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A Patatype fixed point theorem in modular spaces with application
Fixed Point Theory and Applications volume 2013, Article number: 239 (2013)
Abstract
In this paper, we present a Patatype 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 [1] introduced the theory of modular spaces in connection with the theory of ordered spaces. Musielak and Orlicz [2] in 1959 redefined and generalized it to obtain a generalization of the classical function spaces {L}^{p}. Khamsi et al. [3] 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 [15] 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 MusielakOrlicz 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 (=\mathbb{R} or ℂ).

(a)
A function \rho :X\to [0,+\mathrm{\infty}] is called a modular if for all x,y\in X.

(i)
\rho (x)=0 if and only if x=0;

(ii)
\rho (\alpha x)=\rho (x) for every scalar α with \alpha =1;

(iii)
\rho (\alpha x+\beta y)\le \rho (x)+\rho (y) if \alpha +\beta =1 and \alpha \ge 0, \beta \ge 0

(i)

(b)
If (iii) is replaced by we say that ρ is convex modular.

(iv)
\rho (\alpha x+\beta y)\le \alpha \rho (x)+\beta \rho (y) if \alpha +\beta =1 and \alpha \ge 0, \beta \ge 0,

(iv)

(c)
A modular ρ defines a corresponding modular space, i.e., the vector space {X}_{\rho} given by
{X}_{\rho}=\{x\in X:\rho (\lambda x)\to 0\text{as}\lambda \to 0\}.
Example 1.2 Let (X,\parallel \cdot \parallel ) be a norm space, then \parallel \cdot \parallel 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 Fnorm (see [16]).
Definition 1.3 The modular space {X}_{\rho} can be equipped with the Fnorm defined by
Namely, if ρ is convex, then the functional
is a norm called the Luxemburg norm in {X}_{\rho} which is equivalent to the Fnorm \cdot {}_{\rho}.
Definition 1.4 Let {X}_{\rho} be a modular space.

(a)
A sequence {\{{x}_{n}\}}_{n\in \mathbb{N}} in {X}_{\rho} is said to be:

(i)
ρconvergent to x if \rho ({x}_{n}x)\to 0 as n\to \mathrm{\infty}.

(ii)
ρCauchy if \rho ({x}_{n}{x}_{m})\to 0 as n,m\to \mathrm{\infty}.

(i)

(b)
{X}_{\rho} is ρcomplete if every ρCauchy sequence is ρconvergent.

(c)
A subset B\subseteq {X}_{\rho} is said to be ρclosed if {\{{x}_{n}\}}_{n\in \mathbb{N}}\subset B with {x}_{n}\to x, then x\in B.

(d)
A subset B\subseteq {X}_{\rho} is called ρbounded if
{\delta}_{\rho}(B)=sup\{\rho (xy):x,y\in B\}<\mathrm{\infty},
where {\delta}_{\rho}(B) is called the ρdiameter of B.

(e)
We say that ρ has the Fatou property if
\rho (xy)\le lim\hspace{0.17em}inf\rho ({x}_{n}{y}_{n}),
whenever \rho ({x}_{n}x)\to 0, \rho ({y}_{n}y)\to 0 as n\to \mathrm{\infty}.

(f)
ρ is said to satisfy the {\mathrm{\u25b3}}_{2}condition if
\rho ({x}_{n})\to 0\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}\rho (2{x}_{n})\to 0\phantom{\rule{1em}{0ex}}(\text{as}n\to \mathrm{\infty}).
It is easy to check that for every modular ρ and x,y\in {X}_{\rho},

(1)
\rho (\alpha x)\le \rho (\beta x) for each \alpha ,\beta \in {\mathbb{R}}^{+} with \alpha \le \beta,

