SOME FIXED POINT THEOREMS IN MODULAR FUNCTION SPACES [620038]

SOME FIXED POINT THEOREMS IN MODULAR FUNCTION SPACES
ENDOWED WITH A GRAPH
JAAUAD JEDDI1, MUSTAPHA KABIL1& SAMIH LAZAIZ2
Abstract. The aim of this paper is to give xed point theorems for G-monotone-nonexpansive
mappings over -compact or -a.e. compact sets in modular function spaces endowed with
a re
exive digraph not necessarily transitive. Examples are given to support our work.
1.Introduction
LetXbe a non-empty set. We denote by 2Xthe set of subsets of X. An element xofX
is said to be a xed point of a self-mapping TonX, ifTx=x. For a set-valued mapping
T∶X→2X, we call a xed point of the set-valued mapping Tevery element xofXthat
verifyx∈Tx.
The two most important results in xed point theory, are without contest, the Banach
contraction principle (BCP for short) and Tarski's xed point theorem. Since their ap-
pearances, they were subject of many generalizations, either by extending the contractive
condition for the B.C.P., or changing the structure of the space itself. For example, in the
case of B.C.P., Ran and Reurings in [17] have obtained a xed point result for a relaxed
contraction condition in metric spaces endowed with a partially order relation, i.e. a con-
traction only for comparable elements. Jachymski in [9] got a further generalization of Ran
and Reurings result by replacing the partial order by a relaxed type of graph in metric spaces.
In the beginning of 1930's, W. Orlicz and Z. Birnbaum considered the space
L'={f∶R→R∶ ∃>0/integral.dispR'(/divides.alt0f(x)/divides.alt0)dx<∞}
where'∶R+→R+is a convex increasing function, such that lim
x→∞'(x)=∞. So,L'be-
come a generalization of Lpspaces, which corresponds to the particular case '(t)=tpwhere
t∈[0;∞)and 1≤p≤∞. Nevertheless, the formula that gives the norm over Lp, does not
establish a norm over L'. Thereby, H. Nakano in [13, 14, 15], captured the essence of the
good behavior of the quantity ∫R'(/divides.alt0f(x)/divides.alt0)dx, what he called modular function, and gave
some characterization of the geometry of these spaces.
Fixed point theory in modular function spaces was rst studied by M.A. Khamsi et al in
[11], we nd there an outline of a xed point theory for -nonexpansive mappings dened
on some subsets of modular function spaces. Recently, M.R. Alfuraidan in [2], gave some
2010 Mathematics Subject Classication. Primary: 47H09 Secondary: 46B20; 47H10; 47E10.
Key words and phrases. Fixed point, modular function spaces, monotone -nonexpansive mappings,
2-
type condition, -compactness, -a.e. compactness.
1

2 J. JEDDI, M. KABIL & S. LAZAIZ
extensions of xed point theorems in modular function spaces endowed with partial order
relation, namely Ran and Reurings result in this context.
In this paper, we generalize all the results obtained by M.R. Alfuraidan in [2]. We consider
modular function spaces endowed with graph which satisfy a path connectivity instead of
adjacency, and we have got new xed point theorems for G-monotone set-valued mappings.
Examples are given to support our work.
2.Preliminaries
Let(L;)be a modular function space where is a nonzero regular modular function see
[5, 10, 12] for more details. Recall that for all f∈L,satises the following properties:
●(f)=0⇔f=0;
●(f)=(f)if/divides.alt0/divides.alt0=1;
●(f+(1−)g)≤(f)+(1−)(g).
Ifsatises the above conditions, we note then ∈R. Furthermore, the associated norm
/parallel.alt1:/parallel.alt1dened using modular function is called Luxembourg norm and we have :
/parallel.alt1f/parallel.alt1=inf/braceleft.alt3>0;/parenleft.alt3f
/parenright.alt3≤1/braceright.alt3:
The following denitions will be needed in the sequel.
Denition 2.1. [10]Let∈R.
(1) We say that (fn)n∈L-converges to f, and write : fn→f(), if(fn−f)→0, and
a sequence (fn)n∈Lis called-Cauchy if (fn−fm)→0as(n;m)→∞.
(2) A setB⊂Lis called-closed, if for any sequence (fn)n∈B,fn→f()implies
f∈B.
(3) A setB⊂Lis called-bounded, if his diameter (B)=sup{(f−g)/slash.leftf;g∈B}is
nite.
