Abstract

In this paper, we investigate the Hyers-Ulam stability of mixed type additive and quartic functional equations in Random p-normed spaces by direct and fixed-point method.

1. Introduction

In 1940, the stability problems of functional equations about homomorphism of groups was introduced by Ulam ¹. In 1941, Hyers ² gave an affirmative answer to Ulam’s question for additive groups (under the assumption that groups are Banach spaces). Hyers’ theorem was generalized by Aoki ³ for additive mappings and by Rassias ⁴ for linear mappings by considering an unbounded Cauchy difference for all and Following the same approach as Rassias, Gajda ⁵ gave an affirmative solution of this problem for and also proved that it is possible to solve the Rassias-type Also, in 1994, Rassias generalization theorem was delivered by Gavruta ⁶ who replaced by a control function The paper of Rassias has significantly influenced the development of what we now call the Hyers-Ulam-Rassias stability of functional equations. J.M. Rassias ⁴ followed the modern approach of the Th.M. Rassias ⁷ theorem in which he replaced the factor product of norms instead of sum of norms.

The functional equations

(1.1)

are known as additive functional equation. Each additive solution of a functional equation must be an additive mapping. For Stability of additive, quadratic, cubic and quartic functional equations in random normed spaces, we may refer ^{8, 9, 10, 11, 12}.

Some notions and conventions of the theory of random and random p-normed spaces are taken in our paper as in ^{11, 12, 13, 14, 15}.

Throughout the paper ∆⁺ is the distribution functions space, that is, the space of all mappings V: R ∪ {−∞, ∞} → [0,1], such that F is left continuous and increasing on R,V(0) = 0 and V(+∞) = 1. D⁺⊂ ∆⁺ consisting of all functions V ∈ ∆⁺ for which V(+∞) = 1, where (s) denotes (s) = The space ∆⁺ is partially ordered by the usual point wise ordering of functions, i.e., V ≤ W V(t) ≤ W(t) ∀t ∈ R. The maximal element for ∆⁺ in this order is the distribution function given by

In 2012, SA Mohiuddine et al. ¹⁶ are to present a relationship between three various disciplines the theory of functional equation

(1.2)

Recently, in 2019, Senthil Kumar et al. ¹⁷ proved the general solution of Hyers-Ulam stability of mixed type additive and quartic functional equations of the form

(1.3)

We focus on the ensuing mixed type functional equation derived from additive and quartic mappings. We validate the Hyers-Ulam stability of equation (1.3) in p-normed spaces. It is not hard for one for one that the mixed type function is a solution of the equation (1.3).

In this section, we determine the generalized Hyers-Ulam stability of mixed type additive quartic functional equations in random p-normed space, by using direct method and fixed-point method. Following definitions and notions will be usedto prove our main results:

Definition 1.1 ¹¹ (t-norm) is a continuous triangular norm (briefly t - norm) if T satisfies the following conditions:

i) T is commutative and associative;

ii) T is a continuous;

iii) for all

iv) whenever and for all

Examples of continuous t-norm are and

Recall that, if T is a t-norm and are given numbers in then, is defined by recursively by

is defined as

Definition 1.2. ¹¹. A Random Normed space (briefly RN-space) is a triple , where X is a vector space, T is a continuous t-norm and (for all is denoted by ), satisfying the following conditions:

i) for all if and only if ;

ii) for all and with ;

iii) for all and

Definition 1.3. ¹¹ Let () be a RN-space.

RN1) A sequence in X is said to be convergent to a point if .

RN2) A sequence in X is called a Cauchy sequence if , .

RN3) A RN-space () is said to be complete if every Cauchy sequence in X is convergent.

Definition 1.4. ¹³ Let X be a real linear space with and T be a continuous t-norm. The triple () is called a random p-normed space if a mapping (for all is denoted by ), satisfying the following conditions:

i) for all if and only if ;

ii) for all and ;

iii) for all and

Note that every p-normed space defines a random p-normed space where for all and is the minimum t-norm. This space is called the induced random p-normed space.

Definition 1.5. ¹³ Let () is called a random p-normed space.

1. A sequence in X is said to be convergent to a point if for all and there exists a positive integer N such that whenever

2. A sequence in is called a Cauchy convergent if for all and there exists a positive integer N such that whenever .

3. The random p-normed space () is said to be complete if every Cauchy sequence is convergent to a point in X.

Theorem 1.6. ¹⁴. If () is a random normed space and is a sequence of X such that , then

2. Stability Results for Additive Functional Equation by Using Direct Method

In this section, we investigate the generalized Hyers-Ulam stability problem of the functional equation (1.2), in random p-normed spaces under the minimum t-norm by using direct method.

