Ulam-Hyers Stability and Well-Posedness of Fixed Point Problems for α-λ-Contraction Mapping in Metric Spaces

Abstract

We study Ulam-Hyers stability and the well-posedness of the fixed point problem for new type of generalized contraction mapping, so called --contraction mapping. The results in this paper generalize and unify several results in the literature such as the Banach contraction principle.

1. Introduction and Preliminaries

In 1940, the stability problem of functional equations, first initial from a question of Ulam. Among those was the following question concerning the stability of group homomorphisms [1].

Question 1. Let be a group and let be a metric group with a metric . Given , does there exist a such that if a function satisfies the inequality for all , then there is a homomorphism with for all ?

If the answer of this equation is affirmative, we say that the functional equation for group homomorphism is stable.

Next year, some partial answer of Ulam's question about Banach spaces was first given by Hyers [2] which opened an avenue for further development of analysis in this field.

Theorem 1 (see [2]). Let be two Banach spaces and let be a function such that for some and for all . Then the limit exists for each , and is the unique additive function such that for all . Moreover, if is continuous in for each fixed , then the function is linear.

Taking this result into consideration, the additive Cauchy equation is said to have the Ulam-Hyers stability on if, for each function satisfying inequality (3) for some and for all , there exists an additive function such that is bounded on .

There are a number of results studied and extended Ulam-Hyers stability in many directions. In particular, Ulam-Hyers stability for fixed point problems has been studied by many researchers such as Bota et al. [3], Bota-Boriceanu and Petruşel [4], Lazr [5], Rus [6, 7], and F. A. Tişe and I. C. Tişe [8]. Furthermore, there are several remarkable results on the stability of certain classes of functional equations via fixed point approach. Some interesting results in this direction are presented by several mathematicians such as Brzdȩk et al. [9], Brzdek and Ciepliski [10], Brzdȩk and Ciepliński [11], and Cadariu et al. [12].

On the other hand, the concept of well-posedness of a fixed point problem has been of great interest to several researchers, for example, de Blasi and Myjak [13], Reich and Zaslavski [14], Lahiri and Das [15], and Popa [16, 17].

Recently, Samet et al. [18] introduced the concept of -admissible mapping as follows.

Definition 2 (see [18]). Let be a nonempty set and let be a mapping. A mapping is said to be -admissible if it satisfies the following condition:

Example 3. Let . Define and by for all and Then, is -admissible.

Example 4. Let . Define and by for all and Then, is -admissible.

Remark 5. In the setting of Examples 3 and 4 every nondecreasing self-mapping is -admissible.

They also introduced the new type of extension of Banach contraction and proved a fixed point theorem for such mapping via the concept of -admissible mapping. As application, they showed that these results can be utilized to derive fixed point theorems in partially ordered spaces. Furthermore, they apply the main results to the ordinary differential equations. Subsequently, there are a number of results proved for generalized contraction mappings via the concept of -admissible mapping in metric spaces and other spaces (see [19–26] and references therein).

With this work we have two purposes. The first aim of this work is to present new type of contraction mapping. We also establish some existence and uniqueness of fixed point theorems for such mappings in metric spaces via the concept of -admissible mapping. Our second purpose is to present Ulam-Hyers stability and well-posedness of a fixed point problem for this mapping in metric spaces.

2. Main Results

For nonempty set and self-mapping , let be the class of functions for which for all .

First we give the following definition as a generalization of Banach contraction mappings.

Definition 6. Let be a metric space and let be a given mapping. One says that is an --contraction mapping if there exist two functions and such that

Remark 7. It is easy to check that an --contraction mapping is reduced to a Banach contraction mapping if for all and for all , where .

Now, we give some examples which show that the concept of an --contraction mapping is real generalization of Banach contraction mapping.

Example 8. Let , and the metric is defined by for all . Define and by We obtain that Therefore, is not a Banach contraction. It is easy to see that is an --contraction mapping, where is defined by for all .

Next, we give nontrivial example of an --contraction mapping.

Example 9. Let , and the metric is defined by for all . Define and by We obtain that This implies that is not a Banach contraction. Next, we show that is an --contraction mapping, where is defined by Firstly, we show that . For , we get Also, we get From (17) and (18), we conclude that .
Secondarily, we claim that inequality (9) holds. For , we have Otherwise, we see that inequality (9) holds. Therefore, is an --contraction mapping.

Our first main result is the following.

Theorem 10. Let be a complete metric space and let be an --contraction mapping satisfying the following conditions: (i)is -admissible;(ii)there exists such that ;(iii) is continuous.Then the fixed point problem of has a solution; that is, there exists such that .

Proof. Let such that (such a point exists from condition (ii)). Define the sequence in by If for some , then is a fixed point for and the proof finishes. Therefore, we may assume that Since is -admissible and , we get . By induction, we get Applying inequality (9) with and and using (22), we have Again, applying inequality (9) with and and using (22) and the property of function , we get By the same procedure, we get Since , we obtain that is a Cauchy sequence in metric spaces . By the completeness of , we get for some . Since is continuous, that is, is a fixed point of and thus the fixed point problem of has a solution. This completes the proof.

