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Research article

Additive ρ-functional inequalities in non-Archimedean 2-normed spaces

  • Received: 22 October 2020 Accepted: 01 December 2020 Published: 03 December 2020
  • MSC : 39B72, 39B62, 12J25

  • In this paper, we solve the additive ρ-functional inequalities: f(x+y)f(x)f(y)ρ(2f(x+y2)f(x)f(y)),2f(x+y2)f(x)f(y)ρ(f(x+y)f(x)f(y)), where ρ is a fixed non-Archimedean number with |ρ|<1. More precisely, we investigate the solutions of these inequalities in non-Archimedean 2-normed spaces, and prove the Hyers-Ulam stability of these inequalities in non-Archimedean 2-normed spaces. Furthermore, we also prove the Hyers-Ulam stability of additive ρ-functional equations associated with these inequalities in non-Archimedean 2-normed spaces.

    Citation: Zhihua Wang, Choonkil Park, Dong Yun Shin. Additive ρ-functional inequalities in non-Archimedean 2-normed spaces[J]. AIMS Mathematics, 2021, 6(2): 1905-1919. doi: 10.3934/math.2021116

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  • In this paper, we solve the additive ρ-functional inequalities: f(x+y)f(x)f(y)ρ(2f(x+y2)f(x)f(y)),2f(x+y2)f(x)f(y)ρ(f(x+y)f(x)f(y)), where ρ is a fixed non-Archimedean number with |ρ|<1. More precisely, we investigate the solutions of these inequalities in non-Archimedean 2-normed spaces, and prove the Hyers-Ulam stability of these inequalities in non-Archimedean 2-normed spaces. Furthermore, we also prove the Hyers-Ulam stability of additive ρ-functional equations associated with these inequalities in non-Archimedean 2-normed spaces.


    The study of stability problems for functional equations is related to a question of Ulam [23] in 1940 concerning the stability of group homomorphisms.

    The functional equation

    f(x+y)=f(x)+f(y) (1.1)

    is called Cauchy functional equation. Every solution of the Cauchy functional equation is said to be an additive mapping. In 1941, Hyers [10] gave the first affirmative answer to the question of Ulam for Banach spaces. Hyers' result was generalized by Aoki [1] for additive mappings and by Rassias [20] for linear mappings by considering an unbounded Cauchy difference. In 1994, Găvruţă [3] provided a further generalization of the Rassias' theorem in which he replaced the unbounded Cauchy difference by a general control function.

    The following functional equation

    2f(x+y2)=f(x)+f(y) (1.2)

    is called Jensen functional equation. See [15,16,19,22] for more information on functional equations.

    Gilányi [7] and Rätz [21] showed that if f satisfies the functional inequality

    2f(x)+2f(y)f(xy1)f(xy), (1.3)

    then f satisfies the Jordan-Von Neumann functional equation

    2f(x)+2f(y)=f(xy)+f(xy1).

    Fechner [2] and Gilányi [8] proved the generalized Hyers-Ulam stability of the functional inequality (1.1). Park et al. [17] investigated the generalized Hyers-Ulam stability of functional inequalities associated with Jordon-Von Neumann type additive functional equations. Kim et al. [11] solved the additive ρ-functional inequalities in complex normed spaces and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in complex Banach spaces. In 2014, Park [14] considered the following two additive ρ-functional inequalities

    f(x+y)f(x)f(y)ρ(2f(x+y2)f(x)f(y)), (1.4)
    2f(x+y2)f(x)f(y)ρ(f(x+y)f(x)f(y)) (1.5)

    in non-Archimedean Banach spaces and in complex Banach spaces, where ρ is a fixed non-Archimedean number with |ρ|<1 or ρ is a fixed complex number with |ρ|<1.

    In this paper, we establish the solution of the additive ρ-functional inequalities (1.4) and (1.5), and prove the Hyers-Ulam stability of the additive ρ-functional inequalities (1.4) and (1.5) in non-Archimedean 2-Banach spaces. Moreover, we prove the Hyers-Ulam stability of additve ρ-functional equations associated with the additive ρ-functional inequalities (1.4) and (1.5) in non-Archimedean 2-Banach spaces.