(4) A setB⊂Lis called-compact, if for any sequence (fn)n∈Lthere exists a subse-
quence (fkn)nandf∈Bsuch thatfkn-converges to f.
IfC⊂L,C(C)will denote the set of -closed subsets of C, andK(C), the set of-compact
subsets ofC.
Denition 2.2. [10] Let∈R, We say that has the
2-type condition, if there exists
k∈[0;+∞)such that:(2f)≤k(f), for anyf∈L.
It is known that the -convergence and /parallel.alt1:/parallel.alt1-convergence are dierent in general, but if we
assume that has the
2-type condition we have the equivalence. That is, the Luxembourg
norm convergence and -convergence are equivalent. (see [10, Lemma 3.2]).
The following lemma will be very useful along this work.
Lemma 2.1. [7]Letbe convex and satisfy the
2-type condition. Let (fn)n∈L, such that
(fn+1−fn)≤Kn;∀n
whereKis arbitrary nonzero constant and ∈(0;1), then(fn)nis Cauchy for ∥:∥and
-Cauchy.

SOME FIXED POINT RESULTS FOR SET-VALUED MAPPINGS 3
Theorem 2.3. [10]Let∈R.
(i)(L;∥:∥)is a complete normed space, and Lis-complete.
(ii)∥fn∥→0i(fn)→0for every>0.
(iii) Ifhas the
2-property and (fn)→0for>0, then∥fn∥→0.
We recall the denition of digraph, the interested reader can consult the book [18] for more
details.
Denition 2.4. A directed graph or digraph Gis determined by a nonempty set V(G)of its
vertices and the set E(G)⊂V(G)×V(G)of its directed edges. A digraph is re
exive if each
vertex has a loop. Given a digraph G=(V;E).
●If whenever (x;y)∈E(G)⇒(y;x)∉E(G), then the digraph Gis called an oriented
graph.
●A digraph G is transitive whenever [(x;y)∈E(G)and(y;z)∈E(G)]⇒(x;z)∈
E(G), for anyx;y;z∈V(G).
●A dipath of Gis a sequence a0,a1,…,an,… with (ai;ai+1)∈E(G)for eachi∈N.
●A nite dipath of length nfrom x to y is a sequence of n+1vertices (a0;a1;:::;an)
with(ai;ai+1)∈E(G)andx=a0,y=an.
●A closed directed path of length n>1fromxtoy, i.e.,x=y, is called a directed cycle.
●A digraph is connected if there is a nite (di)path joining any two of its vertices and
it is weakly connected if ~Gis connected.
●[x]Gis the set of all vertices which are contained in some path beginning at x.
(i.e.y∈[x]G⇔there exist (a0;a1;:::;an)with(ai;ai+1)∈E(G)andx=a0;y=an:)
It seems that the graph theory when coupled with the classical metric xed point theory
leads to a new interesting theory, following the works of [8, 1, 3, 6, 16] we introduce the
following denition.
Denition 2.5. Let∈R,C⊂La nonempty subset and T∶C→2Ca set-valued mapping.
(i)We say that Tis aG-monotone -contraction, if there exists ∈[0;1)such that
for anyf;h∈Cwithh∈[f]G, and anyF∈T(f); there exists H∈T(h)verifying
H∈[F]Gand(F−H)≤(f−h).
(ii)We say that Tis aG-monotone -nonexpansive mapping, if for any f;h∈Cwith
h∈[f]G, and anyF∈T(f); there exists H∈T(h)verifyingH∈[F]Gand(F−H)≤
(f−h).
Note that if Tis only a single valued and the graph is transitive we get the notion of edge
preserving-contraction (resp. -nonexpansive) mapping.
3.Fixed point results for -contractions mappings
We start this section by the following result which will be useful in the sequel.
Proposition 3.1. Let∈R. LetC⊂Lbe a nonempty -closed subset, Ga re
exive digraph
such thatV(G)=C. LetT∶C→2Cbe aG-monotone -contraction mapping and suppose
thatThas a xed point, then if f∈[f]Gfor somef∈Cwhere fis a xed point for Tsuch
that(f−f)<∞; there exists a sequence (fn)n∈Csuch thatfn+1∈T(fn);∀n; and(fn)n
-converges to f.

