Journal of Mathematical Analysis and Applications 251, 264᎐284 Ž2000. doi:10.1006rjmaa.2000.7046, available online at http:rrwww.idealibrary.com on On the Stability of Functional Equations in Banach Spaces Themistocles M. Rassias Department of Mathematics, National Technical Uni¨ ersity of Athens, Zografou Campus, 15780 Athens, Greece E-mail: trassias@math.ntua.gr Submitted by William F. Ames Received April 26, 2000 DEDICATED TO PROFESSOR GEORGE LEITMANN IN ADMIRATION The paper is devoted to some results on the problem of S. M. Ulam for the stability of functional equations in Banach spaces. The problem was posed by Ulam 60 years ago. 䊚 2000 Academic Press Key Words: generalized Hyers᎐Ulam stability; functional equations. 1. INTRODUCTION In almost all areas of mathematical analysis, we can raise the following fundamental question: When is it true that a mathematical object satisfying a certain property approximately must be close to an object satisfying the property exactly? If we turn our attention to the case of functional equations, we can particularly ask the question when the solutions of an equation differing slightly from a gi¨ en one must be close to the solution of the gi¨ en equation. The stability problem of functional equations originates from such a fundamental question. In connection with the above question, S. M. Ulam w85x raised a question concerning the stability of homomorphisms: Let G 1 be a group and let G 2 be a metric group with a metric d Ž⭈, ⭈ .. Given ) 0, does there exist a ␦ ) 0 such that if a function h : G 1 ª G 2 satisfies the inequality d Ž hŽ xy ., hŽ x . hŽ y .. - ␦ for all x, y g G 1 then there is a homomorphism H : G 1 ª G 2 with d Ž hŽ x ., H Ž x .. - for all x g G 1? 264 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved. STABILITY OF FUNCTIONAL EQUATIONS 265 The first partial solution to Ulam’s question was provided by D. H. Hyers w26x. Indeed, he proved the following celebrated theorem: THEOREM ŽD. H. Hyers.. Assume that E1 and E2 are Banach spaces. If a function f : E1 ª E2 satisfies the inequality f Ž x q y. y f Ž x. y f Ž y. F for some G 0 and for all x, y g E1 , then the limit a Ž x . s lim 2yn f Ž 2 n x . nª⬁ exists for each x in E1 and a : E1 ª E2 is the unique additi¨ e function such that f Ž x . y aŽ x . F for any x g E1. Moreo¨ er, if f Ž tx . is continuous in t for each fixed x g E1 , then a is linear. For the above case, we say that the additive functional equation f Ž x q y . s f Ž x . q f Ž y . has the Hyers᎐Ulam stability on Ž E1 , E2 . or alternatively that it is stable in the sense of Hyers and Ulam. In the Hyers’s theorem, D. H. Hyers explicitly constructed the additive function a : E1 ª E2 directly from the given function f. This method is called a direct method and is a powerful tool for studying the stability of several functional equations. It is often used to construct a solution of a given functional equation. During the last decades, Hyers’s theorem was generalized in various directions Žsee w6, 16, 17, 58, 60, 69, 70, 78x.. In particular, Th. M. Rassias w58x considered a generalized version of the theorem of Hyers which permitted the Cauchy difference to become unbounded. He proved the following theorem by using a direct method. THEOREM ŽTh. M. Rassias.. spaces satisfies the inequality If a function f : E1 ª E2 between Banach f Ž x q y. y f Ž x. y f Ž y. F Ž 5 x5 p q 5 y5 p. for some G 0, 0 F p - 1, and for all x, y g E1 , then there exists a unique additi¨ e function a : E1 ª E2 such that f Ž x . y aŽ x . F 2 2y2p 5 x5 p for any x in E1. Moreo¨ er, if f Ž tx . is continuous in t for each fixed x g E1 , then a is linear. 