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Existence of solutions of integral equations via fixed point theorems
Journal of Inequalities and Applications volume 2014, Article number: 138 (2014)
Existence and uniqueness of fixed points of a mapping defined on partially ordered G-metric spaces is discussed. The mapping satisfies contractive conditions based on certain classes of functions. The results are applied to the problems involving contractive conditions of integral type and to a particular type of initial value problems for the nonhomogeneous heat equation in one dimension. This work is a generalization of the results published recently in (Gordji et al. in Fixed Point Theory Appl. 2012:74, 2012, doi:10.1186/1687-1812-2012-74) to G-metric space.
1 Introduction and preliminaries
One of the most common applications of the fixed point theory is the problem of existence and uniqueness of solutions of initial and boundary value problems for differential and integral equations. The number of studies dealing with such problems has increased considerably in the recent years. An important result in this direction has been reported by JJ Nieto and RR Lopez in 2005 . They studied existence and uniqueness of fixed points on partially ordered metric spaces and applied their results to boundary value problems for ordinary differential equations. The research in this direction is advancing continuously and produces many interesting results; see [2–6].
In 2006, Mustafa and Sims  introduced the concept of a G-metric and G-metric space, which is a generalization of metric space. After this pioneering work, G-metric spaces and particularly fixed points of various maps on G-metric spaces have been intensively studied; see [8–25] and also [26–29].
In this work, we present some fixed point theorems on G-metric spaces and investigate the existence of solutions of an initial value problem for a partial differential equation, more precisely, a nonlinear one dimensional heat equation.
First, we briefly introduce some basic notions of G-metric and G-metric space .
Definition 1.1 Let X be a nonempty set, be a function satisfying the following conditions:
(G1) if ,
(G2) for all with ,
(G3) for all with ,
(G4) (symmetry in all variables),
(G5) for all (rectangle inequality).
Then the function G is called a G-metric on X and the pair is called a G-metric space.
Note that conditions (G4) and (G5) imply that
for all .
Definition 1.2 (see )
Let be a G-metric space and let be a sequence in X.
A point is said to be the limit of the sequence if
and the sequence is said to be G-convergent to x.
A sequence is called a G-Cauchy sequence if for every , there is a positive integer N such that for all ; that is, if as .
is said to be G-complete (or a complete G-metric space) if every G-Cauchy sequence in is G-convergent in X.
Proposition 1.3 (see )
Let be a G-metric space, be a sequence in X and . Then the following are equivalent:
is G-convergent to x,
, as ,
, as ,
, as .
Proposition 1.4 (see )
The following statements are equivalent on a G-metric space :
The sequence is G-Cauchy.
For every , there is such that , for all .
Definition 1.5 (see )
Let and be G-metric spaces. A function is said to be G-continuous at a point if and only if for every , there exists such that and implies . A function f is G-continuous on X if and only if it is G-continuous at all points in X.
Proposition 1.6 Let be a G-metric space. Then the function is jointly continuous in all of its three variables.
Definition 1.7 A G-metric space is said to be symmetric if
holds for arbitrary . Otherwise, the space is called asymmetric.
It is obvious that for every G-metric on the set X, the expression
is a standard metric on X.
Note that on a symmetric G-metric space , but on an asymmetric G-metric space, the inequality
holds for all .
Some examples of G-metric spaces are presented below.
Let be a metric space. Define Gs by
for all . Then clearly, is a symmetric G-metric space. Note that if and d is the Euclidean metric on X, then Gs may be interpreted as the perimeter of the triangle with vertices .
Let . Define
and extend G to by using the symmetry in the variables. Then is an asymmetric G-metric space.
2 The main results
The attempts to generalize the contractive conditions on the maps resulted in definitions of various classes of functions. Altering distance functions defined in , weak ψ-contraction presented in  are some of these classes. In this study we employ contractive conditions based on the following classes of functions.
Let Ψ denote the class of the functions satisfying the following conditions:
ψ is nondecreasing,
ψ is sub-additive, that is, ,
ψ is continuous,
Let S denote the class of the functions such that for any bounded sequence of positive real numbers, .
Before stating our main results, we give the following auxiliary lemma which is going to be needed in the sequel.
Lemma 2.1 Let be a G-metric space and let be a sequence in X such that the sequence of nonnegative real numbers is decreasing and
When the subsequence is not G-Cauchy, then there exist and two sequences and of positive integers such that the sequences
converge to ε as .
