# Approximation of Mixed Euler-Lagrangeσ-Cubic-Quartic Functional Equation in Felbin’s Type f-NLS

Authors: John Michael Rassias; Narasimman Pasupathi; Reza Saadati; Manuel de la Sen
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this research paper, the authors present a new mixed Euler-Lagrangeσ-cubic-quartic functional equation. For this introduced mixed type functional equation, the authors obtain general solution and investigate the various stabilities related to the Ulam problem in Felbin’s type of fuzzy normed linear space (f-NLS) with suitable counterexamples. This approach leads us to approximate the Euler-Lagrange σ-cubic-quartic functional equation with better estimation.


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## Body

## 1. Introduction

One of the famous questions concerning the stability of homomorphisms was raised by Ulam [1] in 1940. The author Hyers [2] provided a partial answer to Ulam’s question in 1941, and then, a generalized solution to Ulam’s question was given by Rassias [3] in 1978, which is called Hyers-Ulam-Rassias stability or generalized Hyers-Ulam stability. The generalization of Hyers stability result by Rassias [4] is called Ulam-Gavruta-Rassias stability. Later, Ravi et al. [5] investigated the stability using mixed powers of norms which is called Rassias stability.Definition 1 (see [6]).
A fuzzy subsetξ on ℝis said to be a fuzzy real number when it satisfies two axioms:
(N1)There exists τ0∈ℝ such that ξτ0=1
(N2)For each γ∈0,1,ξγ=ξγ−,ξγ+, where −∞<ξγ−≤ξγ+<+∞
Note thatξγ=τ:ξτ≥γ is γ-level set. We show the set of all fuzzy real numbers by Λℝ. Also, ξ is said to be a nonnegative fuzzy real number when ξ∈Λℝ and ξτ=0 for τ<0. We show the set of all nonnegative fuzzy real numbers by Λ∗ℝ.
We define0¯ as
(1)0¯τ=1,τ=0,0,τ≠0.Definition 2 (see [6]).
We define⊕,⊖,⊗,⊘onΛℝ×Λℝas(i)
ξ⊗Ξτ=supq∈ℝξq∧Ξτ−q, τ∈ℝ(ii)
ξ⊖Ξτ=supq∈ℝξq∧Ξq−τ, τ∈ℝ(iii)
ξ⊗Ξτ=supq∈ℝ,q≠0ξs∧Ξτ/q, τ∈ℝ(iv)
ξ⊘Ξτ=supq∈ℝξqτ∧Ξq, τ∈ℝ
0¯,1¯∈Λℝ are additive and multiplicative identities, respectively. We also define ⊖ξ as 0¯−ξ; so, ξ⊖Ξ=ξ⊕⊖ξ.Definition 3 (see [6]).
Forl∈ℝ/0, the notation l⊙ξ shows fuzzy scalar multiplication and defied as l⊙ξτ=ξτ/land0⊙ξ=0¯.Definition 4 (see [7]).
Consider the vector spaceS and the left and right norms L,R:0,12⟶0,1 which are symmetric and nondecreasing functions satisfying L0,0=0, R1,1=1. So,∥·∥:S⟶Λ∗R is said to be a fuzzy norm and S,∥·∥,L,R is a fuzzy normed linear space (in short f-NLS) if
(N1)∥s∥=0¯ if and only if s=0
(N2)∥λs∥=λ¯⊙∥s∥ for all s∈S and λ∈−∞,∞
(N3)for all t,s∈S:
(N3L)if q≤∥s∥1−,τ≤∥t∥1− and q+τ≤∥t+s∥1−, then ∥s+t∥q+τ≥L∥s∥q,∥t∥τ
