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Common fixed points for pointwise Lipschitzian semigroups in modular function spaces

Abstract

Let C be a ρ-bounded, ρ-closed, convex subset of a modular function space L ρ . We investigate the existence of common fixed points for asymptotic pointwise nonexpansive semigroups of nonlinear mappings T t :CC, i.e. a family such that T 0 (f)=f, T s + t (f)= T s T t (f) and

ρ ( T ( f ) T ( g ) ) α t (f)ρ(fg),

where lim sup t α t (f)1 for every fC. In particular, we prove that if L ρ is uniformly convex, then the common fixed point is nonempty ρ-closed and convex.

MSC:47H09, 46B20, 47H10, 47E10.

1 Introduction

The purpose of this paper is to prove the existence of common fixed points for semigroups of nonlinear mappings acting in modular function spaces which are natural generalizations of both function and sequence variants of many important, from applications perspective, spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces and many others, see the book by Kozlowski [1] for an extensive list of examples and special cases. Earlier studies of fixed point theory in modular function spaces can be found in [24], see also [5]. Recently, Khamsi and Kozlowski presented a series of fixed point results for pointwise contractions, asymptotic pointwise contractions, pointwise nonexpansive and asymptotic pointwise nonexpansive mappings acting in modular functions spaces [6, 7] (all these should be considered in the modular sense, not in the sense of the corresponding norms). These results are also new and of a big interest, even in a much simpler context of ‘plain’ modular contractions and nonexpansive mappings, i.e., without any pointwise and asymptotic complications.

In many cases, modular type conditions are much more natural as modular type assumptions can be more easily verified than their metric or norm counterparts. Furthermore, there are also important results that can be proved only using the apparatus of modular function spaces. Khamsi et al. demonstrated in [2] that a mapping T is norm-nonexpansive in a modular function space L ρ if and only if

ρ ( λ ( T ( f ) T ( g ) ) ) ρ ( λ ( f g ) ) for any λ0.

They also gave an example of a mapping which is ρ-nonexpansive, but it is not norm-nonexpansive. From this perspective, the fixed point theory in modular function spaces should be considered as complementary to the fixed point theory in normed spaces and in metric spaces.

Let us recall that a family { T t } t 0 of mappings forms a semigroup if T 0 (x)=x and T s + t = T s T t , see Definition 2.6 below for details. Such a situation is quite typical in mathematics and applications. For instance, in the theory of dynamical systems, the modular function space L ρ would define the state space and the mapping (t,f) T t (f) would represent the evolution function of a dynamical system. The question about the existence of common fixed points and about the structure of the set of common fixed points, can be interpreted as a question whether there exist points that are fixed during the state space transformation T t at any given point of time t, and if yes, what the structure of a set of such points may look like. In the setting of this paper, the state space may be an infinite dimensional vector space. Therefore, it is natural to apply these results not only to deterministic dynamical systems but also to stochastic dynamical systems.

The existence of common fixed points for families of contractions and nonexpansive mappings in the Banach spaces have been investigated since the early 1960s, see, e.g., Belluce and Kirk [8, 9], Browder [10], Bruck [11], DeMarr [12], Lim [13]. The asymptotic approach for finding common fixed points of semigroups of Lipschitzian (but not pointwise Lipschitzian) mappings has also been investigated for some time, see, e.g., Tan and Xu [14]. It is worthwhile mentioning the recent studies on the special case, when the parameter set for the semigroup is equal to {0,1,2,3,} and T n = T n , the n th iterate of an asymptotic pointwise nonexpansive mapping, i.e., T:CC such that there exists a sequence of functions α n :C[0,) with

T n ( f ) T n ( g ) α n (f)fgfor any f,gC

and lim sup n α n (f)=1 for any fC. Kirk and Xu [15] proved the existence of fixed points for asymptotic pointwise contractions and asymptotic pointwise nonexpansive mappings in the Banach spaces, while Hussain and Khamsi extended this result to metric spaces [16] and Khamsi and Kozlowski to modular function spaces [6, 7]. Kozlowski in [17] and [18] proved convergence to fixed points of some iterative algorithms, applied to asymptotic pointwise nonexpansive mappings in the Banach spaces, and the existence of common fixed points of semigroups of asymptotic pointwise nonexpansive semigroups in the Banach spaces [19]. Convergence of generalized Mann and Ishikawa algorithms to common points of such semigroups in Banach spaces was established in [20] and [21]. In the context of modular function spaces, convergence to fixed points of some iterative algorithms, applied to asymptotic pointwise nonexpansive mappings, was proven by Bin Dehaish and Kozlowski in [22].

