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A Pata-type fixed point theorem in modular spaces with application
Fixed Point Theory and Applications volume 2013, Article number: 239 (2013)
In this paper, we present a Pata-type fixed point theorem in modular spaces which generalizes and improves some old results. As an application, we study the existence of solutions of integral equations in modular function spaces.
MSC:47H10, 46A80, 45G10.
1 Introduction and preliminaries
In 1950 Nakano  introduced the theory of modular spaces in connection with the theory of ordered spaces. Musielak and Orlicz  in 1959 redefined and generalized it to obtain a generalization of the classical function spaces . Khamsi et al.  investigated the fixed point results in modular function spaces. There exists an extensive literature on the topic of the fixed point theory in modular spaces (see, for instance, [4–14]) and the papers referenced there.
Recently, Pata  improved the Banach principle. Using the idea of Pata, we prove a fixed point theorem in modular spaces. Then we show how our results generalize old ones. Also, we prepare an application of our main results to the existence of solutions of integral equations in Musielak-Orlicz spaces.
In the first place, we recall some basic notions and facts about modular spaces.
Definition 1.1 Let X be an arbitrary vector space over K ( or ℂ).
A function is called a modular if for all .
if and only if ;
for every scalar α with ;
if and ,
If (iii) is replaced by we say that ρ is convex modular.
if and , ,
A modular ρ defines a corresponding modular space, i.e., the vector space given by
Example 1.2 Let be a norm space, then is a convex modular on X. But the converse is not true.
In general the modular ρ does not behave as a norm or a distance because it is not subadditive. But one can associate to a modular the F-norm (see ).
Definition 1.3 The modular space can be equipped with the F-norm defined by
Namely, if ρ is convex, then the functional
is a norm called the Luxemburg norm in which is equivalent to the F-norm .
Definition 1.4 Let be a modular space.
A sequence in is said to be:
ρ-convergent to x if as .
ρ-Cauchy if as .
is ρ-complete if every ρ-Cauchy sequence is ρ-convergent.
A subset is said to be ρ-closed if with , then .
A subset is called ρ-bounded if
where is called the ρ-diameter of B.
We say that ρ has the Fatou property if
whenever , as .
ρ is said to satisfy the -condition if
It is easy to check that for every modular ρ and ,
for each with ,
Now we recall some basic concepts about modular function spaces as formulated by Kozlowski .
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 Σ such that for any and . Let us assume that there is an increasing sequence of sets such that .
In other words, the family plays the role of δ-ring of subsets of finite measure. By ℰ we denote the linear space of all simple functions with supports from .
By ℳ we denote the space of all measurable functions, i.e., all functions such that there exists a sequence , and for all . By we denote the characteristic function of the set A.
Definition 1.5 A function is called a function modular if
for any ;
whenever for any , and ;
is a σ-sub-additive measure for every ;
as α decreases to 0 for every , where ;
for any , is order continuous on , that is, if and decreases to ϕ.
The definition of ρ is then extended to by
For simplicity, we write instead of .
One can verify that the functional is a modular in the sense of Definition 1.1. The modular space determined by ρ will be called a modular function space and will be denoted by . Recall that
Example 1.6 (1) The Orlicz modular is defined for every measurable real function f by the formula
where μ denotes the Lebesgue measure in ℝ and is continuous. We also assume that if and only if and as .
The modular space induced by the Orlicz modular, is a modular function space and is called the Orlicz space. (2) The Musielak-Orlicz modular spaces (see ).
where μ is a σ-finite measure on Ω and satisfy the following:
is a continuous even function of u, which is non-decreasing for , such that , for and as ;
is a measurable function of ω for each ;
is a convex function of u for each .
It is easy to check that ρ is a convex modular function and the corresponding modular space is called the Musielak-Orlicz space and is denoted by .
In the following we give some notions which will be used in the next sections.
Definition 1.7 (Khamsi )
Let C be a subset of a modular function space . A mapping is called ρ-strict contraction if there exists such that
for all .
Theorem 1.8 (Khamsi )
Let C be a ρ-complete, ρ-bounded subset of and let be a ρ-strict contraction. Then T has a unique fixed point . Moreover, z is the ρ-limit of the iterate of any point in C under the action of T.
