RESEARCH Open Access

Geraghty-type theorems in modular metricspaces with an application to partial differentialequationParin Chaipunya1, Yeol Je Cho2* and Poom Kumam1*

* Correspondence: yjcho@gnu.ac.kr;poom.kum@kmutt.ac.th1Department of Mathematics,Faculty of Science, King Mongkut’sUniversity of Technology Thonburi,Bangkok 10140, Thailand2Department of MathematicsEducation and the RINS,Gyeongsang National University,Chinju 660-701, KoreaFull list of author information isavailable at the end of the article

Abstract

In this article, we prove some fixed point theorems of Geraghty-type concerning theexistence 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 anduniqueness of a solution to a nonhomogeneous linear parabolic partial differentialequation in the last section.Mathematics Subject Classification (2010): 47H10, 54H25, 35K15.

Keywords: modular metric space, ordered set, Geraghty’s theorem, initial value pro-blems, parabolic equation.

Introduction and preliminariesThroughout 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 applic-

ability, has became a very popular tool in modern analysis, especially in nonlinear ana-

lysis including its applications to differential and integral equations, variational

inequality theory, complementarity problems, equilibrium problems, minimization pro-

blems 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 b: ℝ+ ® [0, 1) satisfying the following condition:

β(tn) → 1 implies tn → 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 ∈ S satisfying

d(fx, fy) ≤ β(d(x, y))d(x, y) (1:1)

for all x, y Î X. Then the sequence {xn} defined by xn = fxn-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:

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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 nonde-

creasing self-mapping on X which satisfies the inequality (1.1) whenever x, y Î X are

comparable. Assume that f is either continuous or

if a nondecreasing sequence {xn} converges to x∗, then xn � x∗for each n ≥ 1. (1:2)

If, additionally, the following condition is satisfied:

for any x, y ∈ X, there exists z ∈ X which is comparable to both x and y, (1:3)

then the sequence {xn} 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 nonde-

creasing self-mapping on X such that there exists x0 Î X with x0 ⊑ fx0. Suppose that

there exist β ∈ S and ψ Î Ψ such that

ψ(d(fx, fy)) ≤ β(ψ(d(x, y)))ψ(d(x, y)),

whenever x, y Î 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 Î X, the following conditions hold:

(a) ωl(x, y) = 0 for all l >0 if and only if x = y.

(b) ωl(x, y) = ωl(y, x) for all l >0.

(c) ωl+μ(x, y) ≤ ωl(x, z) + ωμ(z, y) for all l, μ >0.

For any xι Î X, the set Xω(xι) = {x Î X: liml®∞ ωl(x, xι) = 0} is called a modular

metric space generated by xι and induced by ω. If its generator xι does not play any

role in the situation (that is, Xω is independent of generators), we write Xω instead of

Xω(xι).

Observe that a metric modular ω on X is nonincreasing with respect to l >0. We

can simply show this assertion using the condition (c). For any x, y Î X and 0 < μ < l,we have

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ωλ(x, y) ≤ ωλ−μ(x, x) + ωμ(x, y) = ωμ(x, y). (1:4)

For any x, y Î X and l >0, we set

ωλ+ (x, y) := limε↓0

ωλ+ε(x, y), ωλ−(x, y) := limε↓0

ωλ−ε(x, y).

Consequently, from (1.4), it follows that

ωλ+(x, y) ≤ ωλ(x, y) ≤ ωλ−(x, y).

For any x, y Î X, if a metric modular ω on X possesses a finite value and ωl(x, y) =

ωμ(x, y) for all l, μ >0, then d(x, y): = ωl(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 Xω be a modular metric space and {xn} be a sequence

in Xω.

(1) A point x Î Xω is called a limit of {xn} if, for each l, � >0, there exists n0 Î N

such that ωl(xn, x) < � for all n ≥ n0. A sequence that has a limit is said to be con-

vergent (or converges to x), which is written as limn®∞ xn = x.

