Generalized Ulam-Hyers Stability, Well-Posedness, and Limit Shadowing of Fixed Point Problems for α-β-Contraction Mapping in Metric Spaces

We study the generalized Ulam-Hyers stability, the well-posedness, and the limit shadowing of the fixed point problem for new type of generalized contraction mapping, the so-called α-β-contraction mapping. Our results in this paper are generalized and unify several results in the literature as the result of Geraghty (1973) and the Banach contraction principle.


Introduction and Preliminaries
The stability problem of functional equations, first initial from a question of Ulam [1] in 1940, concerns the stability of group homomorphisms. In next year, Hyers [2] first gives some partial answer of Ulam's question for Banach spaces and then this type of stability is called the Ulam-Hyers stability. This opened an avenue for further study and development of analysis in this field. Subsequently, many researchers have studied and extended Ulam-Hyers stability in many ways. In particular, there are a number of results that studied and extended Ulam-Hyers stability for fixed point problems as Bota et al. [3], Bota-Boriceanu and Petrusel [4], Brzdȩk et al. [5], Brzdek and Cieplinski [6,7], Cadariu et al. [8], Lazǎr [9], Rus [10], and F. A. Tise and I. C. Tise [11].
On the other hand, the notion of well-posedness and limit shadowing property of a fixed point problem have evoked much interest to many researchers, for example, De Blassi and Myjak [12], Reich and Zaslavski [13], Lahiri and Das [14], and Popa [15,16].
The Scientific World Journal Then, is -admissible.
Samet et al. [17] established fixed point theorems for some type of generalized contraction mapping by using the concept of -admissible mapping. Also, they applied these results to derive fixed point theorems in partially ordered metric spaces. As application, they studied the ordinary differential equations via the main results. Several researchers studied and improved contraction mappings via the concept ofadmissible mapping in metric spaces and other spaces (see [18][19][20][21][22] and references therein).
The first aim of this work is to introduce new type of contraction mapping which generalized several types of mappings in the literature as Geraghty-type contraction mapping [23] and Banach contraction mapping [24]. Also, we establish some existence and uniqueness of fixed point theorems for such mappings in metric spaces by using the concept of -admissible mapping. Our second purpose is to present generalized Ulam-Hyers stability, well-posedness, and limit shadowing of fixed point problems for this mapping in metric spaces.
For any sequence { } of nonnegative real numbers, we have This class is first introduced by Geraghty [23] in 1973. Afterwards, there are many results about fixed point theorems by using such function in this class in many spaces with different contractions; for details we refer the readers to [25][26][27][28] and references therein.
The following are examples of some functions in Υ.
First we give the following definition as a generalization of Banach contraction mappings.
Remark 6. It is easy to check that an --contraction mapping reduces to a Geraghty-type contraction mapping if ( , ) = 1 for all , ∈ .
Next, we introduce the transitive mapping which is useful for our main result. for , , ∈ for which Our first main result is the following.