    Gähler [4,5] has introduced the concept of linear 2-normed spaces in the middle of the 1960s. Then Gähler [6] and White [24,25] introduced the concept of 2-Banach spaces. Following [9,12,13,18], we recall some basic facts concerning non-Archimedean normed space and non-Archimedean 2-normed space and some preliminary results.

    By a non-Archimedean field we mean a field K equipped with a function (valuation) || from K into [0,) such that |r|=0 if and only if r=0, |rs|=|r||s|, and |r+s|max{|r|,|s|} for r,sK. Clearly |1|=|1|=1 and |n|1 for all nN. By the trivial valuation we mean the function || taking everything but 0 into 1 and |0|=0.

    Definition 1.1. (cf. [9,13]) Let X be a linear space over a scalar field K with a non-Archimedean non-trivial valuation ||. A function :XR is called a non-Archimedean norm (valuation) if it satisfies the following conditions:

    (i) x=0 if and only if x=0;

    (ii) rx=|r|x for all rK, xX;

    (iii) the strong triangle inequality; namely,

    x+ymax{x,y}

    for all x,yX. Then (X,) is called a non-Archimedean normed space.

    Definition 1.2. (cf. [12,18]) Let X be a linear space over a scalar field K with a non-Archimedean non-trivial valuation || with dimX>1. A function ,:XR is called a non-Archimedean 2-norm (valuation) if it satisfies the following conditions:

    (NA1) x,y=0 if and only if x,y are linearly dependent;

    (NA2) x,y=y,x;

    (NA3) rx,y=|r|x,y;

    (NA4) x,y+zmax{x,y,x,z};

    for all rK and all x,y,zX. Then (X,,) is called a non-Archimedean 2-normed space.

    According to the conditions in Definition 1.2, we have the following lemma.

    Lemma 1.3. Let (X,,) be a non-Archimedean 2-normed space. If xX and x,y=0 for all yX, then x=0.

    Definition 1.4. A sequence {xn} in a non-Archimedean 2-normed space (X,,) is called a Cauchy sequence if there are two linearly independent points y,zX such that

    limm,nxnxm,y=0andlimm,nxnxm,z=0.

    Definition 1.5. A sequence {xn} in a non-Archimedean 2-normed space (X,,) is called a convergent sequence if there exists an xX such that

    limnxnx,y=0

    for all yX. In this case, we call that {xn} converges to x or that x is the limit of {xn}, write {xn}x as n or limnxn=x.

    By (NA4), we have

    xnxm,ymax{xj+1xj,y:mjn1},(n>m),

    for all yX. Hence, a sequence {xn} is Cauchy in (X,,) if and only if {xn+1xn} converges to zero in a non-Archimedean 2-normed space (X,,).

    Remark 1.6. Let (X,,) be a non-Archimedean 2-normed space. One can show that conditions (NA2) and (NA4) in Definition 1.2 imply that

    x+y,zx,z+y,zand|xzy,z|xy,z

    for all x,y,zX.

    We can easily get the following lemma by Remark 1.6.

    Lemma 1.7. For a convergent sequence {xn} in a non-Archimedean 2-normed space (X,,),

    limnxn,y=limnxn,y

    for all yX.

    Definition 1.8. A non-Archimedean 2-normed space, in which every Cauchy sequence is a convergent sequence, is called a non-Archimedean 2-Banach space.

    Throughout this paper, let X be a non-Archimedean 2-normed space with dimX>1 and Y be a non-Archimedean 2-Banach space with dimY>1. Let N={0,1,2,,}, and ρ be a fixed non-Archimedean number with |ρ|<1.

    In this section, we solve and investigate the additive ρ-functional inequality (1.4) in non-Archimedean 2-normed spaces.

