Ulam stability of radical functional equation in the sense of Hyers, Rassias and Gavruta

The aim of this paper is to obtain the general solution of a radical functional equation of the form s(x+y+2√xy)=s(x)+s(y) and investigate its generalized Hyers-Ulam-Rassias stability. Further, we extend the results pertinent to D.H. Hyers, Th.M. Rassias and J.M. Rassias. We also provide counter-examples for critical values relevant to Hyers-Ulam-Rassias stability, Ulam-Gavruta-Rassias stability and J.M. Rassias stability controlled by mixed product-sum of powers of norms.

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Introduction

The stability of functional equations is an interesting topic that has been dealt for the last six decades. In mathematics, a stipulation in which a slight disturbance in a system does not create a considerable disturbing consequence on that system. An equation is said to be stable if a slightly different solution is close to the exact solution of that equation. In mathematical models of physical problems, the deviations in measurements will result with errors and these deviations can be dealt with the stability of equations. Hence the stability of equations is essential in mathematical models. A stable solution will be sensible in spite of such deviations. An interesting and eminent talk given by S.M. Ulam [28] in 1940, inspired to study the investigation of stability of functional equations. He posed the following question pertaining to the stability of homomorphisms in groups: Let 𝒳𝒳 be a group and 𝒴𝒴 be a metric group with metric 𝑑𝑑(•,•). Given 𝜖𝜖 ≥ 0 does there exist a 𝛿𝛿 ≥ 0 such that if a function 𝑓𝑓: 𝒳𝒳 → 𝒴𝒴 fulfills 𝑑𝑑(𝑓𝑓(𝑥𝑥𝑦𝑦), 𝑓𝑓(𝑥𝑥)𝑓𝑓(𝑦𝑦) ≤ 𝛿𝛿 for all 𝑥𝑥, 𝑦𝑦 ∈ 𝒳𝒳, then there exists a homomorphism 𝑔𝑔: 𝒳𝒳 → 𝒴𝒴 with 𝑑𝑑(𝑓𝑓(𝑥𝑥), 𝑔𝑔(𝑥𝑥)) ≤ 𝜖𝜖 for all 𝑥𝑥 ∈ 𝒳𝒳? If the answer is confirmative, then the functional equation for homomorphism is said to be stable. The foremost answer to the question of S.M. Ulam was provided by D.H. Hyers [12] . He brilliantly answered the question of Ulam by considering 𝒳𝒳 and 𝒴𝒴 as Banach spaces. The result of Hyers is stated in the following celebrated theorem. The stability resulted provided by Hyers in the following theorem is referred as Hyers-Ulam stability.

Theorem 1.1. (D.H. Hyers [12] )Let 𝑓𝑓: 𝒳𝒳 → 𝒴𝒴 be a mapping between Banach spaces such that

for all 𝑥𝑥, 𝑦𝑦 ∈ 𝒳𝒳 and for some 𝛿𝛿 > 0, then the limit

exists for each 𝑥𝑥 ∈ 𝒳𝒳. Then there exist a unique additive mapping 𝐴𝐴: 𝒳𝒳 → 𝒴𝒴 such that

for all 𝑥𝑥 ∈ 𝒳𝒳. In addition if 𝑓𝑓(𝑡𝑡𝑥𝑥) is continuous in 𝑡𝑡 for each fixed 𝑥𝑥 ∈ 𝒳𝒳, then 𝐴𝐴 is linear.

After Hyers gave a positive answer to Ulam's question, a huge number of papers have been published in association with various simplifications of Ulam's problem and Hyers theorem. Hyers theorem was generalized by T. Aoki [1] in 1950 for additive mappings. Since there is no elucidation for the boundedness of Cauchy difference 𝑓𝑓(𝑥𝑥 + 𝑦𝑦) -𝑓𝑓(𝑥𝑥) -𝑓𝑓(𝑦𝑦)in the expression of (1.1), in the year 1978, Th.M. Rassias [26] tried to weaken the stipulation for the Cauchy difference and thrived in proving what is now known to be the Hyers-Ulam-Rassias stability for the Additive Cauchy Equation. This jargon is reasonable because the theorem of Th.M. Rassias has strongly persuaded many mathematicians studying stability problems of functional equation. During 1982-1989, J.M. Rassias ( [18][19][20] ) provided a further generalization of the result of D.H. Hyers and established a theorem using weaker conditions controlled by a product of different powers of norms and this type of stability is termed as Ulam-Gavruta-Rassias stability.

