R E S E A R C H
Open Access
Convergence of Ishikawa iterates of two
mappings in modular function spaces
Afrah Ahmad Noan Abdou
1*, Mohamed Amine Khamsi
2,3and Abdul Rahim Khan
3*Correspondence:
1Department of Mathematics, King
Abdulaziz University, Jeddah, 21589, Saudi Arabia
Full list of author information is available at the end of the article
Abstract
We establish convergence in the modular sense of an iteration scheme associated with a pair of mappings on a nonlinear domain in modular function spaces. In particular, we prove that such a scheme converges to a common fixed point of the mappings. Our results are generalization of known similar results in the non-modular setting. In particular, we avoid smoothness of the norm in the case of Banach spaces and that of the triangle inequality of the distance in metric spaces.
MSC: Primary 47H09; secondary 46B20; 47H10; 47E10
Keywords: fixed point; fixed point iteration process; Ishikawa iterations; modular function space; nonexpansive mapping;
ρ
-convergence; modular function space; Orlicz space; strict convexity1 Introduction and basic definitions
The earliest attempts to generalize the classical function spacesLpof Lebesgue type were
made in the early s by Orlicz and Birnbaum in connection with orthogonal expan-sions. Their approach consisted in considering spaces of functions with some growth properties different from the power type growth control provided by the Lp-norms.
Namely, they considered the function spaces defined as follows:
Lϕ=
f :R→R;∃λ> :
R
ϕλf(x)dx<∞
,
whereϕ: [,∞]→[,∞] was assumed to be a convex function increasing to infinity,i.e.
the function which to some extent behaves similar to power functionsϕ(t) =tp. Later on,
the assumption of convexity for Orlicz functionsϕwas frequently omitted. Let us mention two typical examples of such functions:
ϕ(t) =et– , ϕ(t) =ln( +t).
The possibility of introducing the structure of a linear metric inLϕas well as the interesting properties of these spaces and many applications to differential and integral equations with kernels of nonpower types were among the reasons for the development of the theory of Orlicz spaces, their applications, and generalizations for more than half a century.
We may observe two principal directions of further development. The first is the the-ory of Banach function spaces initiated in by Luxemburg [] and developed further
in a series of joint papers with Zaanen []. The second one is inspired by the theory of Orlicz spaces, based on replacing the particular integral form of the nonlinear functional, which controls the growth of members of the space, by an abstract given functional with some good properties. This idea was the basis of the theory of modular spaces initiated by Nakano [] in connection with the theory of order spaces and redefined and general-ized by Luxemburg and Orlicz in . Such spaces have been studied for almost years and a number of applications of such spaces in various parts of analysis, probability, and mathematical statistics are known.
In this paper, we will consider modular spaces which lie somewhere in between the ab-stract modular theory and Musielak-Orlicz theory,i.e.the class of modular spaces given by modulars not of any particular form but, nevertheless, having much more convenient properties than the abstract modulars can possess. In other words, we present a useful tool for applications whenever there is a need to introduce a function space by means of func-tionals which have some reasonable properties but are far from being norms orF-norms. Let us introduce basic notions in modular function spaces and related notations which will be used in this chapter. For further details we refer the reader to preliminary sections of the recent articles [–] or to the survey article []; see also [–] for the standard framework of modular function spaces.
Letbe a nonempty set and be a nontrivialσ-algebra of subsets of. LetP be a
δ-ring of subsets of, such thatE∩A∈P for anyE∈P andA∈. Let us assume that there exists an increasing sequence of setsKn∈Psuch that=
Kn. ByEwe denote the
linear space of all simple functions with support 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∈. By Awe denote the
characteristic function of the setA.
Definition .[] Letρ:M∞→[,∞] be a nontrivial, convex, and even function. We
say thatρis a regular convex function pseudomodular if: (i) ρ() = ;
(ii) ρis monotone,i.e.|f(ω)| ≤ |g(ω)|for allω∈impliesρ(f)≤ρ(g), where
f,g∈M∞;
(iii) ρis orthogonally subadditive,i.e.ρ(fA∪B)≤ρ(fA) +ρ(fB)for anyA,B∈such
thatA∩B=∅,f ∈M;
(iv) ρhas the Fatou property,i.e.|fn(ω)| ↑ |f(ω)|for allω∈impliesρ(fn)↑ρ(f),
wheref ∈M∞;
(v) ρis order continuous inE,i.e.gn∈Eand|gn(ω)| ↓impliesρ(gn)↓.
