Geraghty-type theorems in modular metric spaces with an application to partial differential equation

Parin Chaipunya; Yeol Je Cho; Poom Kumam

doi:10.1186/1687-1847-2012-83

Research
Open Access

Geraghty-type theorems in modular metric spaces with an application to partial differential equation

Parin Chaipunya¹,
Yeol Je Cho²Email author and
Poom Kumam¹Email author

Advances in Difference Equations20122012:83

https://doi.org/10.1186/1687-1847-2012-83

Received: 3 May 2012
Accepted: 20 June 2012
Published: 20 June 2012

Abstract

In this article, we prove some fixed point theorems of Geraghty-type concerning the existence and uniqueness of fixed points under the setting of modular metric spaces. Also, we give an application of our main results to establish the existence and uniqueness of a solution to a nonhomogeneous linear parabolic partial differential equation in the last section.

Mathematics Subject Classification (2010): 47H10, 54H25, 35K15.

Keywords

modular metric space
ordered set
Geraghty's theorem
initial value problems
parabolic equation.

Introduction and preliminaries

Throughout this article, let ℝ⁺ denote the set of all positive real numbers and let ℝ₊ denote the set of all nonnegative real numbers.

Since the year 1922, Banach's contraction principle, due to its simplicity and applicability, has became a very popular tool in modern analysis, especially in nonlinear analysis including its applications to differential and integral equations, variational inequality theory, complementarity problems, equilibrium problems, minimization problems and many others. Also, many authors have improved, extended and generalized this contraction principle in several ways (see e.g. [1–10]).

In 1973, Geraghty [11] gave an interesting generalization of the contraction principle using the class

S

of the functions β: ℝ₊→ [0, 1) satisfying the following condition:

β (t_{n}) \to 1 implies t_{n} \to 0 .

Theorem 1.1. [11]Let (X, d) be a complete metric space and f be a self-mapping on X such that there exists

β \in S

satisfying

d (f x, f y) \leq β (d (x, y)) d (x, y)

(1.1)

for all x, y ∈ X. Then the sequence {x_n } defined by x_n = fx_{n- 1}for each n ≥ 1 converges to the unique fixed point of f in X.

Later, Amini-Harandini et al. [12] extended Geraghty's fixed point theorem to the setting of partially ordered metric spaces as follows:

Theorem 1.2. [12]Let (X, ⊑) be a partially ordered metric set and suppose that there exists a metric d in X such that (X, d) is a complete metric space. Let f be a nondecreasing self-mapping on X which satisfies the inequality (1.1) whenever x, y ∈ X are comparable. Assume that f is either continuous or

i f a n o n d e c r e a s i n g s e q u e n c e \{x_{n}\} c o n v e r g e s t o x_{*}, t h e n x_{n} ⊑ x_{*} f o r e a c h n \geq 1 .

(1.2)

If, additionally, the following condition is satisfied:

f o r a n y x, y \in X, t h e r e e x i s t s z \in X w h i c h i s c o m p a r a b l e t o b o t h x a n d y,

(1.3)

then the sequence {x_n } converges to the unique fixed point of f in X.

Let Ψ denote the class of functions ψ: ℝ₊→ ℝ₊ satisfying the following conditions:

(a)

ψ is nondecreasing.
(b)

ψ is continuous.
(c)

ψ(t) = 0 if and only if t = 0.

Using this class, Eshaghi Gordji et al. [13] extended the Theorem 1.2 as follows:

Theorem 1.3. [13]Let (X, ⊑) be a partially ordered metric set and suppose that there exists a metric d in X such that (X, d) is a complete metric space. Let f be a nondecreasing self-mapping on X such that there exists x₀ ∈ X with x₀ ⊑ fx₀. Suppose that there exist

β \in S

and ψ ∈ Ψ such that

ψ (d (f x, f y)) \leq β (ψ (d (x, y))) ψ (d (x, y)),

whenever x, y ∈ X are comparable. Assume also that the condition (1.2) holds. Then f has a fixed point.

On the other hand, in 2010, Chistyakov [14] introduced the notion of a modular metric space which is raised in an attempt to avoid some restrictions of the concept of a modular space (for the literature of a modular space, see e.g. [15–21] and references therein). Some of the early investigations on metric fixed point theory in this space refer to [22–24].

For the rest of this section, we present some notions and basic facts of modular metric spaces.

Definition 1.4. [14] Let X be a nonempty set. A function ω: ℝ⁺ × X × X → ℝ₊ ∪ {∞} is said to be a metric modular on X if, for all x, y, z ∈ X, the following conditions hold:

(a)

ω_λ (x, y) = 0 for all λ > 0 if and only if x = y.
(b)

ω_λ (x, y) = ω_λ (y, x) for all λ > 0.
(c)

ω _λ+μ(x, y) ≤ ω_λ (x, z) + ω_μ (z, y) for all λ, μ > 0.

For any x_ι ∈ X, the set X_ω (x_ι ) = {x ∈ X: lim_λ→∞ ω_λ(x, x_ι ) = 0} is called a modular metric space generated by x_ι and induced by ω. If its generator x_ι does not play any role in the situation (that is, X_ω is independent of generators), we write X_ω instead of X_ω (x_ι ).