(2) A sequence {xn} in Xω is said to be a Cauchy sequence if, for each l, � >0, there

exists n0 Î N such that ωl(xn, xm) < � for all m, n ≥ n0.

(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 resultsBefore stating our main results, we first introduce the following classes for a more con-

venience of usage.

For each n Î N, let Sn denote the class of n-tuples of functions (b1, b2, ..., bn), wherefor each i Î {1, 2, ..., n}, bi: ℝ+ ∪ {∞} ® [0, 1) and the following implication holds:

β(tk) := β1(tk) + β2(tk) + · · · + βn(tk) → 1 implies tk → 0.

Actually, Geraghty’s class S is equivalent to the class S1 when ∞ is not considered.

It follows that, for each m Î {1, 2, ..., n}, if (β1, β2, . . . , βm) ∈ Sm , then

(β1, β2, . . . , βm, θ , θ , . . . , θ︸ ︷︷ ︸n−m entries

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

(β , β , . . . , β)︸ ︷︷ ︸n entries

∈ Sn, then we also have the following:

β(tk) → 1n

implies tk → 0.

Besides, if (β1, β2 . . . , βn) ∈ Sn , then π((β1, β2 . . . , βn)) ∈ Sn , where π((b1, b2 ..., bn))is a permutation of (b1, b2 ..., bn). It is also important to know that, if

(β1, β2 . . . , βn) ∈ Sn , then (βn1 , βn2 , . . . , βnm ) ∈ Sm for each m Î {1, 2, ..., n}, where

each βni is selected from {b1, b2, ..., bn} and βni = βnj for all i, j Î {1, 2, ..., m}.

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Let ̄ denote the class of functions ψ: ℝ+ ∪ {∞} ® ℝ+ ∪ {∞} satisfying the following

conditions:

(a) If 0 < t < ∞, then ψ(t) < ∞.

(b) ψ |R+ ∈ .

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

Theorem 2.1. Let Xω be a complete modular metric space with a partial ordering ⊑and f be a self-mapping on Xω such that, for each l >0, there exists h(l) Î (0, l) suchthat

ψ(ωλ(fx, fy)) ≤α(ψ(ωλ(x, y)))ψ(ωλ+η(λ)(x, y)) + β(ψ(ωλ(x, y)))ψ(ωλ(x, fx))

+ γ (ψ(ωλ(x, y)))ψ(ωλ(y, fy)),(2:1)

where ψ ∈ ̄ and (α, β , γ ) ∈ S3 with a(t) + 2 max{supt≥0 b(t), supt≥0 g(t)} <1.

Assume also that the condition (1.2) holds. If there exists x0 Î Xω such that ωl(x0, fx0)

< ∞ for all l >0, then the following hold:

(1) f has a fixed point x∞ Î Xω.

(2) The sequence {fnx0} converges to x∞.

Proof. It is clear that the sequence {fnx0} is nondecreasing. Assume that, for each n ≥ 1,

there exists ln >0 such that ωλn (f nx0, f n+1x0) = 0 . Otherwise, the proof is complete. For

each n ≥ 1, if 0 < l ≤ ln, then we also have ωl(fnx0, f

n+1x0) ≠ 0. Since fnx0⊑f n+1x0, forany 0 < l ≤ ln, we have

ψ(ωλn(f nx0, f n+1x0)) ≤ ψ(ωλ(f nx0, f n+1x0))

≤ α(ψ(ωλ(f n−1x0, f nx0)))ψ(ωλ+η(λ)(f n−1x0, f nx0))

+ β(ψ(ωλ(f n−1x0, f nx0)))ψ(ωλ(f n−1x0, f nx0))

+ γ (ψ(ωλ(f n−1x0, f nx0)))ψ(ωλ(f nx0, f n+1x0))

≤ α(ψ(ωλ(f n−1x0, f nx0)))ψ(ωλ(f n−1x0, f nx0))