Similarly, as in the case of measure spaces, we say that a setA∈isρ-null ifρ(gA) =
for everyg∈E. We say that a property holdsρ-almost everywhere if the exceptional set isρ-null. As usual we identify any pair of measurable sets whose symmetric difference is
ρ-null as well as any pair of measurable functions differing only on aρ-null set. With this in mind, we defineM={f ∈M∞;|f(ω)|<∞ρ-a.e.}, where each element is actually an equivalence class of functions equalρ-a.e. rather than an individual function.
Definition .[] We say that a regular function pseudomodularρis a regular convex
In this paper, we only consider convex function modulars.
Definition .[–] Letρbe a convex function modular. A modular function space is
the vector spaceLρ={f ∈M;ρ(λf)→ asλ→}. In the vector spaceLρ, the following formula:
fρ=inf
α> ;ρ f α
≤
defines a norm, frequently called the Luxemburg norm.
Note that the monographic exposition of the theory of Orlicz spaces may be found in the book of Krasnosel’skii and Rutickii []. For a current review of the theory of Musielak-Orlicz spaces and modular spaces, the reader is referred to Musielak [] and Kozlowski [].
The following definitions will be needed in this paper.
Definition .[]
(a) The sequence{fn} ⊂Lρis said to beρ-convergent tof ∈Lρifρ(fn–f)→as
n→ ∞.
(b) The sequence{fn} ⊂Lρis said to beρ-Cauchy ifρ(fn–fm)→asnandm→ ∞.
(c) We say thatLρisρ-complete if and only if anyρ-Cauchy sequence inLρis
ρ-convergent.
(d) A subsetCofLρis calledρ-closed if theρ-limit of aρ-convergent sequence ofC always belongs toC.
(e) A subsetCofLρis calledρ-compact if every sequence inChas aρ-convergent subsequence inC.
(f ) A subsetCofLρis calledρ-bounded if
δρ(C) =supρ(f –g);f,g∈C<∞.
(g) Letf∈LρandC⊂Lρ. Define theρ-distance betweenf andCas:
dρ(f,C) =infρ(f–g);g∈C.
The above terminology is used because of its similarity to the metric case. Sinceρdoes not behave in general as a distance, one should be very careful when dealing with these notions. In particular,ρ-convergence does not implyρ-Cauchy sinceρ does not satisfy the triangle inequality.
The following proposition brings together a few facts that will be often used.
Proposition .[] Letρ∈ . (i) Lρisρ-complete.
(ii) ρ-ballsBρ(f,r) ={g∈Lρ;ρ(f –g)≤r}areρ-closed.
(iii) Ifρ(αfn)→for anα> then there exists a subsequence{gn}of{fn}such that gn→ρ-a.e.
(iv) ρ(f)≤lim infn→∞ρ(fn)wheneverfn→f ρ-a.e. (Note:this property is equivalent to
(v) Consider the setsL
ρ={f∈Lρ;ρ(f,·)is order continuous},and
Eρ={f ∈Lρ;λf ∈Lρfor anyλ> }.Then we haveEρ⊂Lρ⊂Lρ.
We already pointed out thatρ may not satisfy the triangle inequality, so the modular convergence and norm convergence may not be the same. This will only happen ifρ sat-isfies the so-called-condition.
Definition .[] The modular functionρis said to satisfy the-condition ifρ(fn)→
asn→ ∞, wheneverρ(fn)→ asn→ ∞.
We have the following proposition.
Proposition .[] The following statements are equivalent: (i) ρsatisfies the-condition;
(ii) ρ(fn–f)→if and only ifρ(λ(fn–f))→,for allλ> if and only iffn–f →.
Definition . We will say that the function modularρ is uniformly continuous if for
everyε> andL> , there existsδ> such that
ρ(g) –ρ(h+g)<ε; ifρ(h) <δandρ(g)≤L.
Let us mention that uniform continuity holds for a large class of function modulars. For instance, it can be proved that in Orlicz spaces over a finite atomless measure [] or in sequence Orlicz spaces [], the uniform continuity of the Orlicz modular is equivalent to the-type condition.
Let us recall the definition of different mappings acting in a modular function space. We start with the concept of Lipschitzian mappings.
Definition . Letρ∈ and letC⊂Lρbe a nonempty subset. A mappingT:C→Cis
calledρ-Lipschitzian mapping if there exists a constantL≥ such that
ρT(f) –T(g)≤Lρ(f–g) for anyf,g∈C.
