Quartic Functional Equation: Ulam-Type Stability in <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5269pt" id="M1" height="16.7875pt" version="1.1" viewBox="-0.0657574 -12.2606 39.0757 16.7875" width="39.0757pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,5.937,0)"><path id="g113-224" d="M558 587C558 666 497 712 432 712C379 712 330 691 284 650C212 586 178 508 148 348L71 -65C49 -185 35 -229 23 -235L27 -261C49 -259 101 -251 124 -224C131 -216 138 -178 159 -24C171 66 197 200 227 356C264 550 295 668 393 668C443 668 479 632 479 575C479 516 446 458 383 418C361 404 344 398 318 397L296 350C395 338 460 281 460 192C460 110 411 40 300 40C258 40 215 55 192 69L181 51C191 18 222 -16 266 -16C308 -16 351 1 397 26C471 67 545 142 545 221C545 315 486 365 401 395C469 437 558 498 558 587Z"/></g><g transform="matrix(.017,0,0,-0.017,15.909,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.017,0,0,-0.017,22.697,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g><g transform="matrix(.017,0,0,-0.017,32.876,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>-</span>Banach Space and Non-Archimedean <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5269pt" id="M2" height="16.7875pt" version="1.1" viewBox="-0.0657574 -12.2606 10.1388 16.7875" width="10.1388pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><use xlink:href="#g113-224"/></g></svg>-</span>Normed Space

Sharma, Ravinder Kumar; Chandok, Sumit

doi:https://doi.org/10.1155/2022/9908530

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 9908530 | https://doi.org/10.1155/2022/9908530

Quartic Functional Equation: Ulam-Type Stability in -Banach Space and Non-Archimedean -Normed Space

Ravinder Kumar Sharma¹and Sumit Chandok¹

Academic Editor: G Muhiuddin

Received28 Feb 2022

Accepted11 Apr 2022

Published17 May 2022

Abstract

In this study, we use the alternative fixed-point approach and the direct method to examine the generalized Hyers–Ulam stability of the quartic functional equation with a fixed positive integer in the context of -Banach space. In non-Archimedean -normed space, we also verify Hyers–Ulam stability for the quartic functional equation stated. Many of the findings in the literature are improved and generalized by our findings.

1. Introduction

In the theory of Ulam’s stability, one can find the efficient tool to evaluate the errors that are to study the existence of an exact solution of the perturbed functional equation, which is not far from the given function. Ulam gave the core reason for studying the stability of functional equations in a discussion at the University of Wisconsin in 1940 [1]. Hyers [2] presented a partial affirmative response to the question of Ulam in Banach space. Following that, Aoki [3] generalized Hyers theorem for additive mapping, and Rassias [4] generalized Hyers theorem for linear mapping by taking into account an unbounded Cauchy difference. In 1994, Găvruta [5] generalized Rassias theorem and discussed the stability of linear functional equations. Several mathematicians have obtained numerous results about the stability of functional equations over the last four decades. Hundreds of papers and numerous books have been produced on this vital topic over the previous five decades.

The theory of functional equations is a vast area of nonlinear analysis, which is rather hard to explore. Geometry, economics, game theory, measure theory, dynamics, and a variety of other subjects all use functional equations. In the subject of analysis, the study of solutions and stability results of functional equations is a popular topic. In nonlinear analysis, notably in fixed-point theory (FPT), the stability results of functional equations are used. The stability results are utilised to investigate additive mappings’ asymptotic properties.

The stability of several types of functional equations, such as additive and quadratic, established by Cauchy and D’ Alembert, was extensively addressed in many research articles in the mid-twentieth century. Many academics later looked at the stability of cubic-, quartic-, and mixed-type functional equations, as well as a variety of other properties (see [6–10]).

The equation,is called a quadratic functional equation. It is obvious that the quadratic function is a solution of the quadratic functional equation (1), where is an arbitrary real constant. Skof [11] was the first author who intially worked on the generalized Hyers–Ulam (HU) stability of the quadratic functional equation. Cholewa [12] found that the result of Skof [11] is still valid if the domain normed linear space is replaced by an abelian group. Czerwik [13] generalized Skof’s result.

In computational geometry and related subjects such as computer graphics, computer-aided design, computer-aided manufacturing, and optics, quartic equations are frequently encountered. A quartic equation is the characteristic equation of a fourth-order linear difference equation or differential equation. There are numerous types of quartic equations. In 1999, Rassias [14] investigates stability properties of the following quartic functional equation:

Since , we obtain

Without imposing any regularity criteria on the unknown function, Chung and Sahoo [15] determine the general solution of (3). Furthermore, they showed that the function is a solution of (3) if and only if , where the function is symmetric and additive in each variable. The premise that every solution of (3) is even can be expressed as follows:

The quartic function is easily recognised as a solution of the quartic functional (4), where is an arbitrary real constant.