+ β(ψ(ωλ(f n−1x0, f nx0)))ψ(ωλ(f n−1x0, f nx0))

+ γ (ψ(ωλ(f n−1x0, f nx0)))ψ(ωλ(f nx0, f n+1x0)),

which implies that

ψ(ωλ(f nx0, f n+1x0)) ≤ α(ψ(ωλ(f n−1x0,f nx0)))+β(ψ(ωλ(f n−1x0,f nx0)))1−γ (ψ(ωλ(f n−1x0,f nx0))) ψ(ωλ(f n−1x0, f nx0))

≤ ψ(ωλ(f n−1x0, f nx0))

...

≤ ψ(ωλ(x0, f x0)

< ∞.

Therefore, {ψ(ωl(fnx0, f

n+1x0))} is nonincreasing and bounded below. So, the

sequence converges to some number r ≥ 0. Assume r >0. Observe that

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ψ(ωλ(f nx0, f n+1x0)) ≤ [α(ψ(ωλ(f n−1x0, f nx0))) + β(ψ(ωλ(f n−1x0, f nx0)))

+γ (ψ(ωλ(f n−1x0, f nx0)))]ψ(ωλ(f n−1x0, f nx0)).

Taking n ® ∞, we have

1 ≤ lim infn→∞

[α(ψ(ωλ(f n−1x0, f nx0))) + β(ψ(ωλ(f n−1x0, f nx0)))

+γ (ψ(ωλ(f n−1x0, f nx0)))]

.

So, we have limn®∞ ψ(ωl(fn-1x0, f

nx0)) = 0 and hence

limn→∞ ωλ(f n−1x0, f nx0) = 0,

which is a contradiction of our assumption. Therefore, limn®∞ ψ(ωl(fnx0, f

n+1x0)) = 0 and

so, we have limn®∞ ωl(fnx0, f

n+1x0) = 0. Moreover, we have limn®∞ ωl(fnx0, f

n+1x0) = 0 for

all l >0.

Next, we show that {fnx0} is a Cauchy sequence. Assume the contrary. So, there

exists l0, �0 >0 for which we can define two subsequences {f mkx0} and {f nkx0} of the

sequence {fnx0} such that, for any nk > mk > k, ωλ0(f mkx0, f nk x0) ≥ ε0 , but

ωλ0(f mkx0, f nk−1x0) < ε0 . Now, since f mkx0 � f nkx0 , we observe that

ψ(ε0) ≤ ψ(ωλ0 (f mkx0, f nkx0))

≤ α(ψ(ωλ0 (f mk−1x0, f nk−1x0)))ψ(ωλ0+η(λ0)(f mk−1x0, f nk−1x0))

+ β(ψ(ωλ0(f mk−1x0, f nk−1x0)))ψ(ωλ0(f mk−1x0, f mk x0))

+ γ (ψ(ωλ0(f mk−1x0, f nk−1x0)))ψ(ωλ0(f nk−1x0, f nkx0))

≤ ψ(ωη(λ0)(f mk−1x0, f mkx0) + ωλ0(f mkx0, f nk−1x0))

+ ψ(ωλ0(f mk−1x0, f mk x0)) + ψ(ωλ0(f nk−1x0, f nk x0))

≤ ψ(ωη(λ0)(f mk−1x0, f mkx0) + ε0) + ψ(ωλ0(f mk−1x0, f mk x0))

+ ψ(ωλ0(f nk−1x0, f nk x0)).

Letting k ® ∞, we obtain that limk→∞ψ(ωλ0 (f mkx0, f nk x0)) = ψ(ε0) . So, we have

limk→∞

ωλ0(f mkx0, f nk x0) = ε0.

