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Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 902931, 45 pages
doi:10.1155/2012/902931
Review Article

Nonlinear Random Stability via Fixed-Point Method

Yeol Je Cho,1Β Shin Min Kang,2Β and Reza Saadati3

1Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea
2Department of Mathematics and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea
3Department of Mathematics, Iran University of Science and Technology, Behshahr, Iran

Received 31 October 2011; Accepted 22 December 2011

Academic Editor: Yeong-ChengΒ Liou

Copyright Β© 2012 Yeol Je Cho et al. 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.

Abstract

We prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation 𝑓 ( π‘₯ + 2 𝑦 ) + 𝑓 ( π‘₯ βˆ’ 2 𝑦 ) = 4 𝑓 ( π‘₯ + 𝑦 ) + 4 𝑓 ( π‘₯ βˆ’ 𝑦 ) βˆ’ 6 𝑓 ( π‘₯ ) + 𝑓 ( 2 𝑦 ) + 𝑓 ( βˆ’ 2 𝑦 ) βˆ’ 4 𝑓 ( 𝑦 ) βˆ’ 4 𝑓 ( βˆ’ 𝑦 ) in various complete random normed spaces.

1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for 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. The paper of Rassias [4] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability or as Hyers-Ulam-Rassias stability of functional equations. A generalization of the Rassias theorem was obtained by Găvruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias approach.

The functional equation 𝑓 ( π‘₯ + 𝑦 ) + 𝑓 ( π‘₯ βˆ’ 𝑦 ) = 2 𝑓 ( π‘₯ ) + 2 𝑓 ( 𝑦 ) ( 1 . 1 )

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Cholewa [6] for mappings 𝑓 ∢ 𝑋 β†’ π‘Œ , where 𝑋 is a normed space and π‘Œ is a Banach space. Czerwik [7] proved the generalized Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [8–12]).

In [13], Jun and Kim consider the following cubic functional equation: 𝑓 ( 2 π‘₯ + 𝑦 ) + 𝑓 ( 2 π‘₯ βˆ’ 𝑦 ) = 2 𝑓 ( π‘₯ + 𝑦 ) + 2 𝑓 ( π‘₯ βˆ’ 𝑦 ) + 1 2 𝑓 ( π‘₯ ) . ( 1 . 2 )

It is easy to show that the function 𝑓 ( π‘₯ ) = π‘₯ 3 satisfies the functional equation (1.2), which is called a cubic functional equation, and every solution of the cubic functional equation is said to be a cubic mapping.

Considered the following quartic functional equation 𝑓 ( 2 π‘₯ + 𝑦 ) + 𝑓 ( 2 π‘₯ βˆ’ 𝑦 ) = 4 𝑓 ( π‘₯ + 𝑦 ) + 4 𝑓 ( π‘₯ βˆ’ 𝑦 ) + 2 4 𝑓 ( π‘₯ ) βˆ’ 6 𝑓 ( 𝑦 ) . ( 1 . 3 ) It is easy to show that the function 𝑓 ( π‘₯ ) = π‘₯ 4 satisfies the functional equation, which is called a quartic functional equation, and every solution of the quartic functional equation is said to be a quartic mapping. One can easily show that an odd mapping 𝑓 ∢ 𝑋 β†’ π‘Œ satisfies the additive-quadratic-cubic-quadratic functional equation 𝑓 ( π‘₯ + 2 𝑦 ) + 𝑓 ( π‘₯ βˆ’ 2 𝑦 ) = 4 𝑓 ( π‘₯ + 𝑦 ) + 4 𝑓 ( π‘₯ βˆ’ 𝑦 ) βˆ’ 6 𝑓 ( π‘₯ ) + 𝑓 ( 2 𝑦 ) + 𝑓 ( βˆ’ 2 𝑦 ) βˆ’ 4 𝑓 ( 𝑦 ) βˆ’ 4 𝑓 ( βˆ’ 𝑦 ) ( 1 . 4 ) if and only if it is an additive-cubic mapping, that is, 𝑓 ( π‘₯ + 2 𝑦 ) + 𝑓 ( π‘₯ βˆ’ 2 𝑦 ) = 4 𝑓 ( π‘₯ + 𝑦 ) + 4 𝑓 ( π‘₯ βˆ’ 𝑦 ) βˆ’ 6 𝑓 ( π‘₯ ) . ( 1 . 5 )

It was shown in Lemma  2.2 of [14] that 𝑔 ( π‘₯ ) ∢ = 𝑓 ( 2 π‘₯ ) βˆ’ 2 𝑓 ( π‘₯ ) and β„Ž ( π‘₯ ) ∢ = 𝑓 ( 2 π‘₯ ) βˆ’ 8 𝑓 ( π‘₯ ) are cubic and additive, respectively, and that 𝑓 ( π‘₯ ) = ( 1 / 6 ) 𝑔 ( π‘₯ ) βˆ’ ( 1 / 6 ) β„Ž ( π‘₯ ) .

One can easily show that an even mapping 𝑓 ∢ 𝑋 β†’ π‘Œ satisfies (1.4) if and only if it is a quadratic-quartic mapping, that is, 𝑓 ( π‘₯ + 2 𝑦 ) + 𝑓 ( π‘₯ βˆ’ 2 𝑦 ) = 4 𝑓 ( π‘₯ + 𝑦 ) + 4 𝑓 ( π‘₯ βˆ’ 𝑦 ) βˆ’ 6 𝑓 ( π‘₯ ) + 2 𝑓 ( 2 𝑦 ) βˆ’ 8 𝑓 ( 𝑦 ) . ( 1 . 6 )

Also 𝑔 ( π‘₯ ) ∢ = 𝑓 ( 2 π‘₯ ) βˆ’ 4 𝑓 ( π‘₯ ) and β„Ž ( π‘₯ ) ∢ = 𝑓 ( 2 π‘₯ ) βˆ’ 1 6 𝑓 ( π‘₯ ) are quartic and quadratic, respectively, and 𝑓 ( π‘₯ ) = ( 1 / 1 2 ) 𝑔 ( π‘₯ ) βˆ’ ( 1 / 1 2 ) β„Ž ( π‘₯ ) .

For a given mapping 𝑓 ∢ 𝑋 β†’ π‘Œ , we define 𝐷 𝑓 ( π‘₯ , 𝑦 ) ∢ = 𝑓 ( π‘₯ + 2 𝑦 ) + 𝑓 ( π‘₯ βˆ’ 2 𝑦 ) βˆ’ 4 𝑓 ( π‘₯ + 𝑦 ) βˆ’ 4 𝑓 ( π‘₯ βˆ’ 𝑦 ) + 6 𝑓 ( π‘₯ ) βˆ’ 𝑓 ( 2 𝑦 ) βˆ’ 𝑓 ( βˆ’ 2 𝑦 ) + 4 𝑓 ( 𝑦 ) + 4 𝑓 ( βˆ’ 𝑦 ) ( 1 . 7 )

for all π‘₯ , 𝑦 ∈ 𝑋 .

Let 𝑋 be a set. A function 𝑑 ∢ 𝑋 Γ— 𝑋 β†’ [ 0 , ∞ ] is called a generalized metric on 𝑋 if 𝑑 satisfies(1) 𝑑 ( π‘₯ , 𝑦 ) = 0 if and only if π‘₯ = 𝑦 ,(2) 𝑑 ( π‘₯ , 𝑦 ) = 𝑑 ( 𝑦 , π‘₯ ) for all π‘₯ , 𝑦 ∈ 𝑋 ,(3) 𝑑 ( π‘₯ , 𝑧 ) ≀ 𝑑 ( π‘₯ , 𝑦 ) + 𝑑 ( 𝑦 , 𝑧 ) for all π‘₯ , 𝑦 , 𝑧 ∈ 𝑋 .

We recall the fixed-point alternative of Diaz and Margolis.

Theorem 1.1 (see [15, 16]). Let ( 𝑋 , 𝑑 ) be a complete generalized metric space and let 𝐽 ∢ 𝑋 β†’ 𝑋 be a strictly contractive mapping with Lipschitz constant 𝐿 < 1 , then for each given element π‘₯ ∈ 𝑋 , either 𝑑 ξ€· 𝐽 𝑛 π‘₯ , 𝐽 𝑛 + 1 π‘₯ ξ€Έ = ∞ ( 1 . 8 ) for all nonnegative integers 𝑛 or there exists a positive integer 𝑛 0 such that(1) 𝑑 ( 𝐽 𝑛 π‘₯ , 𝐽 𝑛 + 1 π‘₯ ) < ∞ for all 𝑛 β‰₯ 𝑛 0 ,(2)the sequence { 𝐽 𝑛 π‘₯ } converges to a fixed point 𝑦 βˆ— of 𝐽 ,(3) 𝑦 βˆ— is the unique fixed point of 𝐽 in the set π‘Œ = { 𝑦 ∈ 𝑋 ∣ 𝑑 ( 𝐽 𝑛 0 π‘₯ , 𝑦 ) < ∞ } ,(4) 𝑑 ( 𝑦 , 𝑦 βˆ— ) ≀ ( 1 / ( 1 βˆ’ 𝐿 ) ) 𝑑 ( 𝑦 , 𝐽 𝑦 ) for all 𝑦 ∈ π‘Œ .

In 1996, Isac and Rassias [17] were the first to provide applications of stability theory of functional equations for the proof of new fixed-point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [18–21]).

2. Preliminaries

In the sequel, we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [22–26]. Throughout this paper, Ξ” + is the space of all probability distribution functions, that is, the space of all mappings 𝐹 ∢ ℝ βˆͺ { βˆ’ ∞ , + ∞ } β†’ [ 0 , 1 ] , such taht 𝐹 is left continuous, nondecreasing on ℝ , 𝐹 ( 0 ) = 0 and { 𝐹 ( + ∞ ) = 1 } . 𝐷 + is a subset of Ξ” + consisting of all functions 𝐹 ∈ Ξ” + for which 𝑙 βˆ’ 𝐹 ( + ∞ ) = 1 , where 𝑙 βˆ’ 𝑓 ( π‘₯ ) denotes the left limit of the function 𝑓 at the point π‘₯ , that is, 𝑙 βˆ’ 𝑓 ( π‘₯ ) = l i m 𝑑 β†’ π‘₯ βˆ’ 𝑓 ( 𝑑 ) . The space Ξ” + is partially ordered by the usual pointwise ordering of functions, that is, 𝐹 ≀ 𝐺 if and only if 𝐹 ( 𝑑 ) ≀ 𝐺 ( 𝑑 ) for all 𝑑 in ℝ . The maximal element for Ξ” + in this order is the distribution function πœ€ 0 given by πœ€ 0 ξ‚» ( 𝑑 ) = 0 , i f 𝑑 ≀ 0 , 1 , i f 𝑑 > 0 . ( 2 . 1 )

A triangular norm (shortly 𝑑 -norm) is a binary operation on the unit interval [ 0 , 1 ] , that is, a function 𝑇 ∢ [ 0 , 1 ] Γ— [ 0 , 1 ] β†’ [ 0 , 1 ] , such that for all π‘Ž , 𝑏 , 𝑐 ∈ [ 0 , 1 ] the following four axioms satisfied:(T1) 𝑇 ( π‘Ž , 𝑏 ) = 𝑇 ( 𝑏 , π‘Ž ) (commutativity),(T2) 𝑇 ( π‘Ž , ( 𝑇 ( 𝑏 , 𝑐 ) ) ) = 𝑇 ( 𝑇 ( π‘Ž , 𝑏 ) , 𝑐 ) (associativity),(T3) 𝑇 ( π‘Ž , 1 ) = π‘Ž (boundary condition),(T4) 𝑇 ( π‘Ž , 𝑏 ) ≀ 𝑇 ( π‘Ž , 𝑐 ) whenever 𝑏 ≀ 𝑐 (monotonicity).

