# Some Fixed Point Theorems in Modular Function Spaces Endowed with a Graph

Authors: Jaauad Jeddi; Mustapha Kabil; Samih Lazaiz
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

The aim of this paper is to give fixed point theorems forG-monotone ρ-nonexpansive mappings over ρ-compact or ρ-a.e. compact sets in modular function spaces endowed with a reflexive digraph not necessarily transitive. Examples are given to support our work.


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## Body

## 1. Introduction

LetX be a nonempty set. We denote by 2X the set of subsets of X. An element x of X is said to be a fixed point of a self-mapping T on X, if Tx=x. For a set-valued mapping T:X→2X, we call a fixed point of the set-valued mapping T every element x of X that verify x∈Tx.The two most important results in fixed point theory, are without contest, the Banach contraction principle (BCP for short) and Tarski’s fixed point theorem. Since their appearances, 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 [1] have obtained a fixed point result for a relaxed contraction condition in metric spaces endowed with a partially order relation, i.e., a contraction only for comparable elements. Jachymski in [2] got a further generalization of Ran and Reurings result by replacing the partial order by a relaxed type of graph in metric spaces, where Nadler [3] managed to give an equivalent form of the B.C.P. for multi-valued mappings.In the beginning of 1930’s, Orlicz and Birnbaum considered the space(1)Lφ=f:R→R:∃λ>0∫Rφλfxdx<∞,whereφ:R+→R+ is a convex increasing function, such that limx→∞φx=∞. So, Lφ become a generalization of Lp spaces, which corresponds to the particular case φt=tp where t∈0,∞ and 1≤p≤∞. Nevertheless, the formula that gives the norm over Lp, does not establish a norm over Lφ. Thereby, Nakano in [4, 5], captured the essence of the good behavior of the quantity ∫Rφλ|fx|dx, what he called modular function, and gave some characterization of the geometry of these spaces, see [6].Fixed point theory in modular function spaces was first studied by Khamsi et al. in [7], we find there an outline of a fixed point theory for ρ-nonexpansive mappings defined on some subsets of modular function spaces. Recently, Alfuraidan in [8], gave some extensions of fixed point theorems in modular function spaces endowed with partial order relation, namely Ran and Reurings result in this context.On the other hand, the combination of metric fixed point theory and graph theory allows Jachymski in [2] to give an extension of Banach Contraction Principle in a metric space endowed with a graph. Souayah et al. in [9], after introducing the notion of Gm-contraction they were able to investigate the existence and uniqueness of the fixed point for such contractions in M-metric space endowed with a graph, in particular they generalize the results of Jachymski. Recently, in [10] the authors proved the existence of a unique best proximity point for some contractive type mappings in rectangular metric spaces endowed with a graph structure.In this paper, we generalize all the results obtained by Alfuraidan in [8]. We consider modular function spaces endowed with graph which satisfy a path connectivity instead of adjacency, and we have got new fixed point theorems for G-monotone set-valued mappings. Examples are given to support our work.