Proof From the Proposition 1.4, if is not G-Cauchy, then there exist and two sequences and of ℕ satisfying for which
where is chosen as the smallest integer satisfying (3). In other words,
By (3), (4), and using the symmetry (G4) and the rectangle inequality (G5), we easily derive
Taking the limit in (5) and using (1), we obtain
In addition, from the inequalities
upon taking the limit and using (1) and (6). In a similar way it can be shown that the remaining two sequences in (2) also tend to ε. □
We state next our first main theorem about the existence of fixed points on partially ordered G-metric spaces.
Theorem 2.2 (Existence theorem)
Let be a partially ordered set, be a G-complete metric space and be a nondecreasing function. Suppose that there exist functions and such that
for all with . Assume also that for any increasing sequence in X converging to x,
If there exists such that , then f has a fixed point.
Proof By the assumption, there exists such that . We construct a sequence in the following way:
Since f is nondecreasing, we have for each . Hence, is a nondecreasing sequence. If for some , then is the fixed point of f. Assume that for all . Then, by the definition of ψ, we have for all . Taking and in (7) we get
Thus, the sequence is nonincreasing and bounded below by 0. Consequently, . We will show that . Assume to the contrary that . Due to (7), we have
for each , which yields
However, since , we have and hence, .
We show next that is a G-Cauchy sequence. Suppose that is not G-Cauchy. By Lemma 2.1, there exist and two sequences and of positive integers such that the four sequences
approach ε as k goes to infinity. Setting and in (7) and regarding (9), we get
This inequality implies . Since , we conclude that
By the fact that ψ is a continuous function, and hence , which contradicts the assumption . Therefore, is a G-Cauchy sequence in . Since is a complete G-metric space, there exists such that
Next, we show that z is a fixed point of f. Substituting and in (7), by the virtue of (10), we get
for each . Passing to the limit in the above inequality and regarding (10) and the continuity of ψ, we end up with
This completes the proof of the theorem. □
Next, we discuss the uniqueness conditions for the fixed point of the map in Theorem 2.2. A condition for the uniqueness can be stated as follows:
Every pair of elements in X has a lower bound or an upper bound.
On the other hand, it can be proved that condition (i) is equivalent to condition
For every , there exists which is comparable to both x and y.
Accordingly, we prove the following uniqueness theorem.
Theorem 2.3 (Uniqueness theorem)
Let X satisfies condition (ii) and the hypotheses of Theorem 2.2 hold. If is continuous, then the fixed point of f is unique.
Proof Existence of a fixed point is provided by Theorem 2.2. Assume that y and z are two different fixed points of f. From condition (ii), there exists which is comparable to y and z. The monotonicity of f implies that is comparable to and for . Moreover, using the fact that z is a fixed point of f and condition (7) of Theorem 2.2 we get
Therefore, the sequence defined by is nonnegative and nonincreasing and hence,
To show that , we assume the contrary, that is, . Since β is continuous, . Letting in (13), we get
which results in . Since , we deduce
By similar arguments, we obtain
Employing the rectangle inequality (G4), we have
Since the inequality holds for any in both symmetric and asymmetric G-metric spaces, we have
From the fact that ψ is nondecreasing and sub-additive, we conclude
Letting in the above inequality we obtain , which implies and hence, . This completes the proof. □
If in Theorem 2.2, we take ψ as the identity function on X we deduce the following particular result.
Corollary 2.4 Let be a partially ordered set be a G-complete metric space. Let be a nondecreasing map. Suppose that there exists such that
holds for all with . Assume that either f is continuous or that X satisfies the following condition: if an increasing sequence in X converges to x, then for each . If in addition, there exists such that then f has a fixed point.
Remark 2.5 In a recent paper by Karapınar and Samet  it has been proven that if d is a metric on X and , then the function is also a metric on X. In a similar way, it can be shown that the function , where is also a G-metric. Employing this definition, the contractive condition in Theorem 2.2 can be simplified considerably. More precisely, it becomes
for all with .
As a common application of fixed point theorems one can give integral type contractive conditions. In many articles authors apply their results to maps which are defined by integrals [33, 34]. In what follows, we apply our results to maps defined by Lebesgue integrals. Let Y be a set of functions satisfying the following conditions:
χ is Lebesgue integrable;
χ is summable on each compact of subset of ;
χ is sub-additive;
for each .
A sub-additive integrable function is defined as follows:
Definition 2.6 The function is called a sub-additive integrable function if, for any , we have
For the class of functions in Y, we state the following fixed point theorem.