Observe that a metric modular ω on X is nonincreasing with respect to λ > 0. We can simply show this assertion using the condition (c). For any x, y ∈ X and 0 < μ < λ, we have

ω_{λ} (x, y) \leq ω_{λ - μ} (x, x) + ω_{μ} (x, y) = ω_{μ} (x, y) .

(1.4)

For any x, y ∈ X and λ > 0, we set

ω_{λ^{+}} (x, y) : = lim_{ϵ ↓ 0} ω_{λ + ϵ} (x, y), ω_{λ^{-}} (x, y) : = lim_{ϵ ↓ 0} ω_{λ - ϵ} (x, y) .

Consequently, from (1.4), it follows that

ω_{λ^{+}} (x, y) \leq ω_{λ} (x, y) \leq ω_{λ^{-}} (x, y) .

For any x, y ∈ X, if a metric modular ω on X possesses a finite value and ω_λ (x, y) = ω_μ (x, y) for all λ, μ > 0, then d(x, y): = ω_λ (x, y) is a metric on X.

Later, Chaipunya et al. [23] has altered the notion of convergent and Cauchy sequences in modular metric spaces under the direction of Mongkolkeha et al. [24].

Definition 1.5. [23, 24] Let X_ω be a modular metric space and {x_n } be a sequence in X_ω .

(1)

A point x ∈ X_ω is called a limit of {x_n } if, for each λ, ϵ > 0, there exists n ₀ ∈ ℕ such that ω_λ (x_n , x) < ϵ for all n ≥ n ₀. A sequence that has a limit is said to be convergent (or converges to x), which is written as lim _n→∞ x_n = x.
(2)

A sequence {x_n } in X_ω is said to be a Cauchy sequence if, for each λ, ϵ > 0, there exists n ₀ ∈ ℕ such that ω_λ (x_n , x_m ) < ϵ for all m, n ≥ n ₀.
(3)

If every Cauchy sequences in X converges, X is said to be complete.

In this article, we prove a generalization of Geraghty's theorem which also improves the result of Eshagi Gordji et al. [13] under the influence of a modular metric space. An application to partial differential equation is also provided.

Main results

Before stating our main results, we first introduce the following classes for a more convenience of usage.

For each n ∈ ℕ, let

S_{n}

denote the class of n-tuples of functions (β₁, β₂, ..., β_n ), where for each i ∈ {1, 2, ..., n}, β_i : ℝ₊ ∪ {∞} → [0, 1) and the following implication holds:

β (t_{k}) : = β_{1} (t_{k}) + β_{2} (t_{k}) + \dots + β_{n} (t_{k}) \to 1 implies t_{k} \to 0 .

Actually, Geraghty's class

S

is equivalent to the class

S_{1}

when ∞ is not considered. It follows that, for each m ∈ {1, 2, ..., n}, if

(β_{1}, β_{2}, \dots, β_{m}) \in S_{m}

, then

(β_{1}, β_{2}, \dots, β_{m}, \underset{n - m entries}{\underset{⏟}{θ, θ, \dots, θ}}) \in S_{n}

, where θ denotes the zero function. Also, note that, if

\underset{n entries}{\underset{⏟}{(β, β, \dots, β)}} \in S_{n}

, then we also have the following:

β (t_{k}) \to \frac{1}{n} implies t_{k} \to 0 .

Besides, if $(β_{1}, β_{2} \dots, β_{n}) \in S_{n}$ , then $π ((β_{1}, β_{2} \dots, β_{n})) \in S_{n}$ , where π((β₁, β₂ ..., β_n )) is a permutation of (β₁, β₂ ..., β_n ). It is also important to know that, if $(β_{1}, β_{2} \dots, β_{n}) \in S_{n}$ , then $(β_{n_{1}}, β_{n_{2}}, \dots, β_{n_{m}}) \in S_{m}$ for each m ∈ {1, 2, ..., n}, where each $β_{n_{i}}$ is selected from {β₁, β₂, ..., β_n } and $β_{n_{i}} \neq β_{n_{j}}$ for all i, j ∈ {1, 2, ..., m}.

Let

\bar{Ψ}

denote the class of functions ψ: ℝ₊ ∪ {∞} → ℝ₊ ∪ {∞} satisfying the following conditions:

(a)

If 0 < t < ∞, then ψ(t) < ∞.
(b)

$ψ |_{ℝ_{+}} \in Ψ$ .

Now, we are ready to give our main results in this article.