    Lemma 2.1. A mapping f:XY satisfies

    f(x+y)f(x)f(y),ωρ(2f(x+y2)f(x)f(y)),ω (2.1)

    for all x,yX and all ωY if and only if f:XY is additive.

    Proof. Suppose that f satisfies (2.1). Setting x=y=0 in (2.1), we have f(0),ω0,ω = 0 for all ωY and so f(0),ω=0 for all ωY. Hence we get

    f(0)=0.

    Putting y=x in (2.1), we get

    f(2x)2f(x),ω0,ω (2.2)

    for all xX and all ωY. Thus we have

    f(x2)=12f(x) (2.3)

    for all xX. It follows from (2.1) and (2.3) that

    f(x+y)f(x)f(y),ωρ(2f(x+y2)f(x)f(y)),ω=|ρ|f(x+y)f(x)f(y),ω (2.4)

    for all x,yX and all ωY. Hence, we obtain

    f(x+y)=f(x)+f(y)

    for all x,yX.

    The converse is obviously true. This completes the proof of the lemma.

    The following corollary can be found in [14,Corollary 2.2].

    Corollary 2.2. A mapping f:XY satisfies

    f(x+y)f(x)f(y)=ρ(2f(x+y2)f(x)f(y)) (2.5)

    for all x,yX if and only if f:XY is additive.

    Theorem 2.3. Let φ:X2[0,) be a function such that

    limj1|2|jφ(2jx,2jy)=0 (2.6)

    for all x,yX. Suppose that f:XY be a mapping satisfying

    f(x+y)f(x)f(y),ωρ(2f(x+y2)f(x)f(y)),ω+φ(x,y) (2.7)

    for all x,yX and all ωY. Then there exists a unique additive mapping A:XY such that

    f(x)A(x),ωsupjN{1|2|j+1φ(2jx,2jx)} (2.8)

    for all xX and all ωY.

    Proof. Letting y=x in (2.6), we get

    f(2x)2f(x),ωφ(x,x) (2.9)

    for all xX and all ωY. So

    f(x)12f(2x),ω1|2|φ(x,x) (2.10)

    for all xX and all ωY. Hence

    12lf(2lx)12mf(2mx),ωmax{12lf(2lx)12l+1f(2l+1x),ω,,12m1f(2m1x)12mf(2mx),ω}max{1|2|lf(2lx)12f(2l+1x),ω,,1|2|m1f(2m1x)12f(2mx),ω}supj{l,l+1,}{1|2|j+1φ(2jx,2jx)} (2.11)

    for all nonnegative integers m,l with m>l and for all xX and all ωY. It follows from (2.11) that

    liml,m12lf(2lx)12mf(2mx),ω=0

    for all xX and all ωY. Thus the sequence {f(2nx)2n} is a Cauchy sequence in Y. Since Y is a non-Archimedean 2-Banach space, the sequence {f(2nx)2n} converges for all xX. So one can define the mapping A:XY by

    A(x):=limnf(2nx)2n

    for all xX. That is,

    limnf(2nx)2nA(x),ω=0

    for all xX and all ωY.

    By Lemma 1.7, (2.6) and (2.7), we get

    A(x+y)A(x)A(y),ω=limn1|2|nf(2n(x+y))f(2nx)f(2ny),ωlimn1|2|nρ(2f(2n(x+y)2)f(2nx)f(2ny)),ω+limn1|2|nφ(2nx,2ny)=ρ(2A(x+y2)A(x)A(y)),ω (2.12)

    for all x,yX and all ωY. Thus, the mapping A:XY is additive by Lemma 2.1.

    By Lemma 1.7 and (2.11), we have

    f(x)A(x),ω=limmf(x)f(2mx)2m,ωsupjN{1|2|j+1φ(2jx,2jx)}

    for all xX and all ωY. Hence, we obtain (2.8), as desired.

