequence{fn}inCand a sequence{Fn}such
thatFn ∈ Tfnand limn→ ∞ρfn−Fn 0. Without loss of generality we may assume that
{fn}ρ-a.e. converges tof ∈Cand{Fn}ρ-a.e. converges toF∈C.Lemma 1.10implies
ρf−F≤lim inf
n→ ∞ ρ
fn−Fn
0. 2.21
Hencef F. SinceTis aρ-nonexpansive-type map, then there exists a sequence{Gn} ∈Tf
such that
ρFn−Gn≤ρ
fn−f
, 2.22
for alln ≥ 1. SinceTf isρ-compact, we may assume that{Gn}isρ-convergent to some h∈Tf.Lemma 1.10implies
lim inf
n→ ∞ ρ
fn−f
ρf−hlim inf
n→ ∞ ρ
fn−h
. 2.23
Sinceρsatisfies theΔ2-condition, then
lim inf
n→ ∞ ρ
fn−h
lim inf
n→ ∞ ρ
fn−FnFn−GnGn−h
lim inf
n→ ∞ ρFn−Gn
2.24
see,20. SinceρFn−Gn≤ρfn−f, we get
lim inf
n→ ∞ ρ
fn−h
≤lim inf
n→ ∞ ρ
fn−f
, 2.25
which implies
lim inf
n→ ∞ ρ
fn−f
ρf−h≤lim inf
n→ ∞ ρ
fn−f
Henceρf−h 0 orfh. Hencef∈Tf; that is,fis a fixed point ofT.
Proposition 2.3 and Theorem 2.4 are also hold if we assume that C is starshaped instead of Convex.A setCis called starshaped if there existsf0∈Csuch thatλf0−1−λf∈
Cprovidedf ∈Candλ∈0,1.
3. Fixed Points of
w
-Contractive-Type Maps
In 21 the authors introduced the concept of w-distance in metric spaces which they connected to the existence of fixed point of single and multivalued maps see also 22. Similarly we extend their definition and results to modular spaces. Indeed letρbe a convex,
σ-finite modular function. A functionp : Lρ ×Lρ → 0,∞ is called w-modular on the
modular function spaceLρif the following are satisfied:
1pf, g≤pf, h ph, gfor anyf, g, h∈Lρ;
2for any f ∈ Lρ,pf,· : Lρ → 0,∞is lower semicontinuous; that is, if {gn}ρ
-converges tog, then
pf, g≤lim inf
n→ ∞ p
f, gn
, 3.1
3for anyε >0, there existsδ >0 such thatpf, g≤δandpf, h≤δimplyρg, h≤
ε.
As it was done in21, we need the following technical lemma.
Lemma 3.1. Let p·,· be w-modular on the modular function space Lρ. Let {fn} and {gn} be sequences in Lρ, and let{αn}and {βn}be sequences in 0,∞converging to 0, andf, g, h ∈ Lρ. Then the following hold:
1ifpfn, g≤ αnandpfn, h≤βn, for alln≥1, theng h; in particular ifpf, g 0 andpf, h 0, theng h;
2ifpfn, gn≤αnandpfn, h≤βn, for anyn≥1, then{gn}ρ-converges toh;
3ifpfn, fm≤αnfor anyn, m≥1withm > n, then{fn}is aρ-Cauchy sequence; 4ifpg, fn≤αnfor anyn≥1, then{fn}is aρ-Cauchy sequence.
The proof is easy and similar to the one given in21. Now we are ready to give the first fixed point result in this setting. LetCbe a nonemptyρ-closed subset of the modular function spaceLρ. We say that a multivalued mapT : C → CCis weaklyρ
-contractive-type map if there existsw-modularp·,·onLρandk∈0,1such that for anyf, g∈Cand
anyF∈Tf, there existsG∈Tgsuch thatpF, G≤kpf, g.
Proof. Letp·,·be aw-modular andk ∈ 0,1associated toT, that is, for anyf, g ∈ Cand anyF∈Tf, there existsG∈Tgsuch thatpF, G≤kpf, g. Fixf0 ∈Candf1∈Tf0. By
induction one can construct a sequence{fn}such thatfn1∈Tfnand
pfn, fn1
≤kpfn−1, fn
, 3.2
for everyn ≥ 1. In particular we havepfn, fn1 ≤ knpf0, f1, for everyn ≥ 1. Using the
properties ofp·,·, we get
pfn, fnh
≤ kn
1−kp
f0, f1
, 3.3
for any n, h ≥ 1. Lemma 3.1 implies that the sequence {fn} is ρ-Cauchy. Hence {fn}ρ
-converges to somef∈C. Using the lower semicontinuity ofp, we get
pfn, f
≤lim inf
n→ ∞ p
fn, fnh
≤ kn
1−kp
f0, f1
, 3.4
for anyn≥ 1. Sincefn ∈Tfn−1andT is weaklyρ-contractive-type map, there existsgn ∈ Tfsuch that
pfn, gn
≤kpfn−1, f
≤ kn
1−kp
f0, f1
, 3.5
for anyn ≥2.Lemma 3.1implies that{gn}ρ- converges tof as well. SinceTfisρ-closed,
then f ∈ Tf, that is, f is a fixed point of T. Let us complete the proof by showing that
pf, f 0. Sincef ∈Tf, there existsh1 ∈Tfsuch thatpf, h1≤kpf, f. By induction
we can construct a sequence{hn}inCsuch thathn1 ∈Thnandpf, hn1≤ kpf, hn, for
any n ≥ 1. So we havepf, hn ≤ knpf, f, for anyn ≥ 1. Lemma 3.1implies that{hn}is ρ-Cauchy. Hence{hn}ρ- converges to someh∈ C. Using the lower semicontinuity ofp·,·
we get
pf, h≤lim inf
n→ ∞ p
f, hn
≤0. 3.6
Hencepf, h 0. Then for anyn≥1, we have
pfn, h
≤pfn, f
pf, h≤ k n
1−kp
f0, f1
. 3.7
Lemma 3.1impliesfh, orpf, f 0.