Theorem 2.1. Let X_ω be a complete modular metric space with a partial ordering ⊑ and f be a self-mapping on X_ω such that, for each λ > 0, there exists η(λ) ∈ (0, λ) such that

\begin{gathered} ψ (ω_{λ} (f x, f y)) \leq α (ψ (ω_{λ} (x, y))) ψ (ω_{λ + η (λ)} (x, y)) + β (ψ (ω_{λ} (x, y))) ψ (ω_{λ} (x, f x)) \\ + γ (ψ (ω_{λ} (x, y))) ψ (ω_{λ} (y, f y)), \end{gathered}

(2.1)

where

ψ \in \bar{Ψ}

and

(α, β, γ) \in S_{3}

with α(t) + 2 max{sup_{t≥ 0}β(t), sup_{t≥ 0}γ(t)} < 1. Assume also that the condition (1.2) holds. If there exists x₀ ∈ X_ω such that ω_λ (x₀, fx₀) < ∞ for all λ > 0, then the following hold:

(1)

f has a fixed point x_∞ ∈ X_ω .
(2)

The sequence {fⁿx ₀} converges to x_∞ .

Proof. It is clear that the sequence {fⁿx₀} is nondecreasing. Assume that, for each n ≥ 1, there exists λ_n > 0 such that

ω_{λ_{n}} (f^{n} x_{0}, f^{n + 1} x_{0}) \neq 0

. Otherwise, the proof is complete. For each n ≥ 1, if 0 < λ ≤ λ_n , then we also have ω_λ (fⁿx₀, fⁿ⁺¹x₀) ≠ 0. Since fⁿx₀⊑fⁿ⁺¹x₀, for any 0 < λ ≤ λ_n , we have

\begin{aligned} ψ (ω_{λ_{n}} (f^{n} x_{0}, f^{n + 1} x_{0})) & \leq ψ (ω_{λ} (f^{n} x_{0}, f^{n + 1} x_{0})) \\ \leq α (ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0}))) ψ (ω_{λ + η (λ)} (f^{n - 1} x_{0}, f^{n} x_{0})) \\ + β (ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0}))) ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0})) \\ + γ (ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0}))) ψ (ω_{λ} (f^{n} x_{0}, f^{n + 1} x_{0})) \\ \leq α (ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0}))) ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0})) \\ + β (ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0}))) ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0})) \\ + γ (ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0}))) ψ (ω_{λ} (f^{n} x_{0}, f^{n + 1} x_{0})), \end{aligned}

which implies that

\begin{aligned} ψ (ω_{λ} (f^{n} x_{0}, f^{n + 1} x_{0})) & \leq \frac{α (ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0}))) + β (ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0})))}{1 - γ (ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0})))} ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0})) \\ \leq ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0})) \\ ⋮ \\ \leq ψ (ω_{λ} (x_{0}, f x_{0})) \\ < \infty . \end{aligned}

Therefore, {ψ(ω_λ (fⁿx₀, fⁿ⁺¹x₀))} is nonincreasing and bounded below. So, the sequence converges to some number r ≥ 0. Assume r > 0. Observe that

\begin{aligned} ψ (ω_{λ} (f^{n} x_{0}, f^{n + 1} x_{0})) & \leq [α (ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0}))) + β (ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0}))) \\ + γ (ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0})))] ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0})) . \end{aligned}

Taking n → ∞, we have

\begin{aligned} 1 & \leq \underset{n \to \infty}{lim inf} [α (ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0}))) + β (ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0}))) \\ + γ (ψ (ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0})))] . \end{aligned}

So, we have lim _n→∞ ψ(ω_λ (f^n-1x₀, fⁿx₀)) = 0 and hence

lim_{n \to \infty} ω_{λ} (f^{n - 1} x_{0}, f^{n} x_{0}) = 0,

which is a contradiction of our assumption. Therefore, lim _n→∞ ψ(ω_λ (fⁿx₀, fⁿ⁺¹x₀)) = 0 and so, we have lim_n→∞ ω_λ(fⁿx₀, fⁿ⁺¹x₀) = 0. Moreover, we have lim_n→∞ω_λ (fⁿx₀, fⁿ⁺¹x₀) = 0 for all λ > 0.

Next, we show that {fⁿx₀} is a Cauchy sequence. Assume the contrary. So, there exists λ₀, ϵ₀> 0 for which we can define two subsequences