Basic examples are the Łukasiewicz   𝑑 -norm   𝑇 𝐿 , 𝑇 𝐿 ( π‘Ž , 𝑏 ) = m a x ( π‘Ž + 𝑏 βˆ’ 1 , 0 ) for all π‘Ž , 𝑏 ∈ [ 0 , 1 ] and the 𝑑 -norms   𝑇 𝑃 , 𝑇 𝑀 , 𝑇 𝐷 , where   𝑇 𝑃 ( π‘Ž , 𝑏 ) ∢ = π‘Ž 𝑏 , 𝑇 𝑀 ( π‘Ž , 𝑏 ) ∢ = m i n { π‘Ž , 𝑏 } , 𝑇 𝐷 ξ‚» ( π‘Ž , 𝑏 ) ∢ = m i n ( π‘Ž , 𝑏 ) , i f m a x ( π‘Ž , 𝑏 ) = 1 , 0 , o t h e r w i s e . ( 2 . 2 )

If 𝑇 is a 𝑑 -norm, then π‘₯ 𝑇 ( 𝑛 ) is defined for every π‘₯ ∈ [ 0 , 1 ] and 𝑛 ∈ 𝑁 βˆͺ { 0 } by 1, if 𝑛 = 0 and 𝑇 ( π‘₯ 𝑇 ( 𝑛 βˆ’ 1 ) , π‘₯ ) if 𝑛 β‰₯ 1 . A 𝑑 - n o r m 𝑇 is said to be of Hadžić type (we denote by 𝑇 ∈ β„‹ ) if the family ( π‘₯ 𝑇 ( 𝑛 ) ) 𝑛 ∈ 𝑁 is equicontinuous at π‘₯ = 1 (cf. [27]).

Other important triangular norms are the following (see [28]):(1)The Sugeno-Weber family { 𝑇 S W πœ† } πœ† ∈ [ βˆ’ 1 , ∞ ] is defined by 𝑇 S W βˆ’ 1 = 𝑇 𝐷 , 𝑇 S W ∞ = 𝑇 𝑃 and 𝑇 S W πœ† ξ‚΅ ( π‘₯ , 𝑦 ) = m a x 0 , π‘₯ + 𝑦 βˆ’ 1 + πœ† π‘₯ 𝑦 ξ‚Ά 1 + πœ† ( 2 . 3 ) if   πœ† ∈ ( βˆ’ 1 , ∞ ) .(2)The Domby family { 𝑇 𝐷 πœ† } πœ† ∈ [ 0 , ∞ ] is defined by 𝑇 𝐷 if πœ† = 0 , 𝑇 𝑀 if πœ† = ∞ , and 𝑇 𝐷 πœ† 1 ( π‘₯ , 𝑦 ) = ξ€· 1 + ( ( 1 βˆ’ π‘₯ ) / π‘₯ ) πœ† + ( ( 1 βˆ’ 𝑦 ) / 𝑦 ) πœ† ξ€Έ 1 / πœ† ( 2 . 4 ) if  πœ† ∈ ( 0 , ∞ ) .(3)The Aczel-Alsina family { 𝑇 A A πœ† } πœ† ∈ [ 0 , ∞ ] is defined by 𝑇 𝐷 if πœ† = 0 , 𝑇 𝑀 if πœ† = ∞ and 𝑇 A A πœ† ( π‘₯ , 𝑦 ) = 𝑒 βˆ’ ( | l o g π‘₯ | πœ† + | l o g 𝑦 | πœ† ) 1 / πœ† ( 2 . 5 ) if  πœ† ∈ ( 0 , ∞ ) .

A 𝑑 -norm   𝑇 can be extended (by associativity) in a unique way to an 𝑛 -array operation taking for ( π‘₯ 1 , … , π‘₯ 𝑛 ) ∈ [ 0 , 1 ] 𝑛 the value 𝑇 ( π‘₯ 1 , … , π‘₯ 𝑛 ) defined by 𝑇 0 𝑖 = 1 π‘₯ 𝑖 = 1 , 𝑇 𝑛 𝑖 = 1 π‘₯ 𝑖 ξ€· 𝑇 = 𝑇 𝑛 βˆ’ 1 𝑖 = 1 π‘₯ 𝑖 , π‘₯ 𝑛 ξ€Έ ξ€· π‘₯ = 𝑇 1 , … , π‘₯ 𝑛 ξ€Έ . ( 2 . 6 )

𝑇 can also be extended to a countable operation taking for any sequence ( π‘₯ 𝑛 ) 𝑛 ∈ 𝑁 in [ 0 , 1 ] the value 𝑇 ∞ 𝑖 = 1 π‘₯ 𝑖 = l i m 𝑛 β†’ ∞ 𝑇 𝑛 𝑖 = 1 π‘₯ 𝑖 . ( 2 . 7 ) The limit on the right side of (6.4) exists since the sequence ( 𝑇 𝑛 𝑖 = 1 π‘₯ 𝑖 ) 𝑛 ∈ β„• is nonincreasing and bounded from below.

Proposition 2.1 (see [28]). We have the following.(1)For 𝑇 β‰₯ 𝑇 𝐿 , the following implication holds: l i m 𝑛 β†’ ∞ T ∞ 𝑖 = 1 π‘₯ 𝑛 + 𝑖 = 1 ⟺ ∞  𝑛 = 1 ξ€· 1 βˆ’ π‘₯ 𝑛 ξ€Έ < ∞ . ( 2 . 8 ) (2)If 𝑇 is of Hadžić type, then l i m 𝑛 β†’ ∞ T ∞ 𝑖 = 1 π‘₯ 𝑛 + 𝑖 = 1 ( 2 . 9 ) for every sequence ( π‘₯ 𝑛 ) 𝑛 ∈ 𝑁 in [ 0 , 1 ] such that l i m 𝑛 β†’ ∞ π‘₯ 𝑛 = 1 .(3)If 𝑇 ∈ { 𝑇 A A πœ† } πœ† ∈ ( 0 , ∞ ) βˆͺ { 𝑇 𝐷 πœ† } πœ† ∈ ( 0 , ∞ ) , then l i m 𝑛 β†’ ∞ T ∞ 𝑖 = 1 π‘₯ 𝑛 + 𝑖 = 1 ⟺ ∞  𝑛 = 1 ξ€· 1 βˆ’ π‘₯ 𝑛 ξ€Έ 𝛼 < ∞ . ( 2 . 1 0 ) (4)If 𝑇 ∈ { 𝑇 S W πœ† } πœ† ∈ [ βˆ’ 1 , ∞ ) , then l i m 𝑛 β†’ ∞ T ∞ 𝑖 = 1 π‘₯ 𝑛 + 𝑖 = 1 ⟺ ∞  𝑛 = 1 ξ€· 1 βˆ’ π‘₯ 𝑛 ξ€Έ < ∞ . ( 2 . 1 1 )

Definition 2.2 (see [26]). A Random normed space (briefly, RN-space) is a triple ( 𝑋 , πœ‡ , 𝑇 ) , where 𝑋 is a vector space, 𝑇 is a continuous 𝑑 - norm, and πœ‡ is a mapping from 𝑋 into 𝐷 + such that, the following conditions hold: (RN1) πœ‡ π‘₯ ( 𝑑 ) = πœ€ 0 ( 𝑑 ) for all 𝑑 > 0 if and only if π‘₯ = 0 ,(RN2) πœ‡ 𝛼 π‘₯ ( 𝑑 ) = πœ‡ π‘₯ ( 𝑑 / | 𝛼 | ) for all π‘₯ ∈ 𝑋 , and 𝛼 β‰  0 ,(RN3) πœ‡ π‘₯ + 𝑦 ( 𝑑 + 𝑠 ) β‰₯ 𝑇 ( πœ‡ π‘₯ ( 𝑑 ) , πœ‡ 𝑦 ( 𝑠 ) ) for all π‘₯ , 𝑦 ∈ 𝑋 and 𝑑 , 𝑠 β‰₯ 0 .

Definition 2.3. Let ( 𝑋 , πœ‡ , 𝑇 ) be an RN-space.(1)A sequence { π‘₯ 𝑛 } in 𝑋 is said to be convergent to π‘₯ in 𝑋 if, for every πœ– > 0 and πœ† > 0 , there exists positive integer 𝑁 such that πœ‡ π‘₯ 𝑛 βˆ’ π‘₯ ( πœ– ) > 1 βˆ’ πœ† whenever 𝑛 β‰₯ 𝑁 .(2)A sequence { π‘₯ 𝑛 } in 𝑋 is called a Cauchy sequence if, for every πœ– > 0 and πœ† > 0 , there exists positive integer 𝑁 such that πœ‡ π‘₯ 𝑛 βˆ’ π‘₯ π‘š ( πœ– ) > 1 βˆ’ πœ† whenever 𝑛 β‰₯ π‘š β‰₯ 𝑁 .(3)An RN-space ( 𝑋 , πœ‡ , 𝑇 ) is said to be complete if and only if every Cauchy sequence in 𝑋 is convergent to a point in 𝑋 . A complete RN-space is said to be random Banach space.

Theorem 2.4 (see [25]). If ( 𝑋 , πœ‡ , 𝑇 ) is an RN-space and { π‘₯ 𝑛 } is a sequence such that π‘₯ 𝑛 β†’ π‘₯ , then l i m 𝑛 β†’ ∞ πœ‡ π‘₯ 𝑛 ( 𝑑 ) = πœ‡ π‘₯ ( 𝑑 ) almost everywhere.

The theory of random normed spaces (RN-spaces) is important as a generalization of deterministic result of linear normed spaces and also in the study of random operator equations. The RN-spaces may also provide us with the appropriate tools to study the geometry of nuclear physics and have important application in quantum particle physics. The generalized Hyers-Ulam stability of different functional equations in random normed spaces, RN-spaces, and fuzzy normed spaces has been recently studied [20, 24, 29–39].