(N3R)if q≥∥s∥1−,τ≥∥t∥1− and q+τ≥∥s+t∥1−, then ∥s+t∥q+τ≤R∥s∥q,∥t∥τ,where ∥s∥α=∥s∥α−,∥s∥α+ for s∈S and α∈0;1.Lemma 5 (see [8]).
Consider f-NLSS,·,L,R, and let
(R1)Rc,d≤maxc,d
(R2)∀γ∈0,1,∃ζ0,γ in which Rζ,h≤γ for every h∈0,γ
(R3)limc→0+Rc,c=0
So,R1⇒R2⇒R3. The converse is not true.Lemma 6 (see [8]).
Consider f-NLSS,·,L,R. Then,
(1)
ifRc,d≤maxc,d, then for all γ∈0,1, ∥s+t∥γ+≤∥s∥γ++∥t∥γ+ for all s,t∈S(2)
R2 implies that, for every γ∈0,1, there exists ζ∈0,γ such that ∥s+t∥γ+≤∥s∥ζ++∥t∥γ+ for every s,t∈S(3)
limc→0+Rc,c=0, implies that for every γ∈0,1, there exists ζ∈0,γ such that ∥s+t∥γ+≤∥s∥ζ++∥t∥ζ+ for every s,t∈SLemma 7.
Consider f-NLSS,·,L,R and let
(L1)Lc,d≥minc,d
(L2)∀γ∈0,1,∃ζγ,1 such that Lζ,η≥γ for all η∈γ,1
(L3)limd→1−Ld,d=1
So,L1⇒L2⇒L3, but not conversely.Lemma 8.
Consider f-NLSS,·,L,R, then
(1)
Lc,d≥minc,d, implying that ∀γ∈0,1, ∥s+t∥α−≤∥s∥γ−+∥t∥γ− for every s,t∈S(2)
L2 implies that for every γ∈0,1, there exists ζ∈γ,1 such that ∥s+t∥γ−≤∥s∥ζ−+∥t∥γ− for every s,t∈S(3)
limc→1−Lc,c=1, implying that for every γ∈0,1, there exists ζ∈γ,1 such that ∥s+t∥γ−≤∥s∥ζ−+∥t∥ζ− for every s,t∈SLemma 9 (see [7]).
Consider f-NLSS,·,L,R. Then,
(1)
Rc,d≥maxc,d and for all γ∈0,1, implying that ∥s+t∥γ+≤∥s∥γ++∥t∥γ+ for all s,t∈S, then (N3R)(2)
Lc,d≤minc,d and for all γ∈0,1, implying that ∥s+t∥γ−≤∥s∥γ−+∥t∥γ− for all s,t∈S, so (N3L)Definition 10 (see [7]).
Consider f-NLSS,·,L,Rand letlimc→0+Rc,c=0. A sequence smm=1∞⊆S converges to s∈S, denoted by limm→∞sm=s, iflimm→∞∥sm−s∥γ+=0 for every γ∈0,1, and is called a Cauchy sequence if limm,n→∞∥sm−sn∥γ+=0 for every γ∈0,1. A subset E⊆S is said to be complete if every Cauchy sequence in E converges in E. A f-NLS is called a fuzzy Banach space (f-BS) if it is complete.Lemma 11.
Consider f-NLSS,·,L,R which satisfies R2. Then,
(1)
∥·∥γ+ is continuous from U into ℝ at s∈S for every γ∈0,1(2)
For everym∈ℤ+ and sjj=1m, we have ∀γ∈0,1,∃ζ∈0,γ;∥∑j=1msj∥γ+≤∑j=1m∥sj∥ζ+
Recently, the stability problems of several functional equations (FEs) have been extensively investigated by a number of authors [4, 9–20] in Felbin type f-NLS. Our method helps to solve some new problems of stability and approximation of functional equations [21–28] in Felbin type f-NLS.
Motivated from the above historical developments in the field of FEs, the authors introduce a new mixed Euler-Lagrangeσ-cubic-quartic functional equation (FE)
(2)πt+σs+πσt+s+πt−σs+πs−σt=σ22πt+s+πt−s+πs−t−2σ4−1πt+πs+14σ2σ2−1π2t+π2s,where σ∈ℝ−0,±1. For this mixed type FE, authors obtain the general solution and investigate the various stabilities related to Ulam problem [1] in Felbin’s type f-NLS with suitable counterexamples.