Example 11. Let , and the metric is defined by for all . Define and by We observe that is an --contraction mapping, where is defined by for all . Also, is -admissible and continuous. It is easy to see that there exist such that . Therefore, all hypotheses of Theorem 10 hold. Then we can apply Theorem 10. Indeed, the fixed points of are and .

In the following theorem, we will omit the continuity hypothesis of by adding a new condition.

Theorem 12. Let be a complete metric space and let be an --contraction mapping satisfying the following conditions: (i) is -admissible;(ii)there exists such that ;(iii)if is a sequence in such that for all and as , then for all .Then the fixed point problem of has a solution; that is, there exists such that .

Proof. Following the proof of Theorem 10, we know that is a Cauchy sequence in the complete metric space . Then, there exists such that as .
On the other hand, from (22) and hypothesis (iii), we have Now, using the triangular inequality, (9) and (30), we get for all . According to the proof of Theorem 10, we obtain that for all . Therefore, we have for all . Letting in above relation, we obtain that ; that is, . Therefore, the fixed point problem of has a solution. This completes the proof.

Example 13. Let , and the metric is defined by for all . Define and by We can prove that is an --contraction mapping, where is defined by for all and is also -admissible. It is easy to see that there exist such that . Moreover, condition (iii) in Theorem 12 holds. Therefore, all hypotheses of Theorem 12 hold and thus Theorem 12 assures the existence of fixed points of which are points and .

Remark 14. Theorem 10 cannot be used in case of Example 13 because is not continuous. Also, the Banach contraction principle cannot be used since does not satisfy Banach contractive condition.

We obtain that Theorems 10 and 12 do not claim the uniqueness of fixed point. To assure the uniqueness of the fixed point, we will consider the following hypotheses.: for all fixed points of mapping .: for all there exists such that and .

Theorem 15. Adding condition or to the hypotheses of Theorem 10 (resp., Theorem 12) we obtain uniqueness of the fixed point of .

Proof. Suppose that and are two fixed points of . If condition holds, then we get the uniqueness of the fixed point of from (9). So we only show that the case of holds. From condition , there exists such that Since is -admissible, from (35), we get for all . From (36) and (9), we have for all . Proceeding inductively, we get for all . Letting in previous inequality, we get Similarly, using (36) and (9), we get By the uniqueness of the limit of a sequence , we have . This finishes the proof.

Remark 16. Since a Banach contraction mapping is an --contraction mapping, the Banach contraction principle can be considered as a corollary of our main results.

3. Ulam-Hyers Stability and Well-Posedness Results through the Fixed Point Problems

First we give the notion of Ulam-Hyers stability and well-posedness in sense of a fixed point problem (see also [7]).

Definition 17. Let be a metric space and let be a mapping. The fixed point problem is called Ulam-Hyers stable if and only if there exists such that, for each and for each called an -solution of the fixed point equation (41), that is, satisfies the inequality there exists a solution of (41) such that

Definition 18. Let be a metric space and let be a mapping. The fixed point problem of is said to be well-posed if it satisfies the following conditions: (i) has a unique fixed point in ;(ii)for any sequence in such that , one has .

Theorem 19. Let be a complete metric space. Suppose that all the hypotheses of Theorem 15 hold. Then the following assertion holds: (a)if for all -solutions of the fixed point equation (41), then the fixed point problem is Ulam-Hyers stable;(b)if for all , where with and is a fixed point of , then the fixed point problem is well-posed.

Proof. According to the proof of Theorem 15, we know that has a unique fixed point and so let be a unique fixed point of .
Here, we show that the fixed point problem of is Ulam-Hyers stable under the hypothesis in (a). Let and be a solution of (42); that is, It is obvious that the unique fixed point of is also a solution of (42). From hypothesis in (a), we get . Now we have This implies that Consequently, the fixed point problem of is Ulam-Hyers stable.
Next, we prove that the fixed point problem of is well-posed under the assumption in (b). Let be a sequence in such that . From assumption in (b), we get for all . Now, we obtain that for all . This implies that for all . Since , we have and so the fixed point problem is well-posed.

Interesting Problems. Consider the following.(i)In Theorem 15, can we replace conditions and by other conditions or more general conditions?(ii)In Theorem 19, can we drop some condition in (a) and (b)?(iii)In Theorem 19, can we prove other types of stability of fixed point problem?(iv)Can we extend the result in this paper to other spaces such as complex valued metric space, partial metric space, -metric space, or circular metric space?

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgment

The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. The second author would like to thank the Thailand Research Fund and Thammasat University for financial support, under Grant no. TRG5780013, during the preparation of this manuscript. Moreover, the authors thank the editors and referees for their insightful comments.