In this paper, we extend the definition of asymptotic pointwise nonexpansive mappings to semigroups of mappings and prove some common fixed point results in the context of modular function spaces. Therefore, our results generalize the results of Kozlowski [23], who proved the existence of common fixed points for semigroups of nonexpansive mappings in modular functions spaces, to the pointwise asymptotic semigroups. However, methods used in the current paper are substantially different due to the asymptotic behavior of semigroups in question. It is worth noting that existence of semigroups of nonexpansive mappings in modular function spaces was discussed by Khamsi [24] in the context of Musielak-Orlicz spaces and discussed applications to differential equations.

2 Preliminaries

Let us introduce basic notions related to modular function spaces and related notation, which will be used in this paper. For further details, we refer the reader to preliminary sections of the recent articles [6, 7, 22] or to the survey article [5], see also [1, 25, 26] for the standard framework of modular function spaces.

Let Ω be a nonempty set, and let Σ be a nontrivial σ-algebra of subsets of Ω. Let be a δ-ring of subsets of Ω such that EAP for any EP and AΣ. Let us assume that there exists an increasing sequence of sets K n P such that Ω= K n . By ℰ we denote the linear space of all simple functions with supports from . By M we will denote the space of all extended measurable functions, i.e., all functions f:Ω[,] such that there exists a sequence { g n }E, | g n ||f| and g n (ω)f(ω) for all ωΩ. By 1 A we denote the characteristic function of the set A.

Definition 2.1 [1]

Let ρ: M [0,] be a nontrivial, convex and even function. We say that ρ is a regular convex function pseudomodular if:

  1. (i)

    ρ(0)=0;

  2. (ii)

    ρ is monotone, i.e., |f(ω)||g(ω)| for all ωΩ implies ρ(f)ρ(g), where f,g M ;

  3. (iii)

    ρ is orthogonally subadditive, i.e., ρ(f 1 A B )ρ(f 1 A )+ρ(f 1 B ) for any A,BΣ such that AB, fM;

  4. (iv)

    ρ has the Fatou property, i.e., | f n (ω)||f(ω)| for all ωΩ implies ρ( f n )ρ(f), where f M ;

  5. (v)

    ρ is order continuous in ℰ, i.e., g n E and | g n (ω)|0 implies ρ( g n )0.

Similarly, as in the case of measure spaces, we say that a set AΣ is ρ-null if ρ(g 1 A )=0 for every gE. We say that a property holds ρ-almost everywhere if the exceptional set is ρ-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 a ρ-null set. With this in mind, we define M={f M ;|f(ω)|<ρ-a.e.}, where each element is actually an equivalence class of functions equal ρ-a.e. rather than an individual function.

Definition 2.2 [1]

We say that a regular function pseudomodular ρ is a regular convex function modular if ρ(f)=0 implies f=0 ρ-a.e. The class of all nonzero regular convex function modulars, defined on Ω will be denoted by ℜ.

Definition 2.3 [1, 25, 26]

Let ρ be a convex function modular. A modular function space is the vector space L ρ ={fM;ρ(λf)0 as λ0}. In the vector space L ρ , the following formula

f ρ =inf { α > 0 ; ρ ( f α ) 1 }

defines a norm, frequently called Luxembourg norm.

The following notions will be used throughout the paper.

Definition 2.4 [2]

Let ρ.

  1. (a)

    We say that { f n } is ρ-convergent to f and write f n f(ρ) if and only if ρ( f n f)0.

  2. (b)

    A sequence { f n }, where f n L ρ , is called ρ-Cauchy if ρ( f n f m )0 as n,m.

