Citation: Murali Ramdoss, Divyakumari Pachaiyappan, Inho Hwang, Choonkil Park. Stability of an n-variable mixed type functional equation in probabilistic modular spaces[J]. AIMS Mathematics, 2020, 5(6): 5903-5915. doi: 10.3934/math.2020378
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A mapping f:U→V is called additive if f satisfies the Cauchy functional equation
f(x+y)=f(x)+f(y) | (1.1) |
for all x,y∈U. It is easy to see that the additive function f(x)=ax is a solution of the functional equation (1.1) and every solution of the functional equation (1.1) is said to be an additive mapping. A mapping f:U→V is called quadratic if f satisfies the quadratic functional equation
f(x+y)+f(x−y)=2f(x)+2f(y) | (1.2) |
for all x,y∈U. A mapping f:U→V is quadratic if and only if there exist a symmetric biadditive mapping B:U2→V such that f(x)=B(x,x) and this B is unique, refer (see [1,10]). It is easy to see that the quadratic function f(x)=ax2 is a solution of the functional equation (1.2) and every solution of the functional equation (1.2) is said to be a quadratic mapping.
Mixed type functional equation is an advanced development in the field of functional equations. A single functional equation has more than one nature is known as mixed type functional equation. Further, in the development of mixed type functional equations, atmost only few functional equations have been obtained by many researchers (see [3,6,9,11,12,16,17,22,24]).
Let G be a group and H be a metric group with a metric d(.,.). Given ϵ>0 does there exists a δ>0 such that if a function f:G→H satisfies d(f(xy),f(x)f(y))<δ for all x,y∈G, then is there exist a homomorphism a:G→H with d(f(x),a(x))<ϵ for all x∈G? This problem for the stability of functional equations was raised by Ulam [23] and answerd by Hyers [7]. Later, it was developed by Rassias [20], Rassias [18,21] and G⌣vuruta [5].
The probabilistic modular space was introduced by Nourouzi [14] in 2007. Later, it was developed by K. Nourouzi [4,15].
Definition 1.1. Let V be a real vector space. If μ:V→Δ fulfills the following conditions
(ⅰ) μ(v)(0)=0,
(ⅱ) μ(v)(t)=1 for all t>0, if and only if v=γ (γ is the null vector in V),
(ⅲ) μ(−v)(t)=μ(v)(t),
(ⅳ) μ(au+bv)(r+t)≥μ(v)(r)∧μ(v)(t)
for all u,v∈V, a,b,r,t∈R+, a+b=1, then a pair (V,μ) is called a probabilistic modular space and (V,μ) is b-homogeneous if ρ(av)(t)=μ(v)(t|a|b) for all v∈V,t>0, a∈R∖{0}. Here Δ is g:R→R+ the set of all nondecreasing functions with inft∈Rg(t)=0 and supt∈Rg(t)=1. Also, the function min is denoted by ∧.
Example 1.2. Let V be a real vector space and μ be a modular on X. Then a pair (V,μ) is a probabilistic modular space, where
μ(v)(t)={tt+ρ(x),t>0,v∈V0,t≤0,v∈V. |
In 2002, Rassias [19] studied the Ulam stability of a mixed-type functional equation
g(3∑i=1xi)+3∑i=1g(xi)=∑1≤i≤j≤3g(xi+xj). |
Later, Nakmalachalasint [13] generalized the above functional equation and obtained an n-variable mixed-type functional equation of the form
g(n∑i=1xi)+(n−2)n∑i=1g(xi)=∑1≤i≤j≤ng(xi+xj) |
for n>2 and investigated its Ulam stability.
In 2005, Jun and Kim [8] introduced a generalized AQ-functional equation of the form
g(x+ay)+ag(x−y)=g(x−ay)+ag(x+y) |
for a≠0,±1.