In 2005, Lee et al. [16] found the general solution of (4) and demonstrated the functional equation’s HU stability. Park [17] investigates the stability problem of functional (4) in orthogonality normed space. In 2009, Lee et al. [18] considered the following quartic functional equation, which is a generalization of functional (4):for fixed integer , where . They found the general solution of functional (5) and showed its HU stability. There are several authors (see [14, 19–22]) who investigated the stability of quartic functional equations.

The quasi--normed space is one of the interesting generalization of the normed space (see [22–26]). The variation between a norm and quasi--norm is that the modulus of concavity of a quasi-norm is greater than equal to 1, while that of a norm is equal to 1. In general, the quasi--norm is not continuous, whereas a norm is always continuous. However, every -norm is continuous quasi--norm. Since it is much easier to work with -norm than quasi--norm, henceforth, we restrict our attention mainly to -norm.

In this study, we investigate the generalized HU stability of quartic functional equation of the form:

for a fixed integer and , where is a mapping from a -normed space to a complete -normed space over the same field as by using the directed method and alternative FPT. In non-Archimedean -normed space, we also demonstrate the HU stability for the quartic functional (6). The quartic functional (6) is a generalization of several types of functional equations that depends on the values of .

2. Preliminaries

Now, we will go over the basic definitions and results of quasi -normed space, as well as a number of other results that are required for the main findings.

Definition 1. Let be a real number with , and be a linear space over field with or . A function is called a quasi--norm if it satisfies the following conditions: and = if and only if = for every real number for all and .
The pair is called quasi--normed space [26] if is a quasi--norm on . The smallest possible is called the modulus of concavity of .
A quasi--normed space is called -normed space ifFor some and for all , is called a -norm on .

For a sequence to be convergent and Cauchy in quasi--normed space, see [26].

Example 1. For , we definewhere . Then, is a -norm space (see [24]).

Definition 2. Let be a nonempty set, , and be a function satisfying the following conditions, for all : if and only if , for all There is a constant such that Then, is called a generalized -metric on and the pair is called a generalized -metric space (GBM) [27].

The following result of Diaz et al. [28] will be used in the sequel.

Theorem 1. Let be a complete generalized metric space and be a contraction mapping satisfying the following.

There exists a constant , with , such that whenever , one has

Then, for each , we have either of the following conditions hold.

Either , for all , or the following assertions hold:(i), where is a fixed point of (ii)

The following proposition of Paluszyński et al. [29] will be used in this sequel.

Proposition 1. Let be a -metric space, , andfor all . Then, is a metric on satisfyingfor all . In particular, if is a metric, then and .

The following result of Bodaghi [30] will be used in Section 3.

Theorem 2. Let and be real vector spaces. Then, a mapping satisfies the functional equation (3) if and only if it satisfies the functional equation (6), where . Therefore, every solution of (6) is also a quartic function.

3. Alternative Fixed-Point Method

In this section, we will investigate the stability of a quartic functional (6) on -normed spaces by applying Theorem 1 with Remark 1.

Remark 1. Let and be two vector spaces over the same field and satisfies functional equation (6). Then ,for all and , we have Case 1: if in (6), then . Case 2: if in (6), then . Case 3: if we replace with and with in (6), then . If , so we have . Case 4: if in (6), then .

Let be an integer with ; we use the abbreviation for the given mapping as follows:

Theorem 3. Let be a -normed space, be a complete - normed space over the same field with , be a mapping, and be a mapping with . Assume that the following assumptions are satisfies, for all :Then, there exists a unique mapping satisfying functional (6) andfor all .