3. Non-Archimedean Random Normed Space

By a non-Archimedean field, we mean a field 𝒦 equipped with a function (valuation) | β‹… | from 𝐾 into [ 0 , ∞ ) such that | π‘Ÿ | = 0 if and only if π‘Ÿ = 0 , | π‘Ÿ 𝑠 | = | π‘Ÿ | | 𝑠 | , and | π‘Ÿ + 𝑠 | ≀ m a x { | π‘Ÿ | , | 𝑠 | } for all π‘Ÿ , 𝑠 ∈ 𝒦 . Clearly, | 1 | = | βˆ’ 1 | = 1 and | 𝑛 | ≀ 1 for all 𝑛 ∈ β„• . By the trivial valuation, we mean the mapping | β‹… | taking everything but 0 into 1 and | 0 | = 0 . Let 𝑋 be a vector space over a field 𝒦 with a non-Archimedean nontrivial valuation | β‹… | . A function β€– β‹… β€– ∢ 𝑋 β†’ [ 0 , ∞ ) is called a non-Archimedean norm if it satisfies the following conditions:(NAN1) β€– π‘₯ β€– = 0 if and only if π‘₯ = 0 ,(NAN2) for any π‘Ÿ ∈ 𝒦 and π‘₯ ∈ 𝑋 , β€– π‘Ÿ π‘₯ β€– = | π‘Ÿ | β€– π‘₯ β€– ,(NAN3) the strong triangle inequality (ultrametric), namely, β€– π‘₯ + 𝑦 β€– ≀ m a x { β€– π‘₯ β€– , β€– 𝑦 β€– } ( π‘₯ , 𝑦 ∈ 𝑋 ) , ( 3 . 1 )

then ( 𝑋 , β€– β‹… β€– ) is called a non-Archimedean normed space. Due to the fact that β€– β€– π‘₯ 𝑛 βˆ’ π‘₯ π‘š β€– β€– ξ€½ β€– β€– π‘₯ ≀ m a x 𝑗 + 1 βˆ’ π‘₯ 𝑗 β€– β€– ξ€Ύ ∢ π‘š ≀ 𝑗 ≀ 𝑛 βˆ’ 1 ( 𝑛 > π‘š ) , ( 3 . 2 )

a sequence { π‘₯ 𝑛 } is a Cauchy sequence if and only if { π‘₯ 𝑛 + 1 βˆ’ π‘₯ 𝑛 } converges to zero in a non-Archimedean normed space. By a complete non-Archimedean normed space, we mean one in which every Cauchy sequence is convergent.

In 1897, Hensel [40] discovered the 𝑝 -adic numbers of as a number theoretical analogues of power series in complex analysis. Fix a prime number 𝑝 . For any nonzero rational number π‘₯ , there exists a unique integer 𝑛 π‘₯ ∈ β„€ such that π‘₯ = ( π‘Ž / 𝑏 ) 𝑝 𝑛 π‘₯ , where π‘Ž and 𝑏 are integers not divisible by 𝑝 . Then | π‘₯ | 𝑝 ∢ = 𝑝 βˆ’ 𝑛 π‘₯ defines a non-Archimedean norm on β„š . The completion of β„š with respect to the metric 𝑑 ( π‘₯ , 𝑦 ) = | π‘₯ βˆ’ 𝑦 | 𝑝 is denoted by β„š 𝑝 , which is called the 𝑝 -adic number field.

Throughout the paper, we assume that 𝑋 is a vector space and π‘Œ is a complete non-Archimedean normed space.

Definition 3.1. A non-Archimedean random normed space (briefly, non-Archimedean RN-space) is a triple ( 𝑋 , πœ‡ , 𝑇 ) , where 𝑋 is a linear space over a non-Archimedean field 𝒦 , 𝑇 is a continuous 𝑑 -norm, and πœ‡ is a mapping from 𝑋 into 𝐷 + such that the following conditions hold:(NA-RN1) πœ‡ π‘₯ ( 𝑑 ) = πœ€ 0 ( 𝑑 ) for all 𝑑 > 0 if and only if π‘₯ = 0 ,(NA-RN2) πœ‡ 𝛼 π‘₯ ( 𝑑 ) = πœ‡ π‘₯ ( 𝑑 / | 𝛼 | ) for all π‘₯ ∈ 𝑋 , 𝑑 > 0 , and 𝛼 β‰  0 ,(NA-RN3) πœ‡ π‘₯ + 𝑦 ( m a x { 𝑑 , 𝑠 } ) β‰₯ 𝑇 ( πœ‡ π‘₯ ( 𝑑 ) , πœ‡ 𝑦 ( 𝑠 ) ) for all π‘₯ , 𝑦 , 𝑧 ∈ 𝑋 and 𝑑 , 𝑠 β‰₯ 0 .
It is easy to see that if (NA-RN3) holds, then so is(RN3) πœ‡ π‘₯ + 𝑦 ( 𝑑 + 𝑠 ) β‰₯ 𝑇 ( πœ‡ π‘₯ ( 𝑑 ) , πœ‡ 𝑦 ( 𝑠 ) ) .

As a classical example, if ( 𝑋 , β€– . β€– ) is a non-Archimedean normed linear space, then the triple ( 𝑋 , πœ‡ , 𝑇 𝑀 ) , where πœ‡ π‘₯ ξ‚» ( 𝑑 ) = 0 , 𝑑 ≀ β€– π‘₯ β€– , 1 , 𝑑 > β€– π‘₯ β€– , ( 3 . 3 )

is a non-Archimedean RN-space.

Example 3.2. Let ( 𝑋 , β€– β‹… β€– ) be a non-Archimedean normed linear space. Define πœ‡ π‘₯ ( 𝑑 𝑑 ) = ( 𝑑 + β€– π‘₯ β€– π‘₯ ∈ 𝑋 , 𝑑 > 0 ) , ( 3 . 4 ) then ( 𝑋 , πœ‡ , 𝑇 𝑀 ) is a non-Archimedean RN-space.

Definition 3.3. Let ( 𝑋 , πœ‡ , 𝑇 ) be a non-Archimedean RN-space. Let { π‘₯ 𝑛 } be a sequence in 𝑋 , then { π‘₯ 𝑛 } is said to be convergent if there exists π‘₯ ∈ 𝑋 such that l i m 𝑛 β†’ ∞ πœ‡ π‘₯ 𝑛 βˆ’ π‘₯ ( 𝑑 ) = 1 ( 3 . 5 ) for all 𝑑 > 0 . In that case, π‘₯ is called the limit of the sequence { π‘₯ 𝑛 } .

A sequence { π‘₯ 𝑛 } in 𝑋 is called a Cauchy sequence if for each πœ€ > 0 and each 𝑑 > 0 there exists 𝑛 0 such that for all 𝑛 β‰₯ 𝑛 0 and all 𝑝 > 0 , we have πœ‡ π‘₯ 𝑛 + 𝑝 βˆ’ π‘₯ 𝑛 ( 𝑑 ) > 1 βˆ’ πœ€ .

If each Cauchy sequence is convergent, then the random norm is said to be complete and the non-Archimedean RN-space is called a non-Archimedean random Banach space.

Remark 3.4 (see [41]). Let ( 𝑋 , πœ‡ , 𝑇 𝑀 ) be a non-Archimedean RN-space, then πœ‡ π‘₯ 𝑛 + 𝑝 βˆ’ π‘₯ 𝑛  πœ‡ ( 𝑑 ) β‰₯ m i n π‘₯ 𝑛 + 𝑗 + 1 βˆ’ π‘₯ 𝑛 + 𝑗  ( 𝑑 ) ∢ 𝑗 = 0 , 1 , 2 , … , 𝑝 βˆ’ 1 . ( 3 . 6 ) So, the sequence { π‘₯ 𝑛 } is a Cauchy sequence if for each πœ€ > 0 and 𝑑 > 0 there exists 𝑛 0 such that for all 𝑛 β‰₯ 𝑛 0 , πœ‡ π‘₯ 𝑛 + 1 βˆ’ π‘₯ 𝑛 ( 𝑑 ) > 1 βˆ’ πœ€ . ( 3 . 7 )

4. Generalized Ulam-Hyers Stability for a Quartic Functional Equation in Non-Archimedean RN-Spaces of Functional Equation (1.4): An Odd Case

Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦 , and let ( π‘Œ , πœ‡ , 𝑇 ) be a non-Archimedean random Banach space over 𝒦 .

Next, we define a random approximately AQCQ mapping. Let Ξ¨ be a distribution function on 𝑋 Γ— 𝑋 Γ— [ 0 , ∞ ) such that Ξ¨ ( π‘₯ , 𝑦 , β‹… ) is nondecreasing and ξ‚΅ 𝑑 Ξ¨ ( 𝑐 π‘₯ , 𝑐 π‘₯ , 𝑑 ) β‰₯ Ξ¨ π‘₯ , π‘₯ , ξ‚Ά | 𝑐 | ( π‘₯ ∈ 𝑋 , 𝑐 β‰  0 ) . ( 4 . 1 )

Definition 4.1. A mapping 𝑓 ∢ 𝑋 β†’ π‘Œ is said to be Ξ¨ -approximately AQCQ if πœ‡ 𝐷 𝑓 ( π‘₯ , 𝑦 ) ( 𝑑 ) β‰₯ Ξ¨ ( π‘₯ , 𝑦 , 𝑑 ) ( π‘₯ , 𝑦 ∈ 𝑋 , 𝑑 > 0 ) . ( 4 . 2 )

In this section, we assume that 2 β‰  0 in 𝒦 (i.e., characteristic of 𝒦 is not 2). Our main result, in this section, is the following.

We prove the generalized Hyers-Ulam stability of the functional equation 𝐷 𝑓 ( π‘₯ , 𝑦 ) = 0 in non-Archimedean random spaces, an odd case.

Theorem 4.2. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦 and let ( π‘Œ , πœ‡ , 𝑇 ) be a non-Archimedean random Banach space over 𝒦 . Let 𝑓 ∢ 𝑋 β†’ π‘Œ be an odd mapping and Ξ¨ -approximately AQCQ mapping. If for some 𝛼 ∈ ℝ , 𝛼 > 0 , and some integer π‘˜ , π‘˜ > 3 with | 2 π‘˜ | < 𝛼 , Ξ¨ ξ€· 2 βˆ’ π‘˜ π‘₯ , 2 βˆ’ π‘˜ ξ€Έ 𝑦 , 𝑑 β‰₯ Ξ¨ ( π‘₯ , 𝑦 , 𝛼 𝑑 ) ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) , ( 4 . 3 ) l i m 𝑛 β†’ ∞ T ∞ 𝑗 = 𝑛 𝑀  𝛼 2 π‘₯ , 𝑗 𝑑 | | 8 | | π‘˜ 𝑗 ξƒͺ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) , ( 4 . 4 ) then there exists a unique cubic mapping 𝐢 ∢ 𝑋 β†’ π‘Œ such that πœ‡ 𝑓 ( π‘₯ ) βˆ’ 2 𝑓 ( π‘₯ / 2 ) βˆ’ 𝐢 ( π‘₯ / 2 ) ( 𝑑 ) β‰₯ 𝑇 ∞ 𝑖 = 1 𝑀  𝛼 π‘₯ , 𝑖 + 1 𝑑 | | 8 | | π‘˜ 𝑖 ξƒͺ ( 4 . 5 ) for all π‘₯ ∈ 𝑋 and 𝑑 > 0 , where 𝑀 ( π‘₯ , 𝑑 ) ∢ = 𝑇 π‘˜ βˆ’ 1 ξ‚Έ Ξ¨ ξ‚΅ π‘₯ 2 , π‘₯ 2 , 𝑑 | | 4 | | ξ‚Ά ξ‚€ π‘₯ , Ξ¨ π‘₯ , 2  ξ‚΅ 2 , 𝑑 , … , Ξ¨ π‘˜ βˆ’ 1 π‘₯ 2 , 2 π‘˜ βˆ’ 1 π‘₯ 2 , 𝑑 | | 4 | | ξ‚Ά ξ‚΅ 2 , Ξ¨ π‘˜ βˆ’ 1 2 π‘₯ , π‘˜ βˆ’ 1 π‘₯ 2 , 𝑑 ξ‚Ά ξ‚Ή ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) . ( 4 . 6 )