4 J. JEDDI, M. KABIL & S. LAZAIZ
Proof. Sincef∈[f]Gand f∈T(f)whereTisG-monotone -contraction, there exists
f1∈T(f)such thatf1∈[f]Gand(f−f1)≤(f−f).
By induction we construct a sequence (fn)nsuch thatfn+1∈T(fn)and(f−fn+1)≤(f−fn)
for eachn∈Ni.e.
(f−fn)≤n(f−f);
for alln∈N. And since ∈[0;1), we get (fn)n,-converges to f. 
Ifhas the
2-type condition then (f)<∞for everyf∈L. Thus, we get the following
result which is a generalization of [2, Theorem 3.1].
Theorem 3.2. Let∈Rbe convex and satises the
2-type condition. Let C⊂Lbe
a nonempty -closed subset, Ga re
exive digraph such that V(G)=C, with the property
(P). LetT∶C→C(C)be aG-monotone -contraction mapping and CT∶={f∈C∶g∈
[f]Gfor someg∈T(f)}.
IfCT≠/uni2205, thenThas a xed point, moreover, if f∈CTthenT/divides.alt0[f]Ghas a xed point.
Proof. Letf0∈CT, there exists then f1∈T(f0)such thatf1∈[f0]Gand asTisG-monotone
-contraction, there exists f2∈T(f1)such thatf2∈[f1]Gand
(f2−f1)≤(f1−f0); ∈(0;1)
In the same way, there exists f3∈T(f2)such thatf3∈[f2]Gand(f3−f2)≤(f2−f1)and
by induction we construct a sequence (fn)∈Cwithfn+1∈T(fn)andfn+1∈[fn]Gsuch that
(fn+1−fn)≤(fn−fn−1);∀n≥1, which implies (fn+1−fn)≤n(f1−f0). Hence and
by the lemma 2.1, (fn)is-Cauchy, and as Lis-complete then (fn))-converges to some
f∈CasCis-closed. And since Ghas the(P) property then for each n;f∈[fn]Gand asT
isG-monotone -contraction, there exists gn∈T(f)such that for every n:
gn∈[fn+1]Gand (fn+1−gn)≤(fn−f)
And then clearly, (fn+1−gn)n-converges to zero in Lthus(fn−f)nand(fn+1−gn)nboth
converge to zero in (L;∥:∥)which mean that (gn)nconverges to fin(L;∥:∥)as∥:∥
is a norm on L, therefore (gn)n-converges to fand thenf∈T(f)as(gn)n∈T(f)and
T(f)is-closed i.e. fis a xed point of T.
Note that as f∈[fn]Gfor everyn, in particular f∈[f0]GthusT/divides.alt0[f0]Ghas a xed point. 
Example 3.3. LetL=L∞([0;1])the set of measurable functions f∶[0;1]→Rsuch that
∃M≥0with/divides.alt0f(x)/divides.alt0≤Ma.e. in [0;1]. It is known that L∞([0;1])is a normed linear space
with the norm
(f)=/parallel.alt1f/parallel.alt1∞=inf{M>0∶/divides.alt0f(x)/divides.alt0≤Ma.e. in[0;1]}:
Then,(L;)is a modular function space with the
2-condition. Let C=B(0;1)be the
-closed ball centered at 0with radius 1, that is the set
{f∈L∞∶/parallel.alt1f/parallel.alt1∞≤1}:
Consider the digraph G=(C;E)dened by
(f1;f2)∈E⇔∃c∈[0;1]such that /divides.alt0f2(x)/divides.alt0≤c/divides.alt0f1(x)/divides.alt0
for everyf1;f2∈C. Thus we get,
h∈[f]G∈E⇔∃c∈[0;1]such that /divides.alt0f(x)/divides.alt0≤c/divides.alt0h(x)/divides.alt0:

SOME FIXED POINT RESULTS FOR SET-VALUED MAPPINGS 5
It is clear that this binary relation denes a re
exive digraph without being a partially order
(i.e. without antisymmetric condition).
Letf∈Cand deneT∶C→C(C)by
Tf={1
2nf∶n≥1}∪{0}:
(i)For allF∈Tf, we have /parallel.alt1F/parallel.alt1∞=/parallel.alt11
2nf/parallel.alt1≤/parallel.alt1f/parallel.alt1≤1, which implies that F∈C. Moreover,
Tfis-closed since Tf=/braceleft.alt11
2nf∶n≥1/braceright.alt1.