When L< , T is called ρ-contraction mapping. Moreover, if L≤, then T is called
ρ-nonexpansive mapping. A pointf ∈Cis called a fixed point ofT ifT(f) =f. The set of fixed points ofTwill be denoted byF(T).
As mentioned before, one of the reasons of our interest inρ-behavior of mappings is that the Luxemburg norm associated with the function modular is defined in an indirect way and consequently harder to handle than the function modular. Therefore, one may ask what the relationship is, if there is any, between theF-norm nonexpansiveness and theρ-nonexpansiveness. The following example gives a partial answer.
Example .[] LetX= (,∞), and letbe theσ-algebra of all Lebesgue measurable subsets of X. Let P denote the δ-ring of subsets of finite measures. Define a function modular by
ρ(f) =
e
∞
LetBbe the set of all measurable functionsf : (,∞)→Rsuch that ≤f(x)≤. Define the linear operatorTby the formula
T(f)(x) =
f(x– ) forx≥, forx∈[, ].
Clearly,T(B)⊂B. In [], it is proved that for every fixedλ≤ and for allf,g∈B, we have
ρλT(f) –T(g)≤λρλ(f –g).
In particular, if λ= , then the above inequality shows that T isρ-nonexpansive. It is also easy to see thatBis aρ-a.e.-bounded subset ofLρ. We observe thatT is not · ρ-nonexpansive. Indeed, if we takef = [,], then
T(f)
ρ>e≥ fρ.
In this paper, we will use some geometrical properties of the modular functional.
Definition . Letρ∈ . We say thatρis strictly convex (SC), if for everyf,g∈Lρsuch thatρ(f) =ρ(g) andρ(αf+ ( –α)g) =αρ(f) + ( –α)ρ(g), for someα∈(, ), we havef =g.
It is known that for a wide class of modular function spaces with the -property, geometric properties of the Luxemburg norm are equivalent to the same properties of the modular. For example, in Orlicz spaces these results can be traced in early papers of Luxemburg [], Milnes [], Akimovic [], and Kaminska []. It is also known that, under suitable assumptions, uniform convexity in Orlicz spaces is equivalent to the very convex-ity of the Orlicz function []. Typical examples of Orlicz functions that do not satisfy the
-condition but are uniformly convex (and hence strictly convex) are [, ]
ϕ(t) =e|t|–|t|– and ϕ(t) =et
– .
For the discussion of some geometrical properties of Calderon-Lozanovskii and Orlicz-Lorentz spaces, the reader may consult [].
Recently, special attention was given to the use of the geometric properties in modular function spaces. This is due to recent interest in the Dirichlet energy problem which we discuss in the next example.
Example . Let⊂Rbe an open set and let p:→[,∞) be a measurable
func-tion (called the variable exponent on). We define thevariable exponent Lebesgue space Lp(·)() to consist of all measurable functionsf:→Rsuch that
ρ(λf) =
λf(x)p(x)dx<∞
for some λ> . The functional ρ is called the modular of the space Lp(·)(). The Luxemburg norm on this space is given by the formula
Thevariable exponent Sobolev space W,p(·)() is the space of measurable functionsf :
→Rsuch thatf and the distributional derivativefare inLp(·)(). The function
ρ(f) =ρ(f) +ρf
defines a modular onW,p(·)(). DefineW,p(·)
() as the set off ∈W,p(·)() which can be continuously continued by outside. The energy operator corresponding to the bound-ary value functiongacting on the space
f ∈W,p(·)();f–g∈W,p(·)()
is defined by
Ig(f) =
f(x)p(x)
dx=ρf.
The general Dirichlet energy problem is to find a function that minimizes values of the operatorIg(·). Note that
minIg(g–f);f∈W,p(·)()
=dρ
g,W,p(·)().
For more information on the Dirichlet energy integral problem we refer to [–].
2 Ishikawa iterates for two mappings
In [], the authors introduced the Ishikawa iterative scheme for two mappings and stud-ied the strong convergence of this scheme to a common fixed point of the two mappings. Let us introduce such an iterative scheme in modular function spaces. LetSandTbe two mappings defined on a nonempty closed, convex andρ-bounded subsetCofLρ. Fixf ∈C
and define the sequence{fn}, withf=f, and
fn+=αnS
βnT(fn) + ( –βn)fn
+ ( –αn)fn, n= , , . . . . (.)
WhenS=T, the above iterative scheme collapses into the classical Ishikawa iterative scheme for one map:
fn+=αnT
βnT(fn) + ( –βn)fn
+ ( –αn)fn, n= , , . . . . (.)