In 2013, Zolfaghari et al. [25] investigated the Ulam stability of a mixed type functional equation in probabilistic modular spaces. In the same year, Cho et al. [2] introduced a fixed point method to prove the Ulam stability of AQC-functional equations in β-homogeneous probabilistic modular spaces.
Motivated from the notion of probabilistic modular spaces and by the mixed type functional equations, we introduce a new mixed type functional equation satisfied by the solution f(x)=x+x2 of the form
n−1∑i=1,j=i+1(f(2xi+xj))+f(2xn+x1)−2[n−1∑i=1,j=i+1(f(xi+xj))+f(xn+x1)]=n∑i=1f(−xi), | (1.3) |
for n∈N and investigate its Ulam stability in probabilistic modular spaces.
This paper is organized as follows: In Section 1, we provide a necessary introduction of this paper. In Sections 2 and 3, we obtain the general solution of the functional equation (1.3) in even case and in odd case, respectively. In Sections 4 and 5, we investigate the Ulam stability of (1.3) in probabilistic modular space by using fixed point theory for even and odd cases, respectively and the conclusion is given in Section 6.
Let U and V be real vector spaces. In this section, we obtain the general solution of a mixed type functional equation (1.3) for even case of the form
n−1∑i=1,j=i+1(f(2xi+xj))+f(2xn+x1)−2[n−1∑i=1,j=i+1(f(xi+xj))+f(xn+x1)]=n∑i=1f(xi) | (2.1) |
for n∈N.
Theorem 2.1. Let f:U→V satisfy the functional equation (2.1). If f is an even mapping, then f is quadratic.
Proof. Assume that f:U→V is even and satisfies the functional equation (2.1). Replacing (x1,x2,…,xn) by (0,0,…,0) and by (x1,0,…,0) in (2.1), we obtain f(0)=0 and
f(2x1)=4f(x1) | (2.2) |
for all x1∈U, respectively. Again, replacing (x1,x2,x3,…,xn) by (x1,x1,0,…,0) in (2.1), we have
f(3x1)=9f(x1) | (2.3) |
for all x1∈U. Now, from (2.2) and (2.3), we get
f(nx1)=n2f(x1), |
for all x1∈U. Replacing (x1,x2,x3,x4,…,xn) by (x1,x2,x2,0,…,0) in (2.1), we obtain
f(2x1+x2)+f(x1+2x2)=4f(x1+x2)+f(x1)+f(x2) | (2.4) |
for all x1,x2∈U. Replacing (x1,x2,x3,x4,…,xn) by (x1,x2,0,0,…,0) in (2.1), we get
f(2x1+x2)+f(x2)=2f(x1+x2)+2f(x1) | (2.5) |
for all x1,x2∈U. Replacing x2 by −x2 in (2.5), using the evenness of f and again adding the resultant to (2.5), we get
f(2x1+x2)+f(2x1−x2)+2f(x2)=2f(x1+x2)+2f(x1−x2)+4f(x1) | (2.6) |
for all x1,x2∈U. Replacing (x1,x2) by (x1+x2,x1−x2) in (2.6), we get
f(3x1+x2)+f(x1+3x2)=4f(x1+x2)−2f(x1−x2)+8f(x1)+8f(x2) | (2.7) |
for all x1,x2∈U. Letting (x1,x2) by (x1,x1+x2) in (2.5), we get
f(3x1+x2)+f(x1+x2)=2f(2x1+x2)+2f(x1) | (2.8) |
for all x1,x2∈U. Replacing x1 by x2 and x2 by x1 in (2.8), we have
f(x1+3x2)+f(x1+x2)=2f(x1+2x2)+2f(x2) | (2.9) |
for all x1,x2∈U. Now, adding (2.8) and (2.9), we obtain
f(3x1+x2)+f(x1+3x2)+2f(x1+x2)=2f(2x1+x2)+2f(x1+2x2)+2f(x1)+2f(x2) | (2.10) |
for all x1,x2∈U. Using (2.4), (2.7) and (2.10), we obtain (1.2). Hence the mapping f is quadratic.