  3. (c)

    We say that L ρ is ρ-complete if and only if any ρ-Cauchy sequence in L ρ is ρ-convergent.

  4. (d)

    A set B L ρ is called ρ-closed if for any sequence of f n B, the convergence f n f(ρ) implies that f belongs to B.

  5. (e)

    A set B L ρ is called ρ-bounded if sup{ρ(fg);fB,gB}<.

Since ρ fails in general the triangle identity, many of the known properties of limit may not extend to the ρ-convergence. For example, the ρ-convergence does not necessarily imply the ρ-Cauchy condition. However, it is important to remember that the ρ-limit is unique when it exists. The following proposition brings together few facts that will be often used in the proofs of our results.

Proposition 2.1 [1]

Let ρ.

  1. (i)

    L ρ is ρ-complete.

  2. (ii)

    ρ-balls B ρ (f,r)={g L ρ ;ρ(fg)r} are ρ-closed.

  3. (iii)

    If ρ(α f n )0 for an α>0, then there exists a subsequence { g n } of { f n } such that g n 0 ρ-a.e.

  4. (iv)

    ρ(f) lim inf n ρ( f n ), whenever f n f ρ-a.e. (Note: this property is equivalent to the Fatou property.)

Let us recall the definition of an asymptotic pointwise nonexpansive mapping acting in a modular function space.

Definition 2.5 [7]

Let ρ, and let C L ρ be nonempty and ρ-closed. A mapping T:CC is called

  1. (i)

    a pointwise Lipschitzian mapping, if there exists α:C[0,) such that

    ρ ( T ( f ) T ( g ) ) α(f)ρ(fg)for any f,gC;
  2. (ii)

    an asymptotic pointwise nonexpansive, if there exists a sequence of mappings α n :C[0,) such that

    ρ ( T n ( f ) T n ( g ) ) α n (f)ρ(fg)for any f,gC

and lim sup n α n (f)1 for any f L ρ .

A point fC is called a fixed point of T, whenever T(f)=f. The set of fixed points of T will be denoted by F(T).

This definition is now extended to a one-parameter family of mappings.

Definition 2.6 A one-parameter family F={ T t :t0} of mappings from C into itself is said to be a asymptotic pointwise nonexpansive semigroup on C if ℱ satisfies the following conditions:

  1. (i)

    T 0 (f)=f for fC;

  2. (ii)

    T t + s (f)= T t ( T s (f)) for fC and t,s[0,);

  3. (iii)

    for each t0, T t is an asymptotic pointwise nonexpansive mapping, i.e., there exists a function α t :C[0,) such that

    ρ ( T t ( f ) T t ( g ) ) α t (f)ρ(fg)for all f,gC
    (2.1)

such that lim sup t α t (f)1 for every fC;

  1. (iv)

    for each fC, the mapping t T t (f) is ρ-continuous.

For each t0, let F( T t ) denote the set of its fixed points. Define then the set of all common fixed points set for mappings from ℱ as the following intersection

F(F)= t 0 F( T t ).

Note that without loss of generality, we may assume α t (f)1 for any t0 and fC, and lim sup t α t (f)= lim t α t (f)=1.

3 Existence of common fixed points

The concept ρ-type is a powerful technical tool, which is used in the proofs of many fixed point results. The definition of a ρ-type is based on a given sequence. In this work, we generalize this definition to be adapted to one-parameter family of mappings.

Definition 3.1 Let K L ρ be convex and ρ-bounded.

  1. (1)

    A function τ:K[0,] is called a ρ-type (or shortly a type) if there exists a one-parameter family { h t } t 0 of elements of K such that for any fK there holds

    τ(f)= inf M > 0 ( sup t M ρ ( h t f ) ) .
  2. (2)

    Let τ be a type. A sequence { g n } is called a minimizing sequence of τ if

    lim n τ( g n )=inf { τ ( f ) : f K } .

Note that τ is convex, provided ρ is convex.

Let us recall the modular equivalents of uniform convexity introduced in [7].