Definition 1.9 (Taleb and Hanebaly )
The function , where for all , is said to be continuous at if for and , then as .
If we consider the Musielak-Orlicz modular with -condition, then the continuity of u at is equivalent to
Let be the space of all continuous mappings from into .
Proposition 1.10 (Taleb and Hanebaly )
Suppose that the Musielak-Orlicz modular ρ satisfies -condition and is a ρ-closed and convex subset of . For , let for , then
is a modular space, and is a convex modular satisfying the Fatou property and the -condition;
is a -closed, convex subset of .
2 Main results
Let be a modular function space, C be a nonempty, ρ-complete and ρ-bounded subset of , be an arbitrary point in C and let be an increasing function vanishing with continuity at zero. Also, consider the vanishing sequence depending on , . Let be a mapping. For notational purposes, we define , and inductively by and .
Theorem 2.1 Let , and be fixed constants. If the inequality
is satisfied for every and every , then T has a unique fixed point which is the of the iterate of under the action of T.
Proof We first show existence. Let in (2.1), thus we get
for all . We construct a sequence such that for all . Now we claim is ρ-Cauchy sequence in C. By (2.1), (2.2) for all , we have
Let . Since C is ρ-bounded, M is finite and from (2.3) we have
Letting , we have . Keeping in mind that ψ is an increasing function,
Letting , we have from (2.4)
Taking limit as from both sides of (2.5), we get as . Then is ρ-Cauchy sequence in C. Since C is ρ-complete, there exists such that as . From (2.1) we get
Taking limit as afterwards as , we get
Then . On the other hand, by (2.5), we have
Thus z is the of the iterate of under the action of T.
To show uniqueness, we suppose that y is another fixed point of T. Then from (2.1) we have
Then as , therefore .
If for each strict inequality occurs in (2.6), then
Taking limit as , we get contradiction unless . □
Remark 2.2 Theorem 2.1 is stronger than Theorem 1.8. Indeed, with the hypothesis of Theorem 1.8, if for each and , we have
then by , and
for arbitrary , we get
is satisfied for every . Thus from Theorem 2.1, T has a unique fixed point z which is the of for an arbitrary point in C.
In this section, we study the existence of solution of the following integral equation:
(H1) is ρ-Lipschitz, i.e.,
(H2) B is a ρ-closed, ρ-bounded, convex subset of the Musielak-Orlicz space satisfying the -condition;
(H3) is fixed.
Theorem 3.1 Under the conditions (H1)-(H3), for all , integral equation (3.1) has a solution .
Proof Define the operator S on by
for all .
1st step. First we show that . Let and for all with as . We know u is ρ-continuous thus . From (H1) we get as , thus Tu is ρ-continuous at . By -condition Tu is -continuous at , therefore Su is -continuous at and consequently is ρ-continuous at . Also, we have
where is a closed convex hull of B in .
But B is convex and ρ-closed, then , hence
2nd step. We show that is -complete and -bounded.
By Proposition 1.10, is a -closed subset of -complete space , hence is -complete too.
Now let . By 1st step for all , then
3rd step. For , we have
Let and be any division of .
as , then
Using the Fatou property, we get
By the convexity of ρ, we have
It follows from (3.3) that
On the other hand,
Thus by (3.4), we have
since T is ρ-Lipschitz, we have
for all , which implies (3.2).
4th step. Let , , with
If we have
for all , and a constant K, then (3.2) implies that the inequality (2.1) is satisfied by . To this end, we define
Now imposing the conditions on F, which implies for all , we obtain
Therefore, from steps 1 to 4 and Theorem 2.1, we conclude the existence of a fixed point of S which is the solution of integral equation (3.1). □
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The authors declare that they have no competing interests.
All authors contributed equally. All authors read and approved the final manuscript.
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Paknazar, M., Eshaghi, M., Cho, Y.J. et al. A Pata-type fixed point theorem in modular spaces with application. Fixed Point Theory Appl 2013, 239 (2013). https://doi.org/10.1186/1687-1812-2013-239
- fixed point
- modular spaces
- nonlinear integral equations