Note that in the proof above we did not use theΔ2-condition. The reason behind is
that p·,· satisfies the triangle inequality. If T is single valued, then we have little more information about the fixed point. Indeed, letCbe a nonemptyρ-closed subset of the modular function spaceLρ. The mapT : C → Cis called a weaklyρ-contractive type map if there
Theorem 3.3. Let Cbe a nonempty ρ-closed subset of the modular function space Lρ. Then each weaklyρ-contractive type mapT :C → Chas a unique fixed pointf∈C, andpf, f 0.
Proof. Theorem 3.2ensures the existence of a fixed pointf ∈C, that is,Tf fandpf, f 0. Let us show thatf is the only fixed point ofT. Assume thath∈ Cis another fixed point ofT. Then we must havepf, h 0. Combining this withpf, f 0,Lemma 3.1implies
fh.
Similar extensions of the results as found in21–23may be proved in our setting.
Acknowledgments
The authors thank the referees for their valuable comments and suggestions. The authors would also like to thank Professor M.A. Khamsi for productive discussion and cooperation regarding this work.
References
1 M. Edelstein, “An extension of Banach’s contraction principle,” Proceedings of the American Mathematical Society, vol. 12, no. 1, pp. 7–10, 1961.
2 M. Edelstein, “On fixed and periodic points under contractive mappings,”Journal of the London Mathematical Society, vol. 37, pp. 74–79, 1962.
3 S. Kasahara, “On some generalizations of the Banach contraction theorem,”Publications of the Research Institute for Mathematical Sciences, vol. 12, no. 2, pp. 427–437, 1976.
4 B. E. Rhoades, “A comparison of various definitions of contractive mappings,”Transactions of the American Mathematical Society, vol. 226, pp. 257–290, 1977.
5 A. H. Siddiqi and Q. H. Ansari, “An iterative method for generalized variational inequalities,”
Mathematica Japonica, vol. 34, no. 3, pp. 475–481, 1989.
6 F. E. Browder, “On a theorem of Beurling and Livingston,”Canadian Journal of Mathematics, vol. 17, pp. 367–372, 1965.
7 F. E. Browder, “Fixed-point theorems for noncompact mappings in Hilbert space,”Proceedings of the National Academy of Sciences of the United States of America, vol. 53, no. 6, pp. 1272–1276, 1965.
8 F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,”Proceedings of the National Academy of Sciences of the United States of America, vol. 54, no. 4, pp. 1041–1044, 1965.
9 W. A. Kirk, “A fixed point theorem for mappings which do not increase distances,”The American Mathematical Monthly, vol. 72, no. 9, pp. 1004–1006, 1965.
10 S. B. Nadler Jr., “Multi-valued contraction mappings,”Pacific Journal of Mathematics, vol. 30, no. 2, pp. 475–488, 1969.
11 J. T. Markin, “A fixed point theorem for set valued mappings,”Bulletin of the American Mathematical Society, vol. 74, pp. 639–640, 1968.
12 E. Lami Dozo, “Multivalued nonexpansive mappings and Opial’s condition,” Proceedings of the American Mathematical Society, vol. 38, no. 2, pp. 286–292, 1973.
13 T. Husain and E. Tarafdar, “Fixed point theorems for multivalued mappings of nonexpansive type,”
Yokohama Mathematical Journal, vol. 28, no. 1-2, pp. 1–6, 1980.
14 T. Husain and A. Latif, “Fixed points of multivalued nonexpansive maps,”International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 421–430, 1991.
15 M. A. Khamsi, W. M. Kozłowski, and S. Reich, “Fixed point theory in modular function spaces,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 14, no. 11, pp. 935–953, 1990.
16 M. A. Khamsi, “Fixed point theory in modular function spaces,” inRecent Advances on Metric Fixed Point Theory (Seville, 1995), vol. 48 ofCiencias, pp. 31–57, Universidad de Sevilla, Seville, Spain, 1996. 17 S. Dhompongsa, T. Dom´ınguez Benavides, A. Kaewcharoen, and B. Panyanak, “Fixed point theorems
for multivalued mappings in modular function spaces,”Scientiae Mathematicae Japonicae, vol. 63, no. 2, pp. 161–169, 2006.
19 T. Dominguez Benavides, M. A. Khamsi, and S. Samadi, “Uniformly Lipschitzian mappings in modular function spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 46, no. 2, pp. 267– 278, 2001.
20 T. Dominguez-Benavides, M. A. Khamsi, and S. Samadi, “Asymptotically regular mappings in modular function spaces,”Scientiae Mathematicae Japonicae, vol. 53, no. 2, pp. 295–304, 2001.
21 O. Kada, T. Suzuki, and W. Takahashi, “Nonconvex minimization theorems and fixed point theorems in complete metric spaces,”Mathematica Japonica, vol. 44, no. 2, pp. 381–391, 1996.
22 T. Suzuki and W. Takahashi, “Fixed point theorems and characterizations of metric completeness,”
Topological Methods in Nonlinear Analysis, vol. 8, no. 2, pp. 371–382, 1996.