{f^{m_{k}} x_{0}}

and

{f^{n_{k}} x_{0}}

of the sequence {fⁿx₀} such that, for any n_k > m_k > k,

ω_{λ_{0}} (f^{m_{k}} x_{0}, f^{n_{k}} x_{0}) \geq ϵ_{0}

, but

ω_{λ_{0}} (f^{m_{k}} x_{0}, f^{n_{k} - 1} x_{0}) < ϵ_{0}

. Now, since

f^{m_{k}} x_{0} ⊑ f^{n_{k}} x_{0}

, we observe that

\begin{aligned} ψ (ϵ_{0}) & \leq ψ (ω_{λ_{0}} (f^{m_{k}} x_{0}, f^{n_{k}} x_{0})) \\ \leq α (ψ (ω_{λ_{0}} (f^{m_{k} - 1} x_{0}, f^{n_{k} - 1} x_{0}))) ψ (ω_{λ_{0} + η (λ_{0})} (f^{m_{k} - 1} x_{0}, f^{n_{k} - 1} x_{0})) \\ + β (ψ (ω_{λ_{0}} (f^{m_{k} - 1} x_{0}, f^{n_{k} - 1} x_{0}))) ψ (ω_{λ_{0}} (f^{m_{k} - 1} x_{0}, f^{m_{k}} x_{0})) \\ + γ (ψ (ω_{λ_{0}} (f^{m_{k} - 1} x_{0}, f^{n_{k} - 1} x_{0}))) ψ (ω_{λ_{0}} (f^{n_{k} - 1} x_{0}, f^{n_{k}} x_{0})) \\ \leq ψ (ω_{η (λ_{0})} (f^{m_{k} - 1} x_{0}, f^{m_{k}} x_{0}) + ω_{λ_{0}} (f^{m_{k}} x_{0}, f^{n_{k} - 1} x_{0})) \\ + ψ (ω_{λ_{0}} (f^{m_{k} - 1} x_{0}, f^{m_{k}} x_{0})) + ψ (ω_{λ_{0}} (f^{n_{k} - 1} x_{0}, f^{n_{k}} x_{0})) \\ \leq ψ (ω_{η (λ_{0})} (f^{m_{k} - 1} x_{0}, f^{m_{k}} x_{0}) + ϵ_{0}) + ψ (ω_{λ_{0}} (f^{m_{k} - 1} x_{0}, f^{m_{k}} x_{0})) \\ + ψ (ω_{λ_{0}} (f^{n_{k} - 1} x_{0}, f^{n_{k}} x_{0})) . \end{aligned}

Letting k → ∞, we obtain that

{lim}_{k \to \infty} ψ (ω_{λ_{0}} (f^{m_{k}} x_{0}, f^{n_{k}} x_{0})) = ψ (ϵ_{0})

. So, we have

lim_{k \to \infty} ω_{λ_{0}} (f^{m_{k}} x_{0}, f^{n_{k}} x_{0}) = ϵ_{0} .

Observe again that

\begin{aligned} ψ (ω_{λ_{0}} (f^{m_{k}} x_{0}, f^{n_{k}} x_{0})) & \leq ψ (ω_{λ_{0} + η (λ_{0})} (f^{m_{k} - 1} x_{0}, f^{n_{k} - 1} x_{0})) + ψ (ω_{λ_{0}} (f^{m_{k} - 1} x_{0}, f^{m_{k}} x_{0})) \\ + ψ (ω_{λ_{0}} (f^{n_{k} - 1} x_{0}, f^{n_{k}} x_{0})) \\ \leq ψ (ω_{\frac{η (λ_{0})}{2}} (f^{m_{k} - 1} x_{0}, f^{m_{k}} x_{0}) + ω_{λ_{0}} (f^{m_{k}} x_{0}, f^{n_{k}} x_{0}) \\ + ω_{\frac{η (λ_{0})}{2}} (f^{n_{k}} x_{0}, f^{n_{k} - 1} x_{0})) + ψ (ω_{λ_{0}} (f^{m_{k} - 1} x_{0}, f^{m_{k}} x_{0})) \\ + ψ (ω_{λ_{0}} (f^{n_{k} - 1} x_{0}, f^{n_{k}} x_{0})) . \end{aligned}

Letting k → ∞, we deduce that

{lim}_{k \to \infty} ψ (ω_{λ_{0}} (f^{m_{k} - 1} x_{0}, f^{n_{k} - 1} x_{0})) = ψ (ϵ_{0})

. Similarly, we have

lim_{k \to \infty} ω_{λ_{0}} (f^{m_{k} - 1} x_{0}, f^{n_{k} - 1} x_{0}) = ϵ_{0} .

Thus, it follows that

1 \leq \underset{k \to \infty}{lim inf} α (ψ (ω_{λ_{0}} (f^{m_{k} - 1} x_{0}, f^{n_{k} - 1} x_{0}))) .

Therefore, we conclude that

{lim}_{k \to \infty} ψ (ω_{λ_{0}} (f^{m_{k} - 1} x_{0}, f^{n_{k} - 1} x_{0})) = 0

, which implies that

lim_{k \to \infty} ω_{λ_{0}} (f^{m_{k} - 1} x_{0}, f^{n_{k} - 1} x_{0}) = 0 .

This is a contradiction. Therefore, it follows that {fⁿx₀} is a Cauchy sequence. Due to the completeness of X_ω , {fⁿx₀} converges to some point x_∞ ∈ X_ω .

Now, we show that x_∞ is a fixed point of f. Let λ > 0 be arbitrary. By virtue of the condition (1.2), we consider that

ψ (ω_{λ} (f^{n + 1} x_{0}, f x_{\infty}) \leq ψ (ω_{λ} (f^{n} x_{0}, x_{\infty})) + ψ (ω_{λ} (f^{n} x_{0}, f^{n + 1} x_{0})) .

Letting n → ∞, we obtain that ψ(ω_λ (x_∞ , fx_∞ ) ≤ 0 for all λ > 0. Therefore, x_∞ is a fixed point of f. ■

Theorem 2.2. Additional to the Theorem 2.1, if ψ is subadditive and the following condition holds:

\begin{gathered} f o r a n y x, y \in X_{ω}, t h e r e e x i s t s w \in X_{ω} w i t h w ⊑ f w a n d ω_{λ} (w, f w) < \infty f o r a l l λ > 0 \\ s u c h t h a t w i s c o m p a r a b l e t o b o t h x a n d y, \end{gathered}

(2.2)

then the fixed point in Theorem 2.1 is unique.