(ii)∀f;g∈C, ifh∈[f]G, let for any F∈Tfwe have two cases : F=1
2nforF=0. If
F=1
2nfchooseH=1
2nh, hence we have the G-monotonocity :
/divides.alt0F(x)/divides.alt0=1
2n/divides.alt0f(x)/divides.alt0≤c:1
2n/divides.alt0h(x)/divides.alt0=c/divides.alt0H(x)/divides.alt0
and the-contraction condition for =1
2
(F−H)=1
2n(f−h)≤1
2(f−h):
Now, ifF=0letH=0theG-monotonocity and -contraction conditions are obvious.
For the (P)-property. Let (fn)be a sequence in Csuch that
fn+1∈[fn]G∀nand/parallel.alt1fn−f/parallel.alt1∞→0asn→∞:
fn+1∈[fn]G⇒∃cn∈[0;1]such that /divides.alt0fn(x)/divides.alt0≤cn/divides.alt0fn+1(x)/divides.alt0, and since 0≤cn≤1we get for all
n∈N,
/divides.alt0fn(x)≤/divides.alt0fn+1(x)/divides.alt0;∀n:
Furthermore,
/divides.alt0 /divides.alt0fn(x)/divides.alt0−/divides.alt0f(x)/divides.alt0 /divides.alt0≤/divides.alt0fn(x)−f(x)/divides.alt0≤/parallel.alt1fn−f/parallel.alt1∞
implies that /divides.alt0fn(x)/divides.alt0→/divides.alt0f(x)/divides.alt0, thus/divides.alt0fn(x)/divides.alt0≤/divides.alt0f(x)/divides.alt0for alln∈N, hencef∈[fn]G. That isG
has the (P)- property. Then the set-valued mapping Tsatises all the conditions of Theorem
3.2 hence it has a xed point, namely 0∈T0.
Corollary 3.4. Let∈Rbe convex and satises the
2-type condition. Let C⊂Lbe a
nonempty-closed subset, Ga re
exive digraph such that V(G)=C, with the (P)-property.
LetT∶C→C(C)be aG-monotone -contraction mapping. If Ghas a mother vertex, (a
vertexfsuch that all other vertices in Gcan be reached by a path from f), thenThas a xed
point.
Proof. Indeed for such f,g∈[f]G, for allg∈T(f)and thenf∈CT≠/uni2205, we then get the
result using theorem 3.2. 

6 J. JEDDI, M. KABIL & S. LAZAIZ
4.Fixed point results for -nonexpansive mappings
Recall the notion of approximated xed point sequence.
Denition 4.1. We say that T∶C→2Chas an approximated xed point sequence, if there
exists(fn)n∈Csuch that for every n, there exists Fn∈T(fn)satisfying lim
n(fn−Fn)=0.
Denition 4.2. A digraphGis said to be G-convex if for all f∈[F]Gandh∈[H]Gwe have
f+(1−)h∈[F+(1−)H]G.
If the re
exive digraph Gis transitive and antisymmetric, i.e. being a partially order, then
the most ordered Banach spaces enjoy this property. The following result give a sucient
conditions to obtain an approximated xed point sequence for G-convex digraph.
Proposition 4.3. Let∈Rbe convex and satises the
2-type condition. Let C⊂Lbe a
nonempty convex, -closed, and -bounded subset, GaG-convex re
exive digraph such that
V(G)=C, withe the (P)-property.
LetT∶C→C(C)be aG-monotone -nonexpansive mapping, if CT∶={f∈C∶f∈
[g]Gforsomeg ∈T(f)}is nonempty; then Thas an approximated xed point sequence.
Proof. Letf0∈CTand for each ∈(0;1)dene :
T(f)=f0+(1−)T(f);
for anyf∈C. ThenT(f)is-closed since if
g∈T(f)⇔1
1−(g−f0)∈T(f):
Moreover,CTis nonempty for any ∈(0;1)asf0∈CT.
Letf;g∈Csuch thatg∈[f]G, sinceTisG-monotone -nonexpansive, we get for all
F∈T(f)there exists G∈T(g)such thatG∈[F]Gand
(F−G)≤(f−g);
thus,f0+(1−)G∈[f0+(1−)F]Gand asis convex:
((f0+(1−)F)−(f0+(1−)G))=((1−)(F−G))≤(1−)(f−g)
i.e.TisG-monotone -contraction. Using Theorem 3.2, Thas a xed point f∈C. Thus
∃F∈T(f)such thatf=f0+(1−)Fand then, as is convex :
(f−F)=((f0−F))≤(C):
Choosing=1
nforn≥1 gives the result. 