## 2. Preliminaries

LetLρ,ρ be a modular function space where ρ is a nonzero regular modular function see [6, 11, 12] for more details. Recall that for all f∈Lρ, ρ satisfies the following properties:(i)
ρf=0⇔f=0;(ii)
ραf=ρf if |α|=1;(iii)
ραf+1−αg≤αρf+1−αρg.Ifρ satisfies the above conditions, we note then ρ∈R. Furthermore, the associated norm ‖.‖ρ defined using modular function ρ is called Luxembourg norm and we have:(2)fρ=infα>0;ρfα≤1.The following definitions will be needed in the sequel.Definition 1 (see [11]).
Letρ∈R.(1)
We say that(fn)n∈Lρρ-converges to f, and write: fn→fρ, if ρfn−f→0 , and a sequence (fn)n∈Lρ is called ρ-Cauchy if ρfn−fm→0asn,m→∞.(2)
A setB⊂Lρ is called ρ-closed, if for any sequence (fn)n∈B, fn→fρ implies f∈B.(3)
A setB⊂Lρ is called ρ-bounded, if his diameter δρB=sup{ρf−g/f,g∈B} is finite.(4)
A setB⊂Lρ is called ρ-compact, if for any sequence (fn)n∈Lρ there exists a subsequence (fkn)n and f∈B such that fknρ-converges to f.IfC⊂Lρ, CC will denote the set of ρ-closed subsets of C, and KC, the set of ρ-compact subsets of C.Definition 2 (see [11]).
Letρ∈R, We say that ρ has the Δ2-type condition, if there exists k∈0,+∞ such that: ρ2f≤kρf, for any f∈Lρ.It is known that theρ-convergence and ‖.‖ρ-convergence are different in general, but if we assume that ρ has the Δ2-type condition we have the equivalence. That is, the Luxembourg norm convergence and ρ-convergence are equivalent. (see [11, Lemma 3.2]).The following lemma will be very useful along this work.Lemma 1 (see [13]).
Letρ be convex and satisfy the Δ2-type condition. Let (fn)n∈Lρ, such that(3)ρfn+1−fn≤Kαn,∀n,
whereK is arbitrary nonzero constant and α∈0,1, then (fn)n is Cauchy for ∥.∥ρ and ρ-Cauchy.Theorem 1 (see [11]).
Letρ∈R.(i)
Lρ,‖.‖ρ is a complete normed space, and Lρ is ρ-complete.(ii)
‖fn‖ρ→0 iff ραfn→0 for every α>0.(iii)
Ifρ has the Δ2-property and ραfn→0 for α>0, then ‖fn‖ρ→0.We recall the definition of digraph, the interested reader can consult the book [14] for more details.Definition 3.
A directed graph or digraphG is determined by a nonempty set VG of its vertices and the set EG⊂VG×VG of its directed edges. A digraph is reflexive if each vertex has a loop. Given a digraph G=V,E.(i)
If wheneverx,y∈EG⇒y,x∉EG, then the digraph G is called an oriented graph.(ii)
A digraphG is transitive whenever [(x,y)∈E(G) and (y,z)∈E(G)]⇒(x,z)∈E(G), for any x,y,z∈VG.(iii)
A dipath ofG is a sequence a0, a1,…, an,…with ai,ai+1∈EG for each i∈N.(iv)
A finite dipath of lengthn from x to y is a sequence of n+1 vertices a0,a1,…,an with ai,ai+1∈EG and x=a0, y=an.(v)
A closed directed path of lengthn>1 from x to y, i.e., x=y, is called a directed cycle.(vi)
A digraph is connected if there is a finite (di)path joining any two of its vertices and it is weakly connected ifG̃ is connected.(vii)
[x]G is the set of all vertices which are contained in some path beginning at x. (i.e. y∈[x]G⇔there exista0,a1,…,anwithai,ai+1∈EGandx=a0,y=an).It seems that the graph theory when coupled with the classical metric fixed point theory leads to a new interesting theory, following the works of [2, 8] we introduce the following definitions.Definition 4.
Letρ∈R, C⊂Lρ a nonempty subset and T:C→2C a set-valued mapping.(i)
We say thatT is a G-monotone ρ-contraction, if there exists α∈0,1 such that for any f,h∈C with h∈[f]G, and any F∈Tf; there exists H∈Th verifying H∈[F]G and ρF−H≤ρf−h.(ii)
We say thatT is a G-monotone ρ-nonexpansive mapping, if for any f,h∈C with h∈[f]G, and any F∈Tf; there exists H∈Th verifying H∈[F]G and ρF−H≤ρf−h.Note that ifT is only a single valued and the graph is transitive we get the notion of edge preserving ρ-contraction (resp. ρ-nonexpansive) mapping.Definition 5.
LetC be a nonempty subset of Lρ and G be a digraph such that VG=C. If for any sequence (fn)n in C that ρ-converges to f, we have f∈[fn]G for every n, provided fn+1∈[fn]G for every n; then we say that G has the P-property.