Observe again that

ψ(ωλ0(f mkx0, f nkx0)) ≤ ψ(ωλ0+η(λ0)(f mk−1x0, f nk−1x0)) + ψ(ωλ0(f mk−1x0, f mk x0))

+ ψ(ωλ0(f nk−1x0, f nkx0))

≤ ψ

(ωη(λ0)

2(f mk−1x0, f mkx0) + ωλ0 (f mkx0, f nkx0)

+ωη(λ0)2

(f nkx0, f nk−1x0))

+ ψ(ωλ0(f mk−1x0, f mkx0))

+ ψ(ωλ0(f nk−1x0, f nkx0)).

Letting k ® ∞, we deduce that limk→∞ψ(ωλ0 (f mk−1x0, f nk−1x0)) = ψ(ε0). Similarly,

we have

limk→∞

ωλ0(f mk−1x0, f nk−1x0) = ε0.

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Thus, it follows that

1 ≤ lim infk→∞

α(ψ(ωλ0 (f mk−1x0, f nk−1x0))).

Therefore, we conclude that limk→∞ψ(ωλ0 (f mk−1x0, f nk−1x0)) = 0 , which implies

that

limk→∞

ωλ0(f mk−1x0, f nk−1x0) = 0.

This is a contradiction. Therefore, it follows that {fnx0} is a Cauchy sequence. Due to

the completeness of Xω, {fnx0} converges to some point x∞ Î Xω.

Now, we show that x∞ is a fixed point of f. Let l >0 be arbitrary. By virtue of the

condition (1.2), we consider that

ψ(ωλ(f n+1x0, f x∞) ≤ ψ(ωλ(f nx0, x∞)) + ψ(ωλ(f nx0, f n+1x0)).

Letting n ® ∞, we obtain that ψ(ωl(x∞, fx∞) ≤ 0 for all l >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:

for any x, y ∈ Xω, there exists w ∈ Xωwith w � fw and ωλ

(w, fw

)< ∞ for all λ > 0

such that w is comparable to both x and y,(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∞ Î Xω. Assume that y∞ ÎXω is also another fixed point of f. Thus, we can find w Î Xω with w ⊑ fw and com-

parable to both x∞ and y∞. It follows that fnw is comparable with both x∞ and y∞ for

each n Î N. Observe that, for any l >0,

ψ(ωλ(f n+1w, x∞) = ψ(ωλ(f n+1w, f x∞)

≤ α(ψ(ωλ(f nw, x∞)))ψ(ωλ(f nw, x∞))

+ β(ψ(ωλ(f nw, x∞)))ψ(ωλ(f nw, f n+1w))

≤ α(ψ(ωλ(f nw, x∞)))ψ(ωλ(f nw, x∞))

+ β(ψ(ωλ(f nw, x∞)))ψ(ωλ(f nw, x∞))

+ β(ψ(ωλ(f nw, x∞)))ψ(ωλ(x∞, f n+1w)).

(2:3)

Therefore, without loss of generality, we have

ψ(ωλ(f n+1w, x∞) ≤ α(ψ(ωλ(f nw,x∞)))+β(ψ(ωλ(f nw,x∞)))1−β(ψ(ωλ(f nw,x∞))) ψ(ωλ(f nw, x∞))

≤ ψ(ωλ(f nw, x∞))

...

≤ ψ(ωλ(w, x∞))

< ∞.

Therefore, {ψ(ωl(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 limn®∞ ωl(fnw, f n+1w) = 0 for all l >0. Thus, letting n ® ∞ in the

inequality (2.3), we have

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1 ≤ lim infn→∞ α(ψ(ωλ(f nw, x∞))).

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 Xω is totally ordered, then the fixed

point in Theorem 2.1 is unique.

Proof. Since Xω is totally ordered, the condition (2.2) is satisfied. Thus, applying The-

orem 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 Xω be a complete modular metric space with a partial ordering ⊑and f be a self-mapping on Xω such that, for any l >0, there exists h(l) Î (0, l) suchthat

ψ(ωλ(fx, fy)) ≤ α(ψ(ωλ(x, y)))ψ(ωλ+η(λ)(x, y)),

where α ∈ S and ψ ∈ ̄ . Assume also that f is continuous or the condition (1.2)

holds. Then f has a fixed point in Xω. Moreover, if the condition (2.2) is satisfied, the

fixed point is unique.