Proof. By Theorem 2.1, we know that f has a fixed point x_∞ ∈ X_ω . Assume that y_∞ ∈ X_ω is also another fixed point of f. Thus, we can find w ∈ X_ω with w ⊑ fw and comparable to both x_∞ and y_∞ . It follows that fⁿw is comparable with both x_∞ and y_∞ for each n ∈ ℕ. Observe that, for any λ > 0,

\begin{aligned} ψ (ω_{λ} (f^{n + 1} w, x_{\infty})) & = ψ (ω_{λ} (f^{n + 1} w, f x_{\infty})) \\ \leq α (ψ (ω_{λ} (f^{n} w, x_{\infty}))) ψ (ω_{λ} (f^{n} w, x_{\infty})) \\ + β (ψ (ω_{λ} (f^{n} w, x_{\infty}))) ψ (ω_{λ} (f^{n} w, f^{n + 1} w)) \\ \leq α (ψ (ω_{λ} (f^{n} w, x_{\infty}))) ψ (ω_{λ} (f^{n} w, x_{\infty})) \\ + β (ψ (ω_{λ} (f^{n} w, x_{\infty}))) ψ (ω_{λ} (f^{n} w, x_{\infty})) \\ + β (ψ (ω_{λ} (f^{n} w, x_{\infty}))) ψ (ω_{λ} (x_{\infty}, f^{n + 1} w)) . \end{aligned}

(2.3)

Therefore, without loss of generality, we have

\begin{aligned} ψ (ω_{λ} (f^{n + 1} w, x_{\infty})) & \leq \frac{α (ψ (ω_{λ} (f^{n} w, x_{\infty}))) + β (ψ (ω_{λ} (f^{n} w, x_{\infty})))}{1 - β (ψ (ω_{λ} (f^{n} w, x_{\infty})))} ψ (ω_{λ} (f^{n} w, x_{\infty})) \\ \leq ψ (ω_{λ} (f^{n} w, x_{\infty})) \\ ⋮ \\ \leq ψ (ω_{λ} (w, x_{\infty})) \\ < \infty . \end{aligned}

Therefore, {ψ(ω_λ (fⁿw, x_∞ ))} is nonincreasing and bounded below. So, it converges to some real number h ≥ 0. Assume that h > 0. According to the proof of Theorem 2.1, we know that lim_n→∞ ω_λ(fⁿw, fⁿ⁺¹w) = 0 for all λ > 0. Thus, letting n → ∞ in the inequality (2.3), we have

1 \leq \underset{n \to \infty}{lim inf} α (ψ (ω_{λ} (f^{n} w, x_{\infty}))) .

Thus, we have {fⁿw} converges to x_∞ . Similarly, we obtain that {fⁿw } converges also to y_∞ . Since the limit is unique, we have x_∞ = y_∞ . This contradicts our assumption. Therefore, the theorem is proved. ■

Corollary 2.3. Additional to Theorem 2.1, if X_ω is totally ordered, then the fixed point in Theorem 2.1 is unique.

Proof. Since X_ω is totally ordered, the condition (2.2) is satisfied. Thus, applying Theorem 2.2, we obtain the result. ■

The following two corollaries nicely broaden the results in [24] (see Theorems 3.2 and 3.6 [24]).

Corollary 2.4. Let X_ω be a complete modular metric space with a partial ordering ⊑ and f be a self-mapping on X_ω such that, for any λ > 0, there exists η(λ) ∈ (0, λ) such that

ψ (ω_{λ} (f x, f y)) \leq α (ψ (ω_{λ} (x, y))) ψ (ω_{λ + η (λ)} (x, y)),

where $α \in S$ and $ψ \in \bar{Ψ}$ . Assume also that f is continuous or the condition (1.2) holds. Then f has a fixed point in X_ω. Moreover, if the condition (2.2) is satisfied, the fixed point is unique.

Proof. Since $α \in S$ , we have $(α, θ, θ) \in S_{3}$ . Thus, apply Theorems 2.1 and 2.2, we have the conclusion. ■

Corollary 2.5. Let X_ω be a complete modular metric space with a partial ordering ⊑ and f be a self-mapping on X_ω such that, for any λ > 0, there exist ζ(λ), μ(λ) ∈ (0, λ) such that

ψ (ω_{λ} (f x, f y)) \leq β (ψ (ω_{λ} (x, y))) ψ (ω_{λ} (x, f x)) + γ (ψ (ω_{λ} (x, y))) ψ (ω_{λ} (y, f y)),

where $ψ \in \bar{Ψ}$ and $(β, γ) \in S_{2}$ with max{sup_{t≥ 0}β(t), sup_{t≥ 0}γ(t)} < 1. Assume also that f is continuous or that the condition (1.2) holds. Then f has a fixed point in X_ω. Moreover, if the condition (2.2) is satisfied, the fixed point is unique.