Proof. Letting π‘₯ = 𝑦 in (4.2), we get πœ‡ 𝑓 ( 3 𝑦 ) βˆ’ 4 𝑓 ( 2 𝑦 ) + 5 𝑓 ( 𝑦 ) ( 𝑑 ) β‰₯ Ξ¨ ( 𝑦 , 𝑦 , 𝑑 ) ( 4 . 7 ) for all 𝑦 ∈ 𝑋 and 𝑑 > 0 . Replacing π‘₯ by 2 𝑦 in (4.2), we get πœ‡ 𝑓 ( 4 𝑦 ) βˆ’ 4 𝑓 ( 3 𝑦 ) + 6 𝑓 ( 2 𝑦 ) βˆ’ 4 𝑓 ( 𝑦 ) ( 𝑑 ) β‰₯ Ξ¨ ( 2 𝑦 , 𝑦 , 𝑑 ) ( 4 . 8 ) for all 𝑦 ∈ 𝑋 and 𝑑 > 0 . By (4.7) and (4.8), we have πœ‡ 𝑓 ( 4 𝑦 ) βˆ’ 1 0 𝑓 ( 2 𝑦 ) + 1 6 𝑓 ( 𝑦 ) ξ€· πœ‡ ( 𝑑 ) β‰₯ 𝑇 4 ( 𝑓 ( 3 𝑦 ) βˆ’ 4 𝑓 ( 2 𝑦 ) + 5 𝑓 ( 𝑦 ) ) ( 𝑑 ) , πœ‡ 𝑓 ( 4 𝑦 ) βˆ’ 4 𝑓 ( 3 𝑦 ) + 6 𝑓 ( 2 𝑦 ) βˆ’ 4 𝑓 ( 𝑦 ) ξ€Έ ξ‚΅ πœ‡ ( 𝑑 ) = 𝑇 𝑓 ( 3 𝑦 ) βˆ’ 4 𝑓 ( 2 𝑦 ) + 5 𝑓 ( 𝑦 ) ξ‚΅ 𝑑 | | 4 | | ξ‚Ά , πœ‡ 𝑓 ( 4 𝑦 ) βˆ’ 4 𝑓 ( 3 𝑦 ) + 6 𝑓 ( 2 𝑦 ) βˆ’ 4 𝑓 ( 𝑦 ) ξ‚Ά ξ‚΅ Ξ¨ ξ‚΅ 𝑑 ( 𝑑 ) β‰₯ 𝑇 𝑦 , 𝑦 , | | 4 | | ξ‚Ά ξ‚Ά , Ξ¨ ( 2 𝑦 , 𝑦 , 𝑑 ) ( 4 . 9 ) for all 𝑦 ∈ 𝑋 and 𝑑 > 0 . Letting 𝑦 ∢ = π‘₯ / 2 and 𝑔 ( π‘₯ ) ∢ = 𝑓 ( 2 π‘₯ ) βˆ’ 2 𝑓 ( π‘₯ ) for all π‘₯ ∈ 𝑋 in (4.9), we get πœ‡ 𝑔 ( π‘₯ ) βˆ’ 8 𝑔 ( π‘₯ / 2 ) ξ‚΅ Ξ¨ ξ‚΅ π‘₯ ( 𝑑 ) β‰₯ 𝑇 2 , π‘₯ 2 , 𝑑 | | 4 | | ξ‚Ά ξ‚€ π‘₯ , Ξ¨ π‘₯ , 2  ξ‚Ά , 𝑑 ( 4 . 1 0 ) for all π‘₯ ∈ 𝑋 and 𝑑 > 0 . Now, we show by induction on 𝑗 that for all π‘₯ ∈ 𝑋 , 𝑑 > 0 and 𝑗 β‰₯ 1 , πœ‡ 𝑔 ( 2 𝑗 βˆ’ 1 π‘₯ ) βˆ’ 8 𝑗 𝑔 ( π‘₯ / 2 ) ( 𝑑 ) β‰₯ 𝑀 𝑗 ( π‘₯ , 𝑑 ) ∢ = 𝑇 2 𝑗 βˆ’ 1 ξ‚Έ Ξ¨ ξ‚΅ π‘₯ 2 , π‘₯ 2 , 𝑑 | | 4 | | ξ‚Ά ξ‚€ π‘₯ , Ξ¨ π‘₯ , 2  ξ‚΅ 2 , 𝑑 , … , Ξ¨ 𝑗 βˆ’ 1 π‘₯ 2 , 2 𝑗 βˆ’ 1 π‘₯ 2 , 𝑑 | | 4 | | ξ‚Ά ξ‚΅ 2 , Ξ¨ 𝑗 βˆ’ 1 2 π‘₯ , 𝑗 βˆ’ 1 π‘₯ 2 . , 𝑑 ξ‚Ά ξ‚Ή ( 4 . 1 1 ) Putting 𝑗 = 1 in (4.11), we obtain (4.10). Assume that (4.11) holds for some 𝑗 β‰₯ 1 . Replacing π‘₯ by 2 𝑗 π‘₯ in (4.10), we get πœ‡ 𝑔 ( 2 𝑗 π‘₯ ) βˆ’ 8 𝑔 ( 2 𝑗 βˆ’ 1 π‘₯ ) ξ‚΅ Ξ¨ ξ‚΅ 2 ( 𝑑 ) β‰₯ 𝑇 𝑗 βˆ’ 1 π‘₯ , 2 𝑗 βˆ’ 1 𝑑 π‘₯ , | | 4 | | ξ‚Ά ξ€· 2 , Ξ¨ 𝑗 π‘₯ , 2 𝑗 βˆ’ 1 ξ€Έ ξ‚Ά π‘₯ , 𝑑 . ( 4 . 1 2 ) Since | 8 | ≀ 1 , πœ‡ 𝑔 ( 2 𝑗 π‘₯ ) βˆ’ 8 𝑗 + 1 𝑔 ( π‘₯ / 2 ) ξ€· πœ‡ ( 𝑑 ) β‰₯ 𝑇 𝑔 ( 2 𝑗 π‘₯ ) βˆ’ 8 𝑔 ( 2 𝑗 βˆ’ 1 π‘₯ ) ( 𝑑 ) , πœ‡ 8 𝑔 ( 2 𝑗 βˆ’ 1 π‘₯ ) βˆ’ 8 𝑗 + 1 𝑔 ( π‘₯ / 2 ) ξ€Έ ξ‚΅ πœ‡ ( 𝑑 ) = 𝑇 𝑔 ( 2 𝑗 π‘₯ ) βˆ’ 8 𝑔 ( 2 𝑗 βˆ’ 1 π‘₯ ) ( 𝑑 ) , πœ‡ 𝑔 ( 2 𝑗 βˆ’ 1 π‘₯ ) βˆ’ 8 𝑗 𝑔 ( π‘₯ / 2 ) ξ‚΅ 𝑑 | | 8 | | ξ‚Ά ξ‚Ά β‰₯ 𝑇 2 ξ‚΅ Ξ¨ ξ‚΅ 2 𝑗 βˆ’ 1 π‘₯ , 2 𝑗 βˆ’ 1 𝑑 π‘₯ , | | 4 | | ξ‚Ά ξ€· 2 , Ξ¨ 𝑗 π‘₯ , 2 𝑗 βˆ’ 1 ξ€Έ π‘₯ , 𝑑 , 𝑀 𝑗 ξ‚Ά ( π‘₯ , 𝑑 ) = 𝑀 𝑗 + 1 ( π‘₯ , 𝑑 ) ( 4 . 1 3 ) for all π‘₯ ∈ 𝑋 and 𝑑 > 0 . Thus, (4.11) holds for all 𝑗 β‰₯ 2 . In particular, πœ‡ 𝑔 ( 2 π‘˜ βˆ’ 1 π‘₯ ) βˆ’ 8 π‘˜ 𝑔 ( π‘₯ / 2 ) ( 𝑑 ) β‰₯ 𝑀 ( π‘₯ , 𝑑 ) ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) . ( 4 . 1 4 ) Replacing π‘₯ by 2 βˆ’ ( π‘˜ 𝑛 + π‘˜ βˆ’ 1 ) π‘₯ in (4.14) and using inequality (4.3), we obtain πœ‡ 𝑔 ( π‘₯ / 2 π‘˜ 𝑛 ) βˆ’ 8 π‘˜ 𝑔 ( π‘₯ / 2 π‘˜ ( 𝑛 + 1 ) ) ξ‚΅ ( 𝑑 ) β‰₯ 𝑀 2 π‘₯ 2 π‘˜ ( 𝑛 + 1 ) ξ‚Ά , 𝑑 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 , 𝑛 = 0 , 1 , 2 , … ) . ( 4 . 1 5 ) Then πœ‡ 8 π‘˜ 𝑛 𝑔 ( π‘₯ / 2 π‘˜ 𝑛 ) βˆ’ 8 π‘˜ ( 𝑛 + 1 ) 𝑔 ( π‘₯ / 2 π‘˜ ( 𝑛 + 1 ) )  𝛼 ( 𝑑 ) β‰₯ 𝑀 2 π‘₯ , 𝑛 + 1 | | 8 π‘˜ ( 𝑛 + 1 ) | | 𝑑 ξƒͺ ( π‘₯ ∈ 𝑋 , 𝑑 > 0 , 𝑛 = 0 , 1 , 2 , … ) . ( 4 . 1 6 ) Hence πœ‡ 8 π‘˜ 𝑛 𝑔 ( π‘₯ / 2 π‘˜ 𝑛 ) βˆ’ 8 π‘˜ ( 𝑛 + 𝑝 ) 𝑔 ( π‘₯ / 2 π‘˜ ( 𝑛 + 𝑝 ) ) ( 𝑑 ) β‰₯ 𝑇 𝑛 + 𝑝 𝑗 = 𝑛 ξ€· πœ‡ 8 π‘˜ 𝑗 𝑔 ( π‘₯ / 2 π‘˜ 𝑗 ) βˆ’ 8 π‘˜ ( 𝑗 + 𝑝 ) 𝑔 ( π‘₯ / 2 π‘˜ ( 𝑗 + 𝑝 ) ) ( ξ€Έ 𝑑 ) β‰₯ 𝑇 𝑛 + 𝑝 𝑗 = 𝑛 𝑀 βŽ› ⎜ ⎜ ⎜ ⎝ 𝛼 2 π‘₯ , 𝑗 + 1 | | | ξ€· 8 π‘˜ ξ€Έ 𝑗 + 1 | | | 𝑑 ⎞ ⎟ ⎟ ⎟ ⎠ β‰₯ 𝑇 𝑛 + 𝑝 𝑗 = 𝑛 𝑀 βŽ› ⎜ ⎜ ⎜ ⎝ 𝛼 2 π‘₯ , 𝑗 + 1 | | | ξ€· 8 π‘˜ ξ€Έ 𝑗 + 1 | | | 𝑑 ⎞ ⎟ ⎟ ⎟ ⎠ ( π‘₯ ∈ 𝑋 , 𝑑 > 0 , 𝑛 = 0 , 1 , 2 , … ) . ( 4 . 1 7 ) Since l i m 𝑛 β†’ ∞ 𝑇 ∞ 𝑗 = 𝑛 𝑀 βŽ› ⎜ ⎜ ⎜ ⎝ 𝛼 2 π‘₯ , 𝑗 + 1 | | | ξ€· 8 π‘˜ ξ€Έ 𝑗 + 1 | | | 𝑑 ⎞ ⎟ ⎟ ⎟ ⎠ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) , ( 4 . 1 8 ) then ξ‚» 8 π‘˜ 𝑛 𝑔 ξ‚΅ π‘₯ 2 π‘˜ 𝑛 ξ‚Ά ξ‚Ό 𝑛 ∈ β„• ( 4 . 1 9 ) is a Cauchy sequence in the non-Archimedean random Banach space ( π‘Œ , πœ‡ , 𝑇 ) . Hence we can define a mapping 𝐢 ∢ 𝑋 β†’ π‘Œ such that l i m 𝑛 β†’ ∞ πœ‡ ( 8 8 π‘˜ ) 𝑛 𝑔 ( π‘₯ / 2 π‘˜ 𝑛 ) βˆ’ 𝐢 ( π‘₯ ) ( 𝑑 ) = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) . ( 4 . 2 0 )
Next for each 𝑛 β‰₯ 1 , π‘₯ ∈ 𝑋 and 𝑑 > 0 , πœ‡ 𝑔 ( π‘₯ ) βˆ’ ( 8 8 π‘˜ ) 𝑛 𝑔 ( π‘₯ / 2 π‘˜ 𝑛 ) ( 𝑑 ) = πœ‡ βˆ‘ 𝑛 βˆ’ 1 𝑖 = 0 ( 8 8 π‘˜ ) 𝑖 𝑔 ( π‘₯ / 2 π‘˜ 𝑖 ) βˆ’ ( 8 8 π‘˜ ) 𝑖 + 1 𝑔 ( π‘₯ / 2 π‘˜ ( 𝑖 + 1 ) ) ( 𝑑 ) β‰₯ 𝑇 𝑛 βˆ’ 1 𝑖 = 0 ξ‚€ πœ‡ ( 8 8 π‘˜ ) 𝑖 𝑔 ( π‘₯ / 2 π‘˜ 𝑖 ) βˆ’ ( 8 8 π‘˜ ) 𝑖 + 1 𝑔 ( π‘₯ / 2 π‘˜ ( 𝑖 + 1 ) ) (  𝑑 ) β‰₯ 𝑇 𝑛 βˆ’ 1 𝑖 = 0 𝑀  𝛼 2 π‘₯ , 𝑖 + 1 𝑑 | | 8 π‘˜ | | 𝑖 + 1 ξƒͺ . ( 4 . 2 1 ) Therefore, πœ‡ 𝑔 ( π‘₯ ) βˆ’ 𝐢 ( π‘₯ ) ξ€· πœ‡ ( 𝑑 ) β‰₯ 𝑇 𝑔 ( π‘₯ ) βˆ’ ( 8 8 π‘˜ ) 𝑛 𝑔 ( π‘₯ / 2 π‘˜ 𝑛 ) ( 𝑑 ) , πœ‡ ( 8 8 π‘˜ ) 𝑛 𝑔 ( π‘₯ / 2 π‘˜ 𝑛 ) βˆ’ 𝐢 ( π‘₯ ) ξ€Έ  𝑇 ( 𝑑 ) β‰₯ 𝑇 𝑛 βˆ’ 1 𝑖 = 0 𝑀  𝛼 2 π‘₯ , 𝑖 + 1 𝑑 | | 8 π‘˜ | | 𝑖 + 1 ξƒͺ , πœ‡ ( 8 8 π‘˜ ) 𝑛 𝑔 ( π‘₯ / 2 π‘˜ 𝑛 ) βˆ’ 𝐢 ( π‘₯ ) ξƒͺ . ( 𝑑 ) ( 4 . 2 2 ) By letting 𝑛 β†’ ∞ , we obtain πœ‡ 𝑔 ( π‘₯ ) βˆ’ 𝐢 ( π‘₯ ) ( 𝑑 ) β‰₯ 𝑇 ∞ 𝑖 = 1 𝑀  𝛼 2 π‘₯ , 𝑖 + 1 𝑑 | | 8 π‘˜ | | 𝑖 + 1 ξƒͺ . ( 4 . 2 3 ) So, πœ‡ 𝑓 ( π‘₯ ) βˆ’ 2 𝑓 ( π‘₯ / 2 ) βˆ’ 𝐢 ( π‘₯ / 2 ) ( 𝑑 ) β‰₯ 𝑇 ∞ 𝑖 = 1 𝑀  𝛼 π‘₯ , 𝑖 + 1 𝑑 | | 8 π‘˜ | | 𝑖 + 1 ξƒͺ . ( 4 . 2 4 ) This proves (4.5). From 𝐷 𝑔 ( π‘₯ , 𝑦 ) = 𝐷 𝑓 ( 2 π‘₯ , 2 𝑦 ) βˆ’ 2 𝐷 𝑓 ( π‘₯ , 𝑦 ) , by (4.2), we deduce that πœ‡ 𝐷 𝑓 ( 2 π‘₯ , 2 𝑦 ) πœ‡ ( 𝑑 ) β‰₯ Ξ¨ ( 2 π‘₯ , 2 𝑦 , 𝑑 ) , βˆ’ 2 𝐷 𝑓 ( π‘₯ , 𝑦 ) ( 𝑑 ) = πœ‡ 𝐷 𝑓 ( π‘₯ , 𝑦 ) ξ‚΅ 𝑑 | | 2 | | ξ‚Ά β‰₯ πœ‡ 𝐷 𝑓 ( π‘₯ , 𝑦 ) ( 𝑑 ) β‰₯ Ξ¨ ( π‘₯ , 𝑦 , 𝑑 ) , ( 4 . 2 5 ) and so, by (NA-RN3) and (4.2), we obtain πœ‡ 𝐷 𝑔 ( π‘₯ , 𝑦 ) ξ€· πœ‡ ( 𝑑 ) β‰₯ 𝑇 𝐷 𝑓 ( 2 π‘₯ , 2 𝑦 ) ( 𝑑 ) , πœ‡ βˆ’ 2 𝐷 𝑓 ( π‘₯ , 𝑦 ) ξ€Έ ( 𝑑 ) β‰₯ 𝑇 ( Ξ¨ ( 2 π‘₯ , 2 𝑦 , 𝑑 ) , Ξ¨ ( π‘₯ , 𝑦 , 𝑑 ) ) ∢ = 𝑁 ( π‘₯ , 𝑦 , 𝑑 ) . ( 4 . 2 6 ) It follows that πœ‡ 8 π‘˜ 𝑛 𝐷 𝑔 ( π‘₯ / 2 π‘˜ 𝑛 , 𝑦 / 2 π‘˜ 𝑛 ) ( 𝑑 ) = πœ‡ 𝐷 𝑔 ( π‘₯ / 2 π‘˜ 𝑛 , 𝑦 / 2 π‘˜ 𝑛 )  𝑑 | | 8 | | π‘˜ 𝑛 ξƒͺ  π‘₯ β‰₯ 𝑁 2 π‘˜ 𝑛 , 𝑦 2 π‘˜ 𝑛 , 𝑑 | | 8 | | π‘˜ 𝑛 ξƒͺ  𝛼 β‰₯ β‹― β‰₯ 𝑁 π‘₯ , 𝑦 , 𝑛 βˆ’ 1 𝑑 | | 8 | | π‘˜ ( 𝑛 βˆ’ 1 ) ξƒͺ ( 4 . 2 7 ) for all π‘₯ , 𝑦 ∈ 𝑋 , 𝑑 > 0 , and 𝑛 ∈ β„• . Since l i m 𝑛 β†’ ∞ 𝑁  𝛼 π‘₯ , 𝑦 , 𝑛 βˆ’ 1 𝑑 | | 8 | | π‘˜ ( 𝑛 βˆ’ 1 ) ξƒͺ = 1 ( 4 . 2 8 ) for all π‘₯ , 𝑦 ∈ 𝑋 and 𝑑 > 0 , by Theorem 2.4, we deduce that πœ‡ 𝐷 𝐢 ( π‘₯ , 𝑦 ) ( 𝑑 ) = 1 ( 4 . 2 9 ) for all π‘₯ , 𝑦 ∈ 𝑋 and 𝑑 > 0 . Thus, the mapping 𝐢 ∢ 𝑋 β†’ π‘Œ satisfies (1.4).
Now, we have 𝐢 ( 2 π‘₯ ) βˆ’ 8 𝐢 ( π‘₯ ) = l i m 𝑛 β†’ ∞ ξ‚Έ 8 𝑛 𝑔 ξ‚΅ π‘₯ 2 𝑛 βˆ’ 1 ξ‚Ά βˆ’ 8 𝑛 + 1 𝑔 ξ‚€ π‘₯ 2 𝑛  ξ‚Ή = 8 l i m 𝑛 β†’ ∞ ξ‚Έ 8 𝑛 βˆ’ 1 𝑔 ξ‚΅ π‘₯ 2 𝑛 βˆ’ 1 ξ‚Ά βˆ’ 8 𝑛 𝑔 ξ‚€ π‘₯ 2 𝑛  ξ‚Ή = 0 ( 4 . 3 0 ) for all π‘₯ ∈ 𝑋 . Since the mapping π‘₯ β†’ 𝐢 ( 2 π‘₯ ) βˆ’ 2 𝐢 ( π‘₯ ) is cubic (see Lemma  2.2 of [14]), from the equality 𝐢 ( 2 π‘₯ ) = 8 𝐢 ( π‘₯ ) , we deduce that the mapping 𝐢 ∢ 𝑋 β†’ π‘Œ is cubic.