(2)
\rho (x+y)\le \rho (2x)+\rho (2y).
Now we recall some basic concepts about modular function spaces as formulated by Kozlowski [17].
Let Ω be a nonempty set and let Σ be a nontrivial σalgebra of subsets of Ω. Let be a δring of subsets of Σ such that E\cap A\in \mathcal{P} for any E\in \mathcal{P} and A\in \mathrm{\Sigma}. Let us assume that there is an increasing sequence of sets {K}_{n}\in \mathcal{P} such that \mathrm{\Omega}=\bigcup {K}_{n}.
In other words, the family plays the role of δring 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 f:\mathrm{\Omega}\to \mathbb{R} such that there exists a sequence \{{g}_{n}\}\in \mathcal{E}, {g}_{n}\le f and {g}_{n}(w)\to f(w) for all w\in \mathrm{\Omega}. By {1}_{A} we denote the characteristic function of the set A.
Definition 1.5 A function \rho :\mathcal{E}\times \mathrm{\Sigma}\to [0,+\mathrm{\infty}] is called a function modular if

(i)
\rho (0,E)=0 for any E\in \mathrm{\Sigma};

(ii)
\rho (f,E)\le \rho (g,E) whenever f(w)\le g(w) for any w\in \mathrm{\Omega}, f,g\in \mathcal{E} and E\in \mathrm{\Sigma};

(iii)
\rho (f,\cdot ):\mathrm{\Sigma}\to [0,+\mathrm{\infty}] is a σsubadditive measure for every f\in \mathcal{E};

(iv)
\rho (\alpha ,A)\to 0 as α decreases to 0 for every A\in \mathcal{P}, where \rho (\alpha ,A)=\rho (\alpha {1}_{A},A);

(v)
for any \alpha >0, \rho (\alpha ,\cdot ) is order continuous on , that is, \rho (\alpha ,{A}_{n})\to 0 if \{{A}_{n}\}\in \mathcal{P} and decreases to ϕ.
The definition of ρ is then extended to f\in \mathcal{M} by
For simplicity, we write \rho (f) instead of \rho (f,\mathrm{\Omega}).
One can verify that the functional \rho :\mathcal{M}\to [0,+\mathrm{\infty}] 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 {L}_{\rho}. 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 \phi :\mathbb{R}\to [0,\mathrm{\infty}) is continuous. We also assume that \phi (u)=0 if and only if u=0 and \phi (t)\to \mathrm{\infty} as t\to \mathrm{\infty}.
The modular space induced by the Orlicz modular, is a modular function space and is called the Orlicz space. (2) The MusielakOrlicz modular spaces (see [2]).
Let
where μ is a σfinite measure on Ω and \phi :\mathrm{\Omega}\times \mathbb{R}\to [0,\mathrm{\infty}) satisfy the following:

(i)
\phi (\omega ,u) is a continuous even function of u, which is nondecreasing for u>0, such that \phi (\omega ,0)=0, \phi (\omega ,u)>0 for u\ne 0 and \phi (\omega ,u)\to \mathrm{\infty} as u\to \mathrm{\infty};

(ii)
\phi (\omega ,u) is a measurable function of ω for each u\in \mathbb{R};

(iii)
\phi (\omega ,u) is a convex function of u for each \omega \in \mathrm{\Omega}.
It is easy to check that ρ is a convex modular function and the corresponding modular space is called the MusielakOrlicz space and is denoted by {L}^{\phi}.
In the following we give some notions which will be used in the next sections.
Definition 1.7 (Khamsi [18])
Let C be a subset of a modular function space {L}_{\rho}. A mapping T:C\to C is called ρstrict contraction if there exists \lambda <1 such that
for all f,g\in C.
Theorem 1.8 (Khamsi [18])
Let C be a ρcomplete, ρbounded subset of {L}_{\rho} and let T:C\to C be a ρstrict contraction. Then T has a unique fixed point z\in C. Moreover, z is the ρlimit of the iterate of any point in C under the action of T.
Definition 1.9 (Taleb and Hanebaly [4])
The function u:I\to {L}^{\phi}, where I=[0,A] for all A>0, is said to be continuous at {t}_{0}\in I if for {t}_{n}\in I and {t}_{n}\to {t}_{0}, then \rho (u({t}_{n})u(t))\to 0 as n\to \mathrm{\infty}.
If we consider the MusielakOrlicz modular with {\mathrm{\u25b3}}_{2}condition, then the continuity of u at {t}_{0} is equivalent to
Let {C}^{\phi}=C(I,{L}^{\phi}) be the space of all continuous mappings from I=[0,A] into {L}^{\phi}.
Proposition 1.10 (Taleb and Hanebaly [4])
Suppose that the MusielakOrlicz modular ρ satisfies {\mathrm{\u25b3}}_{2}condition and B\subset {L}^{\phi} is a ρclosed and convex subset of {L}^{\phi}. For a\ge 0, let {\rho}_{a}(u)=sup\{{e}^{at}\rho (u(t)):t\in I\} for u\in {C}^{\phi}, then