Proof. Since $(β, γ) \in S_{2}$ , we have $(θ, β, γ) \in S_{3}$ . Thus, apply Theorems 2.1 and 2.2, we have the conclusion. ■

Applications

In this section, we give an application of our theorems to establish the existence and uniqueness of a solution to a nonhomogeneous linear parabolic partial differential equation satisfying a given initial condition.

Consider the following initial value problem

\{\begin{matrix} u_{t} (x, t) = u_{x x} (x, t) + F (x, t, u (x, t), u_{x} (x, t)), & - \infty < x < \infty, 0 < t \leq T \\ u (x, 0) = φ (x) \geq 0, & - \infty < x < \infty, \end{matrix}

(3.1)

where we assume φ to be continuously differentiable such that φ and φ' are bounded and F is continuous.

By a solution of the system (3.1), we meant a function u ≡ u(x, t) defined on ℝ × I, where I: = [0, T], satisfying the following conditions:

(a)

u, u_t , u_x , u_xx ∈ C(ℝ × I).
(b)

u and u_x are bounded in ℝ × I.
(c)

u_t (x, t) = u_xx (x, t) + F(x, t, u(x, t), u_x (x, t)) for all (x, t) ∈ ℝ × I.
(d)

u(x, 0) = φ(x) for all x ∈ ℝ.

Now, we consider the following space:

Ω : = {u (x, t) : u, u_{x} \in C (ℝ \times I) and || u | | < \infty},

where

∥u∥ : = sup_{x \in ℝ, t \in I} ǀ u (x, t) ǀ + sup_{x \in ℝ, t \in I} ǀ u_{x} (x, t) ǀ .

Obviously, the function ω: ℝ⁺ × Ω × Ω → ℝ₊ given by

ω_{λ} (x, y) : = \frac{1}{1 + λ} | | u - v | |

is a metric modular on Ω. Clearly, the set Ω _ω is a complete modular metric space independent of generators. Define a partial ordering ⊑ on Ω _ω by

u, v \in Ω_{ω}, u ⊑ v \Leftrightarrow u (x, t) \leq v (x, t) and u_{x} (x, t) \leq v_{x} (x, t) at each (x, t) \in ℝ \times I .

Taking a nondecreasing sequence {u_n } in Ω _ω converging to u ∈ Ω_ω. For any (x, t) ∈ ℝ × I,

we have

u_{1} (x, t) \leq u_{2} (x, t) \leq \dots \leq u_{n} (x, t) \leq \dots

and

{(u_{1})}_{x} (x, t) \leq {(u_{2})}_{x} (x, t) \leq \dots \leq {(u_{n})}_{x} (x, t) \leq \dots .

Since the sequences {u_n (x, t)} and {(u_n )_x(x, t)} converges to u(x, t) and u_x (x, t), respectively, it follows that, for any (x, t)∈ℝ × I,

u_{n} (x, t) \leq u (x, t) and {(u_{n})}_{x} (x, t) \leq u_{x} (x, t)

for all n ≥ 1. Therefore, u_n ⊑ u for all n ≥ 1. So, the space Ω _ω satisfies the condition (1.2).

Theorem 3.1. Consider the problem (3.1) and assume the following:

(1)

For any c > 0 with |s| < c and |p| < c, the function F (x, t, s, p) is uniformly Hölder continuous in X and t for each compact subset of ℝ × I.
(2)

There exists a constant $c_{F} \leq {(T + 2 π^{- \frac{1}{2}} T^{\frac{1}{2}})}^{- 1}$ such that, for any λ > 0, there exists η(λ) ∈ (0, λ) such that
$\begin{aligned} 0 & \leq \frac{1}{1 + λ} [F (x, t, s_{2}, p_{2}) - F (x, t, s_{1}, p_{1})] \\ \leq c_{F} [\frac{1}{1 + λ + η (λ)} ρ (Ξ (\frac{s_{2} - s_{1} + p_{2} - p_{1}}{1 + λ})) + \frac{1}{1 + λ} σ (Ξ (\frac{s_{2} - s_{1} + p_{2} - p_{1}}{1 + λ}))] \end{aligned}$

for all (s₁, p₁), (s₂, p₂) ∈ ℝ × ℝ with s₁≤ s₂and p₁≤ p₂, where

Ξ \in \bar{Ψ}

is sublinear with Ξ(x) ≤ t and ρ, σ are nondecreasing functions on ℝ₊such that ρ(t) < (1 -k)t and σ(t) < (1 -k)kt for all t > 0 and for some fixed k ∈ (0, 1).

(3)

The two functions Γ, ϒ: ℝ₊ → [0, 1) given by
$Γ (t) = \{\begin{matrix} 0 & if t = 0, \\ \frac{ρ (t)}{(1 - k) t} & if t > 0, \end{matrix} ϒ (t) = \{\begin{matrix} 0 & if t = 0, \\ \frac{σ (t)}{(1 - k) t} & if t > 0, \end{matrix}$

are such that

(Γ, ϒ, ϒ) \in S_{3}

, Γ + 2ϒ < 1.