Theorem 2.7 Let be a partially ordered set and let be a complete G-metric space. Let be a nondecreasing function. Suppose that there exist functions and such that for
holds for all with . Assume that either f is continuous or X satisfies the condition: if an increasing sequence converges to x, then for each . If there exists such that then f has a fixed point.
Proof For , define the function by . Observe that . The inequality (15) can be written as
Let , where clearly . Then we have
Then the conditions of Theorem 2.2 are satisfied and thus f has a fixed point, which completes the proof. □
The particular case in which the function ψ is the identity function on X can be stated as a corollary.
Corollary 2.8 Let be a partially ordered set and, be a complete G-metric space. Let be a nondecreasing map. Suppose that there exists such that for the inequality
holds for all with . Assume that if an increasing sequence in X converges to x then for each . If there exists such that then f has a fixed point.
As an application of the existence and uniqueness Theorems 2.2 and 2.3, we consider the problem of existence and uniqueness of an initial value problem defined by a nonlinear heat equation in one dimension. Such an initial value problem is defined as follows:
where φ is assumed to be continuously differentiable, φ and bounded, and a continuous function.
Definition 3.1 A solution of the initial value problem (17) is any function defined in , where , satisfying the equation and the condition in (17) and also the conditions:
, and ,
u and are bounded in .
Consider the space Ω defined as
where the norm on this space is defined as
The set Ω endowed with the norm defined in (18) is a Banach space. Define a G-metric on Ω as follows:
Then is a complete G-metric space. Define also a partial order ⪯ on Ω as
for any and . It can easily be observed that satisfies condition (i) of uniqueness, that is, every pair of elements in Ω has a lower bound or an upper bound. Indeed, for any , and are the lower and upper bounds for u and v, respectively. Let be a monotone nondecreasing sequence which converges to v in Ω. Then, for any and , we have
Moreover, since the sequences and of real numbers converge to and , respectively, we have for all , and the inequalities and hold. Therefore, for all and hence, with the G-metric defined above satisfies condition (8).
We next define a lower solution for the initial value problem, which is needed in the existence-uniqueness proof.
Definition 3.2 A lower solution of the initial value problem (17) is a function such that
where the function φ is continuously differentiable and both φ and are bounded, the set Ω is the set defined above and is a continuous function.
We state the following existence-uniqueness theorem for the solution of the initial value problem (17).
Theorem 3.3 Consider the problem (17) and, assume that is a continuous function. Suppose that the following conditions hold:
For any , the function , where and is uniformly Hölder continuous in x and t, for each compact subset of .
There exists a constant such that
for all and in with and .
F is bounded for bounded s and p.
Then the existence of a lower solution for the initial value problem (17) provides the existence of the unique solution of the problem (17).
Proof Observe that the problem (17) is equivalent to the integral equation
for all and , where the function is the Green’s function of the problem defined as
for all and . The initial value problem (17) has a unique solution if and only if the above integral equation has unique solution u such that u and are continuous and bounded for all and . Define a mapping by
for all and . Clearly, the fixed point of f is a solution of the problem (17). We will show that the conditions of Theorems 2.2 and 2.3 are satisfied. Note that the mapping f is nondecreasing since, by condition (2) in the statement of the theorem, for , that is, and , we have
Since for all ,
for all and . In addition, we have
where we have used the facts that
On the other hand, since satisfies condition (i), it also satisfies condition (ii), as they are equivalent. Therefore, either v and fu are comparable or there exists which is comparable to both v and fu. In either case, by means of similar calculations, it can be shown that
for all . Moreover, by using differentiation under integral sign and employing again condition (2) of the theorem, we compute
are satisfied. Combining (19), (22), and (23) with (24), (25), and (26), we deduce
Define and . Obviously, is continuous, sub-additive, nondecreasing, and positive in . Furthermore, and also and hence, . Finally, let be a lower solution for (17). We show that . Upon integrating
over and , we obtain
for all and . Therefore, by Theorems 2.2 and 2.3, f has a unique fixed point. This completes the proof. □
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The authors are thankful to the referees for careful reading of the manuscript and the valuable comments and suggestions for the improvement of the paper.
The authors declare that they have no competing interests.
All authors contributed equally to this work. All authors read and approved the final manuscript.
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Cite this article
Gülyaz, S., Erhan, İ.M. Existence of solutions of integral equations via fixed point theorems. J Inequal Appl 2014, 138 (2014). https://doi.org/10.1186/1029-242X-2014-138
- fixed point theory
- G-metric spaces
- initial value problems