    To prove the uniqueness property of A, Let A:XY be an another additive mapping satisfying (2.8). Then we have

    A(x)A(x),ω=12nA(2nx)12nA(2nx),ωmax{12nA(2nx)12nf(2nx),ω,12nf(2nx)12nA(2nx),ω}supjN{1|2|n+j+1φ(2n+jx,2n+jx)},

    which tends to zero as n for all xX and all ωY. By Lemma 1.3, we can conclude that A(x)=A(x) for all xX. This proves the uniqueness of A.

    Corollary 2.4. Let r,θ be positive real numbers with r>1, and let f:XY be a mapping such that

    f(x+y)f(x)f(y),ωρ(2f(x+y2)f(x)f(y)),ω+θ(xr+yr)ω (2.13)

    for all x,yX and all ωY. Then there exists a unique additive mapping A:XY such that

    f(x)A(x),ω2|2|θxrω (2.14)

    for all xX and all ωY.

    Proof. The proof follows from Theorem 2.3 by taking φ(x,y)=θ(xr+yr)ω for all x,yX and all ωY, as desired.

    Theorem 2.5. Let φ:X2[0,) be a function such that

    limj|2|jφ(x2j,y2j)=0 (2.15)

    for all x,yX. Suppose that f:XY be a mapping satisfying

    f(x+y)f(x)f(y),ωρ(2f(x+y2)f(x)f(y)),ω+φ(x,y) (2.16)

    for all x,yX and all ωY. Then there exists a unique additive mapping A:XY such that

    f(x)A(x),ωsupjN{|2|jφ(x2j+1,x2j+1)} (2.17)

    for all xX and all ωY.

    Proof. It follows from (2.9) that

    f(x)2f(x2),ωφ(x2,x2) (2.18)

    for all xX and all ωY. Hence

    2lf(x2l)2mf(x2m),ωmax{2lf(x2l)2l+1f(x2l+1),ω,,2m1f(x2m1)2mf(x2m),ω}max{|2|lf(x2l)2f(x2l+1),ω,,|2|m1f(x2m1)2f(x2m),ω}supj{l,l+1,}{|2|jφ(x2j+1,x2j+1)} (2.19)

    for all nonnegative integers m,l with m>l and for all xX and all ωY. It follows from (2.19) that

    liml,m2lf(x2l)2mf(x2m),ω=0

    for all xX and all ωY. Thus the sequence {2nf(x2n)} is a Cauchy sequence in Y. Since Y is a non-Archimedean 2-Banach space, the sequence {2nf(x2n)} converges for all xX. So one can define the mapping A:XY by

    A(x):=limn2nf(x2n)

    for all xX. That is,

    limn2nf(x2n)A(x),ω=0

    for all xX and all ωY. By Lemma 1.7 and (2.19), we have

    f(x)A(x),ω=limmf(x)2mf(x2m),ωsupjN{|2|jφ(x2j+1,x2j+1)}

    for all xX and all ωY. Hence, we obtain (2.17), as desired. The rest of the proof is similar to that of Theorem 2.3 and thus it is omitted.

    Corollary 2.6. Let r,θ be positive real numbers with r<1, and let f:XY be a mapping satisfying (2.13) for all x,yX and all ωY. Then there exists a unique additive mapping A:XY such that

    f(x)A(x),ω2|2|rθxrω (2.20)

    for all xX and all ωY.

    Let A(x,y):=f(x+y)f(x)f(y) and B(x,y):=ρ(2f(x+y2)f(x)f(y)) for all x,yX. For x,yX and ωY with A(x,y),ωB(x,y),ω, we have

    A(x,y),ωB(x,y),ωA(x,y)B(x,y),ω.

    For x,yX and ωY with A(x,y),ω>B(x,y),ω, we have

    A(x,y),ω=A(x,y)B(x,y)+B(x,y),ωmax{A(x,y)B(x,y),ω,B(x,y),ω}=A(x,y)B(x,y),ωA(x,y)B(x,y),ω+B(x,y),ω.

    So we can obtain

    f(x+y)f(x)f(y),ωρ(2f(x+y2)f(x)f(y)),ωf(x+y)f(x)f(y)ρ(2f(x+y2)f(x)f(y)),ω.

