In this paper, we prove the generalized nonlinear stability of the first and second of the following -functional equations, , and in latticetic random Banach lattice spaces, where is a fixed real or complex number with .
In 1986 , Alsina investigated the stability of functional equations in random normed spaces. This was a milestone that revealed the role of random theory as a powerful tool for studying stability of functional equations, and many mathematicians attempted to develop and generalize the problem of stability in random normed spaces with a practical approach. In 2011 , Saadati et al. proved the nonlinear stability of a cubic functional equation in non-Archimedean random normed space. They also proved nonlinear stability of a -random additive-cubic-quartic (ACQ) functional equation .
For the first time, in 2012 , Agbeko investigated the stability of a maximum preserving functional equation in latticetic environments. He presented a generalization of the Hyers–Ulam–Aoki stability problem in Banach lattice spaces, by replacing the supremum operation with additive operation in Cauchy’s equation, which is called the maximum preserving functional equation. In addition to supremum and infimum operations, in 2015, Agbeko  developed nonlinear stability for different combinations of these two operations, and proved it using the core of the direct method presented by Forti . On the other hand, in 2017, Park and Jang  introduced -functional equations, and proved the stability of the equations in various spaces.
In the present work, we introduce a latticetic operation-preserving -functional equations and prove the stability of the first and second latticetic operation-preserving -functional equations by the direct method and fixed-point method in latticetic random Banach lattice spaces, which is a generalization of research by Agbeko, Park and Jang.
At first, we describe some known concepts and results, which will be useful in the next section of this study.
(see [8,9]). Assume that is a complete generalized metric space. Assume that is a strictly contractive mapping with Lipschitz constant . If there exist non-negative integers such that
, ; then we have
the sequence convergence to a fixed point of β;
is the unique fixed point of β in the set ;
An ordered set is called a complete lattice if
, A admits supremum and infimum,
Suppose that be the space of lattice random distribution function, i.e.,
It is clear that the space is an ordered set (i.e., if and only if (iff) for all ).
Also, the distribution function given by
is the maximal element for .
Moreover, if denotes the left limit of the function G at the point a and , then obviously .
(see ). Assume that . is a triangular norm brifly (t-norm), iff for all :
if and (monotonicity).
For example, , for all is a t-norm on .
for all , then is called a continous t-norm, where .
(see [9,11]). If there exist a continuous t-norm ⋄ and a continuous t-conorm □ on , we define, for all ,
Then, is called t-representable on .
for all are continuous t-representables.
Define the mapping from to M by
If is a given sequence in M, then is defined recurrently by and for (see ).
Assume that . is called a negation function, iff
, if (monotonically).
A negation function is involutive, iff
A triple is called a latticetic random normed space (briefly, LRN-space) if is a vector space and such that the following conditions hold:
for all iff ;
for all a in , and ;
for all and .
We note that from (L2) it follows that for all and .
Assume that and operation are defined by
Then, is a complete lattice (see ). In this complete lattice, we denote its units by and . Let be a normed space, for all and μ be a mapping defined by
Then, is an LRN-space.
If be an LRN-space, then
are neighborhoods of null vector for linear topology on generalized by the norm G.
Assume that is an LRN-space.
We say if, for every and , there exists a positive integer N such that whenever .
We say is a Cauchy sequence if, for every and , there exists a positive integer N such that whenever .
A LRN-space is said to be complete if every Cauchy sequence in is convergent to a point in .
If is an LRN-space and , then .
The proof is the same as classical RN-spaces, see . ☐
Let be an LRN-space and . If
then and .
Assume that for all . Since , we have and by (L1) we conclude that . ☐
Suppose that triple is an LRN-space. Then, is called latticetic random Banach space (briefly, LRB-space) if is complete with respect to the random metric included by random norm.
Suppose that is a vector lattice and is a Banach lattice with and their respective positive cones. A map is cone-related if
3. Stability of the first -Functional Equation: Direct Method
In this section, using a direct method, we prove nonlinear stability of the first -functional equation in latticetic random Banach lattice space (briefly, LRBL-space).
Assume that is a vector lattice space, is an LRN-space and is a cone-related mapping. Then, the following operator equation is called a Cauchy latticetic operation-preserving functional equation if:
for all , where and are fixed lattice operations.
Note that if the above four lattice operators are all the supremum (join) or the infimum (meet), then the functional Equation (10) is just the definition of a join-homomorphism or a meet-homomorphism. Moreover, if and are the same, then the left-hand side of Equation (10) is the map of the meets or the joints.
Assume that and are vector lattice spaces. If a mapping satisfies
for all , then functional Equation (11) is a Cauchy latticetic operation-preserving functional equation.
Assume that satisfies Equation (11). Letting in Equation (11), then we have .