266 THEMISTOCLES M. RASSIAS This theorem was a generalization of Hyers’s theorem. Since then, many mathematicians began to study the stability of functional equations. The theorem of Rassias was later extended to all p / 1 and generalized by many mathematicians Žsee w18, 19, 30, 31, 35, 67x.. The stability phenomenon that was presented by Th. M. Rassias is called the generalized Hyers᎐Ulam stability. This terminology may also be applied to the cases of other functional equations Žsee w37x.. There is a strong stability phenomenon which is known as a superstability. An equation of a homomorphism is called superstable if each approximate homomorphism is actually a true homomorphism. This property was first observed when the following theorem was proved by J. Baker et al. w3x. THEOREM ŽJ. Baker, J. Lawrence, and F. Zorzitto.. space. If a function f : V ª ⺢ satisfies the inequality Let V be a ¨ ector f Ž x q y. y f Ž x. f Ž y. F for some ) 0 and for all x, y g V, then either f is a bounded function or f Ž x q y . s f Ž x . f Ž y . for all x, y g V. Later, this result was generalized by J. Baker w2x and by L. Szekelyhidi ´ w77x Žcf. w24, 25x.. The superstability can be considered as a special case of the generalized Hyers᎐Ulam stability. Around 1980, the topic of the generalized Hyers᎐Ulam stability of functional equations was taken up by a number of mathematicians, and the study of this area has grown to be one of central subjects in mathematical analysis. The papers w15, 29, 37, 49, 62, 64, 65x may be helpful to get general information on this area. In particular, the text w27x by D. H. Hyers et al., as well as the text w50x by S.-M. Jung give a comprehensive introduction to the theory of the generalized Hyers᎐Ulam stability of functional equations. In this paper, we will give a general introduction to the theory of stability of the functional equations. We will emphasize the direction motivated by Th. M. Rassias w58x. Section 2 will be devoted to the generalized Hyers᎐Ulam stability of the additive equation. The Jensen’s equation, Hosszu’s ´ equation, homogeneous equation, and the logarithmic functional equation will be considered as variations of the additive equation. The exponential equation and the multiplicative equation will be introduced in Section 3. In Section 4, we will discuss the quadratic functional equation. The generalized Hyers᎐Ulam stability of other equations, such as trigonometric equations and the gamma functional equation, will be presented in Section 5. STABILITY OF FUNCTIONAL EQUATIONS 267 2. ADDITIVE FUNCTIONAL EQUATION The additive functional equation f Ž x q y . s f Ž x . q f Ž y . is one of the most famous functional equations. Every solution of the additive equation is called an additi¨ e function. D. H. Hyers w26x proved a theorem for the Hyers᎐Ulam stability of that equation which was the first result concerning this subject. Since then, the theorem of Hyers was generalized in various directions. In particular, Th. M. Rassias w58x extended Hyers’s theorem by permitting the Cauchy difference to become unbounded. THEOREM 2.1. Let E1 and E2 be Banach spaces and suppose f : E1 ª E2 is a function satisfying the inequality f Ž x q y. y f Ž x. y f Ž y. F Ž 5 x5 p q 5 y5 p. Ž 2.1. for some ) 0, p g w0, 1., and for all x, y g E1. Then there exists a unique additi¨ e function a : E1 ª E2 such that 2 5 x5 p f Ž x . y aŽ x . F Ž 2.2. p 2y2 for any x g E1. Moreo¨ er, if f Ž tx . is continuous in t for each fixed x g E1 , then a is linear. Proof. We will use the induction on n to prove 2yn f Ž 2 n x . y f Ž x . F 5 x 5 p ny1 Ý 2 mŽ py1. Ž 2.3. ms0 for all n g ⺞. Putting y s x in Ž2.1. and dividing by 2 yield the validity of Ž2.3. for n s 1. Assume that Ž2.3. holds and prove it for the case n q 1. This is true because by Ž2.3. we obtain 2yn f Ž 2 n 2 x . y f Ž 2 x . F 52 x 5 p ny1 2 mŽ py1. ; Ý ms0 therefore 2yŽ nq1. f Ž 2 nq1 x . y 2y1 f Ž 2 x . F 5 x 5 p n Ý 2 mŽ py1. . ms1 The triangle inequality yields 2yŽ nq1. f Ž 2 nq1 x . y f Ž x . F 2yŽ nq1. f Ž 2 nq1 x . y 2y1 f Ž 2 x . q 2y1 f Ž 2 x . y f Ž x . F 5 x5 p n Ý 2 mŽ py1. , ms0 which completes the proof of Ž2.3.. 