Corollary 4.3. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦 , and let ( π‘Œ , πœ‡ , 𝑇 ) be a non-Archimedean random Banach space over 𝒦 under a t-norm 𝑇 ∈ β„‹ . Let 𝑓 ∢ 𝑋 β†’ π‘Œ be an odd and Ξ¨ -approximately AQCQ mapping. If, for some 𝛼 ∈ ℝ , 𝛼 > 0 , and some integer π‘˜ , π‘˜ > 3 , with | 2 π‘˜ | < 𝛼 , Ξ¨ ξ€· 2 βˆ’ π‘˜ π‘₯ , 2 βˆ’ π‘˜ ξ€Έ 𝑦 , 𝑑 β‰₯ Ξ¨ ( π‘₯ , 𝑦 , 𝛼 𝑑 ) ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) , ( 4 . 3 1 ) then there exists a unique cubic mapping 𝐢 ∢ 𝑋 β†’ π‘Œ such that πœ‡ 𝑓 ( π‘₯ ) βˆ’ 2 𝑓 ( π‘₯ / 2 ) βˆ’ 𝐢 ( π‘₯ / 2 ) ( 𝑑 ) β‰₯ 𝑇 ∞ 𝑖 = 1 𝑀  𝛼 π‘₯ , 𝑖 + 1 𝑑 | | 8 | | π‘˜ 𝑖 ξƒͺ ( 4 . 3 2 ) for all π‘₯ ∈ 𝑋 and 𝑑 > 0 .

Proof. Since l i m 𝑛 β†’ ∞ 𝑀  𝛼 π‘₯ , 𝑗 𝑑 | | 8 | | π‘˜ 𝑗 ξƒͺ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) ( 4 . 3 3 ) and 𝑇 is of Hadžić type, from Proposition 2.1, it follows that l i m 𝑛 β†’ ∞ 𝑇 ∞ 𝑗 = 𝑛 𝑀  𝛼 π‘₯ , 𝑗 𝑑 | | 8 | | π‘˜ 𝑗 ξƒͺ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) . ( 4 . 3 4 ) Now, we can apply Theorem 4.2 to obtain the result.

