# Hyers–Ulam Stability Results for a Functional Inequality ofs,t-Type in Banach Spaces

Authors: Raweerote Suparatulatorn; Tanadon Chaobankoh
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

We introduce an additives,t-functional inequality where s and t are nonzero complex numbers with 2s+t<1. Using the direct method and the fixed point method, we give the Hyers–Ulam stability of such functional inequality in Banach spaces.


---

## Body

## 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 A when any function in A approximately satisfies such equation. One of the well-known functional equations is the (additive) Cauchy functional equation fa+b=fa+fb 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 ρ-functional inequalities (s-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 denoteℂ, ℕ, and ℝ+ the set of complex numbers, the set of positive integers and the set of positive real numbers, respectively, and let ℕ0=ℕ∪0 and ℝ0+=ℝ+∪0.Now, lets,t∈ℂ\0 such that 2s+t<1. Yun and Shin [24] investigated the additive s,t-functional inequality:
(1)2fa+b2−fa−fb≤sfa+b+fa−b−2fa+tfa+b−fa−fb,while Park [25] proposed the additive s,t-functional inequality:
(2)fa+b−fa−fb≤sfa+b+fa−b−2fa+t2fa+b2−fa−fb,and provided the Hyers–Ulam stability results in a Banach space.In this article, motivated by thoses,t-type inequalities mentioned above, we introduce the additive s,t-functional inequality:
(3)2fa+b2+fa−b−2fa≤sfa+b+fa−b−2fa+tfa+b−fa−fb.We first investigate the Hyers–Ulam stability of such functional inequality using the direct method in Section2. 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 s,t-type functional inequalities generalize s-type functional inequalities, our results simply extend existing Hyers–Ulam stability results for functional inequalities of s-type in the literature. These results span alongside those regarding other s,t-type functional inequalities.Throughout this article, letX and B be a normed space and a Banach space, respectively, and let s,t∈ℂ\0 such that 2s+t<1. For convenience, we also require the following classes of mappings:
(4)F0X,B≔g:X⟶B:g0=0,AX,B≔g:X⟶B:gsatisfies1.1,A0X,B≔F0X,B∩AX,B.


## 2. Stability Results: Direct Method

In this section, the stability results of the additives,t-functional inequality (3) are proposed by using the direct method. We begin with the lemma showing that any map g in AX,B is additive.Lemma 1.
Ifg∈AX,B, then g is additive.Proof.
Takinga=b=0 into (3), we obtain that 1−tg0≤0. However, t<1 implies that g0=0. Also, if we let b=0 in (3), then
(5)ga=2ga2,for all a∈X. From (3) and (5),
(6)1−sga+b+ga−b−2ga≤tga+b−ga−gb,for all a,b∈X. Next, taking c=a+b and d=a−b in (3), we have that
(7)1−sgc+gd−2gc+d2≤tgc−gc+d2−gc−d2.
Then, from (5),
(8)1−sgc+d−gc−fd≤t2gc+d+gc−d−2gc,for all c,d∈X. Applying (6) and (8),
(9)1−s2ga+b−ga−gb≤t22ga+b−ga−gb,for all a,b∈X. Finally, since 2s+t<1, we obtain that g is additive.We are now ready to present the main result.Theorem 2.
Letφ:X×X⟶ℝ0+ be a map such that
(10)Φa,b≔∑j=0∞2jφ2−ja,2−jb<∞,for all a,b∈X. For any f∈F0X,B satisfying
(11)2fa+b2+fa−b−2fa≤sfa+b+fa−b−2fa+tfa+b−fa−fb+φa,b,for all a,b∈X, there exists a unique F∈A0X,B such that
(12)fa−Fa≤Φa,0,for all a∈X.Proof.
We first letb=0 in (11). This implies that
(13)2fa2−fa≤φa,0,for all a∈X. It follows that for any m,l∈ℕ0 with m>l,
(14)2lf2−la−2mf2−ma≤∑j=lm−12jf2−ja−2j+1f2−j+1a≤∑j=lm−12jφ2−ja,0,for all a∈X. The completeness of B confirms that the Cauchy sequence 2kf2−ka is convergent for any a∈X. Define F:X⟶B by
(15)Fa=limk⟶∞2kf2−ka,for all a∈X. Clearly, F∈F0X,B. Next, choosing l=0 and letting m⟶∞ in (14), we have that F satisfies (12). Then, from (10) and (11),
(16)2Fa+b2+Fa−b−2Fa=limn⟶∞2n2f2−n+1a+b+f2−na−b−2f2−na≤slimn⟶∞2nf2−na+b+f2−na−b−2f2−na+tlimn⟶∞2nf2−na+b−f2−na−f2−nb+limn⟶∞2nφ2−na,2−nb=sFa+b+Fa−b−2Fa+tFa+b−Fa−Fb,for all a,b∈X. By Lemma 1, F∈A0X,B. Finally, let G be another map in A0X,B satisfying (12). Then, for any a∈X,
(17)Fa−Ga=2pF2−pa−2pG2−pa≤2pF2−pa−2pf2−pa+2pG2−pa−2pf2−pa≤2p+1Φ2−pa,0.
Therefore,Fa−Ga⟶0 as p⟶∞. The uniqueness of F follows.Corollary 3.
Forr,ϑ∈ℝ0+ with r>1, if f∈F0X,B satisfying
(18)2fa+b2+fa−b−2fa≤sfa+b+fa−b−2fa+tfa+b−fa−fb+ϑar+br,for all a,b∈X, then there exists a unique F∈A0X,B such that
(19)fa−Fa≤2rϑ2r−2ar,for all a∈X.Proof.
Letφa,b=ϑar+br for all a,b∈X in Theorem 2. The result immediately follows.Theorem 4.
Letφ:X×X⟶ℝ0+ be a map satisfying
(20)Ψa,b≔∑j=1∞2−jφ2ja,2jb<∞,
for alla,b∈X, and let f∈F0X,B satisfy (11). Then, there exists a unique F∈A0X,B such that
(21)fa−Fa≤Ψa,0,for all a∈X.Proof.
It follows from (13) that fa−1/2f2a≤1/2φ2a,0 for all a∈X. Then, for m,l∈ℕ0 with m>l,(22)2−lf2la−2−mf2ma≤∑j=lm−12−jf2ja−2−j+1f2j+1a≤∑j=l+1m2−jφ2ja,0,for all a∈X. Now, let a∈X. It follows from the completeness of B that 2−nf2na is convergent in X. Next, define a map F:X⟶B by
(23)Fa=limn⟶∞2−nf2na,for all a∈X. Choosing l=0 and taking m⟶∞ in (22), we have that F satisfies (21). The rest is similar to the Proof of Theorem 2.Letφa,b=ϑar+br for all a,b∈X. The following result is straightforward.Corollary 5.
Letr,ϑ∈ℝ0+ with r<1. If f∈F0X,B satisfies (18), then there exists a unique F∈A0X,B such that
(24)fa−Fa≤2rϑ2−2rar,for all a∈X.