## 2. General Solution of Euler-Lagrangeσ-Cubic-Quartic FE

Theorem 12.
Considerπ satisfies (2) and odd that is π−t=−πt, then a mapping π:T⟶S is cubic.Proof.
Assumeπ satisfies (2). Putting t=s=0 in (2), we get π0=0. Setting t,s by t,0 in (2), we obtain
(3)2πt+πσt+π−σt=σ23πt+π−t−2σ4−1πt+14σ2σ2−1π2t,for all t∈T, σ∈ℝ−0,±1 and by assuming π−t=−πt in (3) which leads
(4)π2t=8πt,∀t∈T.
Thus,π is cubic.Theorem 13.
Ifπ satisfies (2) and even that is π−t=πt, then a mapping π:T⟶S is quartic.Proof.
Assumeπ holds (2). Putting t=s=0 in (2), we get π0=0. Setting t,s by t,0 in (2), we arrive
(5)2πσt+2πt=4σ2πt−2σ4−1πt+14σ2σ2−1π2t,∀t∈T.
Allowingσ=2 in (5), we arrive π2t=16πt. Using π−t=πt and π2t=16πt in (5), we get
(6)πσt=σ4πt,for all t∈T,σ∈ℝ−0,±1. Thus, π is quartic.


## 3. Generalized Hyers-Ulam-Rassias Stability of a Euler-Lagrangeσ-Cubic-Quartic FE

Consider the following abbreviation(7)Gπt,s=πt+σs+πσt+s+πt−σs+πs−σt−σ22πt+s+πt−s+πs−t+2σ4−1πt+πs−14σ2σ2−1π2t+π2s,∀t,s∈T,and the integer σ≠0,±1.Theorem 14.
Consider the odd mappingπ:T⟶S for which we can find Φ:T×T⟶Λ∗ℝ for a linear space T and a fuzzy Banach space (f-BS) S where
(8)∑i=0∞Φ2it,2isγ+23i<∞,∀t,s∈T,(9)∥Gπt,s∥≤Φt,s,∀t,s∈T.
So, we can find a unique cubic functionΘ:T⟶S such that
(10)πt−Θtγ+≤48σ2σ2−1∑i=0∞Φ2it,0ζ+23i,∀t∈T,for all γ∈0,1,ζ∈0,γ, where
(11)Θt≔limσ→∞π2σt23σ.Proof.
Puttings=0 in (9) implies that
(12)2σ2σ2−1πt−14σ2σ2−1π2t≤Φt,0,∀t∈T.
Multiply both sides of equation (12) by 4/σ2σ2−1, so we get
(13)π2t−8πt≤4σ2σ2−1⊙Φt,0,∀t∈T.
Again multiplying (13) by 1/23σ+3 and replacing t by 2σt, we obtain
(14)π2σ+1t23σ+1−π2σt23σ≤48σ2σ2−1123σ⊙Φ2σt,0,∀t∈T,and it leads to
(15)π2σ+1t23σ+1−π2lt23lγ+≤48σ2σ2−1∑i=lσ123iΦ2it,0ζ+,∀t∈T with σ≥l, nonnegative integers. Now, (8) and (15) imply that the sequence π2σt/23σ is fuzzy Cauchy in S. So, the sequence π2σt/23σ converges, which let us to define the mapping Θ:T⟶S by
(16)Θt≔limσ→∞π2σt23σ,∀t∈T.
Consideringl=0 and allowing σ⟶∞ in (15), we obtain
(17)πt−Θtγ+≤48σ2σ2−1∑i=0∞Φ2it,0ζ+23i,∀t∈T,and it gives (10). Using (8) and (9), we have
(18)Θt+σs+Θσt+s+Θt−σs+Θs−σt−σ22Θt+s+Θt−s+Θs−t+2σ4−1Θt+Θs−14σ2σ2−1Θ2t+Θ2sγ+≤limσ→∞Φ2σt,2σsγ+23σ=0,∀t,s∈T,which implies that Θ:T⟶S is cubic. Suppose that Θ′:T⟶S is a cubic mapping satisfying (10) and implies
(19)Θt−Θ′t≤limσ→∞123σ4σ2σ2−1∑i=0∞Φ2i2σt,0ζ+23i≤limσ→∞4σ2σ2−1∑i=σ∞Φ2it,0ζ+23i=0,∀t∈T,Θ=Θ′, which shows the uniqueness of Θ.Theorem 15.
Considerπ:T⟶S and let there exist a function Φ:T×S⟶Λ∗ℝ such that