Proof. Let . Defined a function ,for all , where . Firstly, we will prove that is a GBM. Let . It is easy to see that . Now, if , then . Note that , for all . If , then , that is, , for all . Then, . Now, we assert the last property of a GBM. For each , we have , , and .
It follows that, for all ,Then, we can writeSo, we haveTherefore, is a GBM with the coefficient on .
Next, we will show that GBM is complete. Assume that is a Cauchy sequence in . Then, we have . Note that, for all , we haveThen, . It implies that is a Cauchy sequence in . Since is a -Banach, there exists in . Put ; we have the mapping . We will show that in . Indeed, for each , there exists such that for all . So, from (20), for all and ,Taking in (21), we get, for all and ,This implies that for all . So, in . Then, is complete.
Next, letting in (14) and using , we havefor all . It yields thatDefine a map by , for all and for all .
Now, we have to show that . By (13) and , we haveSo, we obtainfor all . Since , using Theorem 1, for the mapping on the complete generalized metric space , we have either , for all , or the following assertions hold:(1), where is a fixed point of (2)By (11), we have, for all ,From (24), we have, for all , . So, . This shows that if we choose , then(1)(2)So, we find thatThen, for all , . That is, (15) holds.
Next, we will show that is quartic by using the continuity. For each , note that . So,So (13) and (14), we have, for all ,It implies that, for all ,So, satisfies functional (6). By Remark 1, is a quartic mapping.
In the last, we show the uniqueness of . Assume that is also a quartic mapping satisfying (15). We need to prove that . It follows from Remark 1 that and . By using (13) and (15), for each , we obtainSince and , letting , we get for all . This proves .

4. Direct Method

In this section, we show the stability of the quartic functional (6) by using the direct method.

Theorem 4. Let be a -normed space, be a complete -normed space over the same field with , and be a mapping for which there exists a function such thatfor all .

Then, there exists a unique quartic mapping such thatfor all .

Proof. Putting in (34), we havefor all . Thus,for all . Replacing by in (37) and continuing this method, we obtainOn the contrary, we can use induction to obtainfor all , and . Thus, the sequence is a Cauchy by (33) and (39). Since is complete, so we assume that there exists a mapping so thatWe can see that inequality (35) holds if we take the limit as in (38) and use (40). Now, we replace by , respectively, in (34); then,Letting the limit as , we obtain , all . Therefore, by Theorem 2, it indicates that is a quartic mapping. Now, let be another quartic mapping which satisfies (35). Then, we havefor all . Taking in the preceding inequality, we can instantly see that G is unique. The proof is now complete.

We have the following result which is analogous to Theorem 4 for the quartic functional (6).

Theorem 5. Let be a -normed space, be a complete -normed space over the same field with , and be a mapping for in which there exists a function such thatfor all . Then, there exists a unique quartic mapping such thatfor all .

Proof. It follows from (43) that . Thus, from (44), we have . Putting in (44), we obtainfor all . In the previous inequality, if we replace by and divide both sides by , we obtainfor all . Using triangular inequality and moving forward in this manner,for all . If we show that the sequence is Cauchy, then the completeness of will imply that it is convergent. For this, if we substitute in (48) with and then multiply both sides by , then we obtainfor all , and . Thus, the above sequence is convergent to the mapping , that is,We now get the result, which is similar to the proof of Theorem 4.

5. Some Consequences

In this section, we give some consequences of the results proved in the previous sections.

Corollary 1. Let be a -normed space, be a complete -normed space space over the same field with , and be a mapping with . Assume that there is a positive real number and a real number such thatfor all .

Then, there exists a unique mapping satisfying (6) andfor all .

Proof. Define a mapping byNext, we will show thatfor all , where . Let . If or , thenIf and , then we haveNow, all the conditions of Theorem 3 hold. Therefore, we obtain the result.

Corollary 2. Let be a -normed space, be a complete -normed space over the same field with , and be a mapping with . Assume that there is a positive real number and a real number such thatfor all .
Then, there exists a unique mapping satisfying (6) andfor all .

Proof. Define a mapping byNext, we will show thatfor all , where . Let . If or , thenIf and , then we haveAll of the conditions in Theorem 3 are now satisfied. Hence, we obtain the result.

Corollary 3. Let be a -normed space, be a complete -normed space over the same field with , and be a mapping with . Assume that there is a positive real number and a negative real number with such thatfor all .

Then, there exists a unique mapping satisfying (6) andfor all .

Proof. Define a mapping byNext, we will show thatfor all , where . Let . If or , thenIf and , then we haveNow, all the conditions of Theorem 3 hold. Therefore, we obtain the result.

Corollary 4. Let , and be nonnegative real numbers such that and . Let be a -normed space, be a complete -normed space over the same field with , and be a mapping fulfillingfor all . Then, there exists a unique quartic mapping such thatfor all and all if .

Proof. If we put in (69), we have . Using in Theorem 4, we obtain the desired result.

Corollary 5. Let , and be nonnegative real numbers such that . Let be a -normed space, be a complete -normed space over the same field with , and be a mapping fulfillingfor all . Then, there exists a unique quartic mapping such thatfor all .

Proof. If we put in (71), we have . Using in Theorem 5, we obtain the desired result.