Definition 3.2 Let ρ. We define the following uniform convexity (UC) type properties of the function modular ρ:

  1. (i)

    Let r>0, ε>0. Define

    D(r,ε)= { ( f , g ) : f , g L ρ , ρ ( f ) r , ρ ( g ) r , ρ ( f g ) ε r } .

    Let

    δ(r,ε)=inf { 1 1 r ρ ( f + g 2 ) : ( f , g ) D ( r , ε ) } if D(r,ε),

    and δ(r,ε)=1 if D(r,ε)=. We say that ρ satisfies (UC) if for every r>0, ε>0, δ(r,ε)>0. Note that for every r>0, D(r,ε), for ε>0 small enough.

  2. (ii)

    We say that ρ satisfies (UUC) if there exists η(s,ε)>0, for every s0, and ε>0 such that

    δ(r,ε)>η(s,ε)>0for r>s.

The following technical lemma is very useful throughout this paper (see [7] for its proof).

Lemma 3.1 Let ρ be (UUC). Let R>0. Assume that { f n } and { g n } are in L ρ such that

lim sup n ρ( f n )R; lim sup n ρ( g n )Rand lim n ρ ( f n + g n 2 ) =R.

Then we must have lim n ρ( f n g n )=0.

The following property plays in the theory of modular function space a role similar to the reflexivity in the Banach spaces, see, e.g., [3].

Definition 3.3 We say that L ρ has property (R) if and only if every nonincreasing sequence { C n } of nonempty, ρ-bounded, ρ-closed and convex subsets of L ρ has a nonempty intersection.

Similarly as in the Banach space case, the modular uniform convexity implies the property (R).

Theorem 3.1 [7]

Let ρ be (UUC), then L ρ has a property (R).

The next lemma is the generalization of the minimizing sequence property for types defined by the sequences in Lemma 4.3 in [6] to the one-parameter semigroup case.

Lemma 3.2 Assume ρ is (UUC). Let C be a nonempty, ρ-bounded, ρ-closed and convex subset of L ρ . Let τ be a type defined by a one-parameter family { h t } t 0 in C.

  1. (i)

    If τ( f 1 )=τ( f 2 )= inf f C τ(f), then f 1 = f 2 .

  2. (ii)

    Any minimizing sequence { f n } of τ is ρ-convergent. Moreover, the ρ-limit of { f n } is independent of the minimizing sequence.

Proof First, let us prove (i). Let f 1 , f 2 C such that τ( f 1 )=τ( f 2 )= inf f C τ(f). Let us consider two cases.

Case 1: inf f C τ(f)=0. Since

ρ ( f 1 f 2 2 ) =ρ ( f 1 h t + h t f 2 2 ) ρ( f 1 h t )+ρ( h t f 2 )

for any t0, we get

ρ ( f 1 f 2 2 ) sup t M ρ( f 1 h t )+ sup t M ρ( h t f 2 )

for any M>0. Since

τ(f)= inf M > 0 ( sup t M ρ ( f h t ) ) = lim M sup t M ρ(f h t )

for any fC, we get

ρ ( f 1 f 2 2 ) τ( f 1 )+τ( f 2 )=0,

which implies f 1 = f 2 as claimed.

Case 2: inf f C τ(f)>0. Assume to the contrary that f 1 f 2 . Set

R= inf f C τ(f)andε= ρ ( f 1 f 2 ) 2 R .

Let ν(0,R). Then ρ( f 1 f 2 )=2Rε(R+ν)ε. Using the definition of τ, we deduce that there exists M ν >0 such that

sup t M ν ρ( f 1 h t )τ( f 1 )+ν=R+νand sup t M ν ρ( f 2 h t )τ( f 2 )+ν=R+ν.

Since ρ is (UUC), there exists η(R,ε)>0 such that

δ(R+ν,ε)η(R,ε)

for any ν(0,R). So for any t M ν , we have

ρ ( f 1 + f 2 2 h t ) (R+ν) ( 1 δ ( R + ν , ε ) ) (R+ν) ( 1 η ( R , ε ) ) .