## 3. Fixed Point Results forρ-Contractions Mappings

We start this section by the following result which will be useful in the sequel.Proposition 1.
Letρ∈R. Let C⊂Lρ be a nonempty ρ-closed subset, G a reflexive digraph such that VG=C. Let T:C→2C be a G-monotone ρ-contraction mapping and suppose that T has a fixed point, then if f∈[f¯]G for some f∈C where f− is a fixed point for T such that ρf−f−<∞ ; there exists a sequence (fn)n∈C such that fn+1∈Tfn,∀n; and (fn)nρ-converges to f−.Proof.
Sincef∈[f¯]G and f¯∈Tf¯ where T is G-monotone ρ-contraction, there exists f1∈Tf such that f1∈[f¯]G and ρf¯−f1≤αρf¯−f.By induction we construct a sequence(fn)n such that fn+1∈Tfn and ρf¯−fn+1≤αρf¯−fn for each n∈N i.e.(4)ρf¯−fn≤αnρf¯−f,for alln∈N. And since α∈0,1, we get (fn)n, ρ-converges to f−.Ifρ has the Δ2-type condition then ρf<∞ for every f∈Lρ. Thus, we get the following result which is a generalization of [8, Theorem 3.1].Theorem 2.
Letρ∈R be convex and satisfies the Δ2-type condition. Let C⊂Lρ be a nonempty ρ-closed subset, G a reflexive digraph such that VG=C , with the (P)-property. Let T:C→CC be a G-monotone ρ-contraction mapping and CT:={f∈C:g∈[f]Gforsomeg∈T(f)}.
IfCT≠∅, then T has a fixed point, moreover, if f∈CT then T|[f]G has a fixed point .Proof.
Letf0∈CT, there exists then f1∈Tf0 such that f1∈[f0]G and as T is G-monotone ρ-contraction, there exists f2∈Tf1 such that f2∈[f1]G and(5)ρf2−f1≤αρf1−f0,α∈0,1.In the same way, there existsf3∈Tf2 such that f3∈[f2]G and ρf3−f2≤αρf2−f1 and by induction we construct a sequence fn∈C with fn+1∈Tfn and fn+1∈[fn]G such that ρfn+1−fn≤αρfn−fn−1,∀n≥1, which implies ρfn+1−fn≤αnρf1−f0. Hence by the Lemma 1, (fn)n is ρ-Cauchy, and as Lρ is ρ-complete then (fn)nρ-converges to some f∈C as C is ρ-closed. And since G has the (P)-property then for each n,f∈[fn]G and as T is G-monotone ρ-contraction, there exists gn∈Tf such that for every n:(6)gn∈fn+1Gandρfn+1−gn≤αρfn−f.And then clearly,fn+1−gnnρ-converges to zero in Lρ thus (fn−f)n and (fn+1−gn)n both converge to zero in Lρ,‖.‖ρ which mean that (gn)n converges to f in Lρ,‖.‖ρ as ‖.‖ρ is a norm on Lρ, therefore (gn)nρ-converges to f and then f∈Tf as (gn)n∈T(f) and Tf is ρ-closed i.e., f is a fixed point of T.Note that asf∈[fn]G for every n, in particular f∈[f0]G thus T|[f0]G has a fixed point.Example 1.
