Stability of a Generalized Trigonometric-Quadratic Functional Equation
A stability result for the generalized trigonometric-quadratic functional equation over the domain of a normed space and the range of a Banach space is derived. It states that under appropriate conditions on the function , the four functions , , , and are approximately equal to combinations of certain quadratic and additive functions.
A generalized trigonometric functional equation is a functional equation of the form which is so named because the two best known trigonometric functions, sine and cosine, are solutions of special cases of this equation. In , the authors investigated the stability of (1), where , , , and are nonzero functions from an abelian group to the complex field. Later in the same year, a generalized trigonometric functional equation with either the function or being bounded was investigated to complement the above earlier work where such function boundedness was not treated. A generalized quadratic functional equation is a functional equation of the form which is so named because the quadratic function is a solution of one of its particular cases, namely, and any solution of (3) is referred to as a quadratic function. The stability of the quadratic functional equation was first proved by Skof  in 1983 for functions from a normed space to a Banach space. In 1984, Cholewa  demonstrated that Skof’s theorem is also valid if a normed space is replaced by an abelian group. Later, Czerwik  extended Cholewa’s result by changing a control function and considering functions from a normed space to a Banach space. In another paper, , Czerwik investigated the stability of the “partially pexiderized” quadratic functional equation of the form . In 2000, Jung  considered (2), where, , , and are functions from a normed space to a Banach space . In the paper , the authors considered the stability of a combined generalized trigonometric-quadratic functional equation of the form over the domain of an abelian group and the range , except the control function whose range is taken to be the nonnegative real numbers . There, the functions and are explicitly determined under certain restrictions on the remaining functions , , , and . It is natural to ask whether the restrictions on the functions , , , and can be removed and/or altered to obtain some stability result of (4). We answer this question affirmatively using Jung’s techniques elaborated in [6, 8].
We collect in this section the notation and terminology that will be kept standard throughout. A normed space is the domain and a Banach space is the range of all functions involved. Let . Define Let be a given function satisfying the following three conditions (i) , (ii) , and (iii) there is an integer such that For , define
Define a real sequence by and let Clearly, the functions , satisfy the conditions (i), (ii), and (iii).
3. Auxiliary Results
We start with an important lemma.
Lemma 1. If , , satisfies then one has, for integer , and ,
Proof. We prove the lemma by induction on . To verify the case , we use induction on . Putting in (10), we obtain that is, the assertion is true for . Assume that the statement is true for all . Replacing and in (10) by and , we get and so and we are done for . Assuming that the lemma is true for some , we have verifying the lemma in general.
We come now to our first main proposition.
Proposition 2. Assume that satisfies When for all , one further assumes that . Then there exists a unique quadratic function satisfying provided that and for all , or provided that and for all . Moreover, if is measurable or is continuous in for every fixed , then the function satisfies
Proof. From Lemma 1, we have
For with , from (20), we obtain
In (21), since the last term can be made as small as we wish by choosing sufficiently large, we deduce that is a Cauchy sequence in . Since is a Banach space, the limit
is well defined. Using and (20), the validity of the inequality (17) is easily verified. Next, replacing , in (16) by , , respectively, dividing the resulting inequality by , and using , , we get
which implies that is a quadratic function. If is another quadratic function satisfying the inequality (17), since and are quadratic functions, we have
It thus follows from (24), (17), , and that
that is, .
The other provision and is treated analogously by replacing by in (11) to obtain an upper bound for , which is then used to show that is a Cauchy sequence in so that the limit is well defined and satisfies (18). Finally, the proof that (19) holds when is measurable or is continuous is the same as that in .
We now proceed to our second main proposition.
Proposition 3. If satisfies then there exists a unique additive function such that either provided that for , one has and or provided that for , one has and Moreover, if is continuous in for every fixed , then is linear.
Proof. First, we show that
Putting in (27), we obtain
This shows that (32) holds for . Assume that (32) is valid for ; that is,
Replacing by in (27) and using (34), we have
verifying the case , and so the assertion (32) holds for all and all .
Next, we use induction on to show that The case follows from (32). Assuming that this holds for , putting , and then substituting for in (36), we obtain verifying (36) generally. Rewriting (36), we get Using (38), , and the inequality and proceeding as in the proof of Proposition 2, we deduce that is a Cauchy sequence. Thus, the limit is well defined. To show that is additive, we substitute and for and in (27), respectively, to get From and it follows that Using this last inequality, dividing both sides of (41) by and letting , we obtain .
According to (38), we have To show that is unique, let be another additive mapping satisfying (44). Thus,