In this paper, let X be a linear space, () be a random p-normed space and () be a complete random p-normed space. We determine the stability of the additive functional equation defined by

(2.1)

in random p-normed spaces by using Direct method.

Theorem 2.1. Let be amapping such that for some

(2.2)

and , for all and all . If is an odd mapping with such that

(2.3)

for all and all , then there exists a unique additive mapping such that

(2.4)

for all and all .

Proof. Replacing y by and z by x in (2.3), we obtain

(2.5)

for all and all .

Replacing x by in (2.5), we obtain

(2.6)

for all and all .

Since

(2.7)

for all and all .

(2.8)

for all and all .

Replacing x by in (2.8), we get

(2.9)

for all and all with It follows from

that the sequence is a Cauchy in (), and so it converges to some point . We can define a mapping by for all and all . Fix and put in (2.9). Then, we obtain

for all and all . For every

(2.10)

for all and all Taking the limit in (2.10), we obtain

(2.11)

Thus, since s is arbitrary and taking in (2.11), we have

for all and all Thus, the condition (2.4) holds for all and all . If we replace by in (2.3), then

(2.12)

for all and all Letting in (2.12), we find that , for all , which implies for all . Therefore, the mapping is additive. Now, we prove that the additive mapping is unique. Let us assume that there exists another mapping which satisfies (2.4). For fixed and all It follows from (2.4) that

, we have for all . Thus, , for all . Hence, it is completed.

Theorem 2.2. Let be a mapping such that for some

(2.13)

and , for all and all . If be an odd mapping with which satisfies (2.3), then there exists a unique additive mapping such that

(2.14)

for all and all .

Proof. Putting x = z = and in (2.3), we obtain

(2.15)

for all and all

Replacing by in (2.15), we obtain

(2.16)

for all and all

Since

(2.17)

for all and all

From inequality (2.16) and (2.17), we get

(2.18)

for all and all

Replacing x by in (2.18), we get

(2.19)

for all and all with . Then, the sequence is a Cauchy in (), and so it converges to some point . Now, we can define a mapping by , for all and all The remaining part goes through in a similar method to the corresponding Theorem 2.1.

Corollary 2.3. Let be a linear space, () be a random p-normed space and () be a complete random p-normed space. Assume is a positive real number and If is a mapping with which satisfies

(2.20)

for all and all , then there exists a unique additive mapping such that

(2.21)

for all and all .

Proof. Let a mapping be defined by . Then, the proof follows from Theorem 2.1 by . This completes the proof.

Corollary 2.4. Let X be a linear space, () be a random p-normed space and () be a complete random p-normed space. Assume r is a positive real number with and . If is a mapping with which satisfies

(2.22)

for all and all , then there exists a unique mapping such that

(2.23)

for all and all .

Proof. Let a mapping be defined by . Then, the proof follows from Theorem 2.1 and Theorem 2.2 by . This completes the proof.

3. Stability Results for Mixed Type Functional Equations

In this section, we prove the generalized Hyers-Ulam stability of mixed type Additive Quartic functional equations in random p-normed space, by using direct method. For any mapping defined by

(3.1)

Theorem 3.1. Let be a mapping such that for some .

(3.2)

and for all and all . If an even mapping with satisfying

(3.3)

for all and all then there exists a unique quartic mapping such that

(3.4)

for all and all .

Proof. Replacing y by 0, in (3.3), we obtain

(3.5)

for all and all .

Replacing by in (3.5) we obtain

(3.6)

for all and all .

Since

So,

(3.7)

for all and all .

Replacing x by in (3.7), we get

(3.8)

for all and all with . It follows from

that the sequence is a Cauchy in () and so it converges to some point . We can define a mapping by , for all and all Fix and put in (3.8). Then, we obtain

for all and all For every

(3.9)

for all and all Taking the limit in (3.9), we obtain

(3.10)

Thus, since s is arbitrary and taking limit in (3.10), we have

for all and all . Thus, the condition (3.4), holds for all and all If we replace by in (3.3), then

(3.11)

for all and all . Letting in (3.11), we find that for all , which implies , for all . Therefore, the mapping is quartic. Now, we prove that the quartic mapping is unique. Let us assume that there exists another mapping which satisfies (3.3). For fixed and , all . It follows from (3.3) that

, we have for all . Thus, , for all . Therefore, the proof is completed.

Theorem 3.2. Let be a mapping such that for some ,

(3.12)

and for all and all . If an even mapping with which satisfies (3.3), then there exists a unique quartic mapping such that

(3.13)

for all and all .