    As corollaries of Theorems 2.3 and 2.5, we obtain the Hyers-Ulam stability results for the additive ρ-functional equation associated with the additive ρ-functional inequality (1.4) in non-Archimedean 2-Banach spaces.

    Corollary 2.7. Let φ:X2[0,) be a function and let f:XY be a mapping satisfying (2.6) and

    f(x+y)f(x)f(y)ρ(2f(x+y2)f(x)f(y)),ωφ(x,y) (2.21)

    for all x,yX and all ωY. Then there exists a unique additive mapping A:XY satisfying (2.8) for all xX and all ωY.

    Corollary 2.8. Let r,θ be positive real numbers with r>1, and let f:XY be a mapping such that

    f(x+y)f(x)f(y)ρ(2f(x+y2)f(x)f(y)),ωθ(xr+yr)ω (2.22)

    for all x,yX and all ωY. Then there exists a unique additive mapping A:XY satisfying (2.14) for all xX and all ωY.

    Corollary 2.9. Let φ:X2[0,) be a function and let f:XY be a mapping satisfying (2.15) and (2.21) for all x,yX and all ωY. Then there exists a unique additive mapping A:XY satisfying (2.17) for all xX and all ωY.

    Corollary 2.10. Let r,θ be positive real numbers with r<1, and let f:XY be a mapping satisfying (2.22) for all x,yX and all ωY. Then there exists a unique additive mapping A:XY satisfying (2.20) for all xX and all ωY.

    In this section, we solve and investigate the additive ρ-functional inequality (1.5) in non-Archimedean 2-normed spaces.

    Lemma 3.1. A mapping f:XY satisfies f(0)=0 and

    2f(x+y2)f(x)f(y),υρ(f(x+y)f(x)f(y)),υ (3.1)

    for all x,yX and all υY if and only if f:XY is additive.

    Proof. Suppose that f satisfies (3.1). Letting y=0 in (3.1), we have

    2f(x2)f(x),υ0,υ=0 (3.2)

    for all xX and all υY. Thus we have

    f(x2)=12f(x) (3.3)

    for all xX. It follows from (3.1) and (3.3) that

    f(x+y)f(x)f(y),υ=2f(x+y2)f(x)f(y),υ|ρ|f(x+y)f(x)f(y),υ (3.4)

    for all x,yX and all υY. Hence, we obtain

    f(x+y)=f(x)+f(y)

    for all x,yX.

    The converse is obviously true. This completes the proof of the lemma.

    The following corollary can be found in [14,Corollary 3.2].

    Corollary 3.2. A mapping f:XY satisfies f(0)=0 and

    2f(x+y2)f(x)f(y)=ρ(f(x+y)f(x)f(y)) (3.5)

    for all x,yX if and only if f:XY is additive.

    Theorem 3.3. Let ϕ:X2[0,) be a function such that

    limj1|2|jϕ(2jx,2jy)=0 (3.6)

    for all x,yX. Suppose that f:XY be a mapping satisfying f(0)=0 and

    2f(x+y2)f(x)f(y),υρ(f(x+y)f(x)f(y)),υ+ϕ(x,y) (3.7)

    for all x,yX and all υY. Then there exists a unique additive mapping A:XY such that

    f(x)A(x),υsupjN{1|2|j+1ϕ(2j+1x,0)} (3.8)

    for all xX and all υY.

    Proof. Letting y=0 in (3.6), we get

    2f(x2)f(x),υϕ(x,0) (3.9)

    for all xX and all υY. So

    f(x)12f(2x),υ1|2|ϕ(2x,0) (3.10)

    for all xX and all υY. Hence

    12lf(2lx)12mf(2mx),υmax{12lf(2lx)12l+1f(2l+1x),υ,,12m1f(2m1x)12mf(2mx),υ}max{1|2|lf(2lx)12f(2l+1x),υ,,1|2|m1f(2m1x)12f(2mx),υ}supj{l,l+1,}{1|2|j+1ϕ(2j+1x,0)} (3.11)

    for all nonnegative integers m,l with m>l and for all xX and all υY. It follows from (3.11) that

    liml,m12lf(2lx)12mf(2mx),υ=0

    for all xX and all υY. Thus the sequence {f(2nx)2n} is a Cauchy sequence in Y. Since Y is a non-Archimedean 2-Banach space, the sequence {f(2nx)2n} converges for all xX. So one can define the mapping A:XY by

    A(x):=limnf(2nx)2n

    for all xX. That is,

    limnf(2nx)2nA(x),υ=0

    for all xX and all υY.