Now with more requirement on the graph G, we obtain a stronger version of the above
result (Proposition 4.3).
Denition 4.4. Let∈R, a digraph Gis said to be compatible with the vector structure of
L, if for every f;g; ~f;∈Lsuch that ~f∈[f]Gthen ~f+g∈[f+g]Gand⋅~f∈[⋅f]G;∀∈R+.
Notice that if Gis compatible with the vector structure of Lthen it isG-convex. Indeed,
for everyf;g; ~f;~g∈Lsuch that ~f∈[f]Gand ~g∈[g]Gthen ~f+~g∈[f+g]G, and thus for
every∈[0;1];~f+(1−)~g∈[f+(1−g)]G.

SOME FIXED POINT RESULTS FOR SET-VALUED MAPPINGS 7
Lemma 4.1. Let∈Rbe convex and satises the
2-type condition. Let C⊂Lbe a
nonempty convex, -closed, and -bounded subset, Ga re
exive digraph compatible with the
vector structure of Lsuch thatV(G)=C, with the (P)-property. Let T∶C→C(C)be
aG-monotone -nonexpansive mapping, if CT∶={f∈C∶f∈[g]Gforsomeg ∈T(f)}is
nonempty; then there exists two sequences (fn)nand(Fn)n∈Lsuch that
Fn∈T(fn); fn+1∈[fn]G; Fn+1∈[fn+1]G;for everyn;
andlim
n(Fn−fn)=0.
Proof. Letf0∈CTwe deneT1(f)=1
2f0+1
2T(f);f∈L, as seen in the proof of Proposition 4.3
T1is aG-monotone -contraction mapping, T1(f)is-closed for every f∈LthenT1has
a xed point f1such thatf1∈[f0]Gas we have already seen in Theorem 3.2. Then, there
existsF1∈T(f1)such that
f1=1
2f0+1
2F1
thus, it comes clearly that F1∈[f1]G. In addition:
(f1−F1)=(1
2(f0−F1))≤1
2(f1−F1)≤1
2(C)
asis convex.
Clearlyf1∈CT. Let dene T2(f)=1
3f1+2
3T(f), for allf∈L. As above T2admits a xed
pointf2such thatf2∈[f1]Gand then there exists F2∈T(f2)such thatf2=1
3f1+2
3F2, thus
F2∈[f2]Gand(f2−F2)≤1
3(C).
By induction on n≥1, if we dene the set-valued mapping Tnto be
Tn(f)=1
n+1fn+(1−1
n+1)T(f);∀f∈L
we get the existence of a xed point fnofTnsuch thatfn∈[fn−1]G, thus there exists
Fn∈T(fn)such thatfn=1
n+1fn+1+(1−1
n+1Fn)and thenFn∈[fn]Gand(fn−Fn)≤1
n+1(C).
SinceCis- bounded we get the result. 
We then apply Lemma 4.1 to get a new xed point result for G-monotone nonexpansive
mapping.
Theorem 4.5. Let∈Rbe convex, satises the
2-type condition. Let C⊂Lbe a nonempty
convex,-compact, and -bounded subset, Ga re
exive digraph compatible with the vector
structure of Lwith the (P)-property. Let T∶C→K(C)be aG-monotone -nonexpansive
mapping, if CT∶={f∈C∶f∈[g]Gforsomeg ∈T(f)}is nonempty, then Thas a xed point.
Proof. The preceding lemma (Lemma 4.1) establish the existence of sequences (fn)nand
(Fn)nsuch thatFn∈T(fn),fn+1∈[fn]G; Fn∈[fn]Gfor everyn, and lim
n(Fn−fn)=0.
SinceCis-compact, there exists a sub-sequence (fkn)nof(fn)nthat-converges to some
f∈C, but as lim
n(Fkn−fkn)=0 and the-convergence is equivalent to the convergence in
(L;∥:∥)(due to the
2-type condition, see [10, Proposition 3.13]); then (Fkn)n-converges
tof. Thus one can suppose that (fn)nand(Fn)nboth-converge to f, and asfn+1∈[fn]G

8 J. JEDDI, M. KABIL & S. LAZAIZ
for everyn, using the (P)-property we get f∈[fn]Gfor everyn.