(20)∑i=1∞23iΦt2i,s2iγ+<∞,Gπt,s≤Φt,s,∀t,s∈T,for a linear space T and a fuzzy Banach space (f-BS) S. So, we can find a unique cubic mapping Θ:T⟶S, such that
(21)πt−Θtγ+≤4σ2σ2−1∑i=1∞23iΦt2i,0ζ+,∀t∈T,where
(22)Θt≔limσ→∞23σπt2σ.
The following corollary gives the Hyers-Ulam, Hyers-Ulam-Rassias, and Rassias stabilities of (2).Corollary 16.
Considerπ:T⟶S and let there be real numbers δ and ρ such that
(23)Gπt,sγ+≺¯δ,δ⊗tρ⊕sρ,ρ≠3,δ⊗tρ⊗sρ+t2ρ⊕s2ρ,2ρ≠3,∀t,s∈T,then there is a unique cubic mapping Q:T⟶S such that
(24)πt−Qtγ+≺¯4δζ+7σ2σ2−1,4δζ+tρζ+σ2σ2−1∣23−2ρ∣,4δζ+t2ρζ+σ2σ2−1∣23−22ρ∣,∀t∈T.
In the next example, we consider the unstability of FE (2) for p=3 in Corollary 16.Example 17.
Define the mappingΦ:T×T⟶Λ∗ℝ as
(25)Φt=δ⊗t3,if∣t∣<1,δ,o.w.,in which δ>0 is a fuzzy real number. Define π:T⟶S as
(26)πt=∑σ=0∞Φ2σt23σ,∀t∈T.
So,(27)∣πt+σs+πσt+s+πt−σs+πs−σt−σ22πt+s+πt−s+πs−t+2σ4−1πt+πs−14σ2σ2−1π2t+π2s∣≤32σ2−1δt3+s3,∀t,s∈T.
As a result, there does not exist a cubic mappingΘ:T⟶S and a constant ζ>0 such that
(28)πt−Θt≤ζ⊗t3,∀t∈T.Proof.
The below inequality(29)πt≤∑σ=0∞Φ2σt23σ=∑σ=0∞δ23σ=23δ23−1,showing the boundedness of π. Now, we show that π satisfies (27).
Lett=s=0, then (27) is trivial. If t3+s3≥1/m3, then the left-hand side of (27) is less than 28σ2−1/7δ. If 0<t3+s3<1/m3. So, we can find a positive integer r such that
(30)123r+1≤t3+s3<123r,so that
(31)23r−1t3<123,23r−1s3<123,and therefore, for each σ=0,1,⋯,r−1, we have
(32)2σt+σs,2σσt+s,2σt−σs,2σs−σt,2σt+s,2σt−s,mσs−t,mσt,mσs,mσ2t,mσ2s∈−1,1,Φ2σt+σs+Φ2σσt+s+Φ2σt−σs+Φ2σs−σt−σ22Φ2σt+s+Φ2σt−s+Φ2σs−t+2σ4−1Φ2σt+Φ2σs−14σ2σ2−1Φ2σ2t+Φ2σ2s,for σ=0,1,⋯,r−1. From (26) and (30), we have
(33)∣πt+σs+πσt+s+πt−σs+πs−σt−σ22πt+s+πt−s+πs−t+2σ4−1πt+πs−14σ2σ2−1π2t+π2s∣≤∑σ=r∞123σ∣Φ2σt+σs+Φ2σσt+s+Φ2σt−σs+Φ2σs−σt−σ22Φ2σt+s+Φ2σt−s+Φ2σs−t+2σ4−1Φ2σt+Φ2σs−14σ2σ2−1Φ2σ2t+Φ2σ2s∣=32σ2−1δt3+s3.
Thus,π satisfies (27) for all t,s∈T with 0<t3+s3<1/23. Corollary 16 shows Θt=rt3 for any t in T, and so,
(34)πt≤ζ⊕r⊗t3.But we can choose a positive integer l with lδ>ζ⊕r. If t∈0,1/2l−1, then 2σt∈0,1 for all σ=0,1,⋯,l−1. For this t, we have
(35)πt=∑σ=0∞Φ2σt23σ≥∑σ=0l−1δ2σt323σ=lδt3>ζ⊕r⊗t3,which contradicts (34). Therefore, the functional equation (2) is not stable in the sense of Ulam, Hyers, and Rassias if ρ=3.Theorem 18.
Consider the even mappingπ:T⟶S for which we can find Φ:T×T⟶Λ∗ℝ such that
(36)∑i=0∞Φσit,σisγ+σ4i<∞,∀t,s∈T,(37)Gπt,s≤Φt,s,∀t,s∈T,and all γ∈0,1. So, we can find a unique quartic mapping Q:T⟶S and ∀γ∈0,1,∃ζ∈0,γ, such that
(38)πt−Qtγ+≤12σ4∑i=0∞Φσit,0ζ+σ4i,∀t,∈T,where
(39)Qt≔limm→∞πσmtσ4m.Proof.