Hence

τ ( f 1 + f 2 2 ) sup t M ν ρ ( f 1 + f 2 2 h t ) (R+ν) ( 1 η ( R , ε ) ) .

Since C is convex, we get

Rτ ( f 1 + f 2 2 ) (R+ν) ( 1 η ( R , ε ) ) .

If we let ν0, we will get

RR ( 1 η ( R , ε ) ) ,

which is impossible, since R>0 and η(R,ε)>0. Therefore, we must have f 1 = f 2 .

Next, we prove (ii). Denote R= inf g C τ(g). For any n1, let us set

K n = conv ¯ ρ { h t ;tn},

where conv ¯ ρ (A) is the intersection of all ρ-closed convex subset of C, which contains AC. Since C is itself ρ-closed and convex, we get K n C for any n1. Property (R) will then imply K n . Let us fix then arbitrary f K n , gC and ε>0. By definition of τ(g), there exists M ε >0 such that sup t M ε ρ(g h t )τ(g)+ε. Let n M ε . Then for any tn, we have ρ(g h t )τ(g)+ε, i.e., h t B ρ (g,τ(g)+ε). Since B ρ (g,τ(g)+ε) is ρ-closed and convex, we get K n B ρ (g,τ(g)+ε). Hence f B ρ (g,τ(g)+ε), i.e.,

ρ(gf)τ(g)+ε.
(3.1)

Since ε was taken arbitrarily greater than 0, we get ρ(gf)τ(g) for any gC. Let { f n } be a minimizing sequence for τ. If R=0, then, since { f n } is a minimizing sequence, we get lim n τ( f n )=R=0. Using (3.1), we can see that ρ( f n f)τ( f n ) for any n1. Hence { f n } is ρ-convergent to f. Since selection of f was independent of { f n }, it follows that any minimizing sequence is ρ-convergent to f if R=0. We can assume, therefore, that R>0. For any n1, let us set

d n = sup i , j n ρ( f i f j ).
(3.2)

We claim that { f n } is ρ-Cauchy. Assume to the contrary that this is not the case. Since the sequence { d n } is decreasing and { f n } is not ρ-Cauchy, we get d:= inf n 1 d n >0. Set ε= d 4 R >0. Let us fix arbitrary ν(0,R). Since lim n τ( f n )=R, there exists n 0 1 such that for any n n 0 , we have

τ( f n )R+ ν 2 .
(3.3)

Let n n 0 . By (3.2), there exists i n , j n 1 such that

ρ( f i n f j n )> d n d 2 d 2 =2Rε>(R+ν)ε.

Using the definition of τ and (3.3), we deduce the existence of M>0 such that

sup t M ρ( f i n h t )τ( f i n )+ ν 2 R+ν

and

sup t M ρ( f j n h t )τ( f j n )+ ν 2 R+ν.

Hence

ρ ( f i n + f j n 2 h t ) (R+ν) ( 1 δ ( R + ν , ε ) )

for any tM. Since ρ is (UUC), there exists η 1 (R,ε)>0 such that δ 1 (R+ν,ε) η 1 (R,ε). Hence

ρ ( f i n + f j n 2 h t ) (R+ν) ( 1 η ( R , ε ) )

for any tM. Hence

Rτ ( f i n + f j n 2 ) sup t M ρ ( f j n + f j n 2 h t ) (R+ν) ( 1 η ( R , ε ) ) <R.

Using the definition of R, we get

R(R+ν) ( 1 η ( R , ε ) )

for any ν(0,R). If we let ν0, we get RR(1 η 1 (R,ε)). This contradiction implies that { f n } is ρ-Cauchy. Since L ρ is ρ-complete, we deduce that { f n } is ρ-convergent as claimed.