Let U and V be real vector spaces. In this section, we obtain the general solution of a mixed type functional equation (1.3) for even case of the form
n−1∑i=1,j=i+1(f(2xi+xj))+f(2xn+x1)−2[n−1∑i=1,j=i+1(f(xi+xj))+f(xn+x1)]=−n∑i=1f(xi) | (3.1) |
for n∈N.
Theorem 3.1. Let f:U→V satisfy the functional equation (3.1). If f is an odd mapping, then f is additive.
Proof. Assume that f is odd and satisfies the functional equation (3.1). Replacing (x1,x2,…,xn) by (0,0,…,0) and (x1,0,…,0) in (3.1), we obtain f(0)=0 and
f(2x1)=2f(x1) | (3.2) |
for all x1∈U, respectively. Again, replacing (x1,x2,x3,…,xn) by (x1,x1,0,…,0) in (3.1), we have
f(3x1)=9f(x1) | (3.3) |
for all x1∈U. Now, from (3.2) and (3.3), we get
f(nx1)=nf(x1) |
for all x1∈U. Replacing (x1,x2,x3,x4,…,xn) by (x1,x2,0,0,⋯,0) in (3.1), we get
f(2x1+x2)−2f(x1+x2)=−f(x2) | (3.4) |
for all x1,x2∈U. Replacing x2 by −x2 in (3.4), using the oddness of g and again adding the resultant to (3.4), we get
f(2x1+x2)+f(2x1−x2)=2f(x1+x2)+2f(x1−x2) | (3.5) |
for all x1,x2∈U. Replacing (x1,x2) by (x1+x2,x1−x2) in (3.5), we get
f(3x1+x2)+f(x1+3x2)=4f(x1)+4f(x2) | (3.6) |
for all x1,x2∈U. Replacing x1 by x2 and x2 by x1 in (3.4), we have
f(2x1+x2)+f(x1+2x2)=4f(x1+x2)−f(x1)−f(x2) | (3.7) |
for all x1,x2∈U. Replacing (x1,x2) by (x1,x1+x2) in (3.4), we get
f(3x1+x2)−2f(2x1+x2)=−f(x1+x2) | (3.8) |
for all x1,x2∈U. Replacing x1 by x2 and x2 by x1 in (3.8) and adding the resultant equation to (3.8), we obtain
f(3x1+x2)+f(x1+3x2)−2f(2x1+x2)−2f(x1+2x2)=−2f(x1+x2) | (3.9) |
for all x1,x2∈U. Using (3.6), (3.7) and (3.9), we obtain (1.1). Hence the mapping f is additive.
In this section, we prove the Ulam stability of the n-variablel mixed type functional equation (1.3) for even case in probabilistic modular spaces (PM-spaces) by using fixed point technique.
For a mapping f:M→(V,μ), consider
Se(x,y)=n−1∑i=1,j=i+1(f(2xi+xj))+f(2xn+x1)−2[n−1∑i=1,j=i+1(f(xi+xj))+f(xn+x1)]−n∑i=1f(xi) |
for n∈N.
Theorem 4.1. Let M be a linear space and (V,μ) be a μ-complete b-homogeneous PM-space. Suppose that a mapping f:M→(V,μ) satisfies an inequality
μ(Se(x1,x2,…,xn))≥ρ(x1,x2,…,xn)(t) | (4.1) |
for all x1,x2,…,xn∈M and a given mapping ρ:M×M→Δ such that
ρ(2ax,0,…,0)(22baNt)≥ρ(x,0,…,0)(t) | (4.2) |
for all x∈M and
ρ(2amx1,2amx2,…,2amxn)(22bamt)=1 | (4.3) |
for all x1,x2,…,xn∈M and a constant 0<N<12b. Then there exists a unique quadratic mapping T:M→(V,μ) satisfying (2.1) and
μ(T(x)−f(x))(t22bNa−12(1−2bN))≥ρ(x,0,…,0)(t) | (4.4) |
for all x∈M.