LetLρ=L∞0,1 the set of measurable functions f:0,1→R such that ∃M≥0 with |fx|≤M a.e. in 0,1. It is known that L∞0,1 is a normed linear space with the norm(7)ρf=f∞=infM>0:fx≤Ma.e.in0,1.
Then,Lρ,ρ is a modular function space with the Δ2-condition. Let C=Bρ0,1 be the ρ-closed ball centered at 0 with radius 1, that is the set(8)f∈L∞:f∞≤1.
Consider the digraphG=C,E defined by (9)f1,f2∈E⇔∃c∈0,1suchthatf2x≤cf1x,a.e.in0,1,
for everyf1,f2∈C. Thus we get,(10)h∈fG⇔∃c∈0,1suchthatfx≤chxa.e.in0,1.
It is clear that this binary relation defines a reflexive digraph without being a partially order (i.e., without antisymmetric condition).
Letf∈C and define T:C→CC by(11)Tf=12nf:n≥1∪0.
(i)
For allF∈Tf, we have F∞=1/2nf≤f≤1, which implies that F∈C. Moreover, Tf is ρ-closed since Tf=1/2nf:n≥1¯.
(ii)
∀
f
∈
C, if h∈[f]G, let for any F∈Tf we have two cases: F=1/2nf or F=0. If F=1/2nf choose H=1/2nh, hence we have the G-monotonocity:(12)Fx=12nfx≤c.12nhx=cHx,
and theρ-contraction condition for α=1/2(13)ρF−H=12nρf−h≤12ρf−h.
Now, ifF=0 let H=0 the G-monotonocity and ρ-contraction conditions are obvious.
For the (P)-property. Let fn be a sequence in C such that(14)fn+1∈fnG∀nandfn−f∞→0asn→∞.
f
n
+
1
∈
[
f
n
]
G
⇒
∃
c
n
∈
[
0
,
1
]
such that
|
f
n
(
x
)
|
≤
c
n
|
f
n
+
1
(
x
)
|, and since 0≤cn≤1 we get for all n∈N,(15)fnx≤fn+1x,∀n.
Furthermore,(16)fnx−fx≤fnx−fx≤‖fn−f‖∞,
implies that|fnx|→|fx|, thus |fnx|≤|fx| for all n∈N, hence f∈[fn]G. That is G has the (P)-property. Then the set-valued mapping T satisfies all the conditions of Theorem 2 hence it has a fixed point, namely 0∈T0.Corollary 1.
Letρ∈R be convex and satisfies the Δ2-type condition. Let C⊂Lρ be a nonempty ρ-closed subset, G a reflexive digraph such that VG=C, with the (P)-property. Let T:C→CC be a G-monotone ρ-contraction mapping. If G has a mother vertex, (a vertex f such that all other vertices in G can be reached by a path from f), then T has a fixed point.Proof.
Indeed for suchf, g∈[f]G, for all g∈Tf and then f∈CT≠∅, we then get the result using Theorem 2.