LetS,T:C→Cbe twoρ-nonexpansive mappings. Assume thatF=F(S)∩F(T)=∅. Letf ∈Candh∈F. Setr=ρ(f –h). Then
C(f) =C∩B(h,r) =g∈C;ρ(h–g)≤r
is a nonempty closed and convex subset ofCand invariant under bothSandT. Therefore one may always assume thatCisρ-bounded onceSandT have a common fixed point. Moreover, if{fn}is the sequence generated by (.), withf=f, then we have
ρ(fn+–h) =ρ
αnS(gn) + ( –αn)fn–h
≤αnρ
S(gn) –h
≤αnρ(gn–h) + ( –αn)ρ(fn–h)
=αnρ
βnT(fn) + ( –βn)fn–h
+ ( –αn)ρ(fn–h)
≤αn
βnd
T(fn) –h
+ ( –βn)ρ(fn–h)
+ ( –αn)ρ(fn–h)
≤ρ(fn–h),
wheregn=βnT(fn) + ( –βn)fn. This proves that{ρ(fn–h)}is decreasing, which implies
thatlimn→∞ρ(fn–h) exists. Using the above inequalities, we get
lim
n→∞ρ(fn–h) =nlim→∞ρ
αnS(gn) + ( –αn)fn–h
= lim n→∞αnρ
S(gn) –h
+ ( –αn)ρ(fn–h)
= lim
n→∞αnρ(gn–h) + ( –αn)ρ(fn–h)
= lim n→∞αnαnρ
βnT(fn) + ( –βn)fn–h
+ ( –αn)ρ(fn–h)
= lim n→∞αn
βnd
T(fn) –h
+ ( –βn)ρ(fn–h)
+ ( –αn)ρ(fn–h).
Our first result discusses the convergence behavior of the sequence generated by (.).
Theorem . Assume ρ∈ is strictly convex and uniformly continuous. Let C be a
nonempty ρ-bounded, closed and convex subset of X.Let S,T :C→C be twoρ -non-expansive mappings.Assume that F=F(S)∩F(T)=∅.Let f∈C and{fn}be given by(.). Then the following hold:
(i) Ifαn∈[a,b]andβn∈[,b],with <a≤b< ,then for any subsequence{fni}of{fn} whichρ-converges tof,we havef ∈F(S).
(ii) Ifαn∈[a, ]andβn∈[a,b],with <a≤b< ,then for any subsequence{fni}of{fn} whichρ-converges tof,we havef ∈F(T).
(iii) Ifαn,βn∈[a,b],with <a≤b< ,then for any subsequence{fni}of{fn}which
ρ-converges tof,we havef ∈F.In this case,we have that{fn}ρ-converges tof.
Proof Assume that{fni}ρ-converges tof. Leth∈F. Without loss of generality, we may assumelimn→∞αni=α, andlimn→∞βni=β. Since{ρ(fn–h)}is decreasing andρis uni-formly continuous, we get
lim
n→∞ρ(fn–h) =nlim→∞ρ(fni–h) =ρ(f –h).
Since T is ρ-nonexpansive, {T(fni)} ρ-converges to T(f). Moreover, asρ is uniformly continuous,{βniT(fni) + ( –βni)fni}ρ-converges toβT(f) + ( –β)f. Using theρ -non-expansiveness ofS, we get{S(βniT(fni) + ( –βni)fni)}ρ-converges toS(βT(f) + ( –β)f). Finally, sinceρis uniformly continuous, we get{αniS(βniT(fni) + ( –βni)fni) + ( –αni)fni}
ρ-converges toαS(βT(f) + ( –β)f) + ( –α)f. The above inequalities imply
ρ(f–h) =ραSβT(f) + ( –β)f+ ( –α)f–h
=αρSβT(f) + ( –β)f–h+ ( –α)ρ(f–h)
=αρβT(f) + ( –β)f –h+ ( –α)ρ(f –h)
Setr=ρ(f –h). Without loss of generality we may assumer> (otherwise most of the conclusions in the theorem are trivial). Assume thatlim infn→∞αn> . Thenα= . Hence
ρSβT(f) + ( –β)f–h=ρβT(f) + ( –β)f –h
=βρT(f) –h+ ( –β)r
=r,
which impliesβρ(T(f) –h) =βr. If we assume thatlim infn→∞βn> , thenβ= which
impliesρ(T(f) –h) =r.
() Ifα∈(, )andβ> , then
ρSβT(f) + ( –β)f–h=ραSβT(f) + ( –β)f+ ( –α)f–h
=ρ(f–h).