Proof. Replacing (x1,x2,…,xn) by (x,0,…,0) in (4.1), we obtain
μ(f(2x)−22f(x))(t)≥ρ(x,0,…,0)(t) | (4.5) |
for all x∈M. This implies
μ(f(2x)22−f(x))(t)=μ(f(2x)−22f(x))(22bt)≥ρ(x,0,…,0)(22bt) | (4.6) |
for all x∈M. Replacing x by 2−1x in (4.6), we obtain
μ(f(2−1x)2−2−f(x))(t)=μ(f(x)22−f(2−1x))(t22b)≥ρ(2−1x,0,…,0)(22bN−1Nt22b)≥ρ(x,0,…,0)(22bN−1t). | (4.7) |
From (4.6) and (4.7), we obtain
μ(f(2ax)22a−f(x))(t)≥Ψ(x)(t):=ρ(x,0,…,0)(22bNa−12t) | (4.8) |
for all x∈M.
Consider P:={f:M→(U,μ)|f(0)=0} and define η on P as follows:
η(f)=inf{l>0:μ(f(x))(lt)≥Ψ(x)(t),∀x∈M}. |
It is simple to prove that η is modular on N and indulges the Δ2-condition with 2b=κ and Fatou property. Also, N is η-complete (see [25]). Consider the mapping Q:Pη→Pη defined by QT(x):=T(2ax)22a for all T∈Pη.
Let f,j∈Pη and l>0 be an arbitrary constant with η(f−j)≤l. From the definition of η, we get
μ(f(x)−j(x))(lt)≥Ψ(x)(t) |
for all x∈M. This implies
μ(Qf(x)−Qj(x))(Nlt)=μ(2−2af(2ax)−2−2aj(2ax))(Nlt)=μ(f(2ax)−j(2ax))(22baNlt)≥Ψ(2ax)(22baNt)≥Ψ(x)(t) |
for all x∈M. Hence η(Qf−Qj)≤Nη(f−j) for all f,j∈Pη, which means that Q is an η-strict contraction. Replacing x by 2ax in (4.8), we have
μ(f(22ax)22a−f(2ax))(t)≥Ψ(2ax)(t) | (4.9) |
for all x∈M and therefore
μ(2−2(2a)f(22ax)−2−2af(2ax))(Nt)=μ(2−2af(22ax)−f(2ax))(22baNt)≥Ψ(2ax)(22baNt)≥Ψ(x)(t) | (4.10) |
for all x∈E. Now
μ(f(22ax)22(2a)−f(x))(2b(Nt+t))≥μ(f(22ax)22(2a)−f(2ax)22a)(Nt)∧μ(f(2ax)22a−f(x))(t)≥Ψ(x)(t) | (4.11) |
for all x∈M. In (4.11), replacing x by 2ax and 2b(Nt+t) by 22βa2b(N2t+Nt), we obtain
μ(f(23ax)22(2a)−f(2ax))(22ba2b(N2t+Nt))≥Ψ(2ax)(22bjNt)≥Ψ(x)(t) | (4.12) |
for all x∈M. Therefore,
μ(f(23ax)23(2a)−f(2ax)22a)(2b(N2t+Nt))≥Ψ(x)(t) | (4.13) |
for all x∈M. This implies
μ(f(23ax)23(2a)−f(x))(2b(2b(N2t+Nt)+t))≥μ(f(23ax)23(2a)−f(2ax)22a)(2b(N2t+Nt))∧μ(f(2ax)22a−f(x))(t)≥Ψ(x)(t) | (4.14) |
for all x∈M. Generalizing the above inequality, we get
μ(f(2amx)22(am)−f(x))((2bN)m−1t+2bm−1∑i=1(2bN)i−1t)≥Ψ(x)(t) | (4.15) |
for all x∈M and a positive integer m. Hence we have
η(Qmf−f)≤(2bN)m−1+2bm−1∑i=1(2bN)i−1≤2bm∑i=1(2bN)i−1≤2b1−2bN. | (4.16) |
Now, one can easily prove that {Qm(f)} is η−converges to T∈Pη (see [25]). Thus (4.16) becomes
η(T−f)≤2b1−2bN, | (4.17) |
which implies
μ(T(x)−f(x))(2b1−2bNt)≥Ψ(x)(t)=ρ(x,0,…,0)(2b22bNa−12t) | (4.18) |
for all x∈M and hence we have
μ(T(x)−f(x))(t22bNa−12(1−2bN))≥ρ(x,0,…,0)(t) | (4.19) |
for all x∈M and hence the inequality (4.4) holds. One can easily prove the uniqueness of T (see [25]).