## 4. Fixed Point Results forρ-Nonexpansive Mappings

Recall the notion of approximated fixed point sequence.Definition 6.
We say thatT:C→2C has an approximated fixed point sequence, if there exists (fn)n∈C such that for every n, there exists Fn∈Tfn satisfying limnρfn−Fn=0.Definition 7.
A digraphG is said to be G-convex if for all f∈[F]G and h∈[H]G we have λf+(1−λ)h∈[λF+(1−λ)H]G.If the reflexive digraphG is transitive and antisymmetric, i.e., being a partially order, then the most ordered Banach spaces enjoy this property. The following result gives a sufficient condition to obtain an approximated fixed point sequence for G-convex digraph.Proposition 2.
Letρ∈R be convex and satisfies the Δ2-type condition. Let C⊂Lρ be a nonempty convex, ρ-closed, and ρ-bounded subset, G a G-convex reflexive digraph such that VG=C, withe the (P)-property.LetT:C→CC be a G-monotone ρ-nonexpansive mapping, if CT:={f∈C:f∈[g]Gforsomeg∈T(f)} is nonempty; then T has an approximated fixed point sequence.Proof.
Letf0∈CT and for each λ∈0,1 define:(17)Tλf=λf0+1−λTf,
for anyf∈C. Then Tλf is ρ-closed since if(18)g∈Tλf⇔11−λg−λf0∈Tf.
Moreover,CTλ is nonempty for any λ∈0,1 as f0∈CTλ.
Letf,g∈C such that g∈[f]H, since T is H-monotone ρ-nonexpansive, we get for all F∈Tf there exists H∈Tg such that H∈[F]H and(19)ρF−H≤ρf−g,
thus,λf0+(1−λ)H∈[λf0+(1−λ)F]H and as ρ is convex:(20)ρλf0+1−λF−λf0+1−λH=ρ1−λF−H≤1−λρf−g,
i.e.,Tλ is H-monotone ρ-contraction. Using Theorem 2, Tλ has a fixed point fλ∈C. Thus ∃Fλ∈Tfλ such that fλ=λf0+1−λFλ and then, as ρ is convex :(21)ρfλ−Fλ=ρλf0−Fλ≤λδρC.
Choosingλ=1/n for n≥1 gives the result.Now with more restrictions on the graphH, we obtain a stronger version of the above result (Proposition 2).Definition 8.
Letρ∈R, a digraph G is said to be compatible with the vector structure of Lρ, if for every f,g,f˜,∈Lρ such that f˜∈[f]G then f˜+g∈[f+g]G and α⋅f˜∈α⋅fG,∀α∈R+.Notice that ifG is compatible with the vector structure of Lρ then it is G-convex. Indeed, for every f,g,f˜,g˜∈Lρ such that f˜∈[f]G and g˜∈[g]G then f˜+g˜∈[f+g]G, and thus for every α∈[0,1],αf˜+(1−α)g˜∈[αf+(1−αg)]G.Lemma 2.
Letρ∈R be convex and satisfies the Δ2-type condition. Let C⊂Lρ be a nonempty convex, ρ-closed, and ρ-bounded subset, G a reflexive digraph compatible with the vector structure of Lρ such that VG=C, with the (P)-property. Let T:C→CC be a G-monotone ρ-nonexpansive mapping, if CT:={f∈C:f∈[g]Gforsomeg∈T(f)} is nonempty; then there exist two sequences (fn)n and (Fn)n∈Lρ such that(22)Fn∈Tfn,fn+1∈fnG,Fn+1∈fn+1G,foreveryn,
andlimnρFn−fn=0.Proof.
Letf0∈CT we define T1f=1/2f0+1/2Tf,f∈Lρ, as seen in the proof of Proposition 2T1 is a G-monotone ρ-contraction mapping, T1f is ρ-closed for every f∈Lρ then T1 has a fixed point f1 such that f1∈[f0]G as we have already seen in Theorem 2. Then, there exists F1∈Tf1 such that(23)f1=12f0+12F1,
thus, it comes clearly thatF1∈[f1]G. In addition:(24)ρf1−F1=ρ12f0−F1≤12ρf1−F1≤12δC,