In order to finish the proof of (ii), let us show that the ρ-limit of { f n } is independent of the minimizing sequence. Indeed, let { g n } be another minimizing sequence of τ. The previous proof will show that { g n } is also ρ-convergent. In order to prove that the ρ-limits of { f n } and { g n } are equal, let us show that lim n ρ( f n g n )=0. Assume not, i.e., lim n ρ( f n g n )0. Without loss of generality, we may assume that there exists d>0 such that ρ( f n g n )d for any n1. Set ε= d 2 R >0. Let ν(0,R). Since lim n τ( f n )= lim n τ( g n )=R, there exists n 0 1 such that for any n1, we have τ( f n )R+ ν 2 and τ( g n )R+ ν 2 . Fix n n 0 . Then

ρ( f n g n )d=2Rε>(R+ν)ε.

Using the definition of τ, we deduce the existence of M>0 such that

sup t M ρ( f n h t )τ( f n )+ ν 2 R+ν

and

sup t M ρ( g n h t )τ( g n )+ ν 2 R+ν.

Hence

ρ ( f n + g n 2 h t ) (R+ν) ( 1 δ ( R + ν , ε ) )

for any tM. Since ρ is (UUC), there exists η(R,ε)>0 such that δ(R+ν,ε)η(R,ε) for any ν>0. Hence

ρ ( f n + g n 2 h t ) (R+ν) ( 1 η ( R , ε ) )

for any tM. So

τ ( f n + g n 2 ) sup t M ρ ( f n + g n 2 h t ) (R+ν) ( 1 η ( R , ε ) ) .

Using the definition of R, we get

R(R+ν) ( 1 η ( R , ε ) )

for any ν(0,R). If we let ν0, we get RR(1η(R,ε)). This contradiction implies lim n ρ( f n g n )=0. The Fatou property will finally imply that

ρ(fg) lim inf n ρ( f n g n ),

where f is the ρ-limit of { f n } and g is the ρ-limit of { g n }. Hence ρ(fg)=0, i.e., f=g. □

Using Lemma 3.2, we are ready to prove our common fixed point result for asymptotic pointwise nonexpansive semigroups.

Theorem 3.2 Assume ρ is (UUC). Let C be a ρ-closed, ρ-bounded convex nonempty subset. Let F={ T t :t0} be an asymptotic pointwise nonexpansive semigroup on C. Thenhas a common fixed point, and the set F(F) of common fixed points is ρ-closed and convex.

Proof Let us fix fC and define the function

τ(g)= inf M > 0 ( sup t M ρ ( T t ( f ) g ) ) .

Since C is ρ-bounded, we have τ(g) diam ρ (C)<+ for any gC. Hence τ 0 =inf{τ(g):gC} exists and is finite. For any n1, there exists g n C, such that

τ 0 τ( g n )< τ 0 + 1 n .

Therefore, lim n τ( z n )= τ 0 , i.e., { g n } is a minimizing sequence for τ. By Lemma 3.2, there exists gC such that { g n } ρ-converges to g. Let us now prove that gF(F). Note that

ρ ( T s + t ( f ) T s ( h ) ) α s (h)ρ ( T t ( f ) h )

for s,t>0 and hC. Using the definition of τ, we get

τ ( T s ( h ) ) sup t + s M ρ ( T s + t ( f ) T s ( h ) ) α s (h) sup t M s ρ ( T t ( f ) h )

for any M>s, which implies

τ ( T s ( h ) ) α s (h)τ(h).
(3.4)

Since lim s α s ( g 1 )=1, there exists s 1 >0 such that for any s s 1 , we have α s ( g 1 )<1+1. Repeating this argument, one will find s 2 > s 1 +1 such that for any s s 2 , we have α s ( g 2 )<1+ 1 2 . By induction, we will construct a sequence { s n } of positive numbers such that s n + 1 < s n + 1 n and for any s s n , we have α s ( g n )<1+ 1 n . Let us fix t0. Then inequality (3.4) will imply

τ ( T s n + t ( g n ) ) α s n + t ( g n )τ( g n ) ( 1 + 1 n ) τ( g n )

for any n1. In particular, we get that { T s n + t ( g n )} is a minimizing sequence of τ. Therefore, Lemma 3.2 implies that { T s n + t ( g n )} ρ-converges to g for any t0. In particular, we have { T s n ( g n )} ρ-converges to g. Since