The strict convexity ofρwill implyS(βT(f) + ( –β)f) =f. () Ifα∈(, )andβ= , then
ρ(f–h) =ρS(f) –h=ραS(f) + ( –α)f–h.
The strict convexity ofρwill implyS(f) =f. () Ifβ∈(, )andα> , then
ρ(f–h) =ρT(f) –h=ρβT(f) + ( –β)f–h.
The strict convexity ofρwill implyT(f) =f.
() Ifα,β∈(, ), thenT(f) =f andS(βT(f) + ( –β)f) =f. HenceS(f) =f.
Let us finish the proof of Theorem .. Note that (i) impliesα∈[a,b] andβ∈[,b]. If
β= , then the conclusion () above impliesf ∈F(S). Otherwise the conclusion () will implyf ∈F. This proves (i).
For (ii), notice thatα∈[a, ] andβ∈[a,b]. Hence the conclusion () will implyf ∈F(T), which proves (ii).
For (iii), notice thatα,β∈[a,b]. Hence the conclusion () will implyf ∈F. Since
lim
n→∞ρ(fn–f) =nlim→∞ρ(fni–f) = ,
we find that{fn}ρ-converges tof, which completes the proof of (iii).
If we assume compactness of the domain, Theorem . will imply the following result.
Theorem . Assume ρ∈ is strictly convex and uniformly continuous. Let C be a
Proof SinceCisρ-compact,{fn}has aρ-convergent subsequence{fni},i.e.,{fni}ρ -con-verges toz. By Theorem ., we havez∈Fand{fn}ρ-converges toz.
The existence of a common fixed pointT andSis crucial to the conclusion of Theo-rems . and .. Note that each mapping has a nonempty fixed point set. Indeed, letC
andTbe as in Theorem .. Fixε∈(, ) andf∈C. Define the mappingTε:C→Cby
Tε(f) =εf+ ( –ε)T(f).
Then, for anyf,g∈C, we have
ρTε(f) –Tε(g)≤( –ε)ρ(f –g).
Fixf∈C. Since isρ-bounded, we get
ρTεn+h(f) –Tεn(f)≤( –ε)nρTεh(f) –f≤( –ε)nδρ(C)
for anyn,h∈N. SinceCisρ-compact, there exists a subsequence{Tni
ε (f)}which isρ -con-vergent toh∈C. We claim thathis a fixed point ofTε. Indeed, since
ρTεni+(f) –Tε(h)
≤( –ε)ρTni ε (f) –h
and
ρTεni+(f) –Tεni(f)
≤( –ε)niρTε(f) –f≤( –ε)niδρ(C),
we conclude thatTε(h) =h. Indeed, we have
ρ h–Tε(h)
≤
ρh–Tni ε (f)
+ρTni
ε (f) –Tεni+(f)
+ρTεni+(f) –Tε(h)
.
If we letni→ ∞, we getρ(h–Tε(h)) = , which implies Tε(h) =h. In fact, one can now
easily show that{Tεn(f)}ρ-converges tohand thathis independent off and is the only fixed point ofTε. Clearly,
ρT(h) –h=ρT(h) – ( –ε)T(h) –εf
≤ερT(h) –f
≤εδρ(C).
Henceinff∈Cρ(T(f) –f) = . AsCisρ-compact, the fixed point set ofTis nonempty. As
ρ is strictly convex,F(T) is convex. ClearlyF(T) isρ-closed subset ofC. HenceF(T) is
ρ-compact. If we assume thatTandScommute,i.e.,S◦T=T◦S, thenS(F(T))⊂F(T). Consequently,SandThave a common fixed point. In general, it is not the case thatSand
T have a common fixed point.
If we takeS=Tin Theorem ., we get the following result.
Theorem .[] Assumeρ∈ is strictly convex and uniformly continuous.Let C be a
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors participated in the design of this work and performed equally. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia.2Department of Mathematical
Sciences, University of Texas at El Paso, El Paso, TX 79968, USA. 3Department of Mathematics and Statistics, King Fahd
University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia.
Acknowledgements
The authors MA Khamsi and AR Khan acknowledge gratefully KACST, Riyad, Saudi Arabia, for supporting Research Project ARP-32-34. The author AAN Abdou gratefully acknowledges the Deanship of Scientific Research at King Abdulaziz University, Jeddah, Saudi Arabia, for supporting this research.
Received: 27 December 2013 Accepted: 7 March 2014 Published:25 Mar 2014
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