Abstract
We introduce an additive -functional inequality where and are nonzero complex numbers with Using the direct method and the fixed point method, we give the Hyers–Ulam stability of such functional inequality in Banach spaces.
1. Introduction and Preliminaries
A problem regarding the stability of homomorphisms was mentioned by Ulam [1] in 1940. The first answer was then found by Hyers in [2] which motivating the study of the stability problems of functional equations. We may roughly say that a given functional equation is stable on a class of functions when any function in approximately satisfies such equation. One of the well-known functional equations is the (additive) Cauchy functional equation which is a useful tool in natural and social sciences. The stability of functional equations has been widely acknowledged as Hyers–Ulam stability. It was notably weakened by Rassias in [3] by making use of a direct method. The result was later extended in [4] which uses a general control function instead of the unbounded Cauchy difference. The concept of stability has been also developed for functional inequalities. Recently, Park introduced additive -functional inequalities (-type functional inequalities) and investigated the Hyers–Ulam stability in [5, 6]. Over the last decades, stability of functional equations and functional inequalities have been extensively studied, see [7–13], for example.
Not only the direct method, the fixed point method is also one of the most popular methods of proving the stability of functional equations and functional inequalities. Applications of stability of functional equations in a fixed point theory and in nonlinear analysis were introduced in [14]. It was known that Hyers–Ulam stability results can be derived using fixed point theorems while the latter can often be obtained from the former, see [15–20] and there references.
The Hyers–Ulam stability concept is very useful in many applications (i.e., optimization, numerical analysis, biology, and economics), since it can be very difficult to find the exact solutions for those physical problems. It is remarkably used in the field of differential equations. For some recent works, see [21–23] (and references therein) where the Hyers–Ulam stability results concerning (fractional) stochastic functional differential equations were given.
We denote , , and the set of complex numbers, the set of positive integers and the set of positive real numbers, respectively, and let and
Now, let such that Yun and Shin [24] investigated the additive -functional inequality: while Park [25] proposed the additive -functional inequality: and provided the Hyers–Ulam stability results in a Banach space.
In this article, motivated by those -type inequalities mentioned above, we introduce the additive -functional inequality:
We first investigate the Hyers–Ulam stability of such functional inequality using the direct method in Section 2. Then, in Section 3, we use the fixed point method to prove the Hyers–Ulam stability of such inequality. We also include some example and remarks in the last section. Note that, since -type functional inequalities generalize -type functional inequalities, our results simply extend existing Hyers–Ulam stability results for functional inequalities of -type in the literature. These results span alongside those regarding other -type functional inequalities.
Throughout this article, let and be a normed space and a Banach space, respectively, and let such that For convenience, we also require the following classes of mappings:
2. Stability Results: Direct Method
In this section, the stability results of the additive -functional inequality (3) are proposed by using the direct method. We begin with the lemma showing that any map in is additive.
Lemma 1. If then is additive.
Proof. Taking into (3), we obtain that However, implies that . Also, if we let in (3), then for all From (3) and (5), for all . Next, taking and in (3), we have that Then, from (5), for all . Applying (6) and (8), for all . Finally, since we obtain that is additive.
We are now ready to present the main result.
Theorem 2. Let be a map such that for all For any satisfying for all , there exists a unique such that for all .
Proof. We first let in (11). This implies that for all . It follows that for any with , for all The completeness of confirms that the Cauchy sequence is convergent for any Define by for all . Clearly, Next, choosing and letting in (14), we have that satisfies (12). Then, from (10) and (11), for all . By Lemma 1, Finally, let be another map in satisfying (12). Then, for any , Therefore, as The uniqueness of follows.
Corollary 3. For with , if satisfying for all then there exists a unique such that for all .
Proof. Let for all in Theorem 2. The result immediately follows.
Theorem 4. Let be a map satisfying for all and let satisfy (11). Then, there exists a unique such that for all .
Proof. It follows from (13) that for all . Then, for with for all . Now, let It follows from the completeness of that is convergent in . Next, define a map by for all . Choosing and taking in (22), we have that satisfies (21). The rest is similar to the Proof of Theorem 2.
Let for all The following result is straightforward.
Corollary 5. Let with If satisfies (18), then there exists a unique such that for all .
3. Stability Results: Fixed Point Method
In this section, we apply the fixed point method to present the Hyers–Ulam stability of the functional inequality (3).
We first state a useful tool in the field of fixed point theory.
Proposition 6. [26, 27]. Let be a complete generalized metric space, and let be a strict Lipschitz contraction with the Lipschitz constant Then, for either (a) for all or(b) for all for some ; where is a unique fixed point of in and for all
Theorem 7. Let be a function such that for all for some with Then, for satisfying (11), there exists a unique such that for all .
Proof. Firstly, let us equip with the generalized metric defined by
Then, from [28], is complete. Next, define a map by for all . Let where . Then, for all . Consequently, for all Then, which means that for all By (13), we have that
Now, let From Proposition 6, there exists satisfying the following: (i) is a unique fixed point of , i.e., for all (ii) as
It follows that (a) and
Using the same method as in the proof of Theorem 2, we can prove that
Corollary 8. Let with If satisfies (18), then there exists a unique such that for all .
Proof. By taking and for all in Theorem 7, the result follows.
Theorem 9. Let be a map such that for all , for some with Then, for any satisfying (11), there exists a unique such that for all .
Proof. We first consider the complete metric space given as in the proof of Theorem 7. Define a mapping by for all . It follows from (13) and (36) that for all . As in the proof of Theorem 2 and Theorem 7, there exists a unique satisfying (37).
Corollary 10. Let with and let be a map satisfying (18). Then, there exists a unique such that for all .
Proof. Taking and for all in Theorem 9, the result follows.
4. Conclusions and Final Remarks
We have obtained several Hyers–Ulam stability results for the functional inequality (3) using the direct method and the fixed point method. We now discuss some example for Theorem 2 (via Corollary 3). Consider the sequence space equipped with the -norm. Define by for all . Let . Then, satisfies (18). By Corollary 3, there exists a unique such that for all . This example is also valid for the other corollaries in the paper.
There could also be other -type functional inequalities to be investigated, and thus, of course, their stability results to be examined. Moreover, these functional inequalities can still be possibly generalized in several ways.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was partially supported by the Chiang Mai University and by the Faculty of Science, Chiang Mai University, Thailand.