asρ is convex.Clearlyf1∈CT. Let define T2f=1/3f1+2/3Tf, for all f∈Lρ. As above T2 admits a fixed point f2 such that f2∈[f1]G and then there exists F2∈Tf2 such that f2=1/3f1+2/3F2, thus F2∈[f2]G and ρf2−F2≤1/3δC.By induction onn≥1, if we define the set-valued mapping Tn to be(25)Tnf=1n+1fn+1−1n+1Tf,∀f∈Lρ,we get the existence of a fixed pointfn of Tn such that fn∈[fn−1]G, thus there exists Fn∈Tfn such that fn=1/n+1fn+1+1−1/n+1Fn and then Fn∈[fn]G and ρfn−Fn≤1/n+1δC. Since C is ρ- bounded we get the result.We then apply Lemma2 to get a new fixed point result for G-monotone nonexpansive mapping.Theorem 3.
Letρ∈R be convex, satisfies the Δ2-type condition. Let C⊂Lρ be a nonempty convex, ρ-compact, and ρ-bounded subset, G a reflexive digraph compatible with the vector structure of Lρ with the (P)-property. Let T:C→KC be a G-monotone ρ-nonexpansive mapping, if CT:={f∈C:f∈[g]Gforsomeg∈T(f)} is nonempty, then T has a fixed point.Proof.
The preceding lemma (Lemma2) establish the existence of sequences (fn)n and (Fn)n such that Fn∈Tfn, fn+1∈[fn]G,Fn∈[fn]G for every n, and limnρFn−fn=0.SinceC is ρ-compact, there exists a sub-sequence (fkn)n of (fn)n that ρ-converges to some f∈C, but as limnρFkn−fkn=0 and the ρ-convergence is equivalent to the convergence in Lρ,‖.‖ρ (due to the Δ2-type condition, see [11, Proposition 3.13]); then Fknnρ-converges to f. Thus one can suppose that (fn)n and (Fn)n both ρ-converge to f, and as fn+1∈[fn]G for every n, using the (P)-property we get f∈[fn]G for every n.Now, sinceT is G-monotone ρ-nonexpansive mapping, we have for all n there exists gn∈Tf such that gn∈[Fn]G and ρFn−gn≤ρfn−f and thus gn→fρ as n→∞. Since Tf is ρ-closed because it is ρ-compact, we obtain f∈Tf, i.e., f is a fixed point for T.Example 2.
Consider the modular function spaceLρ=L2(]−1,1[) the space of all square-integrable functions over ]−1,1[, and let(26)ρf=∫−11fx2dx,
for everyf∈Lρ. It is clear that ρ is convex modular function satisfying the Δ2-type condition.
LetC be the set of all constant maps f of Lρ such that ρf≤1. Thus, C is ρ-bounded and convex subset of Lρ. For the ρ-compactness it is enough to show that C satisfies the Riesz-Fréchet-Kolmogorov’s conditions (see [15, Theorem IV.25]) which is obvious. Let G=C,E be the directed graph defined by:(27)f,g∈E⇔∃c∈0,1,gx≤cfxa.e.in−1,1.
It is clear thatG is a reflexive digraph compatible with the vector structure of Lρ. Moreover, it has the (P)-property. Indeed, let fnn be a sequence that ρ-converges to f in Lρ with the condition(28)fn+1∈fnG∀n∈N.
Then there exists a subsequencefnkk which is pointwise convergent almost everywhere to f. Thus,(29)fnkx≤fxa.e.
Moreover, for eachn there exists cn∈0,1 such that(30)fnx≤cnfn+1x≤fn+1x,
then for alln, fnx≤fx a.e., hence G has the (P)-property. Note that the digraph G is not even tranisitive.
LetT be the mapping defined by Tf=f+=max{0,f}. It is clear that T is G-monotone ρ-nonexpansive mapping. Thus all the conditions of Theorem 3 hold, so T has a fixed point namely T0=0.