In this section, we prove the Ulam stability of the n-variable mixed type functional equation (1.3) for odd case in probabilistic modular spaces (PM-spaces) by using fixed point technique.
For a mapping f:M→(U,μ), consider
So(x1,x2,…,xn)=n−1∑i=1,j=i+1(f(2xi+xj))+f(2xn+x1)−2[n−1∑i=1,j=i+1(f(xi+xj))+f(xn+x1)]+n∑i=1f(xi) |
for n∈N.
Theorem 5.1. Let M be a linear space and (V,μ) be a μ-complete b-homogeneous PM-space. Suppose that a mapping f:M→(V,μ) satisfies an inequality
μ(So(x1,x2,…,xn))≥ρ(x1,x2,…,xn)(t) | (5.1) |
for all x1,x2,…,xn∈M and a given mapping ρ:M×M→Δ such that
ρ(2ax,0,…,0)(2baNt)≥ρ(x,0,…,0)(t) | (5.2) |
for all x∈M and
ρ(2amx1,2amx2,…,2amxn)(2bamt)=1 | (5.3) |
for all x1,x2,…,xn∈M and a constant 0<N<12b. Then there exists a unique additive mapping A:M→(V,μ) satisfying (3.1) and
μ(A(x)−f(x))(t2bNa−12(1−2bN))≥ρ(x,0,…,0)(t) | (5.4) |
for all x∈M.
Proof. Replacing (x1,x2,…,xn) by (x,0,…,0) in (5.1), we obtain
μ(f(2x)−2f(x))(t)≥ρ(x,0,…,0)(t) | (5.5) |
for all x∈M. This implies
μ(f(2x)2−f(x))(t)=μ(f(2x)−2f(x))(2bt)≥ρ(x,0,…,0)(2bt) | (5.6) |
for all x∈M. Replacing x by 2−1x in (5.6), we obtain
μ(f(2−1x)2−1−f(x))(t)=μ(f(x)2−f(2−1x))(t2b)≥ρ(2−1x,0,…,0)(2bN−1Nt2b)≥ρ(x,0,…,0)(2bN−1t). | (5.7) |
From (5.6) and (5.7), we obtain
μ(f(2ax)2a−f(x))(t)≥Ψ(x)(t):=ρ(x,0,…,0)(2bNa−12t) | (5.8) |
for all x∈M.
Consider P:={f:M→(U,μ)|f(0)=0} and define η on P as follows:
η(f)=inf{l>0:μ(f(x))(lt)≥Ψ(x)(t),∀x∈M}. |
It is simple to prove that η is modular on N and indulges the Δ2-condition with 2b=κ and Fatou property. Also, N is η-complete (see [25]). Consider a mapping Q:Pη→Pη defined by QA(x):=A(2ax)2a for all A∈Pη.