Proof. Since α ∈ S , we have (α, θ , θ) ∈ S3 . Thus, apply Theorems 2.1 and 2.2, we

have the conclusion. ■Corollary 2.5. Let Xω be a complete modular metric space with a partial ordering ⊑

and f be a self-mapping on Xω such that, for any l >0, there exist ζ(l), μ(l) Î (0, l)such that

ψ(ωλ(fx, fy)) ≤ β(ψ(ωλ(x, y)))ψ(ωλ(x, fx)) + γ (ψ(ωλ(x, y)))ψ(ωλ(y, fy)),

where ψ ∈ ̄ and (β , γ ) ∈ S2 with max{supt≥0 b(t), supt≥0 g(t)} <1. Assume also

that f is continuous or that the condition (1.2) holds. Then f has a fixed point in Xω.

Moreover, if the condition (2.2) is satisfied, the fixed point is unique.

Proof. Since (β , γ ) ∈ S2 , we have (θ , β , γ ) ∈ S3 . Thus, apply Theorems 2.1 and 2.2,

we have the conclusion. ■

ApplicationsIn 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{ut(x, t) = uxx(x, t) + F(x, t, u(x, t), ux(x, t)), −∞ < x < ∞, 0 < t ≤ Tu(x, 0) = ϕ(x) ≥ 0, −∞ < x < ∞,

(3:1)

where we assume � to be continuously differentiable such that � and �’ 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:

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(a) u, ut, ux, uxx Î C(ℝ × I).

(b) u and ux are bounded in ℝ × I.

(c) ut(x, t) = uxx(x, t) + F(x, t, u(x, t), ux(x, t)) for all (x, t) Î ℝ × I.

(d) u(x, 0) = �(x) for all x Î ℝ.

Now, we consider the following space:

� := {u(x, t) : u, ux ∈ C(R × I) and||u|| < ∞},

where

‖u‖ := supx∈R,t∈I

|u(x, t)| + supx∈R,t∈I

|ux(x, t)|.

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

ωλ(x, y) :=1

1 + λ||u − v||

is a metric modular on Ω. Clearly, the set Ωω is a complete modular metric space

independent of generators. Define a partial ordering ⊑ on Ωω by

u, v ∈ �ω, u � v ⇔ u(x, t) ≤ v(x, t) and ux(x, t) ≤ vx(x, t) at each (x, t) ∈ R × I.

Taking a nondecreasing sequence {un} in Ωω converging to u Î Ωω. For any (x, t) Îℝ × I,

we have

u1(x, t) ≤ u2(x, t) ≤ · · · ≤ un(x, t) ≤ · · ·

and

(u1)x(x, t) ≤ (u2)x(x, t) ≤ · · · ≤ (un)x(x, t) ≤ · · · .

Since the sequences {un(x, t)} and {(un)x(x, t)} converges to u(x, t) and ux(x, t),

respectively, it follows that, for any (x, t)Îℝ × I,

un(x, t) ≤ u(x, t) and (un)x(x, t) ≤ ux(x, t)

for all n ≥ 1. Therefore, un ⊑ u for all n ≥ 1. So, the space Ωω 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 constantcF ≤ (T + 2π

−12 T

12 )−1 such that, for any l > 0, there

exists h(l) Î (0, l) such that

0 ≤ 11+λ

[F(x, t, s2, p2) − F(x, t, s1, p1)]

≤ cF

[1

1+λ+η(λ)ρ

(�

(s2 − s1 + p2 − p1

1 + λ

))+ 1

1+λσ

(�

(s2 − s1 + p2 − p1

1 + λ

))]

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for all (s1, p1), (s2, p2) Î ℝ × ℝ with s1 ≤ s2 and p1 ≤ p2, where � ∈ ̄ is sublinear

with Ξ(x) ≤ t and r, s are nondecreasing functions on ℝ+ such that r(t) <(1 -k)t

and s(t) <(1 -k)kt for all t >0 and for some fixed k Î (0, 1).

