Assume that X is a real inner product space and is a solution of the orthogonal Cauchy functional equation , where . By the Pythagorean theorem, is a solution of the conditional equation. Of course, this function does not satisfy the additivity equation everywhere. Thus the orthogonal Cauchy equation is not equivalent to the classic Cauchy equation on the whole inner product space.
Pinsker  characterized orthogonally additive functionals on an inner product space when the orthogonality is the ordinary one in such spaces. Sundaresan  generalized this result to arbitrary Banach spaces equipped with the Birkhoff-James orthogonality. The orthogonal Cauchy functional equation
in which ⊥ is an abstract orthogonality relation, was first investigated by Gudder and Strawther . They defined ⊥ by a system consisting of five axioms and described the general semi-continuous real-valued solution of the conditional Cauchy functional equation. In 1985, Rätz  introduced a new definition of orthogonality by using more restrictive axioms than Gudder and Strawther. Moreover, he investigated the structure of orthogonally additive mappings. Rätz and Szabó  investigated the problem in a rather more general framework.
Let us recall the orthogonality in the sense of Rätz .
Suppose that X is a real vector space (algebraic module) with , and ⊥ is a binary relation on X with the following properties:
() Totality of ⊥ for zero: and for all ;
() Independence: if and , then x and y are linearly independent;
() Homogeneity: if and , then for all ;
() Thalesian property: if P is a 2-dimensional subspace of X, and , which is the set of nonnegative real numbers, then there exists such that and .
The pair is called an orthogonality space (resp., module). By an orthogonality normed space (normed module) we mean an orthogonality space (resp., module) having a normed (resp., normed module) structure.
Some interesting examples are as follows:
The trivial orthogonality on a vector space X defined by () and, for any non-zero elements , if and only if x, y are linearly independent.
The ordinary orthogonality on an inner product space given by if and only if .
The Birkhoff-James orthogonality on a normed space defined by if and only if for all .
The relation ⊥ is called symmetric if implies that for all . Clearly, Examples (1) and (2) are symmetric, but Example (3) is not. It is remarkable to note, however, that a real normed space of a dimension greater than 2 is an inner product space if and only if the Birkhoff-James orthogonality is symmetric. There are several orthogonality notions on a real normed space such as Birkhoff-James, Boussouis, Singer, Carlsson, unitary-Boussouis, Roberts, Phythagorean, isosceles and Diminnie (see [6–12]).
The stability problem of functional equations originated from the following question of Ulam : Under what condition is there an additive mapping near an approximately additive mapping? In 1941, Hyers  gave a partial affirmative answer to the question of Ulam in the context of Banach spaces. In 1978, Rassias  extended the theorem of Hyers by considering the unbounded Cauchy difference
During the last decades, several stability problems of functional equations have been investigated in the spirit of Hyers-Ulam-Rassias. The readers refer to [16–20] and references therein for detailed information on the stability of functional equations.
Ger and Sikorska  investigated the orthogonal stability of the Cauchy functional equation , namely they showed that, if f is a mapping from an orthogonality space X into a real Banach space Y and
for all with and for some , then there exists exactly one orthogonally additive mapping such that for all .
The first author treating the stability of the quadratic equation was Skof  by proving that, if f is a mapping from a normed space X into a Banach space Y satisfying
for some , then there is a unique quadratic mapping such that . Cholewa  extended Skof’s theorem by replacing X by an abelian group G. Skof’s result was later generalized by Czerwik  in the spirit of Hyers-Ulam-Rassias. The stability problem of functional equations has been extensively investigated by some mathematicians (see [25–31]).
The orthogonally quadratic equation
was first investigated by Vajzović  when X is a Hilbert space, Y is the scalar field, f is continuous and ⊥ means the Hilbert space orthogonality. Later, Drljević , Fochi , Moslehian [35, 36] and Szabó  generalized this result (see also [38–40]).
In , Jun and Kim considered the following cubic functional equation:
It is easy to show that the function satisfies the functional equation (1.1), which is called a cubic functional equation, and every solution of the cubic functional equation is said to be a cubic mapping.
Let X be an orthogonality space and Y be a real Banach space. A mapping is called orthogonally cubic if it satisfies the orthogonally cubic functional equation
for all x, y with .
In , Lee et al. considered the following quartic functional equation:
It is easy to show that the function satisfies the functional equation (1.3), which is called a quartic functional equation, and every solution of the quartic functional equation is said to be a quartic mapping (for the stability of the ACQ and quartic functional equations, see [26, 31] and others).
Let X be an orthogonality space and Y be a Banach space. A mapping is called orthogonally quartic if it satisfies the orthogonally quartic functional equation
for all x, y with .
Let X be a set. A function is called a generalized metric on X if d satisfies the following conditions:
if and only if ;
for all ;
for all .
We recall a fundamental result in a fixed point theory.
Theorem 1.1 ([43, 44])
Let be a complete generalized metric space and be a strictly contractive mapping with the Lipschitz constant . Then, for each given element , either
for all nonnegative integers
or there exists a positive integer
for all ;
the sequence converges to a fixed point of J;
is the unique fixed point of J in the set ;
for all .
In 1996, Isac and Rassias  were the first to provide applications of the 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 [46–55]).
This paper is organized as follows. In Section 2, we prove the Hyers-Ulam stability of the orthogonally Jensen additive functional equation in orthogonality modules over a unital Banach algebra. In Section 3, we prove the Hyers-Ulam stability of the orthogonally Jensen quadratic functional equation in orthogonality modules over a unital Banach algebra. In Section 4, we prove the Hyers-Ulam stability of the orthogonally cubic functional equation (1.2) in orthogonality modules over a unital Banach algebra. In Section 5, we prove the Hyers-Ulam stability of the orthogonally quartic functional equation (1.4) in orthogonality modules over a unital Banach algebra.
Throughout this paper, assume that is an orthogonality module over a unital Banach algebra A and that is a real Banach module over A. Let , and e be the unity of A.