(1)
({C}^{\phi},{\rho}_{a}) is a modular space, and {\rho}_{a} is a convex modular satisfying the Fatou property and the {\mathrm{\u25b3}}_{2}condition;

(2)
{C}^{\phi} is {\rho}_{a}complete;

(3)
{C}_{0}^{\phi}=C(I,B) is a {\rho}_{a}closed, convex subset of {C}^{\phi}.
2 Main results
Let {X}_{\rho} be a modular function space, C be a nonempty, ρcomplete and ρbounded subset of {X}_{\rho}, {x}_{0} be an arbitrary point in C and let \psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) be an increasing function vanishing with continuity at zero. Also, consider the vanishing sequence depending on \alpha \ge 1, {w}_{n}(\alpha )={(\frac{\alpha}{n})}^{\alpha}{\sum}_{k=1}^{n}\psi (\frac{\alpha}{k}). Let T:C\to C be a mapping. For notational purposes, we define {T}^{n}(x), x\in {X}_{\rho} and n\in \{0,1,2,\dots \} inductively by {T}^{0}(x)=x and {T}^{n+1}(x)=T({T}^{n}(x)).
Theorem 2.1 Let \alpha \ge 1, \beta >0 and k\ge 0 be fixed constants. If the inequality
is satisfied for every \u03f5\in [0,1] and every x,y\in C, then T has a unique fixed point z=T(z) which is the \rho \text{}lim of the iterate of {x}_{0} under the action of T.
Proof We first show existence. Let \u03f5=0 in (2.1), thus we get
for all x,y\in C. We construct a sequence {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}} such that {x}_{n}=T({x}_{n1}) for all n\in \mathbb{N}. Now we claim \{{x}_{n}\} is ρCauchy sequence in C. By (2.1), (2.2) for all m,n\in \mathbb{N}, we have
Let M:={({\delta}_{\rho}(C)+k)}^{\beta}. Since C is ρbounded, M is finite and from (2.3) we have
Letting \u03f5=1{(\frac{n}{n+1})}^{\alpha}, we have \u03f5\le \frac{\alpha}{n+1}. Keeping in mind that ψ is an increasing function,
Letting {r}_{n}:={n}^{\alpha}\rho ({x}_{n+m}{x}_{n}), we have from (2.4)
Therefore
Taking limit as n\to \mathrm{\infty} from both sides of (2.5), we get \rho ({x}_{n+m}{x}_{n})\to 0 as n\to \mathrm{\infty}. Then \{{x}_{n}\} is ρCauchy sequence in C. Since C is ρcomplete, there exists z\in C such that \rho ({x}_{n}z)\to 0 as n\to \mathrm{\infty}. From (2.1) we get
Taking limit as \u03f5\to 0 afterwards as n\to \mathrm{\infty}, we get
Then Tz=z. On the other hand, by (2.5), we have
Thus z is the \rho \text{}lim of the iterate of {x}_{0} 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 \rho (zy)\le {\u03f5}^{\alpha 1}\psi (\u03f5){(\rho (zy)+k)}^{\beta}\to 0 as \u03f5\to 0, therefore z=y.
If for each \u03f5\in (0,1] strict inequality occurs in (2.6), then
Taking limit as \u03f5\to 0, we get contradiction unless \rho (zy)=0. □
Remark 2.2 Theorem 2.1 is stronger than Theorem 1.8. Indeed, with the hypothesis of Theorem 1.8, if for each f,g\in C and \lambda \in (0,1), we have
then by \alpha =\beta =1, k=0 and
for arbitrary \gamma >0, we get
is satisfied for every \u03f5\in [0,1]. Thus from Theorem 2.1, T has a unique fixed point z which is the \rho \text{}lim of {T}^{n}{f}_{0} for an arbitrary point {f}_{0} in C.
