A new type fixed point theorem for a contraction on partially ordered generalized complete metric spaces with applications
© Eshaghi Gordji et al.; licensee Springer. 2014
Received: 20 May 2013
Accepted: 29 October 2013
Published: 20 January 2014
In this paper, we prove a fixed point theorem for a contraction in generalized complete metric spaces endowed with partial order. As an application, we use the fixed point theorem to prove the Hyers-Ulam stability of the Cauchy functional equation in Banach spaces endowed with a partial order.
MSC:54H25, 47H10, 39B52.
In 1940, Ulam gave a wide ranging talk in front of the mathematics club of University of Wisconsin in which he discussed a number of important unsolved problems (see ). One of the problems was the question concerning the stability of homomorphisms:
Let be a group and be a metric group with a metric . Given , does there exist such that, if a mapping satisfies the inequality for all , then there exists a homomorphism with for all ?
In 1941, Hyers  affirmatively answered the question of Ulam for the case where and are Banach spaces. Taking this fact into account, the additive Cauchy functional equation is said to satisfy the Hyers-Ulam stability.
On the other hand, Banach’s contraction principle is one of the pivotal results of analysis. It is widely considered as the source of metric fixed point theory. Also, its significance lies in its vast applicability in a number of branches of mathematics. Many kinds of generalizations of the above principle have been a heavily investigated branch of research. In particular, Diaz and Margolis  presented the following definition and fixed point theorem in a ‘generalized complete metric space’.
if and only if ;
every d-Cauchy sequence in X is d-convergent, i.e., for a sequence in X implies the existence of an element with (the point x is unique by (D1) and (D3)).
Then we call a generalized complete metric space.
Theorem 1.2 Suppose that is a generalized complete metric space and the function is a contraction, that is, T satisfies the following condition:
Let and consider a sequence of successive approximations with initial element . Then the following alternative holds: either
(A) for all , one has
(B) the sequence is d-convergent to a fixed point of T.
Recently, Nieto and Rodriguez-Lopez  proved a fixed point theorem in partially ordered sets as follows.
for all . If there exists with , then f has a fixed point.
In 2003, Cǎdariu and Radu  applied the fixed point method to investigate the Jensen functional equation (see also [6–9]) and presented a short and simple proof (different from the direct method initiated by Hyers in 1941) for the Hyers-Ulam stability of the Jensen functional equation  for proving properties of generalized Hyers-Ulam stability for some functional equations in a single variable  for the stability of some nonlinear equations . Recently, Brzdek , Brzdek and Cieplinski [11, 12] reported some interesting results in this direction (see also [13–16]).
In this paper, we prove a fixed point theorem for self-mappings on a partially ordered set X which has a generalized metric d. Moreover, we give a generalization of the Hyers-Ulam stability of the conditional Cauchy equation as an important result of our fixed point theorem.
2 Main results
We start our work by the following fixed point theorem in generalized complete metric spaces.
for all with . If there exists , then the following alternative holds: either
(B) the sequence of is d-convergent to a fixed point of f.
If, for all , , then (A) holds;
If, for some integer l, , then denotes the smallest nonnegative integer such that .
for all .
and we can write for all .
On the other hand, since X is a complete generalized metric space, there exists such that .
and hence . This completes the proof. □
Theorem 2.2 In Theorem 2.1, we can replace the following condition with the continuity of f:
If is a nondecreasing sequence and in X, then for all .
Then f has a fixed point.
This shows that . This completes the proof. □
Theorem 2.4 If, for all , there exists z which is comparable to x and y and , , then, in Theorems 2.1 and 2.2, the uniqueness of the fixed point of f follows.
whenever and so we have . This completes the proof. □
for all , for all ;
for all , there exists such that z is comparable to x and y.
for all , there exists such that z is an upper bound of ;
if is a nondecreasing sequence in and , then for all .
As a simple example, we can show that ℝ satisfies the conditions (a), (b), (c) and (d). Also, in this section, we consider .
Now, we prove the main result of this section as follows.
for all .
for all . It is easy to show that is a complete generalized metric space.
It is easy to show that J is a nondecreasing mapping.
This implies inequality (3.4).
for all . By using (3.6) and (3.7), it follows that T is a Cauchy mapping.
for all . Since , . This completes the proof. □
for all .
Proof Set for all with , , and let in Theorem 3.1. Then we get the desired result. □
for all .
Proof Set for all , and let in Theorem 3.1. Then we get the desired result. □
YJ Cho and C Park were supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012-0008170) and (NRF-2012R1A1A2004299), respectively.
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