    By Lemma 1.7, (3.6) and (3.7), we get

    2A(x+y2)A(x)A(y),υ=limn1|2|n2f(2n(x+y)2)f(2nx)f(2ny),υlimn1|2|nρ(f(2n(x+y))f(2nx)f(2ny)),υ+limn1|2|nϕ(2nx,2ny)=ρ(A(x+y)A(x)A(y)),υ (3.12)

    for all x,yX and all υY. Thus, the mapping A:XY is additive by Lemma 3.1.

    By Lemma 1.7 and (3.11), we have

    f(x)A(x),υ=limmf(x)f(2mx)2m,υsupjN{1|2|j+1ϕ(2j+1x,0)}

    for all xX and all υY. Hence, we obtain (3.8), as desired.

    To prove the uniqueness property of A, Let A:XY be an another additive mapping satisfying (3.8). Then we have

    A(x)A(x),υ=12nA(2nx)12nA(2nx),υmax{12nA(2nx)12nf(2nx),υ,12nf(2nx)12nA(2nx),υ}supjN{1|2|n+j+1ϕ(2n+j+1x,0)},

    which tends to zero as n for all xX and all υY. By Lemma 1.3, we can conclude that A(x)=A(x) for all xX. This proves the uniqueness of A.

    Corollary 3.4. Let s,δ be positive real numbers with s>1, and let f:XY be a mapping satisfying f(0)=0 and

    2f(x+y2)f(x)f(y),υρ(f(x+y)f(x)(y)),υ+δ(xs+ys)υ (3.13)

    for all x,yX and all υY. Then there exists a unique additive mapping A:XY such that

    f(x)A(x),υ|2|s|2|δxsυ (3.14)

    for all xX and all υY.

    Proof. The proof follows from Theorem 3.3 by taking ϕ(x,y)=δ(xs+ys)υ for all x,yX and all υY, as desired.

    Theorem 3.5. Let ϕ:X2[0,) be a function such that

    limj|2|jϕ(x2j,y2j)=0 (3.15)

    for all x,yX. Suppose that f:XY be a mapping satisfying f(0)=0 and

    2f(x+y2)f(x)f(y),υρ(f(x+y)f(x)f(y)),υ+ϕ(x,y) (3.16)

    for all x,yX and all υY. Then there exists a unique additive mapping A:XY such that

    f(x)A(x),υsupjN{|2|jϕ(x2j,0)} (3.17)

    for all xX and all υY.

    Proof. It follows from (3.9) that

    f(x)2f(x2),υϕ(x,0) (3.18)

    for all xX and all υY. Hence

    2lf(x2l)2mf(x2m),υmax{2lf(x2l)2l+1f(x2l+1),υ,,2m1f(x2m1)2mf(x2m),υ}max{|2|lf(x2l)2f(x2l+1),υ,,|2|m1f(x2m1)2f(x2m),υ}supj{l,l+1,}{|2|jϕ(x2j,0)} (3.19)

    for all nonnegative integers m,l with m>l and for all xX and all υY. It follows from (3.19) that

    liml,m2lf(x2l)2mf(x2m),υ=0

    for all xX and all υY. Thus the sequence {2nf(x2n)} is a Cauchy sequence in Y. Since Y is a non-Archimedean 2-Banach space, the sequence {2nf(x2n)} converges for all xX. So one can define the mapping A:XY by

    A(x):=limn2nf(x2n)

    for all xX. That is,

    limn2nf(x2n)A(x),υ=0

    for all xX and all υY. By Lemma 1.7 and (3.19), we have

    f(x)A(x),υ=limmf(x)2mf(x2m),υsupjN{|2|jϕ(x2j,0)}

    for all xX and all υY. Hence, we obtain (3.17), as desired. The rest of the proof is similar to that of Theorem 3.3 and thus it is omitted.

