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- 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
https://doi.org/10.1186/1687-1847-2012-83
© Chaipunya et al; licensee Springer. 2012
- 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]).
for all x, y ∈ X. Then the sequence {x n } defined by x n = fxn- 1for 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:
then the sequence {x n } converges to the unique fixed point of f in X.
- (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:
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.
- (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 ).
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].
- (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.
Besides, if , then , where π((β1, β2 ..., β n )) is a permutation of (β1, β2 ..., β n ). It is also important to know that, if , then for each m ∈ {1, 2, ..., n}, where each is selected from {β1, β2, ..., β n } and for all i, j ∈ {1, 2, ..., m}.
- (a)
If 0 < t < ∞, then ψ(t) < ∞.
- (b)
.
Now, we are ready to give our main results in this article.
- (1)
f has a fixed point x ∞ ∈ X ω .
- (2)
The sequence {f n x 0} converges to x ∞ .
which is a contradiction of our assumption. Therefore, lim n→∞ ψ(ω λ (f n x0, fn+1x0)) = 0 and so, we have lim n→∞ ω λ (f n x0, fn+1x0) = 0. Moreover, we have lim n→∞ ω λ (f n x0, fn+1x0) = 0 for all λ > 0.
This is a contradiction. Therefore, it follows that {f n x0} is a Cauchy sequence. Due to the completeness of X ω , {f n x0} converges to some point x ∞ ∈ X ω .
Letting n → ∞, we obtain that ψ(ω λ (x ∞ , fx ∞ ) ≤ 0 for all λ > 0. Therefore, x ∞ is a fixed point of f. ■
then the fixed point in Theorem 2.1 is unique.
Thus, we have {f n w} converges to x ∞ . Similarly, we obtain that {f n w } 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]).
whereand. 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 , we have . Thus, apply Theorems 2.1 and 2.2, we have the conclusion. ■
whereandwith max{supt≥ 0β(t), supt≥ 0γ(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 , we have . 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.
where we assume φ to be continuously differentiable such that φ and φ' are bounded and F is continuous.
- (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.
Taking a nondecreasing sequence {u n } in Ω ω converging to u ∈ Ω ω . For any (x, t) ∈ ℝ × I,
for all n ≥ 1. Therefore, u n ⊑ u for all n ≥ 1. So, the space Ω ω satisfies the condition (1.2).
- (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 such that, for any λ > 0, there exists η(λ) ∈ (0, λ) such that
- (3)The two functions Γ, ϒ: ℝ+ → [0, 1) given by
- (4)
- (5)
F is bounded for bounded s and p.
Then, the existence and uniqueness of the solution of the system (3.1) is affirmative.
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.
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.
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 .
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
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