268 THEMISTOCLES M. RASSIAS It follows then that 2yn f Ž 2 n x . y f Ž x . F 2 2y2p 5 x 5 p. Ž 2.4. However, for m ) n ) 0, 2ym f Ž 2 m x . y 2yn f Ž 2 n x . s 2yn 2yŽ myn. f Ž 2 myn 2 n x . y f Ž 2 n x . F 2 nŽ py1. yn 2 2y2 p 5 x 5 p. n Therefore, 2 f Ž2 x .4 is a Cauchy sequence for each x in E1. Since E2 is complete, we can define a function a : E1 ª E2 by the formula a Ž x . s lim 2yn f Ž 2 n x . . Ž 2.5. nª⬁ If we replace x and y in Ž2.1. by 2 n x and 2 n y, respectively, then f Ž2nŽ x q y. . y f Ž2n x. y f Ž2n y. F 2n p Ž 5 x5 p q 5 y5 p . . Dividing by 2 n the last expression and letting n ª ⬁, together with Ž2.5., yield that a is an additive function. The inequality Ž2.2. is an immediate consequence of Ž2.4. and Ž2.5.. Now we prove that a is the unique such additive function. Assume that there exists another one, denoted by a⬘ : E1 ª E2 . Then there exists a constant 1 G 0 and q Ž0 F q - 1. with a⬘ Ž x . y f Ž x . F 1 5 x 5 q . Ž 2.6. By the triangle inequality, Ž2.2., and Ž2.6. we have a Ž x . y a⬘ Ž x . s ny1 a Ž nx . y a⬘ Ž nx . F ny1 ž s n py1 2 2y2 p 2 2y2p 5 nx 5 p q 1 5 nx 5 q / 5 x 5 p q n qy1 1 5 x 5 q for all n g ⺞. By letting n ª ⬁ we get aŽ x . s a⬘Ž x . for any x g E1. Now, assume that f Ž tx . is continuous in t for each fixed x g E1. We know that aŽ qx . s qaŽ x . holds for all rational numbers q. Fix x 0 in E1 and in E2U Žthe dual space of E2 .. Define a function : ⺢ ª ⺢ by Ž t . s Ž a Ž tx 0 . . STABILITY OF FUNCTIONAL EQUATIONS 269 for all t g ⺢. Then is additive. Moreover, is a Borel function, because of the following reasoning. Let Ž t . s lim nª⬁ 2yn Ž f Ž2 n tx 0 .. and set nŽ t . s 2yn Ž f Ž2 n tx 0 ... Then the nŽ t . are continuous functions. But Ž t . is the pointwise limit of continuous functions; thus Ž t . is a Borel function. Therefore, is linear and hence it is continuous. Let r g ⺢. Then there is a sequence Ž qn . of rational numbers with r s lim qn . nª⬁ Hence, Ž rt . s t lim qn s lim Ž tqn . s lim qn Ž t . s r Ž t . . ž nª⬁ / nª⬁ nª⬁ Therefore, Ž rt . s r Ž t . for any r g ⺢. Thus aŽ rx . s raŽ x . for any r g ⺢. Hence, a is a linear function. It is found in w59x that the proof of this theorem also works for p - 0, and asked whether such a theorem is true for p G 1. Z. Gajda w18x affirmatively answered the question of Rassias for the case of p ) 1 by replacing n by yn in the formula Ž2.5.. It turned out that 1 is the only critical value of p to which Theorem 2.1 cannot be extended. Z. Gajda w18x showed that this theorem is false for p s 1 by constructing a counterexample. For a fixed ) 0 and s Ž1r6. define a function f : ⺢ ª ⺢ by f Ž x. s ⬁ Ý 2yn Ž 2 n x . , x g ⺢, ns0 where the function : ⺢ ª ⺢ is gi¨ en by ¡ ¢y Ž x . s~ x for x g 1, ⬁ . , for x g Ž y1, 1 . , for x g Ž y⬁, y1 . This construction gives an example for which Theorem 2.1 does not hold when p s 1, as we see in the following theorem: THEOREM 2.2. The function f defined abo¨ e satisfies the inequality f Ž x q y . y f Ž x . y f Ž y . F Ž < x < q < y <. Ž 2.7. for all x, y g ⺢. But there is no constant ␦ g w0, ⬁. and no additi¨ e function a : ⺢ ª ⺢ satisfying the inequality f Ž x . y aŽ x . F ␦ < x < for all x, y g ⺢. 