Example 4.4. Let ( 𝑋 , πœ‡ , 𝑇 𝑀 ) be non-Archimedean random normed space in which πœ‡ π‘₯ ( 𝑑 𝑑 ) = ( 𝑑 + β€– π‘₯ β€– π‘₯ ∈ 𝑋 , 𝑑 > 0 ) . ( 4 . 3 5 ) And let ( π‘Œ , πœ‡ , 𝑇 𝑀 ) be a complete non-Archimedean random normed space (see Example 3.2). Define 𝑑 Ξ¨ ( π‘₯ , 𝑦 , 𝑑 ) = . 1 + 𝑑 ( 4 . 3 6 ) It is easy to see that (4.3) holds for 𝛼 = 1 . Also, since 𝑑 𝑀 ( π‘₯ , 𝑑 ) = , 1 + 𝑑 ( 4 . 3 7 ) we have l i m 𝑛 β†’ ∞ 𝑇 ∞ 𝑀 , 𝑗 = 𝑛 𝑀  𝛼 π‘₯ , 𝑗 𝑑 | | 8 | | π‘˜ 𝑗 ξƒͺ = l i m 𝑛 β†’ ∞  l i m π‘š β†’ ∞ 𝑇 π‘š 𝑀 , 𝑗 = 𝑛 𝑀  𝑑 π‘₯ , | | 8 | | π‘˜ 𝑗 ξƒͺ ξƒͺ = l i m 𝑛 β†’ ∞ l i m π‘š β†’ ∞  𝑑 | | 8 𝑑 + π‘˜ | | 𝑛 ξƒͺ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) . ( 4 . 3 8 ) Let 𝑓 ∢ 𝑋 β†’ π‘Œ be an odd and Ξ¨ -approximately AQCQ mapping. Thus, all the conditions of Theorem 4.2 hold, and so there exists a unique cubic mapping 𝐢 ∢ 𝑋 β†’ π‘Œ such that πœ‡ 𝑓 ( π‘₯ ) βˆ’ 2 𝑓 ( π‘₯ / 2 ) βˆ’ 𝐢 ( π‘₯ / 2 ) ( 𝑑 𝑑 ) β‰₯ | | 8 𝑑 + π‘˜ | | . ( 4 . 3 9 )

Theorem 4.5. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦 , and let ( π‘Œ , πœ‡ , 𝑇 ) be a non-Archimedean random Banach space over 𝒦 . Let 𝑓 ∢ 𝑋 β†’ π‘Œ be an odd mapping and Ξ¨ -approximately AQCQ mapping. If for some 𝛼 ∈ ℝ , 𝛼 > 0 , and some integer π‘˜ , π‘˜ > 1 with | 2 π‘˜ | < 𝛼 , Ξ¨ ξ€· 2 βˆ’ π‘˜ π‘₯ , 2 βˆ’ π‘˜ ξ€Έ 𝑦 , 𝑑 β‰₯ Ξ¨ ( π‘₯ , 𝑦 , 𝛼 𝑑 ) ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) , l i m 𝑛 β†’ ∞ 𝑇 ∞ 𝑗 = 𝑛 𝑀  𝛼 2 π‘₯ , 𝑗 𝑑 | | 2 | | π‘˜ 𝑗 ξƒͺ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) , ( 4 . 4 0 ) then there exists a unique additive mapping 𝐴 ∢ 𝑋 β†’ π‘Œ such that πœ‡ 𝑓 ( π‘₯ ) βˆ’ 8 𝑓 ( π‘₯ / 2 ) βˆ’ 𝐴 ( π‘₯ / 2 ) ( 𝑑 ) β‰₯ 𝑇 ∞ 𝑖 = 1 𝑀  𝛼 π‘₯ , 𝑖 + 1 𝑑 | | 2 | | π‘˜ 𝑖 ξƒͺ ( 4 . 4 1 ) for all π‘₯ ∈ 𝑋 and 𝑑 > 0 , where 𝑀 ( π‘₯ , 𝑑 ) ∢ = 𝑇 π‘˜ βˆ’ 1 ξ‚Έ Ξ¨ ξ‚΅ π‘₯ 2 , π‘₯ 2 , 𝑑 | | 4 | | ξ‚Ά ξ‚€ π‘₯ , Ξ¨ π‘₯ , 2  ξ‚΅ 2 , 𝑑 , … , Ξ¨ π‘˜ βˆ’ 1 π‘₯ 2 , 2 π‘˜ βˆ’ 1 π‘₯ 2 , 𝑑 | | 4 | | ξ‚Ά ξ‚΅ 2 , Ξ¨ π‘˜ βˆ’ 1 2 π‘₯ , π‘˜ βˆ’ 1 π‘₯ 2 , 𝑑 ξ‚Ά ξ‚Ή ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) ( 4 . 4 2 )

Proof. Letting 𝑦 ∢ = π‘₯ / 2 and 𝑔 ( π‘₯ ) ∢ = 𝑓 ( 2 π‘₯ ) βˆ’ 8 𝑓 ( π‘₯ ) for all π‘₯ ∈ 𝑋 in (4.9), we get πœ‡ 𝑔 ( π‘₯ ) βˆ’ 2 𝑔 ( π‘₯ / 2 ) ξ‚΅ Ξ¨ ξ‚΅ π‘₯ ( 𝑑 ) β‰₯ 𝑇 2 , π‘₯ 2 , 𝑑 | | 4 | | ξ‚Ά ξ‚€ π‘₯ , Ξ¨ π‘₯ , 2  ξ‚Ά , 𝑑 ( 4 . 4 3 ) for all π‘₯ ∈ 𝑋 and 𝑑 > 0 .
The rest of the proof is similar to the proof of Theorem 4.2.

Corollary 4.6. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦 , and let ( π‘Œ , πœ‡ , 𝑇 ) be a non-Archimedean random Banach space over 𝒦 under a t-norm 𝑇 ∈ β„‹ . Let 𝑓 ∢ 𝑋 β†’ π‘Œ be an odd and Ξ¨ -approximately AQCQ mapping. If, for some 𝛼 ∈ ℝ , 𝛼 > 0 , and some integer π‘˜ , π‘˜ > 1 , with | 2 π‘˜ | < 𝛼 , Ξ¨ ξ€· 2 βˆ’ π‘˜ π‘₯ , 2 βˆ’ π‘˜ ξ€Έ 𝑦 , 𝑑 β‰₯ Ξ¨ ( π‘₯ , 𝑦 , 𝛼 𝑑 ) ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) , ( 4 . 4 4 ) then there exists a unique additive mapping 𝐴 ∢ 𝑋 β†’ π‘Œ such that πœ‡ 𝑓 ( π‘₯ ) βˆ’ 8 𝑓 ( π‘₯ / 2 ) βˆ’ 𝐴 ( π‘₯ / 2 ) ( 𝑑 ) β‰₯ 𝑇 ∞ 𝑖 = 1 𝑀  𝛼 π‘₯ , 𝑖 + 1 𝑑 | | 2 | | π‘˜ 𝑖 ξƒͺ ( 4 . 4 5 ) for all π‘₯ ∈ 𝑋 and 𝑑 > 0 .

Proof. Since l i m 𝑛 β†’ ∞ 𝑀  𝛼 π‘₯ , 𝑗 𝑑 | | 2 | | π‘˜ 𝑗 ξƒͺ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) ( 4 . 4 6 ) and 𝑇 is of Hadžić type, from Proposition 2.1, it follows that l i m 𝑛 β†’ ∞ 𝑇 ∞ 𝑗 = 𝑛 𝑀  𝛼 π‘₯ , 𝑗 𝑑 | | 2 | | π‘˜ 𝑗 ξƒͺ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) . ( 4 . 4 7 ) Now, we can apply Theorem 4.5 to obtain the result.

Example 4.7. Let ( 𝑋 , πœ‡ , 𝑇 𝑀 ) non-Archimedean random normed space in which πœ‡ π‘₯ = 𝑑 ( 𝑑 ) 𝑑 + β€– π‘₯ β€– ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) , ( 4 . 4 8 ) and let ( π‘Œ , πœ‡ , 𝑇 𝑀 ) be a complete non-Archimedean random normed space (see Example 3.2). Define 𝑑 Ξ¨ ( π‘₯ , 𝑦 , 𝑑 ) = . 1 + 𝑑 ( 4 . 4 9 ) It is easy to see that (4.3) holds for 𝛼 = 1 . Also, since 𝑑 𝑀 ( π‘₯ , 𝑑 ) = , 1 + 𝑑 ( 4 . 5 0 ) we have l i m 𝑛 β†’ ∞ 𝑇 ∞ 𝑀 , 𝑗 = 𝑛 𝑀  𝛼 π‘₯ , 𝑗 𝑑 | | 2 | | π‘˜ 𝑗 ξƒͺ = l i m 𝑛 β†’ ∞  l i m π‘š β†’ ∞ 𝑇 π‘š 𝑀 , 𝑗 = 𝑛 𝑀  𝑑 π‘₯ , | | 2 | | π‘˜ 𝑗 ξƒͺ ξƒͺ = l i m 𝑛 β†’ ∞ l i m π‘š β†’ ∞  𝑑 | | 2 𝑑 + π‘˜ | | 𝑛 ξƒͺ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) . ( 4 . 5 1 ) Let 𝑓 ∢ 𝑋 β†’ π‘Œ be an odd and Ξ¨ -approximately AQCQ mapping. Thus, all the conditions of Theorem 4.2 hold, and so there exists a unique additive mapping 𝐴 ∢ 𝑋 β†’ π‘Œ such that πœ‡ 𝑓 ( π‘₯ ) βˆ’ 8 𝑓 ( π‘₯ / 2 ) βˆ’ 𝐴 ( π‘₯ / 2 ) ( 𝑑 𝑑 ) β‰₯ | | 2 𝑑 + π‘˜ | | . ( 4 . 5 2 )

5. Generalized Hyers-Ulam Stability of the Functional Equation (1.4) in Non-Archimedean Random Normed Spaces: An Even Case

Now, we prove the generalized Hyers-Ulam stability of the functional equation 𝐷 𝑓 ( π‘₯ , 𝑦 ) = 0 in non-Archimedean Banach spaces, an even case.

