On the Stability of Cubic and Quadratic Mapping in Random Normed Spaces under Arbitrary T-Norms

The study of stability problems for functional equations is related to a question of Ulam [1] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [2]. Subsequently, the result of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [4] for linearmappings by considering an unbounded Cauchy difference.The paper [4] of Rassias has provided a lot of influence in the development of what we now call HyersUlam-Rassias stability of functional equations. We refer the interested readers for more information on such problems to the papers [5–9]. In addition, some authors investigated the stability in the settings of fuzzy, probabilistic, and random normed spaces, and for more details see [10–22]. The functional equation


Introduction and Preliminaries
The study of stability problems for functional equations is related to a question of Ulam [1] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [2].Subsequently, the result of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference.The paper [4] of Rassias has provided a lot of influence in the development of what we now call Hyers-Ulam-Rassias stability of functional equations.We refer the interested readers for more information on such problems to the papers [5][6][7][8][9].In addition, some authors investigated the stability in the settings of fuzzy, probabilistic, and random normed spaces, and for more details see [10][11][12][13][14][15][16][17][18][19][20][21][22].
The functional equation  ( + 2) +  ( − 2) = 4 ( + ) + 4 ( − ) − 6 () is said to be the cubic functional equation since the function () =  3 is its solution.Every solution of the cubic functional equation is said to be a cubic mapping.The stability problem for the cubic functional equation was solved by Skof [23] for mappings  :  → , where  is a normed space and  is a Banach space.
is said to be the quadratic functional equation since the function () =  2 is its solution.Every solution of the quadratic functional equation is said to be a quadratic mapping (see [8,9]).Throughout this paper, the space of all probability distribution functions is denoted by Δ + = { :  ∪ {−∞, +∞} → [0, 1] :  is left-continuous and non decreasing on  and (0) = 0, (+∞) = 1} and the subset  + ⊆ Δ + is the set  + = { ∈ Δ + :  − (+∞) = 1}, where  − () denotes the left limit of the function  at the point .The space is partially ordered by the usual pointwise ordering of functions; that is,  ⩽  if and only if () ⩽ () for all  in .The maximal element for Δ + in this order is the distribution function given by 2
(1) A sequence {  } in  is said to be convergent to  in  if, for every  > 0 and  > 0, there exists a positive integer  such that    − () > 1 −  whenever  ≥ .
(3) An RN-space (, , ) is said to be complete if every Cauchy sequence in  is convergent to a point in .

On the Stability of Cubic Mappings in RN-Spaces
Theorem 5. Let  be a normed space, let ( + ,   , ) be an RNspace, and let  :  ×  → [0, +∞) be a mapping such that for some  < 1, and let (, , ) be a complete RN-space.If  :  →  is an odd mapping such that (0) = 0 and then there exists a unique cubic mapping  :  →  such that Proof.Letting  =  in ( 6) we get thus since  is an odd mapping and (0) = 0 then we have Replacing  by 2 in ( 6) we get from Definition 2 we have Therefore by ( 10) and ( 12), we get Replacing  by  and letting () = (2) − 2() in ( 14) we get Then Replacing  by 2   in ( 16) we obtain then )) . ( Whence and using (4) we have therefore and from (21) we have also, from above inequality we have )) . ( Then By replacing  by 2   in (25) we observe that then We know when  tend ∞, then (2  )/8  → (), and Therefore taking the lim  → ∞ and using (30), and also by using Definition 1(iii) and Definition 2 and (RN2) we get and since  was arbitrary, by taking  → 0 in (31), we have which implies that the Inequality ( 7) holds.Replacing  and  by 2   and 2   in ( 6), respectively, we get that by division by the 8  we have then for all ,  ∈  and for all  > 0. Since lim  → ∞   (,) (/  ) = 1, we conclude that  fulfills (1).To prove the uniqueness of the cubic function , let us assume that there exists a quadratic function  :  →  which satisfies (7).Obviously we have (2  ) = 8  () and (2  ) = 8  () for all  ∈  and  ∈ N. It follows from ( 7) and (32) that for each  ∈ ,  ()−() () then  ()−() () = 1, for all  > 0. And so () = ().This completes the proof.