    Corollary 3.6. Let s,δ be positive real numbers with s<1, and let f:XY be a mapping satisfying f(0)=0 and (3.13) for all x,yX and all υY. Then there exists a unique additive mapping A:XY such that

    f(x)A(x),υδxsυ (3.20)

    for all xX and all υY.

    Let ˜A(x,y):=2f(x+y2)f(x)f(y) and ˜B(x,y):=ρ(f(x+y)f(x)f(y)) for all x,yX. For x,yX and υY with ˜A(x,y),υ˜B(x,y),υ, we have

    ˜A(x,y),υ˜B(x,y),υ˜A(x,y)˜B(x,y),υ.

    For x,yX and υY with ˜A(x,y),υ>˜B(x,y),υ, we have

    ˜A(x,y),υ=˜A(x,y)˜B(x,y)+˜B(x,y),υmax{˜A(x,y)˜B(x,y),υ,˜B(x,y),υ}=˜A(x,y)˜B(x,y),υ˜A(x,y)˜B(x,y),υ+˜B(x,y),υ.

    So we can obtain

    2f(x+y2)f(x)f(y),υρ(f(x+y)f(x)f(y)),υ2f(x+y2)f(x)f(y)ρ(f(x+y)f(x)f(y)),υ.

    As corollaries of Theorems 3.3 and 3.5, we obtain the Hyers-Ulam stability results for the additive ρ-functional equation associated with the additive ρ-functional inequality (1.5) in non-Archimedean 2-Banach spaces.

    Corollary 3.7. Let ϕ:X2[0,) be a function and let f:XY be a mapping satisfying f(0)=0, (3.6) and

    2f(x+y2)f(x)f(y)ρ(f(x+y)f(x)f(y)),υϕ(x,y) (3.21)

    for all x,yX and all υY. Then there exists a unique additive mapping A:XY satisfying (3.8) for all xX and all υY.

    Corollary 3.8. Let s,δ be positive real numbers with s>1, and let f:XY be a mapping satisfying f(0)=0 and

    2f(x+y2)f(x)f(y)ρ(f(x+y)f(x)(y)),υδ(xs+ys)υ (3.22)

    for all x,yX and all υY. Then there exists a unique additive mapping A:XY satisfying (3.14) for all xX and all υY.

    Corollary 3.9. Let ϕ:X2[0,) be a function and let f:XY be a mapping satisfying f(0)=0, (3.15) and (3.21) for all x,yX and all υY. Then there exists a unique additive mapping A:XY satisfying (3.17) for all xX and all υY.

    Corollary 3.10. Let s,δ be positive real numbers with s<1, and let f:XY be a mapping satisfying f(0)=0 and (3.22) for all x,yX and all υY. Then there exists a unique additive mapping A:XY satisfying (3.20) for all xX and all υY.

    In this paper, we have solved the additive ρ-functional inequalities:

    f(x+y)f(x)f(y)ρ(2f(x+y2)f(x)f(y)),2f(x+y2)f(x)f(y)ρ(f(x+y)f(x)f(y)),

    where ρ is a fixed non-Archimedean number with |ρ|<1. More precisely, we have investigated the solutions of these inequalities in non-Archimedean 2-normed spaces, and have proved the Hyers-Ulam stability of these inequalities in non-Archimedean 2-normed spaces. Furthermore, we have also proved the Hyers-Ulam stability of additive ρ-functional equations associated with these inequalities in non-Archimedean 2-normed spaces.

    We would like to express our sincere gratitude to the anonymous referees for their helpful comments that will help to improve the quality of the manuscript.

    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.



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