- Open Access
A Pata-type fixed point theorem in modular spaces with application
© Paknazar et al.; licensee Springer. 2013
- Received: 28 November 2012
- Accepted: 17 May 2013
- Published: 31 October 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.
- fixed point
- modular spaces
- nonlinear integral equations
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.
- (a)A function is called a modular if for all .
if and only if ;
for every scalar α with ;
if and ,
- (b)If (iii) is replaced by we say that ρ is convex modular.
if and , ,
- (c)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 ).
is a norm called the Luxemburg norm in which is equivalent to the F-norm .
- (a)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 .
- (d)A subset is called ρ-bounded if
- (e)We say that ρ has the Fatou property if
- (f)ρ is said to satisfy the -condition if
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.
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.
For simplicity, we write instead of .
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 ).
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 )
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 .
Let be the space of all continuous mappings from into .
Proposition 1.10 (Taleb and Hanebaly )
is a modular space, and is a convex modular satisfying the Fatou property and the -condition;
is a -closed, convex subset of .
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 .
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.
Thus z is the of the iterate of under the action of T.
Then as , therefore .
Taking limit as , we get contradiction unless . □
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.
(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 .
for all .
where is a closed convex hull of B in .
2nd step. We show that is -complete and -bounded.
By Proposition 1.10, is a -closed subset of -complete space , hence is -complete too.
Let and be any division of .
for all , which implies (3.2).
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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