Puttings=0 in (37), we get
(40)2πσt−σ4πt≤Φt,0,∀t∈T.
Multiply (40) by 1/2, we obtain
(41)πσt−σ4πt≤12⊙Φt,0,∀t∈T.
Replacingt by σmt and multiplying (41) by 1/σ4m+4, we obtain
(42)πσm+1tσ4m+1−πσmtσ4m≤12σ41σ4meΦσmt,0,∀t∈T.
Therefore, for allγ∈0,1, there is ζ∈0,γ such that
(43)πσm+1tσ4m+1−πσltσ4lγ+≤12σ4∑i=lm1σ4iΦσit,0ζ+,∀t∈T,with m≥l. From (36) and (43) and because S is a f-BS, we have the sequence πσmt/σ4m which is a fuzzy Cauchy in S and converges ∀t∈T. Now, we define Q:T⟶S by
(44)Qt≔limm→∞πσmtσ4m,∀t∈T.
Assumingl=0 and allowing the limit as m⟶∞ in (43), we have
(45)πt−Qtγ+≤12σ4∑i=0∞Φσit,0ζ+σ4i,∀t∈T.
Therefore, we obtain (38). From (36) and (37), we have
(46)Qt+σs+Qσt+s+Qt−σs+Qs−σt−σ22Qt+s+Qt−s+Qs−t+2σ4−1Qt+Qs−14σ2σ2−1Q2t+Q2sγ+≤limm→∞Φσmt,kmsγ+σ4m=0,∀t,s∈T,and hence, the mapping Q:T⟶S is quartic. Letting Q′:T⟶S be a quartic mapping fulfills (38), and we have
(47)Qt−Q′t≤limσ→∞1σ4m12σ4∑i=0∞Φσiσmt,0ζ+σ4i≤limm→∞12σ4∑i=m∞Φσit,0ζ+σ4i=0,for all t∈T,Q=Q′, and hence, Q is unique.Theorem 19.
Considerπ:T⟶S for which we can find a mapping Φ:T×T⟶Λ∗ℝ such that
(48)∑i=1∞σ4iΦtσi,sσiγ+<∞,∀t,s∈T,Gπt,s≤Φt,s,∀t,s∈T,and all γ∈0,1. So, we can find a unique quartic mapping π:T⟶S and ∀γ∈0,1,∃ζ∈0,γ, such that
(49)πt−Qtγ+≤12σ4∑i=1∞σ4iΦtσi,0ζ+,∀t∈T,where
(50)Qt≔limm→∞σ3mπtσm.Corollary 20.
Considerπ:T⟶S and let there be real numbers δ and ρ such that
(51)Gπt,sγ+≺¯δ,δ⊗tρ⊕sρ,ρ≠4,δ⊗tρ⊗sρ+t2ρ⊕s2ρ,ρ≠2,∀t,s∈T,so we can find a unique quartic mapping Q:T⟶S satisfying
(52)πt−Qtγ+≺¯δζ+2σ4−1,δζ+tρζ+σ4−σρ,δζ+t2ρζ+2σ4−σ2ρ,∀t∈T.
In the next example, we show that the FE (2) is not stable for ρ=4 in Corollary 20.Example 21.
LettingΦ:T×T⟶Λ∗ℝ be a mapping defined by
(53)Φt=δ⊗t4,if∣t∣<1,δ,o.w.,where δ>0 is a fuzzy real number and π:T⟶S is defined by
(54)πt=∑σ=0∞Φσmtσ4m,∀t∈T,a linear space T and a fuzzy Banach space (f-BS) S. Then π fulfills the functional inequality
(55)πt+σs+πσt+s+πt−σs+πs−σt−σ22πt+s+πt−s+πs−t+2σ4−1πt+πs−14σ2σ2−1π2t+π2s≤7σ10+9σ122σ4−1δt4+s4,∀t,s∈T.
So, we cannot find a quartic mappingQ:T⟶S and a constant ζ>0 such that
(56)πt−Qt≤ζ⊗t4,∀t∈T.


## 4. Conclusion

In our work, we have obtained the general solution of a new generalized mixed Euler-Lagrangeσ-cubic-quartic functional equation and studied its generalized Hyers-Ulam-Rassias, Hyers-Ulam, Hyers-Ulam-Rassias, and Rassias stabilities in fuzzy normed linear space using Felbin’s concept. Moreover, some counterexamples show both stability and unstability of FE (2) in f-BS.



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