Remark 2. If in Theorems 4 and 5 and Corollaries 4 and 5, we get the corresponding results of [30].

6. Non-Archimedean -Normed Space

In this section, we demonstrate the stability of functional (6) in the context of non-Archimedean -normed space.

In 1897, Hensel [31] presented a normed space without the Archimedean property. Later on, it was discovered that non-Archimedean spaces offer a wide range of useful applications (see [32–34]).

Definition 3. A valuation mapping on a field is a function such that, for any , we have and the equality holds if and only if . . . A field having a valuation mapping is referred to as a valued field. The valuation is said to be non-Archimedean if condition (A3) in the definition of a valuation mapping is substituted by Condition is called the strict triangle inequality. By , we have . Thus, by induction, it follows from that , for each . We always assume in addition that is nontrivial, that is, there is a such that . The adic numbers are the most notable examples of non-Archimedean spaces.

Example 2. Assume that is a prime number. Define the -adic absolute value , for any nonzero rational number such that and are coprime to the prime number . Then, on , is a non-Archimedean norm. The -adic number field is the completion of with respect to and is denoted by .

Definition 4. Let be a real number with and be a linear space over field with or . A function is called a non-Archimedean -norm if it satisfies the following conditions: and = if and only if = for every real number The strong triangle inequality (ultrametric): For all , the pair is called a non-Archimedean -normed space if is a non-Archimedean -norm on .

A sequence in a non-Archimedean -normed space is a Cauchy sequence if and only if converge to zero.

Theorem 6. Let be a non-Archimedean -normed space and be a complete non-Archimedean -normed space, and let such thatSuppose that is a mapping satisfying equalityfor all . Then, there exists a unique quartic mapping such thatfor all , where .

Proof. Putting in (74), we obtainfor all . Thus, we havefor all . Replacing by in (77) and then dividing both sides by , we havefor all and all nonnegative integers . Thus, the sequence is Cauchy by (73) and (78). Because of the completeness of as a non-Archimedean -normed space, there exists a mapping so thatFor each and nonnegative integers , we haveTaking in (80) and using (79), we can see that inequality (75) holds when . It follows from (73), (74), and (79) that, for all ,Hence, the mapping satisfies (6). Now, let be another quartic map satisfying (75). Then, we havefor all . This shows the uniqueness of .

We have the following results which is analogous to Theorem 6 for (6).

Theorem 7. Let be a non-Archimedean -normed space and be a complete non-Archimedean -normed space, and let such thatSuppose that is a mapping satisfying equalityfor all . Then, there exists a unique quartic mapping such thatfor all , where .

Proof. In the same way as Theorem 6, we havefor all . Replacing by in (86) and then multiplying both sides by , we obtainfor all and all nonnegative integers . Thus, the sequence is Cauchy by (83) and (87). Because of the completeness of as a non-Archimedean -normed space, there exists a map so thatFor each and nonnegative integers , we havefor all and nonnegative integers . Since the right-hand side of inequality (89) tends to 0 as , by using (88), we deduce inequality (85). We can now complete the rest of the proof in the similar manner as in Theorem 6.

6.1. Consequences

In this section, we give some consequences of Theorems 6 and 7.

Corollary 6. Let , be a non-Archimedean -normed space, and be a complete non-Archimedean -normed space, and let be a function satisfying , for all , for which . Assume that is a mapping satisfying the inequalityfor all . Then, there exists a unique quartic mappings such thatfor all .

Proof. Defining by , we havefor all . We also havefor all . Now, Theorem 6 implies the desired result.

The proof of the following consequence is a direct consequence of Theorem 7 and similar to the proof of Corollary 6.

Corollary 7. Let , be a non-Archimedean -normed space, and be a complete non-Archimedean -normed space, and let be a function satisfying for all for which . Assume that is a mapping satisfying the inequalityfor all . Then, there exists a unique quartic mappings such thatfor all .

Remark 3. If in Theorems 6 and 7 and Corollaries 6 and 7, we get the corresponding results of Bodagi ([30] Theorems 12, 14 and Corollaries 13, 15).

7. Conclusion

In Banach space and non-Archimedean -normed space, we investigate the Hyers–Ulam–Rassias stability for the quartic functional equation using fixed-point approach and direct method. Also, we obtain some interesting consequences.

Data Availability

No data are used to support the finding of this study.

Conflicts of Interest

The authors declare that they have not any conflict of interest.

Acknowledgments

This research work was supported by the ASEAN-DST (Government of India) (Grant CRD/2018/000017).

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Copyright © 2022 Ravinder Kumar Sharma and Sumit Chandok. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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