(4)

$F (x, t, s, 0) \geq \frac{s}{\int_{0}^{t} \int_{\infty}^{\infty} k (x - ξ, t - τ) d ξ d τ} f o r a l l s \geq 0 .$
(5)

F is bounded for bounded s and p.

Then, the existence and uniqueness of the solution of the system (3.1) is affirmative.

It is essential to note that the problem (3.1) is equivalent (under the assumption of Theorem 3.1) to the integral equation:

u (x, t) = \int_{- \infty}^{\infty} k (x - ξ, t) φ (ξ) d ξ + \int_{0}^{t} \int_{- \infty}^{\infty} k (x - ξ, t - τ) F (ξ, τ, u (ξ, τ), u_{x} (ξ, τ)) d ξ d τ

(3.2)

for all x ∈ ℝ and 0 < t ≤ T, where

k (x, t) : = \frac{1}{\sqrt{4 π t}} e^{- \frac{x^{2}}{4 t}}

for all x ∈ ℝ and t > 0. The system (3.1) possesses a unique solution if and only if the equation (3.2) possesses a unique solution u such that u and u_x are both continuous and bounded for all x ∈ ℝ and 0 < t ≤ T.

Define a mapping Λ: Ω _ω → Ω _ω by

(Λ u) (x, t) : = \int_{- \infty}^{\infty} k (x - ξ, t) φ (ξ) d ξ + \int_{0}^{t} \int_{- \infty}^{\infty} k (x - ξ, t - τ) F (ξ, τ, u (ξ, τ), u_{x} (ξ, τ)) d ξ d τ

for all (x, t) ℝ × I. Then the problem of finding the solution to the equation (3.2) is equivalent to the problem of finding the fixed point of Λ.

Proof (Theorem 3.1). It is easy to see that the mapping Λ is nondecreasing by the definition. Let u, v ∈ Ω _ω with u ⊑ v. Suppose that u ≠ v. Besides, we have

\begin{aligned} \frac{1}{1 + λ} & |(Λ v) (x, t) - (Λ u) (x, t)| \\ \leq \frac{1}{1 + λ} \int_{0}^{t} \int_{- \infty}^{\infty} k (x - ξ, t - τ) | F (ξ, τ, v (ξ, τ), v_{x} (ξ, τ)) - F (ξ, τ, u (ξ, τ), u_{x} (ξ, τ)) | d ξ d τ \\ \leq \int_{0}^{t} \int_{- \infty}^{\infty} k (x - ξ, t - τ) c_{F} [\frac{1}{1 + λ + η (λ)} ρ (Ξ (\frac{1}{1 + λ} (v (ξ, τ) - u (ξ, τ) + v_{x} (ξ, τ) - u_{x} (ξ, τ)))) \\ + \frac{1}{1 + λ} σ (Ξ (\frac{1}{1 + λ} (v (ξ, τ) - u (ξ, τ) + v_{x} (ξ, τ) - u_{x} (ξ, τ))))] d ξ d τ \\ \leq c_{F} [\frac{1}{1 + λ + η (λ)} ρ (Ξ (ω_{λ} (u, v))) + \frac{1}{1 + λ} σ (Ξ (ω_{λ} (u, v)))] \int_{0}^{t} \int_{- \infty}^{\infty} k (x - ξ, t - τ) d ξ d τ \\ \leq c_{F} [\frac{1}{1 + λ + η (λ)} ρ (Ξ (ω_{λ} (u, v))) + \frac{1}{1 + λ} σ (Ξ (ω_{λ} (u, v)))] T . \end{aligned}

(3.3)

Similarly, we have

\begin{aligned} \frac{1}{1 + λ} & |{(Λ v)}_{x} (x, t) - {(Λ u)}_{x} (x, t)| \\ \leq c_{F} [\frac{1}{1 + λ + η (λ)} ρ (Ξ (ω_{λ} (u, v))) + \frac{1}{1 + λ} σ (Ξ (ω_{λ} (u, v)))] \int_{0}^{t} \int_{- \infty}^{\infty} | k_{x} (x - ξ, t - τ) | d ξ d τ \\ \leq 2 π^{- \frac{1}{2}} T^{- \frac{1}{2}} c_{F} [\frac{1}{1 + λ + η (λ)} ρ (Ξ (ω_{λ} (u, v))) + \frac{1}{1 + λ} σ (Ξ (ω_{λ} (u, v)))] . \end{aligned}

(3.4)