Now, since TisG-monotone -nonexpansive mapping, we have for all nthere exists gn∈
T(f)such thatgn∈[Fn]Gand(Fn−gn)≤(fn−f)and thusgn→f()asn→∞. Since
T(f)is-closed because it is -compact, we obtain f∈T(f), i.e.fis a xed point for T.
Example 4.6. Consider the modular function space L=L2(]−1;1[)the space of all square-
integrable functions over ]−1;1[, and let
(f)=/integral.disp1
−1/divides.alt0f(x)/divides.alt02dx
for everyf∈L. It is clear that is convex modular function satisfying the
2-type condition.
LetCbe the set of all constant maps fofLsuch that(f)≤1. Thus,Cis-bounded and
convex subset of L. For the-compactness it is enough to show that Csatises the Riesz-
Frchet-Kolmogorov's conditions (see [4, Theorem IV.25] ) which is obvious. Let G=(C;E)
be the directed graph dened by :
(1) (f;g)∈E⇔(∃c∈[0;1]); g(x)≤cf(x)a.e in]−1;1[:
It is clear that Gis a re
exive digraph compatible with the vector structure of L. Moreover,
it has the (P)-property. Indeed, let (fn)nbe a sequence that -converges to finLwith the
property
fn+1∈[fn]G∀n∈N:
Then there exists a subsequence (fnk)kwhich is pointwise convergent almost everywhere to
f. Thus,
fnk(x)≤f(x)a.e.
Moreover, for each nthere exists cn∈[0;1]such that
fn(x)≤cnfn+1(x)≤fn+1(x);
then for all n,fn(x)≤f(x)a.e., hence Ghas the (P)-property. Note that the di-graph Gis
not even tranisitive.
LetTbe the mapping dened by Tf=f+=max{0;f}. It is clear that TisG-monotone
-nonexpansive mapping. Thus all the conditions of Theorem 4.5 hold, so Thas a xed point
namelyT0=0.
5.Fixed point theorem for -a.e. compact subsets
The-compactness assumption maybe relaxed using the weak concept of -a.e. compact-
ness. Of course, if a subset set CofLis-compact then it is -a.e. compact, see [10,
Proposition 3.13] for more details.
Denition 5.1. [10]Let∈R.
(i)A setB⊂Lis called-a.e. compact, if for any sequence (fn)n∈Lthere exists a
subsequence (fkn)nandf∈Bsuch thatfkn-a.e. converges to f.
(ii)A setB⊂Lis called-a.e. closed, if for any sequence (fn)n∈B,fn→f,-a.e.
impliesf∈B.
Theorem 5.2. [10]Let∈R.

SOME FIXED POINT RESULTS FOR SET-VALUED MAPPINGS 9
(i)If(fn−f)→0there exists (fnk)ksubsequence of (fn)nsuch thatfnk→f,-a.e.
(ii)Iffn→f-a.e, then(f)≤lim inf
n→+∞(fn)(The Fatou property).
We need the following denition of the growth function.
Denition 5.3. [10]Letbe a function modular, the function !∶[0;+∞]→[0;+∞]dened
by:
!(t)=sup/braceleft.alt4(tf)
(f)∶f∈Land0<(f)<∞/braceright.alt4
is called the growth of .
The growth function has the following properties:
Proposition 5.4. [10]Let∈Rthat has the
2-type condition, and !its growth function,
then:
(i)!(t)<∞;∀t∈[0;+∞[.
(ii)!∶[0;+∞[→[0;+∞[is convex, and strictly increasing, it is then also continuous.
We then get the following lemma.
Lemma 5.1. Let∈Rthat has the
2-type condition, and (fn)nand(gn)nbe two sequences
inL, such that gn→0(), then :
lim inf
n(fn+gn)=lim inf
n(fn)
Proof. It is obvious that for ∈]0;+∞[andf∈Lwe have :(f)≤!()(f). Let"∈]0;1[,
then for every nwe have :
(fn+gn)≤/parenleft.alt3fn
1−"/parenright.alt3+/parenleft.alt3gn
"/parenright.alt3≤!/parenleft.alt31
1−"/parenright.alt3(fn)+!(1
")(gn)
and then,
lim inf
n(fn+gn)≤!(1
1−")lim inf(fn);
as!is continuous lim
"→0!(1
1−")=!(1)=1 we then get : lim inf
n(fn+gn)≤lim inf
n(fn)
letting→0.