A generalized and modified form of the theorem evolved by Th.M. Rassias was promoted by P. Gavruta [10] who replaced the unbounded Cauchy difference by motivating into study a general control function within the viable approach designed by Th.M. Rassias and this type of stability is celebrated as Generalized Hyers-Ulam stability.

The investigation of stability of functional equations involving with the mixed type product-sum of powers of norms is introduced by K. Ravi et al. [23] is called J.M. Rassias stability controlled by mixed product-sum of powers of norms.

For the last four decades, many mathematicians have investigated various stability results of several functional equations like additive, quadratic, cubic, quartic, quintic, sextic, septic, octic and nonic, functional equations (refer [2-4, [7, 8, 14, 16, 17, 21, 22, 29, 30] ). There are many interesting results concerning this problem and many research monographs are also available in functional equations, one can see ( [5,6,13,15,27] ). K. Ravi and P. Narasimman [24] obtained the generalized Hyers-Ulam stability of a radical reciprocal-quadratic functional equation

𝑓𝑓(𝑥𝑥)+𝑓𝑓(𝑦𝑦) (1.4) in Banach spaces. The reciprocal-quadratic function 𝑓𝑓(𝑥𝑥) = 𝑐𝑐 𝑥𝑥 2 is a solution of the functional equation (1.4). K. Ravi et al. [25] investigated the generalized Hyers-Ulam stability of a generalized radical-quadrtic functional equation

in Felbin's type fuzzy normed linear spaces.

In this paper, we introduce a new radical type functional equation of the form 𝑠𝑠�𝑥𝑥 + 𝑦𝑦 + 2�𝑥𝑥𝑦𝑦� = 𝑠𝑠(𝑥𝑥) + 𝑠𝑠(𝑦𝑦).

The radical function 𝑠𝑠(𝑥𝑥) = √𝑥𝑥 is a solution of the functional equation (1.6). We achieve the general solution of the functional equation (1.6) and investigate its generalized Hyers-Ulam stability, Hyers-Ulam stability, Hyers-Ulam-Rassias stability, Ulam-Gavruta-Rassias stability and J.M. Rassias stability controlled by mixed product-sum of powers of norms. We also prove that the functional equation (1.6) is not stable for critical values associated with the results of Th.M. Rassias and J.M. Rassias via counter-examples. Throughout this paper, let us assume that 𝑋𝑋 be the space of non-negative real numbers.

Preliminaries

In this section, we introduce the following definition and theorem in order to proceed further to our main results.

Definition 2.1. A mapping 𝑠𝑠: 𝑋𝑋 → ℝ is called square root mapping, if the functional equation (1.6) holds for all 𝑥𝑥, 𝑦𝑦 ∈ 𝑋𝑋. Note that the mapping 𝑠𝑠 is called square root mapping because the following algebraic identity

holds for all 𝑥𝑥, 𝑦𝑦 ∈ 𝑋𝑋 and 𝑠𝑠 satisfies the functional equations

2) for all 𝑥𝑥 ∈ 𝑋𝑋 and 𝑛𝑛 ∈ ℕ. Theorem 2.2. Let 𝑠𝑠: 𝑋𝑋 → ℝ be a square root mapping satisfying (1.6). Then 𝑠𝑠 satisfies the general functional equations (2.1) and ( 2.2) for all 𝑥𝑥 ∈ 𝑋𝑋 and 𝑛𝑛 ∈ ℕ.

Proof. Letting 𝑥𝑥 = 0 and 𝑦𝑦 = 0 in (1.6), we obtain 𝑠𝑠(0) = 0.

for all 𝑥𝑥 ∈ 𝑋𝑋. Now, letting 𝑥𝑥 as 𝑥𝑥 4 in (2.6), we obtain𝑠𝑠(2 -6 𝑥𝑥) = 2 -3 𝑠𝑠(𝑥𝑥), for all 𝑥𝑥 ∈ 𝑋𝑋. Proceeding further and using induction on a positive integer 𝑛𝑛, we obtain (2.2), which completes the proof.