## 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 X,d be a complete generalized metric space, and let L:X⟶X be a strict Lipschitz contraction with the Lipschitz constant α<1. Then, for a∈X, either
(a)
dLna,Ln+1a=∞ for all n∈ℕ0 or(b)
dLna,Ln+1a<∞ for all n≥n0 for some n0∈ℕ; Lna⟶b∗ where b∗ is a unique fixed point of L in X0≔b∈X∣dLn0a,b<∞ and db,b∗≤1/1−αdb,Lb for all b∈X0Theorem 7.
Letφ:X×X⟶ℝ0+ be a function such that
(25)φa2,b2≤L2φa,b,for all a,b∈X for some L∈ℝ0+ with L<1. Then, for f∈F0X,B satisfying (11), there exists a unique F∈A0X,B such that
(26)fa−Fa≤11−Lφa,0,for all a∈X.Proof.
Firstly, let us equipF0X,B with the generalized metric d defined by
(27)dg,h=infμ∈ℝ+:ga−ha≤μφa,0,foralla∈X.Then, from [28], F0X,B,d is complete. Next, define a map J:F0X,B⟶F0X,B by
(28)Jga=2ga2,for all a∈X. Let g,h∈F0X,B where dg,h=ε. Then,
(29)ga−ha≤εφa,0,for all a∈X. Consequently,
(30)Jga−Jha=2ga2−2ha2≤2εφa2,0≤2εL2φa,0=Lεφa,0,for all a∈X. Then, dJg,Jh≤Lε which means that
(31)dJg,Jh≤Ldg,h,for all g,h∈F0X,B. By (13), we have that df,Jf≤1.Now, leta∈X. From Proposition 6, there exists F:X⟶B satisfying the following:
(i)
F is a unique fixed point of J, i.e., Fa=2Fa/2 for all a∈X(ii)
dJlf,F⟶0 as l⟶∞(32)df,F≤11−Ldf,Jf.It follows that(33)fa−Fa≤μφa,0(a)
liml⟶∞2lf2−la=Fa and
(34)fa−Fa≤11−Lφa,0Using the same method as in theproof of Theorem 2, we can prove that F∈A0X,B.Corollary 8.
Letr,ϑ∈ℝ0+ with r>1.Iff∈F0X,B satisfies (18), then there exists a unique F∈A0X,B such that
(35)fa−Fa≤2rϑ2r−2ar,for all a∈X.Proof.
By takingL=21−r and φa,b=ϑar+br for all a,b∈X in Theorem 7, the result follows.Theorem 9.
Letφ:X×X⟶ℝ0+ be a map such that
(36)φa,b≤2Lφa2,b2,for all a,b∈X, for some L∈ℝ0+ with L<1. Then, for any f∈F0X,B satisfying (11), there exists a unique F∈A0X,B such that
(37)fa−Fa≤L1−Lφa,0,for all a∈X.Proof.
We first consider the complete metric spaceF0X,B,d given as in the proof of Theorem 7. Define a mapping J:F0X,B⟶F0X,B by
(38)Jga=12g2a,for all a∈X. It follows from (13) and (36) that
(39)fa−12f2a≤12φ2a,0≤Lφa,0,for all a∈X. As in the proof of Theorem 2 and Theorem 7, there exists a unique F∈A0X,B satisfying (37).Corollary 10.
Letr,ϑ∈ℝ0+ with r<1, and let f∈F0X,B be a map satisfying (18). Then, there exists a unique F∈A0X,B such that
(40)fa−Fa≤2rϑ2−2rar,for all a∈X.Proof.
TakingL=2r−1 and φa,b=ϑar+br for all a,b∈X 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 l2 equipped with the 2-norm. Define f:l2⟶l2 by
(41)fa=a1+a2,a1−a2,2a3,0,0,0,⋯,for all a=a1,a2,a3,⋯∈l2. Let ϑ=r=2. Then, f∈F0X,B satisfies (18). By Corollary 3, there exists a unique F∈A0X,B such that fa−Fa2≤4a22 for all a∈l2. This example is also valid for the other corollaries in the paper.There could also be others,t-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.



---

*Source: 2195754-2022-04-11.xml*