Proof. Replacing x by and y by 0 in (3.3), we obtain

(3.14)

for all and all .

Replacing by in (3.14), we obtain

(3.15)

for all and all .

Since

(3.16)

for all and all .

From inequality (3.15) and (3.16), we get

(3.17)

for all and all .

Replacing x by in (3.17), we get

(3.18)

for all and all with . Then, the sequence is a Cauchy in (), and so it converges to some point . Now, we can define a quartic mapping by , for all and all . The remaining part goes through in a similar method to the corresponding Theorem 3.1.

Corollary 3.3. Let X be a linear space, () be a random p-normed space and () be complete a random p-normed space. Assume is a positive real number and . If an even mapping with which satisfies

(3.19)

for all and all then there exists a unique quartic mapping such that

(3.20)

for all and all .

Proof. Let a mapping be defined by . Then, the proof follows from Theorem 3.1 by . This completes the proof.

Corollary 3.4. Let be a linear space, () be a random p-normed space and () be a complete random p-normed space. Assume r is a positive real number with and If is a mapping which satisfies

(3.21)

for all and all , then there exists a unique quartic mapping such that

(3.22)

for all and all .

Proof. Let a mapping be defined by . Then, the proof follows from Theorem 3.1 and Theorem 3.2 by . This completes the proof.

Theorem 3.5. Let be a mapping such that for some .

(3.23)

and for all and all If is an odd mapping with such that

(3.24)

for all and all then there exists a unique additive mapping such that

(3.25)

for all and all

Proof. Replacing y by 0 in, (3.24), we obtain

(3.26)

for all and all .

Replacing by in (3.26), we obtain

(3.27)

for all and all .

Since

(3.28)

for all and all .

for all and all .

Replacing x by in (3.29), we get

(3.30)

for all and all with It follows from

that the sequence is Cauchy in (), and so it converges to some point . We can define a mapping by for all and all . Fix and put in (3.30), Then, we obtain

for all and all . For every ,

(3.31)

for all and all . Taking the limit in (3.31), we obtain

(3.32)

Thus, since s is arbitrary and taking in (3.32), we have

for all and all . Thus, the condition (3.25), holds for all and all . If we replace by in (3.24), then

(3.33)

for all and all . Letting in (3.33), we find that , for all , which implies , for all . Therefore, the mapping is additive. Now, we prove that the additive mapping is unique. Let us assume that there exists another mapping which satisfies (3.25). For fixed and , all . It follows from (3.25) that

, we have for all . Thus, for all Hence, the proof is complete.

Theorem 3.6. Let be a mapping such that for some ,

(3.34)

and , for all and all . If is an odd mapping with which satisfies (3.24), then there exists a unique additive mapping such that

(3.35)

for all and all .

Proof. Replacing x by and y by 0 in (3.24), we obtain

(3.36)

for all and all

Replacing x by in (3.36), we obtain

(3.37)

for all and all

Since

(3.38)

for all and all .

From inequality (3.37) and (3.38), we get

(3.39)

for all and all .

Replacing x by in (3.39), we get

(3.40)

for all and all with . Then, the sequence is a Cauchy in , and so it converges to some point . Now, we can define a mapping by , for all and all . The remaining part goes through in a similar method to the corresponding Theorem 3.5.

Corollary 3.7. Let be a linear space, () be a random p-normed space and () be a completerandom p-normed space. Assume is a positive real number and in Z. If is a mapping with which satisfies

(3.41)

for all and all , then there exists a unique mapping such that

(3.42)

for all and all .

Proof. Let a mapping be defined by Then, the proof follows from Theorem 3.5 by . This completes the proof.

Corollary 3.8. Let X be a linear space, () be a random p-normed space and () be complete a random p-normed space. Assume r is a positive real number with and . If is an odd mapping with which satisfies

(3.43)

for all and all , then there exists a unique additive mapping such that

(3.44)

for all and all

Proof. Let a mapping be defined by . Then, the proof follows from Theorem 3.5 and Theorem 3.6 by . This completes the proof.

4. Stability Results by Fixed Point Method

In this section, we give the generalized Hyers-Ulam stability of mixed type additive quartic functional equations in random p-normed spaces. Let us recall that a mapping is a called a metric on a non-empty set X if

i) if and only if

ii)

iii)

for all Before proceeding to the main results in this section, we give the fixed-point theorem which plays an important role in proving our theorems.