## 5. Fixed Point Theorem forρ-A.E. Compact Subsets

Theρ-compactness assumption maybe relaxed using the weak concept of ρ-a.e. compactness. Of course, if a subset set C of Lρ is ρ-compact then it is ρ-a.e. compact, see [11, Proposition 3.13] for more details.Definition 9 (see [11]).
Letρ∈R.(i)
A setB⊂Lρ is called ρ-a.e. compact, if for any sequence fnn∈Lρ there exists a subsequence fknn and f∈B such that fknρ-a.e. converges to f.(ii)
A setB⊂Lρ is called ρ-a.e. closed, if for any sequence fnn∈B,fn→f, ρ-a.e. implies f∈B.Theorem 4 (see [11]).
Letρ∈R.(i)
Ifρfn−f→0 there exists fnkk subsequence of fnn such that fnk→f, ρ-a.e.(ii)
Iffn→fρ-a.e, then ρf≤liminfn→+∞ρfn (The Fatou property).We need the following definition of the growth function.Definition 10 (see [11, 13]).
Letρ be a function modular, the function ωρ:0,+∞→0,+∞ defined by:(31)ωρt=supρtfρf:f∈Lρand0<ρf<∞,
is called the growth ofρ.The growth function has the following properties.Proposition 3 (see [11, 16]).
Letρ∈R that has the Δ2-type condition, and ωρ its growth function, then:(i)
ωρ(t)<∞,∀t∈[0,+∞[.(ii)
ωρ:[0,+∞[→[0,+∞[ is convex, and strictly increasing, it is then also continuous.We then get the following lemma.Lemma 3.
Letρ∈R that has the Δ2-type condition, and fnn and gnn be two sequences in Lρ, such that gn→0ρ, then:(32)liminfnρfn+gn=liminfnρfn.Proof.
It is obvious that forα∈]0,+∞[ and f∈Lρ we have: ραf≤ωραρf. Let ε∈]0,1[, then for every n we have:(33)ρfn+gn≤ρfn1−ε+ρgnε≤ωρ11−ερfn+ωρ1ερgn,
and then,(34)liminfnρfn+gn≤ωρ11−εliminfρfn,
asωρ is continuous limε→0ωρ1/1−ε=ωρ1=1 we then get: liminfnρfn+gn≤liminfnρfn letting ϵ→0.
The same arguments give:liminfnρfn=liminfnρfn+gn−gn≤liminfnρfn+gn which ends the proof.Definition 11 (Opial property).
Letρ∈R, Lρ is said satisfying the ρ-a.e. Opial property if for every fnn⊂Lρ which ρ-a.e. converges to 0, and such that supnρβfn<+∞ for some β>1, then we have:(35)liminfnρfn<liminfnρfn+f,
for everyf∈Lρ not equal to 0.The following relaxed definition replace the (P)-property.Definition 12.
LetC⊂Lρ be a nonempty set and G a digraph such that VG=C.
If for any sequencefnn in C which ρ-a.e. converges to f, and such that fn+1∈fnG for every n; we have f∈fnG for every n; then we say that G has the P′-property.Remark 1.
If we substitute “(P)-Property” by “P′-Property” in the Theorem 2, Proposition 2, and Lemma 2; then all these results hold. This is due to the fact that for every ρ-convergence sequence fnn there exists a sub-sequence fknn that ρ-a.e. converges.We conclude this paper by theρ-a.e. compact version of Theorem 3.Theorem 5.
Letρ∈R be convex, satisfies the Δ2-type condition and the ρ-a.e. Opial property. Let C⊂Lρ be a nonempty convex, ρ-a.e. compact, and ρ-bounded subset, G a reflexive digraph compatible with the vector structure of Lρ such that VG=C and satisfies the P′-property. Let T:C→KC be a G-monotone ρ-nonexpansive mapping, if CT:={f∈C:f∈gGforsomeg∈Tf} is nonempty, then T has a fixed point.Proof.
By Lemma2 and Remark 1, there exist two sequences fnn and Fnn in C such that:(36)Fn∈Tfn,fn+1∈fnG,Fn∈fnG,
for everyn, and Fn−fnnρ-a.e. converges to 0.
AsC is ρ-a.e compact, there exists a sub-sequence fknn that ρ-a.e converges to some f in C. But then as limnρfkn−Fkn=0, there exists a sub-sequence fln−Flnn of fkn−Fknn that ρ-a.e. converges to 0. Thus both flnn and Flnnρ-a.e. converge to f. So, without loss of generality, we may suppose that both fnn and Fnnρ-a.e. converge to f.
Now theP′-property gives f∈fnG,∀n. Then for every n, there exists hn∈Tf such that hn∈FnG and(37)ρhn−Fn≤ρfn−f.
Moreover, ashnn⊂Tf which is ρ-compact (since Tf∈KC), it admits a sub-sequence that we will keep noting hnn that ρ-converges to some h∈C. Applying Lemma 3 we get,(38)liminfnρfn−h=liminfnρfn−Fn+Fn−hn+hn−h=liminfnρFn−hn.
So, by lettingn→∞ in the inequality (37), we obtain(39)liminfnρfn−h≤liminfnρfn−f.
Then iff≠h the ρ-a.e. Opial property gives(40)liminfnρfn−f<liminfnρfn−f+f−h=liminfnρfn−h,
which is a contradiction. Then necessarilyf=h and as hnn⊂Tf which is ρ-compact, thus ρ-closed, hence h=f∈Tf, i.e., f is a fixed point for T.



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*Source: 2135859-2020-01-23.xml*