Theorem 8. Let ( , ) be a complete metric space and : →
an --contraction mapping satisfying the following conditions: (iv) is continuous.
Then the fixed point problem of has a solution; that is, there exists * ∈ such that * = ( * ).
Proof. Let 0 ∈ such that ( 0 , ( 0 )) ≥ 1 (such a point exists from condition (iii)). Define the sequence { } in by If = −1 for some ∈ N, then = ( ); that is, is a fixed point of and thus the proof ends. Therefore, we may assume that Since is -admissible and ( 0 , 1 ) = ( 0 , ( 0 )) ≥ 1, we get ( 1 , 2 ) = ( ( 0 ), ( 1 )) ≥ 1. By induction, we get For ∈ N, we have The Scientific World Journal 3 This implies that for all ∈ N. Therefore, the sequence { ( −1 , )} is strictly decreasing and so ( −1 , ) → as → ∞ for some ≥ 0. Next, we claim that = 0. Assume on the contrary that > 0. On taking limit as → ∞ in (12), we obtain that Since ∈ Υ, we have lim → ∞ ( −1 , ) = 0, which is a contradiction. Therefore, = 0 and thus Next, we show that { } is a Cauchy sequence. On the contrary, assume that { } is not a Cauchy sequence. Then there exists > 0 and subsequence of integers and with > ≥ such that for all ∈ N. Further, corresponding to , we can choose in such a way that it is the smallest integer with > ≥ and satisfying (15). Then we have From (16) and the triangle inequality, we have Letting → ∞ and using (14), we have Since is transitive and > , we can conclude that Now we have This implies that That is, ≤ ( ( , )) < 1.
On taking limit as → ∞ and using (14) and (18) Since ∈ Υ, we have lim → ∞ ( , ) = 0 which contradicts with (18). Therefore, { } is a Cauchy sequence. By the completeness of , we get lim → ∞ = * 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.
In the next theorem, we omit the continuity hypothesis of by adding some condition. an --contraction mapping satisfying the following conditions: for all ∈ N and → ∈ as → ∞, then ( , ) ≥ 1 for all ∈ N.
Then the fixed point problem of has a solution; that is, there exists * ∈ such that * = ( * ).
We obtain that Theorems 8 and 9 do not claim the uniqueness of fixed point. To assure the uniqueness of the fixed point, we will add some properties. Proof. Suppose that * and * are two fixed points of . If condition (H 0 ) holds, then we get the uniqueness of the fixed point of from (6). So we only show that the case of (H 1 ) holds. From condition (H 1 ), there exists ∈ such that ( * , ) ≥ 1, ( * , ) ≥ 1.
Using the fact that ∈ Υ, we obtain that which is a contradiction. Therefore, we can conclude that Remark 11. Since Geraghty-type contraction mapping is an --contraction mapping, Geraghty's fixed point results [23] can be considered as a corollary of our main results. Also, the Banach contraction principle [24] can be derived from our main results.

Generalized Ulam-Hyers Stability, Well-Posedness, and Limit Shadowing Results through the Fixed Point Problems
For the beginning of this section, we give the notion of generalized Ulam-Hyers stability in sense of a fixed point problem and also give the notion of well-posedness and limit shadowing property for fixed point problem.
The Scientific World Journal there exists a solution * ∈ of (39) such that Remark 13. If the function is defined by ( ) = for all ≥ 0, where > 0, then the fixed point equation (39) is said to be Ulam-Hyers stable.
Definition 14 (see [12]). Let ( , ) be a metric space and : → 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 lim → ∞ ( , ( )) = 0, one has lim → ∞ ( , * ) = 0.
Definition 15. Let ( , ) be a metric space and : → a mapping. We say that the fixed point problem of has the limit shadowing property in if, for any sequence { } in satisfying lim → ∞ ( , ( )) = 0, it follows that there exists ∈ such that lim → ∞ ( ( ), ) = 0.
Concerning the generalized Ulam-Hyers stability, wellposedness, and limit shadowing property of the fixed point problem for a self-map of a complete metric space satisfying the conditions of Theorem 10, we have the following results.  (c) if ( * , ) ≥ 1 for all ∈ N such that { } is sequence in in which lim → ∞ ( , ( )) = 0 and * is a fixed point of , then the fixed point problem of has the limit shadowing property in .
Proof. From the proof of Theorem 10, we obtain that has a unique fixed point and so let * be a unique fixed point of .
From the hypothesis in (a), we claim that the fixed point problem of is generalized Ulam-Hyers stability. Let > 0 and * ∈ be a solution of (40); that is, It is obvious that the fixed point * of satisfies inequality (40). From hypothesis in (a), we get ( * , * ) ≥ 1. Now we have This implies that ( * , * ) ≤ ( ( * , * )) ( * , * ) + and then That is, Therefore, It is easy to see that −1 is increasing, continuous at 0 and −1 (0) = 0. Consequently, the fixed point problem of is generalized Ulam-Hyers stability.
(51) Therefore, has the limit shadowing property. (iv) Can we extend the result in this paper to other spaces as cone metric space, complex valued metric space, partial metric space, -metric space, and circular metric space?