3 Application
In this section, we study the existence of solution of the following integral equation:
where
(H_{1}) T:B\to B is ρLipschitz, i.e.,
(H_{2}) B is a ρclosed, ρbounded, convex subset of the MusielakOrlicz space {L}^{\phi} satisfying the {\mathrm{\u25b3}}_{2}condition;
(H_{3}) {f}_{0}\in B is fixed.
Theorem 3.1 Under the conditions (H_{1})(H_{3}), for all A>0, integral equation (3.1) has a solution u\in {C}^{\phi}=C([0,A],{L}^{\phi}).
Proof Define the operator S on {C}_{0}^{\phi} by
for all t\in I:=[0,A].
1st step. First we show that S:{C}_{0}^{\phi}\to {C}_{0}^{\phi}. Let u\in {C}_{0}^{\phi} and {t}_{n},{t}_{0}\in I for all n\in \mathbb{N} with {t}_{n}\to {t}_{0} as n\to \mathrm{\infty}. We know u is ρcontinuous thus \rho (u({t}_{n})u({t}_{0}))\to 0. From (H_{1}) we get \rho (Tu({t}_{n})Tu({t}_{0}))\to 0 as n\to \mathrm{\infty}, thus Tu is ρcontinuous at {t}_{0}. By {\mathrm{\u25b3}}_{2}condition Tu is {\parallel \cdot \parallel}_{\rho}continuous at {t}_{0}, therefore Su is {\parallel \cdot \parallel}_{\rho}continuous at {t}_{0} and consequently is ρcontinuous at {t}_{0}. Also, we have
where \overline{\mathit{co}}B is a closed convex hull of B in ({L}^{\phi},{\parallel \cdot \parallel}_{\rho}).
But B is convex and ρclosed, then \overline{\mathit{co}}B=B\subseteq {\overline{B}}_{\rho}=B, hence
2nd step. We show that {C}_{0}^{\phi} is {\rho}_{a}complete and {\rho}_{a}bounded.
By Proposition 1.10, {C}_{0}^{\phi} is a {\rho}_{a}closed subset of {\rho}_{a}complete space {C}^{\phi}, hence {C}_{0}^{\phi} is {\rho}_{a}complete too.
Now let u,v\in {C}_{0}^{\phi}. By 1st step u(t),v(t)\in B for all t\in I, then
therefore
3rd step. For u,v\in {C}_{0}^{\phi}, we have
Let w\in {C}^{\phi} and \{{t}_{0},{t}_{1},\dots ,{t}_{n}\} be any division of [0,t].
Now suppose
as n\to \mathrm{\infty}, then
By {\mathrm{\u25b3}}_{2}condition,
Using the Fatou property, we get
Furthermore,
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 ρLipschitz, we have
Therefore
for all t\in I, which implies (3.2).
4th step. Let \alpha =\beta =1, k=0, a>0 with
If we have
for all \gamma >0, \u03f5\in [0,1] and a constant K, then (3.2) implies that the inequality (2.1) is satisfied by \psi (\u03f5)=K{\u03f5}^{\gamma}. To this end, we define
Now imposing the conditions on F, which implies 0\le F(\u03f5) for all \u03f5\in [0,1], 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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Paknazar, M., Eshaghi, M., Cho, Y.J. et al. A Patatype fixed point theorem in modular spaces with application. Fixed Point Theory Appl 2013, 239 (2013). https://doi.org/10.1186/168718122013239
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DOI: https://doi.org/10.1186/168718122013239
Keywords
 fixed point
 modular spaces
 nonlinear integral equations