(3) The two functions Γ, ϒ: ℝ+ ® [0, 1) given by

�(t) =

{0 if t = 0,

ρ(t)(1−k)t if t > 0,

ϒ(t) =

{0 if t = 0,

σ (t)(1−k)t if t > 0,

are such that (�, ϒ , ϒ) ∈ S3 , Γ + 2ϒ <1.

(4) F(x, t, s, 0) ≥ s∫ t0

∫ ∞∞ k(x − ξ , t − τ )dξdτ

for all s ≥ 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) =

∞∫−∞

k(x− ξ , t)ϕ(ξ)dξ +

t∫0

∞∫−∞

k(x− ξ , t − τ )F(ξ , τ , u(ξ , τ ), ux(ξ , τ ))dξdτ (3:2)

for all x Î ℝ and 0 <t ≤ T, where

k(x, t) :=1√4π t

e− x2

4t

for all x Î 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 ux are both continuous

and bounded for all x Î ℝ and 0 < t ≤ T.

Define a mapping Λ: Ωω ® Ωω by

(�u)(x, t) :=

∞∫−∞

k(x−ξ , t)ϕ(ξ)dξ+

t∫0

∞∫−∞

k(x−ξ , t−τ )F(ξ , τ , u(ξ , τ ), ux(ξ , τ ))dξdτ

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 Î Ωω with u ⊑ v. Suppose that u ≠ v. Besides, we have

11+λ

∣∣(�v)(x, t) − (�u)(x, t)∣∣

≤ 11+λ

t∫0

∞∫−∞

k(x − ξ , t − τ )|F(ξ , τ , v(ξ , τ ), vx(ξ , τ )) − F(ξ , τ , u(ξ , τ ), ux(ξ , τ ))|dξdτ

≤t∫

0

∞∫−∞

k(x − ξ , t − τ )cF

[1

1+λ+η(λ)ρ(�

( 11+λ

(v(ξ , τ ) − u(ξ , τ ) + vx(ξ , τ ) − ux(ξ , τ )

)))+ 1

1+λσ

(�

( 11+λ

(v(ξ , τ ) − u(ξ , τ ) + vx(ξ , τ ) − ux(ξ , τ )

)))]dξdτ

≤ cF

[1

1+λ+η(λ)ρ(�(ωλ(u, v))) + 11+λ

σ (�(ωλ(u, v)))] t∫

0

∞∫−∞

k(x − ξ , t − τ )dξdτ

≤ cF

[1

1+λ+η(λ)ρ(�(ωλ(u, v))) + 11+λ

σ (�(ωλ(u, v)))]

T.

(3:3)

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Similarly, we have

11+λ

∣∣(�v)x(x, t) − (�u)x(x, t)∣∣

≤ cF

[1

1+λ+η(λ)ρ(�(ωλ(u, v))) + 11+λ

σ (�(ωλ(u, v)))] t∫

0

∞∫−∞

|kx(x − ξ , t − τ )|dξdτ

≤ 2π− 1

2 T− 12 cF

[1

1+λ+η(λ)ρ(�(ωλ(u, v))) + 11+λ

σ (�(ωλ(u, v)))]

.