The same arguments gives: lim inf
n(fn)=lim inf
n(fn+gn−gn)≤lim inf
n(fn+gn)witch ends
the proof.. 
Denition 5.5 (Opial property) .Let∈R,Lis said satisfying the -a.e. Opial property
if for every (fn)n⊂Lwhich-a.e. converges to 0, and such that sup
n(fn)<+∞for some
>1, then we have :
lim inf
n(fn)<lim inf
n(fn+f)
for everyf∈Lnot equal to 0.
The following relaxed denition replace the (P)-property.
Denition 5.6. LetC⊂Lbe a nonempty set and Ga digraph such that V(G)=C.
If for any sequence (fn)ninCwhich-a.e. converges to f, and such that fn+1∈[fn]Gfor
everyn; we havef∈[fn]Gfor everyn; then we say that Ghas the (P′)-property

10 J. JEDDI, M. KABIL & S. LAZAIZ
Remark 5.7. If we substitute " (P)-Property" by " (P′)-Property" in the Theorem 3.2, Propo-
sition 4.3 and Lemma 4.1; then all these results hold. This is due to the fact that for every
-convergence sequence (fn)nthere exists a sub-sequence (fkn)nthat-a.e. converges.
We conclude this paper by the -a.e. compact version of Theorem 4.5.
Theorem 5.8. Let∈Rbe convex, satises the
2-type condition and the -a.e. Opial
property. Let C⊂Lbe a nonempty convex, -a.e. compact, and -bounded subset, Ga
re
exive digraph compatible with the vector structure of Lsuch thatV(G)=Cand satises
the(P′)-property. Let T∶C→K(C)be aG-monotone -nonexpansive mapping, if CT∶=
{f∈C∶f∈[g]Gforsomeg ∈T(f)}is nonempty, then Thas a xed point.
Proof. By Lemma 4.1 and Remark 5.7, there exist two sequences (fn)nand(Fn)ninCsuch
that :
Fn∈T(fn); fn+1∈[fn]G; Fn∈[fn]G
for everyn, and(Fn−fn)n-a.e. converges to 0.
AsCis-a.e compact, there exists a sub-sequence (fkn)nthat-a.e converges to some f
inC. But then as lim
n(fkn−Fkn)=0, there exists a sub-sequence (fln−Fln)nof(fkn−Fkn)n
that-a.e. converges to 0. Thus both (fln)nand(Fln)n-a.e. converge to f. So, without
loss of generality, we may suppose that both (fn)nand(Fn)n-a.e. converge to f.
Now the (P′)-property gives f∈[fn]G;∀n. Then for every n, there exists hn∈T(f)such
thathn∈[Fn]Gand
(2) (hn−Fn)≤(fn−f):
Moreover, as (hn)n⊂T(f)which is-compact (since Tf∈K(C)), it admits a sub-sequence
that we will keep noting (hn)nthat-converges to some h∈C. Applying Lemma 5.1 we get,
lim inf
n(fn−h)=lim inf
n(fn−Fn+Fn−hn+hn−h)=lim inf
n(Fn−hn):
So, by letting n→∞in the inequality (2), we obtain
lim inf
n(fn−h)≤lim inf
n(fn−f):
Then iff≠hthe-a.e. Opial property gives
lim inf
n(fn−f)<lim inf
n(fn−f+f−h)=lim inf
n(fn−h);
which is a contradiction. Then necessarily f=hand as(hn)n⊂T(f)which is-compact,
thus-closed, hence h=f∈T(f), i.e.fis a xed point for T. 
Data Availability
No data were used to support this study.

SOME FIXED POINT RESULTS FOR SET-VALUED MAPPINGS 11
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1Laboratory of Mathematics and Applications, Faculty of Sciences and Technologies
Mohammedia, University Hassan II Casablanca, Morocco,
Email address :jaauadjeddi@gmail.com; kabilfstm@gmail.com
2Laboratory of Mathematical Analysis and Applications, Faculty of Sciences Dhar El
Mahraz, University Sidi Mohamed Ben Abdellah, Fes, Morocco,
Email address :samih.lazaiz@usmba.ac.ma