Note that by (2), we have sup_{t≥ 0}ϒ(t) ≤ k < 1. Together with (3.3) and (3.4), we obtain

\begin{aligned} ω_{λ} (Λ u, Λ v) & \leq (T + 2 π^{- \frac{1}{2}} T^{\frac{1}{2}}) c_{F} [\frac{1}{1 + λ + η (λ)} ρ (Ξ (ω_{λ} (u, v))) + \frac{1}{1 + λ} σ (Ξ (ω_{λ} (u, v)))] \\ \leq \frac{1}{1 + λ + η (λ)} ρ (Ξ (ω_{λ} (u, v))) + \frac{1}{1 + λ} σ (Ξ (ω_{λ} (u, v))) \\ = \frac{| | u - v | |}{1 + λ + η (λ)} \frac{ρ (Ξ (ω_{λ} (u, v)))}{| | u - v | |} + \frac{| | u - v | |}{1 + λ} \frac{σ (Ξ (ω_{λ} (u, v)))}{| | u - v | |} \\ \leq ω_{λ + η (λ)} (u, v) \frac{ρ (Ξ (ω_{λ} (u, v)))}{ω_{λ} (u, v)} + ω_{λ} (u, Λ u) \frac{σ (Ξ (ω_{λ} (u, v)))}{ω_{λ} (u, v))} \\ + ω_{λ} (v, Λ v) \frac{σ (Ξ (ω_{λ} (u, v)))}{ω_{λ} (u, v)} + ω_{λ} (Λ u, Λ v) \frac{σ (Ξ (ω_{λ} (u, v)))}{ω_{λ} (u, v)} \\ \leq ω_{λ + η (λ)} (u, v) \frac{ρ (Ξ (ω_{λ} (u, v)))}{ω_{λ} (u, v)} + ω_{λ} (u, Λ u) \frac{σ (Ξ (ω_{λ} (u, v)))}{ω_{λ} (u, v)} \\ + ω_{λ} (v, Λ v) \frac{σ (Ξ (ω_{λ} (u, v)))}{ω_{λ} (u, v)} + ω_{λ} (Λ u, Λ v) ϒ (ω_{λ} (u, v)) \\ \leq ω_{λ + η (λ)} (u, v) \frac{ρ (Ξ (ω_{λ} (u, v)))}{ω_{λ} (u, v)} + ω_{λ} (u, Λ u) \frac{σ (Ξ (ω_{λ} (u, v)))}{ω_{λ} (u, v)} \\ + ω_{λ} (v, Λ v) \frac{σ (Ξ (ω_{λ} (u, v)))}{ω_{λ} (u, v)} + k ω_{λ} (Λ u, Λ v) \\ \leq ω_{λ + η (λ)} (u, v) \frac{ρ (Ξ (ω_{λ} (u, v)))}{Ξ (ω_{λ} (u, v))} + ω_{λ} (u, Λ u) \frac{σ (Ξ (ω_{λ} (u, v)))}{Ξ (ω_{λ} (u, v))} \\ + ω_{λ} (v, Λ v) \frac{σ (Ξ (ω_{λ} (u, v)))}{Ξ (ω_{λ} (u, v))} + k ω_{λ} (Λ u, Λ v) . \end{aligned}

Further, we obtain

\begin{aligned} ω_{λ} (Λ u, Λ v) & \leq ω_{λ + η (λ)} (u, v) Γ (Ξ (ω_{λ} (u, v))) + ω_{λ + ζ (λ)} (u, Λ u) ϒ (Ξ (ω_{λ} (u, v))) \\ + ω_{λ + μ (λ)} (v, Λ v) ϒ (Ξ (ω_{λ} (u, v))) . \end{aligned}

Moreover, since ξ is a sublinear (real) function in the class Ψ, it follows that

\begin{aligned} Ξ (ω_{λ} (Λ u, Λ v)) & \leq Ξ (ω_{λ + η (λ)} (u, v)) Γ (Ξ (ω_{λ} (u, v))) + Ξ (ω_{λ + ζ (λ)} (u, Λ u)) ϒ (Ξ (ω_{λ} (u, v))) \\ + Ξ (ω_{λ + μ (λ)} (v, Λ v)) ϒ (Ξ (ω_{λ} (u, v))) . \end{aligned}

For the case u = v, it is obvious that the above inequality is satisfied. Thus, we now have the inequality (2.1) holds for any comparable u, v ∈ Ω_ω.

Note that any constant functions are contained in Ω. Now, for any u, v ∈ Ω_ω, we may choose a constant function w ∈ Ω _ω for which u, v ⊑ w and θ ⊑ w. Consequently, we have w(x, t) = ||w|| for all (x, t) ∈ ℝ × I. Also, observe that this w attains the following:

\begin{aligned} (Λ w) (x, t) & = \int_{- \infty}^{\infty} k (x - ξ, t) φ (ξ) d ξ + \int_{0}^{t} \int_{- \infty}^{\infty} k (x - ξ, t - τ) F (ξ, τ, | | w | |, 0) d ξ d τ \\ \geq \int_{- \infty}^{\infty} k (x - ξ, t) φ (ξ) d ξ + | | w | | \\ \geq | | w | | \\ = w (x, t) . \end{aligned}

Hence, the condition (2.2) is satisfied. Therefore, by applying the Theorems 2.1 and 2.2, the result follows. This completes the proof. ■

Declarations

Acknowledgements

The authors were supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC No.55000613). This research was partially finished at Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju, Korea, while the first and third authors visit here. Also, the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (Grant no. 2011-0021821). Furthermore, the authors are grateful for the reviewers for the careful reading of the article and for the suggestions which improved the quality of this work.

Authors’ Affiliations

(1)

Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, Bangkok, 10140, Thailand

(2)

Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju, 660-701, Korea

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