Theorem 4.1. ¹⁸. (Alternative fixed-point theorem) Let (,) be a generalized complete metric space and be a strictly contractive function with Lipschitz constant L. Then, for each , either for all non-negative integer or there exists a natural number such that

i) for all ;

ii) the sequence converges to a fixed-point of ;

iii) y is the unique fixed point of in the set ;

iv)

Theorem 4.2. Let ( is denoted by ) be a mapping such that, for some .

(4.1)

for all and all . If is an odd mapping with such that

(4.2)

for all and all then there exists a unique mapping such that

(4.3)

for all and all

Proof. Replacing y by x in (4.2), we obtain

(4.4)

for all and all .

Consider a general metric d on , here be a set of all mappings from X into Y and introduce a generalized metric on as follows:

whereas inf . It is easy to show that () is a complete metric space ¹⁰. Now, let us consider a mapping defined by

for all and for all . Let in and be an arbitrary constant with . Then, we have

for all and all , hence

(4.5)

for all and all , and so, if , then

for all . Then is a strictly contractive self-mapping on with Lipschitz constant L Also, it follows from (4.2) that

(4.6)

for all and all , which implies that

Using Theorem 4.1, there exists a mapping , which is a unique fixed point of in the set such that

for all since Again, it follows from Theorem 4.1 that

which implies

for all and all . Replacing x and y by and in (4.2), respectively,

for all and all . It follows from that . Hence, the mapping is additive. Now, we show that mapping is unique. To prove this, we assume that there exists an additive mapping , which satisfies (4.3). Then, is a fixed point of in However, it follows from Theorem 4.1 that has only one fixed point in Hence, we deduce that

Theorem 4.3. Let be a mapping such that, for some

(4.7)

for all and all If is a mapping with which satisfies (4.2), then there exists a unique mapping such that

(4.8)

for all and all

Proof. Let and d be as in the proof of Theorem 4.2. Then becomes a complete metric space and the mapping defined by

for all and . Then,

for all . Then, is a strictly contractive self-mapping on with Lipschitz constant L It follows from (4.3) that we get

which implies the inequality (4.2) holds for all and all . The remaining assertion goes through in a similar method to the corresponding part of Theorem 4.3. This completes the proof.

Corollary 4.4. Let be a real p-Banach spaces, and define for all and all . Then, () is a complete random p-normed space. Define

for all and all in which . Assume that is a mapping with which satisfies (4.2), then there exists a unique mapping such that

(4.9)

for all and all , where . Hence, we have

(4.10)

for all .

Theorem 4.5. Let ( is denoted by ) be a mappingsuch that, for some .

(4.11)

for all and all . If an even mapping with such that

(4.12)

for all and all , then there exists a unique quartic mapping such that

(4.13)

for all and all .

Proof. Replacing y by 0 in (4.12), we obtain

(4.14)

for all and all .

Consider a general metric d on , here be a set of all mappings from X into Y and introduce a generalized metric on as follows:

whereas inf . It is easy to show that () is a complete metric space ¹⁰. Now, let us consider a mapping defined by

for all and for all . Let in and be an arbitrary constant with . Then, we have

for all and all hence

(4.15)

for all and all , and so, if , then

for all Then is a strictly contractive self-mapping on with Lipschitz constant L Also, it follows from (4.12) that

(4.16)

for all and all which implies that

Using Theorem 4.1, there exists a mapping , which is a unique fixed point of in the set such that

for all since Again, it follows from Theorem 4.1 that

which implies

for all and all . Replacing x and y by and in (4.12), respectively,

for all and all . It follows from that . Hence, the mapping is quartic. Now, we show that mapping is unique. To prove this, we assume that there exists a quartic mapping which satisfies (4.13). Then, is a fixed point of J in . However, it follows from Theorem 4.1 that J has only one fixed point in . Hence, we deduce that .

Theorem 4.6. Let be a mapping such that, for some .

(4.17)

for all and all If is a mapping with which satisfies (4.12), then there exists a unique mapping such that

(4.18)

for all and all .

Proof. Let and d be as in the proof of Theorem 4.5. Then becomes a complete metric space and the mapping defined by

for all and . Then,

for all .

Then, is a strictly contractive self-mapping on with Lipschitz constant

It follows from (4.12) that we get

which implies the inequality (4.12) holds for all and all . The remaining assertion goes through in a similar method to the corresponding part of Theorem 4.5.

Corollary 4.7. Let X be a real p-Banach spaces, and define for all and all . Then, () is a complete random p-normed space. Define

for all and all in which . Assume that is a mapping which satisfies (4.12), then there exists a unique mapping such that

(4.19)

for all and all , where . Hence, we have

(4.20)

for all .

References