General Solution of Functional Equation (1.6)

In this section, we obtain the general solution of functional equation (1.6) in the setting of space of real numbers.

Generalized Hyers-Ulam Stability of Equation (1.6)

In this section, we investigate the generalized Hyers-Ulam stability of the functional equation (1.6) in the setting of space of real numbers. We also extend the stability results in the spirit of Hyers, Th.M. Rassias and J.M. Rassias.

Theorem 4.1. Let 𝜙𝜙: 𝑋𝑋 × 𝑋𝑋 → ℝ be a mapping satisfying

for all 𝑥𝑥, 𝑦𝑦 ∈ 𝑋𝑋. Let 𝑓𝑓: 𝑋𝑋 → ℝ be a mapping such that

for all 𝑥𝑥, 𝑦𝑦 ∈ 𝑋𝑋. Then there exists a unique square root mapping 𝑠𝑠: 𝑋𝑋 → ℝ which satisfies (1.6) and the inequality

for all 𝑥𝑥 ∈ 𝑋𝑋. The mapping 𝑠𝑠(𝑥𝑥) is defined by This shows that the sequence {2

for all 𝑥𝑥, 𝑦𝑦 ∈ 𝑋𝑋. Let 𝑓𝑓: 𝑋𝑋 → ℝ be a mapping such that (4.2) holds for all 𝑥𝑥, 𝑦𝑦 ∈ 𝑋𝑋. Then there exists a unique square root mapping 𝑠𝑠: 𝑋𝑋 → ℝ which satisfies (1.6) and the inequality

|𝑓𝑓(𝑥𝑥) -𝑠𝑠(𝑥𝑥)| ≤ Φ(𝑥𝑥, 𝑥𝑥)

for all 𝑥𝑥 ∈ 𝑋𝑋. The mapping 𝑠𝑠(𝑥𝑥) is defined by in (4.9), multiplying by 2 and summing the resulting inequality with (4.9), one gets

for all 𝑥𝑥 ∈ 𝑋𝑋. Proceeding further and using induction on a positive integer 𝑛𝑛, we conclude that

for all 𝑥𝑥 ∈ 𝑋𝑋. The rest of the proof is obtained by similar arguments as in Theorem 4.1. Proof. By choosing 𝜙𝜙(𝑥𝑥, 𝑦𝑦) = 𝑐𝑐 3 �|𝑥𝑥| 𝛼𝛼 |𝑦𝑦| 𝛼𝛼 + (|𝑥𝑥| 2𝛼𝛼 + |𝑦𝑦| 2𝛼𝛼 )�, for all 𝑥𝑥, 𝑦𝑦 ∈ 𝑋𝑋 in Theorems 4.1 and 4.2, the proof is obtained.

Counter-Examples for Singular Cases

In this section, we present counter-examples to prove that the functional equation (1.6) isnot stable for singular cases in Corollaries 4.4, 4.5 and 4.6.

Using the idea of Z. Gajda [9] , we construct the following counter-example to prove that the Corollary 4.4 is false for 𝑝𝑝 =

Define a mapping 𝑓𝑓: 𝑋𝑋 → ℝ by

for all 𝑥𝑥 ∈ 𝑋𝑋, where the function 𝜑𝜑: 𝑋𝑋 → ℝ is given by 𝜑𝜑(𝑥𝑥) = � 𝜇𝜇√𝑥𝑥, for all 𝑥𝑥, 𝑦𝑦 ∈ (0,1) 𝜇𝜇, otherwise.

Then the function 𝑓𝑓 serves as a counter-example for 𝑝𝑝 = (5.4) for all 𝑥𝑥 ∈ 𝑋𝑋.

, then the left-hand side of (5.3) is less than 6𝜇𝜇. Now, suppose that0

. Then there exists a positive integer 𝑘𝑘 such that Therefore, the inequality (5.3) holds true. We claim that the square root functional equation (1.6) is not stable for 𝑝𝑝 = Inspired by the counter-example provided by P. Gavruta in [11] , we present the following example which establishes that the Corollary 4.5 is false for 𝜌𝜌 = 𝑎𝑎 + 𝑏𝑏 = -𝑎𝑎� in (5.12), it follows that

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