Theorem 5.1. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦 , and let ( π‘Œ , πœ‡ , 𝑇 ) be a non-Archimedean random Banach space over 𝒦 . Let 𝑓 ∢ 𝑋 β†’ π‘Œ be an even mapping, 𝑓 ( 0 ) = 0 , and Ξ¨ -approximately AQCQ mapping. If for some 𝛼 ∈ ℝ , 𝛼 > 0 , and some integer π‘˜ , π‘˜ > 4 with | 2 π‘˜ | < 𝛼 , Ξ¨ ξ€· 2 βˆ’ π‘˜ π‘₯ , 2 βˆ’ π‘˜ ξ€Έ 𝑦 , 𝑑 β‰₯ Ξ¨ ( π‘₯ , 𝑦 , 𝛼 𝑑 ) ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) , l i m 𝑛 β†’ ∞ T ∞ 𝑗 = 𝑛 𝑀  𝛼 2 π‘₯ , 𝑗 𝑑 | | | | 1 6 π‘˜ 𝑗 ξƒͺ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) , ( 5 . 1 ) then there exists a unique quartic mapping 𝑄 ∢ 𝑋 β†’ π‘Œ such that πœ‡ 𝑓 ( π‘₯ ) βˆ’ 4 𝑓 ( π‘₯ / 2 ) βˆ’ 𝑄 ( π‘₯ / 2 ) ( 𝑑 ) β‰₯ T ∞ 𝑖 = 1 𝑀  𝛼 π‘₯ , 𝑖 + 1 𝑑 | | | | 1 6 π‘˜ 𝑖 ξƒͺ ( 5 . 2 ) for all π‘₯ ∈ 𝑋 and 𝑑 > 0 , where 𝑀 ( π‘₯ , 𝑑 ) ∢ = 𝑇 π‘˜ βˆ’ 1 ξ‚Έ Ξ¨ ξ‚΅ π‘₯ 2 , π‘₯ 2 , 𝑑 | | 4 | | ξ‚Ά ξ‚€ π‘₯ , Ξ¨ π‘₯ , 2  ξ‚΅ 2 , 𝑑 , … , Ξ¨ π‘˜ βˆ’ 1 π‘₯ 2 , 2 π‘˜ βˆ’ 1 π‘₯ 2 , 𝑑 | | 4 | | ξ‚Ά ξ‚΅ 2 , Ξ¨ π‘˜ βˆ’ 1 2 π‘₯ , π‘˜ βˆ’ 1 π‘₯ 2 , 𝑑 ξ‚Ά ξ‚Ή ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) . ( 5 . 3 )

Proof. Letting π‘₯ = 𝑦 in (4.2), we get πœ‡ 𝑓 ( 3 𝑦 ) βˆ’ 6 𝑓 ( 2 𝑦 ) + 1 5 𝑓 ( 𝑦 ) ( 𝑑 ) β‰₯ Ξ¨ ( 𝑦 , 𝑦 , 𝑑 ) ( 5 . 4 ) for all 𝑦 ∈ 𝑋 and 𝑑 > 0 . Replacing π‘₯ by 2 𝑦 in (4.2), we get πœ‡ 𝑓 ( 4 𝑦 ) βˆ’ 4 𝑓 ( 3 𝑦 ) + 4 𝑓 ( 2 𝑦 ) + 4 𝑓 ( 𝑦 ) ( 𝑑 ) β‰₯ Ξ¨ ( 2 𝑦 , 𝑦 , 𝑑 ) ( 5 . 5 ) for all 𝑦 ∈ 𝑋 and 𝑑 > 0 . By (5.4) and (5.5), we have πœ‡ 𝑓 ( 4 𝑦 ) βˆ’ 2 0 𝑓 ( 2 𝑦 ) + 6 4 𝑓 ( 𝑦 ) ξ€· πœ‡ ( 𝑑 ) β‰₯ 𝑇 4 ( 𝑓 ( 3 𝑦 ) βˆ’ 4 𝑓 ( 2 𝑦 ) + 5 𝑓 ( 𝑦 ) ) ( 𝑑 ) , πœ‡ 𝑓 ( 4 𝑦 ) βˆ’ 4 𝑓 ( 3 𝑦 ) + 6 𝑓 ( 2 𝑦 ) βˆ’ 4 𝑓 ( 𝑦 ) ξ€Έ ξ‚΅ πœ‡ ( 𝑑 ) = 𝑇 𝑓 ( 3 𝑦 ) βˆ’ 4 𝑓 ( 2 𝑦 ) + 5 𝑓 ( 𝑦 ) ξ‚΅ 𝑑 | | 4 | | ξ‚Ά , πœ‡ 𝑓 ( 4 𝑦 ) βˆ’ 4 𝑓 ( 3 𝑦 ) + 6 𝑓 ( 2 𝑦 ) βˆ’ 4 𝑓 ( 𝑦 ) ξ‚Ά ξ‚΅ Ξ¨ ξ‚΅ 𝑑 ( 𝑑 ) β‰₯ 𝑇 𝑦 , 𝑦 , | | 4 | | ξ‚Ά ξ‚Ά , Ξ¨ ( 2 𝑦 , 𝑦 , 𝑑 ) ( 5 . 6 ) for all 𝑦 ∈ 𝑋 and 𝑑 > 0 . Letting 𝑦 ∢ = π‘₯ / 2 and 𝑔 ( π‘₯ ) ∢ = 𝑓 ( 2 π‘₯ ) βˆ’ 4 𝑓 ( π‘₯ ) for all π‘₯ ∈ 𝑋 in (5.6), we get πœ‡ 𝑔 ( π‘₯ ) βˆ’ 1 6 𝑔 ( π‘₯ / 2 ) ξ‚΅ Ξ¨ ξ‚΅ π‘₯ ( 𝑑 ) β‰₯ 𝑇 2 , π‘₯ 2 , 𝑑 | | 4 | | ξ‚Ά ξ‚€ π‘₯ , Ξ¨ π‘₯ , 2  ξ‚Ά , 𝑑 ( 5 . 7 ) for all π‘₯ ∈ 𝑋 and 𝑑 > 0 .
The rest of the proof is similar to the proof of Theorem 4.2.

Corollary 5.2. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦 , and let ( π‘Œ , πœ‡ , 𝑇 ) be a non-Archimedean random Banach space over 𝒦 under a t-norm 𝑇 ∈ β„‹ . Let 𝑓 ∢ 𝑋 β†’ π‘Œ be an even, 𝑓 ( 0 ) = 0 , and Ξ¨ -approximately AQCQ mapping. If,   for some 𝛼 ∈ ℝ ,  𝛼 > 0 , and some integer π‘˜ , π‘˜ > 4 , with | 2 π‘˜ | < 𝛼 , Ξ¨ ξ€· 2 βˆ’ π‘˜ π‘₯ , 2 βˆ’ π‘˜ ξ€Έ 𝑦 , 𝑑 β‰₯ Ξ¨ ( π‘₯ , 𝑦 , 𝛼 𝑑 ) ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) , ( 5 . 8 ) then there exists a unique quartic mapping 𝑄 ∢ 𝑋 β†’ π‘Œ such that πœ‡ 𝑓 ( π‘₯ ) βˆ’ 4 𝑓 ( π‘₯ / 2 ) βˆ’ 𝑄 ( π‘₯ / 2 ) ( 𝑑 ) β‰₯ T ∞ 𝑖 = 1 𝑀  𝛼 π‘₯ , 𝑖 + 1 𝑑 | | | | 1 6 π‘˜ 𝑖 ξƒͺ ( 5 . 9 ) for all π‘₯ ∈ 𝑋 and 𝑑 > 0 .

Proof. Since l i m 𝑛 β†’ ∞ 𝑀  𝛼 π‘₯ , 𝑗 𝑑 | | | | 1 6 π‘˜ 𝑗 ξƒͺ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) ( 5 . 1 0 ) and 𝑇 is of Hadžić type, from Proposition 2.1, it follows that l i m 𝑛 β†’ ∞ 𝑇 ∞ 𝑗 = 𝑛 𝑀  𝛼 π‘₯ , 𝑗 𝑑 | | | | 1 6 π‘˜ 𝑗 ξƒͺ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) . ( 5 . 1 1 ) Now, we can apply Theorem 5.1 to obtain the result.

Example 5.3. Let ( 𝑋 , πœ‡ , 𝑇 𝑀 ) be non-Archimedean random normed space in which πœ‡ π‘₯ ( 𝑑 𝑑 ) = ( 𝑑 + β€– π‘₯ β€– π‘₯ ∈ 𝑋 , 𝑑 > 0 ) . ( 5 . 1 2 ) And let ( π‘Œ , πœ‡ , 𝑇 𝑀 ) be a complete non-Archimedean random normed space (see Example 3.2). Define 𝑑 Ξ¨ ( π‘₯ , 𝑦 , 𝑑 ) = . 1 + 𝑑 ( 5 . 1 3 ) It is easy to see that (4.3) holds for 𝛼 = 1 . Also, since 𝑑 𝑀 ( π‘₯ , 𝑑 ) = , 1 + 𝑑 ( 5 . 1 4 ) we have l i m 𝑛 β†’ ∞ 𝑇 ∞ 𝑀 , 𝑗 = 𝑛 𝑀  𝛼 π‘₯ , 𝑗 𝑑 | | | | 1 6 π‘˜ 𝑗 ξƒͺ = l i m 𝑛 β†’ ∞  l i m π‘š β†’ ∞ 𝑇 π‘š 𝑀 , 𝑗 = 𝑛 𝑀  𝑑 π‘₯ , | | | | 1 6 π‘˜ 𝑗 ξƒͺ ξƒͺ = l i m 𝑛 β†’ ∞ l i m π‘š β†’ ∞  𝑑 | | 𝑑 + 1 6 π‘˜ | | 𝑛 ξƒͺ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) . ( 5 . 1 5 ) Let 𝑓 ∢ 𝑋 β†’ π‘Œ be an even, 𝑓 ( 0 ) = 0 , and Ξ¨ -approximately AQCQ mapping. Thus all the conditions of Theorem 5.1 hold, and so there exists a unique quartic mapping 𝑄 ∢ 𝑋 β†’ π‘Œ such that πœ‡ 𝑓 ( π‘₯ ) βˆ’ 4 𝑓 ( π‘₯ / 2 ) βˆ’ 𝑄 ( π‘₯ / 2 ) ( 𝑑 𝑑 ) β‰₯ | | 𝑑 + 1 6 π‘˜ | | . ( 5 . 1 6 )

Theorem 5.4. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦 and let ( π‘Œ , πœ‡ , 𝑇 ) be a non-Archimedean random Banach space over 𝒦 . Let 𝑓 ∢ 𝑋 β†’ π‘Œ be an even mapping, 𝑓 ( 0 ) = 0 and Ξ¨ -approximately AQCQ mapping. If for some 𝛼 ∈ ℝ , 𝛼 > 0 , and some integer π‘˜ , π‘˜ > 2 with | 2 π‘˜ | < 𝛼 , Ξ¨ ξ€· 2 βˆ’ π‘˜ π‘₯ , 2 βˆ’ π‘˜ ξ€Έ 𝑦 , 𝑑 β‰₯ Ξ¨ ( π‘₯ , 𝑦 , 𝛼 𝑑 ) ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) , l i m 𝑛 β†’ ∞ T ∞ 𝑗 = 𝑛 𝑀  𝛼 2 π‘₯ , 𝑗 𝑑 | | 4 | | π‘˜ 𝑗 ξƒͺ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) , ( 5 . 1 7 ) then there exists a unique quadratic mapping 𝑄 ∢ 𝑋 β†’ π‘Œ such that πœ‡ 𝑓 ( π‘₯ ) βˆ’ 1 6 𝑓 ( π‘₯ / 2 ) βˆ’ 𝑄 ( π‘₯ / 2 ) ( 𝑑 ) β‰₯ T ∞ 𝑖 = 1 𝑀  𝛼 π‘₯ , 𝑖 + 1 𝑑 | | 4 | | π‘˜ 𝑖 ξƒͺ ( 5 . 1 8 ) for all π‘₯ ∈ 𝑋 and 𝑑 > 0 , where 𝑀 ( π‘₯ , 𝑑 ) ∢ = 𝑇 π‘˜ βˆ’ 1 ξ‚Έ Ξ¨ ξ‚΅ π‘₯ 2 , π‘₯ 2 , 𝑑 | | 4 | | ξ‚Ά ξ‚€ π‘₯ , Ξ¨ π‘₯ , 2  ξ‚΅ 2 , 𝑑 , … , Ξ¨ π‘˜ βˆ’ 1 π‘₯ 2 , 2 π‘˜ βˆ’ 1 π‘₯ 2 , 𝑑 | | 4 | | ξ‚Ά ξ‚΅ 2 , Ξ¨ π‘˜ βˆ’ 1 2 π‘₯ , π‘˜ βˆ’ 1 π‘₯ 2 , 𝑑 ξ‚Ά ξ‚Ή ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) . ( 5 . 1 9 )

Proof. Letting 𝑦 ∢ = π‘₯ / 2 and 𝑔 ( π‘₯ ) ∢ = 𝑓 ( 2 π‘₯ ) βˆ’ 1 6 𝑓 ( π‘₯ ) for all π‘₯ ∈ 𝑋 in (5.6), we get πœ‡ 𝑔 ( π‘₯ ) βˆ’ 4 𝑔 ( π‘₯ / 2 ) ξ‚΅ Ξ¨ ξ‚΅ π‘₯ ( 𝑑 ) β‰₯ 𝑇 2 , π‘₯ 2 , 𝑑 | | 4 | | ξ‚Ά ξ‚€ π‘₯ , Ξ¨ π‘₯ , 2  ξ‚Ά , 𝑑 ( 5 . 2 0 ) for all π‘₯ ∈ 𝑋 and 𝑑 > 0 .
The rest of the proof is similar to the proof of Theorem 5.1.