ρ ( T s n + t ( g n ) T t ( g ) ) α t (g)ρ ( T s n ( g n ) g ) ,

we get { T s n + t ( g n )} ρ-converges to T t (g). Finally, using

ρ ( T t ( g ) g 2 ) ρ ( T t ( g ) T s n + t ( g n ) ) +ρ ( T s n + t ( g n ) g ) ,

we get T t (g)=g. Since t was arbitrarily positive, we get gF(F), i.e., F(F) is not empty. Next, let us prove that F(F) is ρ-closed. Let { f n } in F(F) ρ-convergent to f. Since

ρ ( T s ( f n ) T s ( f ) ) α s (f)ρ( f n f)

for any n1 and s>0, we get { T s ( f n )} ρ-converges to T s (f). Since f n F(F), we get { T s ( f n )}={ f n }. In other words, { f n } ρ-converges to T s (f) and f. The uniqueness of the ρ-limit implies then that T s (f)=f for any s0, i.e., fF(F). Therefore, F(F) is ρ-closed. Let us finish the proof of Theorem 3.2 by showing that F(F) is convex. It is sufficient to show that

h= f + g 2 F(F)

for any f,gF(F). Without loss of generality, we will assume that fg. Let s>0. We have

ρ ( f T s ( h ) ) =ρ ( T s ( f ) T s ( h ) ) α s (f)ρ(fh)

and

ρ ( g T s ( h ) ) =ρ ( T s ( g ) T s ( h ) ) α s (g)ρ(gh).

Since ρ(fh)=ρ(gh)=ρ( f g 2 ) and

ρ ( f g 2 ) 1 2 ρ ( f T s ( h ) ) + 1 2 ρ ( g T s ( h ) ) ,

we conclude that

lim s ρ ( f T s ( h ) ) = lim s ρ ( g T s ( h ) ) =ρ ( f g 2 ) .

Similarly, we have

ρ ( f h + T s ( h ) 2 ) 1 2 ρ(fh)+ 1 2 ρ ( f T s ( h ) )

and

ρ ( g h + T s ( h ) 2 ) 1 2 ρ(gh)+ 1 2 ρ ( g T s ( h ) ) .

Since

ρ ( f g 2 ) 1 2 ρ ( f h + T s ( h ) 2 ) + 1 2 ρ ( g h + T s ( h ) 2 ) ,

we conclude that

lim s ρ ( f h + T s ( h ) 2 ) = lim s ρ ( g h + T s ( h ) 2 ) =ρ ( f g 2 ) .

Therefore, we have

lim s ρ ( f T s ( h ) ) = lim s ρ ( f h + T s ( h ) 2 ) =ρ(fh).

Lemma 3.1, applied to A t =f T s (h) and B t = T s (h)g, implies that ρ( A t B t )0. Hence

lim s ρ ( h T s ( h ) ) = lim s ρ ( A t B t 2 ) lim s ρ( A t B t )=0.

Clearly, we will get lim s ρ(h T s + t (h))=0, for any t0. Since

ρ ( T t ( h ) T s + t ( h ) ) α t (h)ρ ( h T s ( h ) ) ,

we get lim s ρ( T t (h) T s + t (h))=0. Finally, using the inequality

ρ ( h T t ( h ) 2 ) 1 2 ρ ( h T s + t ( h ) ) + 1 2 ρ ( T t ( h ) T s + t ( h ) ) ,

by letting s, we get T t (h)=h for any t0, i.e., hF(F). □

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Acknowledgements

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (247-006-D1433). The authors, therefore, acknowledge with thanks technical and financial support of DSR.

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Correspondence to Buthinah A Bin Dehaish.

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Bin Dehaish, B.A., Khamsi, M.A. & Kozlowski, W.M. Common fixed points for pointwise Lipschitzian semigroups in modular function spaces. Fixed Point Theory Appl 2013, 214 (2013). https://doi.org/10.1186/1687-1812-2013-214

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Keywords

  • fixed point
  • modular function space
  • nonexpansive mapping
  • Orlicz space
  • pointwise Lipschitzian mapping
  • pointwise nonexpansive mapping
  • semigroup
  • uniform convexity