Existence of solutions of integral equations via fixed point theorems
© Gülyaz and Erhan; licensee Springer. 2014
Received: 28 November 2013
Accepted: 14 March 2014
Published: 3 April 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.
for all .
Definition 1.2 (see )
- 1.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 )
is G-convergent to x,
, as ,
, as ,
, as .
Proposition 1.4 (see )
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.
holds for arbitrary . Otherwise, the space is called asymmetric.
is a standard metric on X.
holds for all .
Some examples of G-metric spaces are presented below.
- (1)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 .
- (2)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.
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.
converge to ε as .
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)
If there exists such that , then f has a fixed point.
However, since , we have and hence, .
This completes the proof of the theorem. □
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.
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.
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.
for all with .
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:
For the class of functions in Y, we state the following fixed point theorem.
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.
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.
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.
where φ is assumed to be continuously differentiable, φ and bounded, and a continuous function.
, and ,
u and are bounded in .
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.
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).
For any , the function , where and is uniformly Hölder continuous in x and t, for each compact subset of .
- (2)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).
for all and . Therefore, by Theorems 2.2 and 2.3, f has a unique fixed point. This completes the proof. □
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.
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