Corollary 5.5. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦 , and let ( π‘Œ , πœ‡ , 𝑇 ) be a non-Archimedean random Banach space over 𝒦 under a t-norm 𝑇 ∈ β„‹ . Let 𝑓 ∢ 𝑋 β†’ π‘Œ be an even, 𝑓 ( 0 ) = 0 , and Ξ¨ -approximately AQCQ mapping. If,   for some 𝛼 ∈ ℝ , 𝛼 > 0 , and some integer π‘˜ , π‘˜ > 2 , with | 2 π‘˜ | < 𝛼 , Ξ¨ ξ€· 2 βˆ’ π‘˜ π‘₯ , 2 βˆ’ π‘˜ ξ€Έ 𝑦 , 𝑑 β‰₯ Ξ¨ ( π‘₯ , 𝑦 , 𝛼 𝑑 ) ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) , ( 5 . 2 1 ) then there exists a unique quadratic mapping 𝑄 ∢ 𝑋 β†’ π‘Œ such that πœ‡ 𝑓 ( π‘₯ ) βˆ’ 1 6 𝑓 ( π‘₯ / 2 ) βˆ’ 𝑄 ( π‘₯ / 2 ) ( 𝑑 ) β‰₯ T ∞ 𝑖 = 1 𝑀  𝛼 π‘₯ , 𝑖 + 1 𝑑 | | 4 | | π‘˜ 𝑖 ξƒͺ ( 5 . 2 2 ) for all π‘₯ ∈ 𝑋 and 𝑑 > 0 .

Proof. Since l i m 𝑛 β†’ ∞ 𝑀  𝛼 π‘₯ , 𝑗 𝑑 | | 4 | | π‘˜ 𝑗 ξƒͺ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) ( 5 . 2 3 ) and 𝑇 is of Hadžić type, from Proposition 2.1, it follows that l i m 𝑛 β†’ ∞ 𝑇 ∞ 𝑗 = 𝑛 𝑀  𝛼 π‘₯ , 𝑗 𝑑 | | 4 | | π‘˜ 𝑗 ξƒͺ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) . ( 5 . 2 4 ) Now, we can apply Theorem 5.4 to obtain the result.

Example 5.6. Let ( 𝑋 , πœ‡ , 𝑇 𝑀 ) be a non-Archimedean random normed space in which πœ‡ π‘₯ ( 𝑑 𝑑 ) = ( 𝑑 + β€– π‘₯ β€– π‘₯ ∈ 𝑋 , 𝑑 > 0 ) . ( 5 . 2 5 ) And let ( π‘Œ , πœ‡ , 𝑇 𝑀 ) be a complete non-Archimedean random normed space (see Example 3.2). Define 𝑑 Ξ¨ ( π‘₯ , 𝑦 , 𝑑 ) = . 1 + 𝑑 ( 5 . 2 6 ) It is easy to see that (4.3) holds for 𝛼 = 1 . Also, since 𝑑 𝑀 ( π‘₯ , 𝑑 ) = , 1 + 𝑑 ( 5 . 2 7 ) we have l i m 𝑛 β†’ ∞ 𝑇 ∞ 𝑀 , 𝑗 = 𝑛 𝑀  𝛼 π‘₯ , 𝑗 𝑑 | | 4 | | π‘˜ 𝑗 ξƒͺ = l i m 𝑛 β†’ ∞  l i m π‘š β†’ ∞ 𝑇 π‘š 𝑀 , 𝑗 = 𝑛 𝑀  𝑑 π‘₯ , | | 4 | | π‘˜ 𝑗 ξƒͺ ξƒͺ = l i m 𝑛 β†’ ∞ l i m π‘š β†’ ∞  𝑑 | | 4 𝑑 + π‘˜ | | 𝑛 ξƒͺ = 1 ( π‘₯ ∈ 𝑋 , 𝑑 > 0 ) . ( 5 . 2 8 ) Let 𝑓 ∢ 𝑋 β†’ π‘Œ be an even, 𝑓 ( 0 ) = 0 , and Ξ¨ -approximately AQCQ mapping. Thus, all the conditions of Theorem 5.4 hold, and so there exists a unique quadratic mapping 𝑄 ∢ 𝑋 β†’ π‘Œ such that πœ‡ 𝑓 ( π‘₯ ) βˆ’ 1 6 𝑓 ( π‘₯ / 2 ) βˆ’ 𝑄 ( π‘₯ / 2 ) ( 𝑑 𝑑 ) β‰₯ | | 4 𝑑 + π‘˜ | | . ( 5 . 2 9 )

6. Latticetic Random Normed Space

Let β„’ = ( 𝐿 , β‰₯ 𝐿 ) be a complete lattice, that is, a partially ordered set in which every nonempty subset admits supremum and infimum, and 0 β„’ = i n f 𝐿 , 1 β„’ = s u p 𝐿 . The space of latticetic random distribution functions, denoted by Ξ” + 𝐿 , is defined as the set of all mappings 𝐹 ∢ ℝ βˆͺ { βˆ’ ∞ , + ∞ } β†’ 𝐿 such that 𝐹 is left continuous and nondecreasing on ℝ , 𝐹 ( 0 ) = 0 β„’ , 𝐹 ( + ∞ ) = 1 β„’ .

𝐷 + 𝐿 βŠ† Ξ” + 𝐿 is defined as 𝐷 + 𝐿 = { 𝐹 ∈ Ξ” + 𝐿 ∢ 𝑙 βˆ’ 𝐹 ( + ∞ ) = 1 β„’ } , where 𝑙 βˆ’ 𝑓 ( π‘₯ ) denotes the left limit of the function 𝑓 at the point π‘₯ . The space Ξ” + 𝐿 is partially ordered by the usual pointwise ordering of functions, that is, 𝐹 β‰₯ 𝐺 if and only if 𝐹 ( 𝑑 ) β‰₯ 𝐿 𝐺 ( 𝑑 ) for all 𝑑 in ℝ . The maximal element for Ξ” + 𝐿 in this order is the distribution function given by πœ€ 0 ξ‚» 0 ( 𝑑 ) = β„’ 1 , i f 𝑑 ≀ 0 , β„’ , i f 𝑑 > 0 . ( 6 . 1 )

In Section 2, we defined 𝑑 -norms on [ 0 , 1 ] , and now we extend 𝑑 -norms on a complete lattice.

Definition 6.1 (see [42]). A triangular norm ( 𝑑 -norm) on 𝐿 is a mapping 𝒯 ∢ ( 𝐿 ) 2 β†’ 𝐿 satisfying the following conditions:(a) ( f o r a l l π‘₯ ∈ 𝐿 ) ( 𝒯 ( π‘₯ , 1 β„’ ) = π‘₯ ) (boundary condition);(b) ( f o r a l l ( π‘₯ , 𝑦 ) ∈ ( 𝐿 ) 2 ) ( 𝒯 ( π‘₯ , 𝑦 ) = 𝒯 ( 𝑦 , π‘₯ ) ) (commutativity);(c) ( f o r a l l ( π‘₯ , 𝑦 , 𝑧 ) ∈ ( 𝐿 ) 3 ) ( 𝒯 ( π‘₯ , 𝒯 ( 𝑦 , 𝑧 ) ) = 𝒯 ( 𝒯 ( π‘₯ , 𝑦 ) , 𝑧 ) ) (associativity);(d) ( f o r a l l ( π‘₯ , π‘₯ ξ…ž , 𝑦 , 𝑦 ξ…ž ) ∈ ( 𝐿 ) 4 ) ( π‘₯ ≀ 𝐿 π‘₯ ξ…ž a n d 𝑦 ≀ 𝐿 𝑦 ξ…ž β‡’ 𝒯 ( π‘₯ , 𝑦 ) ≀ 𝐿 𝒯 ( π‘₯ ξ…ž , 𝑦 ξ…ž ) ) (monotonicity).

Let { π‘₯ 𝑛 } be a sequence in 𝐿 converges to π‘₯ ∈ 𝐿 (equipped order topology). The 𝑑 -norm 𝒯 is said to be a continuous 𝑑 -norm if l i m 𝑛 β†’ ∞ 𝒯 ξ€· π‘₯ 𝑛 ξ€Έ , 𝑦 = 𝒯 ( π‘₯ , 𝑦 ) ( 6 . 2 ) for all 𝑦 ∈ 𝐿 .

A 𝑑 -norm 𝒯 can be extended (by associativity) in a unique way to an 𝑛 -array operation taking for ( π‘₯ 1 , … , π‘₯ 𝑛 ) ∈ 𝐿 𝑛 the value 𝒯 ( π‘₯ 1 , … , π‘₯ 𝑛 ) defined by 𝒯 0 𝑖 = 1 π‘₯ 𝑖 = 1 , 𝒯 𝑛 𝑖 = 1 π‘₯ 𝑖 ξ€· 𝒯 = 𝒯 𝑛 βˆ’ 1 i = 1 π‘₯ 𝑖 , π‘₯ 𝑛 ξ€Έ ξ€· π‘₯ = 𝒯 1 , … , π‘₯ 𝑛 ξ€Έ . ( 6 . 3 )

𝒯 can also be extended to a countable operation taking for any sequence ( π‘₯ 𝑛 ) 𝑛 ∈ 𝑁 in 𝐿 the value 𝒯 ∞ 𝑖 = 1 π‘₯ 𝑖 = l i m 𝑛 β†’ ∞ 𝒯 𝑛 𝑖 = 1 π‘₯ 𝑖 . ( 6 . 4 ) The limit on the right side of (6.4) exists since the sequence ( 𝒯 𝑛 𝑖 = 1 π‘₯ 𝑖 ) 𝑛 ∈ β„• is nonincreasing and bounded from below.

Note that we put 𝒯 = 𝑇 whenever 𝐿 = [ 0 , 1 ] . If