(3:4)

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

obtain

ωλ(�u, �v) ≤ (T + 2π− 1

2 T12 )cF

[1

1+λ+η(λ)ρ(�(ωλ(u, v))) + 11+λ

σ (�(ωλ(u, v)))]

≤ 11+λ+η(λ)ρ(�(ωλ(u, v))) + 1

1+λσ (�(ωλ(u, v)))

= ||u−v||1+λ+η(λ)

ρ(�(ωλ(u, v)))||u − v|| + ||u−v||

1+λ

σ (�(ωλ(u, v)))||u − v||

≤ ωλ+η(λ)(u, v)ρ(�(ωλ(u, v)))

ωλ(u, v)+ ωλ(u, �u)

σ (�(ωλ(u, v)))

ωλ(u, v))

+ ωλ(v, �v)σ (�(ωλ(u, v)))

ωλ(u, v)+ ωλ(�u, �v)

σ (�(ωλ(u, v)))ωλ(u, v)

≤ ωλ+η(λ)(u, v)ρ(�(ωλ(u, v)))

ωλ(u, v)+ ωλ(u, �u)

σ (�(ωλ(u, v)))ωλ(u, v)

+ ωλ(v, �v)σ (�(ωλ(u, v)))

ωλ(u, v)+ ωλ(�u, �v)ϒ(ωλ(u, v))

≤ ωλ+η(λ)(u, v)ρ(�(ωλ(u, v)))

ωλ(u, v)+ ωλ(u, �u)

σ (�(ωλ(u, v)))ωλ(u, v)

+ ωλ(v, �v)σ (�(ωλ(u, v)))

ωλ(u, v)+ kωλ(�u, �v)

≤ ωλ+η(λ)(u, v)ρ(�(ωλ(u, v)))

�(ωλ(u, v))+ ωλ(u, �u)

σ (�(ωλ(u, v)))�(ωλ(u, v))

+ ωλ(v, �v)σ (�(ωλ(u, v)))

�(ωλ(u, v))+ kωλ(�u, �v).

Further, we obtain

ωλ(�u, �v) ≤ ωλ+η(λ)(u, v)�(�(ωλ(u, v))) + ωλ+ζ (λ)(u, �u)ϒ(�(ωλ(u, v)))

+ ωλ+μ(λ)(v, �v)ϒ(�(ωλ(u, v))).

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

�(ωλ(�u, �v)) ≤ �(ωλ+η(λ)(u, v))�(�(ωλ(u, v))) + �(ωλ+ζ (λ)(u, �u))ϒ(�(ωλ(u, v)))

+ �(ωλ+μ(λ)(v, �v))ϒ(�(ωλ(u, v))).

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 Î Ωω.

Note that any constant functions are contained in Ω. Now, for any u, v Î Ωω, we

may choose a constant function w Î Ωω for which u, v ⊑ w and θ ⊑ w. Consequently,

we have w(x, t) = ||w|| for all (x, t) Î ℝ × I. Also, observe that this w attains the

following:

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(�w)(x, t) =

∞∫−∞

k(x − ξ , t)ϕ(ξ)dξ+

t∫0

∞∫−∞

k(x − ξ , t − τ )F(ξ , τ , ||w||, 0)dξdτ

≥∞∫

−∞k(x − ξ , t)ϕ(ξ)dξ + ||w||

≥ ||w||= w(x, t).

Hence, the condition (2.2) is satisfied. Therefore, by applying the Theorems 2.1 and

2.2, the result follows. This completes the proof. ■

AcknowledgementsThe authors were supported by the Higher Education Research Promotion and National Research University Project ofThailand, Office of the Higher Education Commission (NRU-CSEC No.55000613). This research was partially finished atDepartment of Mathematics Education and the RINS, Gyeongsang National University, Chinju, Korea, while the firstand third authors visit here. Also, the second author was supported by the Basic Science Research Program throughthe 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 articleand for the suggestions which improved the quality of this work.

Author details1Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok 10140,Thailand 2Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Korea

Authors’ contributionsAll authors read and approved the final manuscript.

Competing interestsThe authors declare that they have no competing interests.

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

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doi:10.1186/1687-1847-2012-83Cite this article as: Chaipunya et al.: Geraghty-type theorems in modular metric spaces with an application topartial differential equation. Advances in Difference Equations 2012 2012:83.

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