Let f,j∈Pη and l>0 be an arbitrary constant with η(f−j)≤l. From the definition of η, we get
μ(f(x)−j(x))(lt)≥Ψ(x)(t) |
for all x∈M. This implies
μ(Qf(x)−Qj(x))(Nlt)=μ(2−af(2ax)−2−aj(2ax))(Nlt)=μ(f(2ax)−j(2ax))(2baNlt)≥Ψ(2ax)(2baNt)≥Ψ(x)(t) |
for all x∈M. Hence η(Qf−Qj)≤Nη(f−j) for all f,j∈Pη, which means that Q is an η-strict contraction. Replacing x by 2ax in (5.8), we get
μ(f(22ax)2a−f(2ax))(t)≥Ψ(2ax)(t) | (5.9) |
for all x∈M and thus
μ(2−2af(22ax)−2−af(2ax))(Nt)=μ(2−af(22ax)−f(2ax))(2baNt)≥Ψ(2ax)(2baNt)≥Ψ(x)(t), | (5.10) |
for all x∈E. Now
μ(f(22ax)22a−f(x))(2b(Nt+t))≥μ(f(22ax)22a−f(2ax)2a)(Nt)∧μ(f(2ax)2a−f(x))(t)≥Ψ(x)(t) | (5.11) |
for all x∈M. In (5.11), replacing x by 2ax and 2b(Nt+t) by 2ba2b(N2t+Nt), we obtain
μ(f(23ax)22a−f(2ax))(2ba2b(N2t+Nt))≥Ψ(2ax)(2bjNt)≥Ψ(x)(t) | (5.12) |
for all x∈M. Therefore,
μ(f(23ax)23a−f(2ax)2a)(2b(N2t+Nt))≥Ψ(x)(t) | (5.13) |
for all x∈M. This implies
μ(f(23ax)23a−f(x))(2b(2b(N2t+Nt)+t))≥μ(f(23ax)23a−f(2ax)2a)(2b(N2t+Nt))∧μ(f(2ax)2a−f(x))(t)≥Ψ(x)(t) | (5.14) |
for all x∈M. Generalizing the above inequality, we have
μ(f(2amx)2am−f(x))((2bN)m−1t+2bm−1∑i=1(2bN)i−1t)≥Ψ(x)(t) | (5.15) |
for all x∈M and a positive integer m. Hence we have
η(Qmf−f)≤(2bN)m−1+2bm−1∑i=1(2bN)i−1≤2bm∑i=1(2bN)i−1≤2b1−2bN. | (5.16) |
Now, one can easily prove that {Qm(f)} is η-convergent to A∈Pη (see [25]). Thus (4.16) becomes
η(A−f)≤2b1−2bN, | (5.17) |
which leads
μ(A(x)−f(x))(2b1−2bNt)≥Ψ(x)(t)=ρ(x,0,…,0)(2b2bNa−12t) | (5.18) |
for all x∈M and hence we have
μ(A(x)−f(x))(t2bNa−12(1−2bN))≥ρ(x,0,…,0)(t) | (5.19) |
for all x∈M and hence the inequality (5.4) holds. One can easily prove the uniqueness of A (see [25]).
In this paper, we introduced a new n-variable mixed type functional equation satisfied by the solution f(x)=ax+bx2. Mainly, we obtained its general solution and investigated its Ulam stability in PM-spaces by using fixed point theory and we hope that this research work is a further improvement in the field of functional equations.
This work was supported by Incheon National University Research Grant 2020-2021.
The authors equally conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
The authors declare that they have no competing interests.
[1] | J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, Cambridge, 1989. |
[2] | Y. Cho, M. B. Ghaemi, M. Choubin, et al. On the Hyers-Ulam stability of sextic functional equations in β-homogeneous probabilistic modular spaces, Math. Inequal. Appl., 16 (2013), 1097- 1114. |
[3] |
I. EL-Fassi, Generalized hyperstability of a Drygas functional equation on a restricted domain using Brzdek's fixed point theorem, J. Fixed Point Theory Appl., 19 (2017), 2529-2540. doi: 10.1007/s11784-017-0439-8
![]() |
[4] |
K. Fallahi and K. Nourouzi, Probabilistic modular spaces and linear operators, Acta Appl. Math., 105 (2009), 123-140. doi: 10.1007/s10440-008-9267-6
![]() |
[5] |
P. Găvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436. doi: 10.1006/jmaa.1994.1211
![]() |
[6] | V. Govindan, C. Park, S. Pinelas, et al. Solution of a 3-D cubic functional equation and its stability, AIMS Math., 5 (2020), 1693-1705. |
[7] |
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222-224. doi: 10.1073/pnas.27.4.222
![]() |
[8] |
K. Jun and H. Kim, On the Hyers-Ulam-Rassias stability of a generalized quadratic and additive functional equation, Bull. Korean Math. Soc., 42 (2005), 133-148. doi: 10.4134/BKMS.2005.42.1.133
![]() |
[9] |
S. Jung, D. Popa and M. Th. Rassias, On the stability of the linear functional equation in a single variable on complete metric spaces, J. Global Optim., 59 (2014), 165-171. doi: 10.1007/s10898-013-0083-9
![]() |
[10] |
Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math., 27 (1995), 368-372. doi: 10.1007/BF03322841
![]() |
[11] | Y. Lee, S. Jung and M. Th. Rassias, Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation, J. Math. Inequal., 12 (2018), 43-61. |
[12] | J. Lee, J. Kim and C. Park, A fixed point approach to the stability of an additive-quadratic-cubicquartic functional equation, Fixed Point Theory Appl., 2010 (2010), 185780. |
[13] | P. Nakmahachalasint, On the generalized Ulam-Găvruta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equation, Int. J. Math. Math. Sci., 2007 (2007), 63239. |
[14] | K. Nourouzi, Probabilistic modular spaces, Proceedings of the 6th International ISAAC Congress, Ankara, Turkey, 2007. |
[15] | K. Nourouzi, Probabilistic modular spaces, WSPC Proceedings, (2009), 814-818. |
[16] | C. Park, S. Jang, J. Lee, et al. On the stability of an AQCQ-functional equation in radom normed spaces, J. Inequal. Appl., 2011 (2011), 34. |
[17] | M. Ramdoss, J. M. Rassias and D. Pachaiyappan, Stability of generalized AQCQ-functional equation in modular space, J. Eng. Appl. Sci., 15 (2020), 1148-1157. |
[18] |
J. M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal., 46 (1982), 126-130. doi: 10.1016/0022-1236(82)90048-9
![]() |
[19] |
J. M. Rassias, On the Ulam stability of the mixed type mappings on restricted domains, J. Math. Anal. Appl., 276 (2002), 747-762. doi: 10.1016/S0022-247X(02)00439-0
![]() |
[20] |
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc., 72 (1978), 297-300. doi: 10.1090/S0002-9939-1978-0507327-1
![]() |
[21] | K. Ravi, M. Arunkumar and J. M. Rassias, On the Ulam stability for the orthogonally general Euler-Lagrange type functional equation, Int. J. Math. Sci., 3 (2008), 36-47. |
[22] | K. Ravi, J. M. Rassias, M. Arunkumar, et al. Stability of a generalized mixed type additive, quadratic, cubic and quartic functional equational equation, J. Inequal. Pure Appl. Math., 10 (2009), 1-29. |
[23] | S. M. Ulam, Problems in Modern Mathematics, Science Ed., Wiley, New York, 1940. |
[24] | T. Z. Xu, J. M. Rassias and W. X. Xu, Generalized Ulam-Hyers stability of a general mixed AQCQfunctional equation in multi-Banach spaces: a fixed point approach, Eur. J. Pure Appl. Math., 3 (2010), 1032-1047. |
[25] | S. Zolfaghari, A. Ebadian, S. Ostadbashi, et al. A fixed point approach to the Hyers-Ulam stability of an AQ functional equation in probabilistic modular spaces, Int. J. Nonlinear Anal. Appl., 4 (2013), 89-101. |
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