270 THEMISTOCLES M. RASSIAS For more information on this subject, we may refer to w22, 23, 72, 80, 81x. Similarly, Th. M. Rassias and P. ˇ Semrl w66x constructed a simple counterexample to Theorem 2.1 for p s 1 as follows. The continuous real-¨ alued function defined by f Ž x. s ½ x log 2 Ž x q 1 . x log 2 < x y 1 < for x G 0, for x - 0 satisfies the inequality Ž2.7. with s 1 and < f Ž x . y cx <r< x < ª ⬁, as x ª ⬁, for any real number c. During the last two decades, the condition Ž2.1. for the Cauchy difference has continuously been weakened by many mathematicians Žsee, e.g., w19, 30, 35, 56, 57, 67x.. Among them we will introduce the result presented w19x. by P. Gavruta ˇ THEOREM 2.3. Let G and E be an abelian group and a Banach space, respecti¨ ely, and let : G = G ª w0, ⬁. be a function satisfying ⬁ Ý 2yŽ kq1. Ž2 k x, 2 k y . - ⬁ ⌽ Ž x, y . s ks0 for all x, y g G. If a function f : G ª E satisfies the inequality f Ž x q y . y f Ž x . y f Ž y . F Ž x, y . for any x, y g G, then there exists a unique additi¨ e function a : G ª E such that f Ž x . y a Ž x . F ⌽ Ž x, x . for all x g G. If moreo¨ er f Ž tx . is continuous in t for each fixed x g G, then a is linear. So far, we studied the stability of additive functions defined on a whole space. Now, it is natural to ask about the stability of the additive equation on a restricted domain. More precisely, we may ask whether there always exists a true additive function near a function which satisfies the additive equation approximately in a restricted domain. F. Skof w73, 74x and Z. Kominek w52x answered this question in the case of functions defined on certain subsets of ⺢ N with values in a Banach space. First, we will introduce a theorem of Z. Kominek w52x: THEOREM 2.4. Let E be a Banach space and let N be a positi¨ e integer. Suppose D is a bounded subset of ⺢ N containing zero in its interior. Assume, moreo¨ er, that there exist a non-negati¨ e integer n and a positi¨ e number c ) 0 such that Ži. Ž1r2. D ; D; Žii. Žyc, c . N ; D; Žiii. D ; Žy2 n c, 2 n c . N . STABILITY OF FUNCTIONAL EQUATIONS 271 If a function f : D ª E satisfies the inequality f Ž x q y. y f Ž x. y f Ž y. F for some G 0 and for all x, y g D with x q y g D, then there exists an additi¨ e function a : ⺢ N ª E such that f Ž x . y a Ž x . F Ž 2 n ⭈ 5N y 1 . for all x g D. The following theorem was presented by F. Skof and was applied to the proof of an asymptotic behavior of additive functions Žsee w74x.. THEOREM 2.5. Let E1 and E2 be a normed space and a Banach space, respecti¨ ely. Gi¨ en d ) 0, suppose a function f : E1 ª E2 satisfies the inequality f Ž x q y. y f Ž x. y f Ž y. F for some ) 0 and for all x, y g E1 with 5 x 5 q 5 y 5 ) d. Then there exists a unique additi¨ e function a : E1 ª E2 such that f Ž x . y a Ž x . F 9 for all x g E1. D. H. Hyers et al. w28x proved a generalized Hyers᎐Ulam stability result of the additive equation on a restricted domain and applied it to the asymptotic derivability which is important in nonlinear analysis. THEOREM 2.6. Gi¨ en a real normed space E1 and a real Banach space E2 , let numbers M ) 0, ) 0, and p with 0 - p - 1 be chosen. Let a function f : E1 ª E2 satisfy the inequality f Ž x q y. y f Ž x. y f Ž y. F Ž 5 x5 p q 5 y5 p. for all x, y g E1 satisfying 5 x 5 p q 5 y 5 p ) M p. Then there exists an additi¨ e function a : E1 ª E2 such that f Ž x . y aŽ x . F 2 2y2p 5 x5 p for all x g E1 with 5 x 5 ) 2y1 r p M. There are many variations of the additive functional equation. But the simplest and most elegant variation of the additive equation is the Jensen’s functional equation 2f ž xqy 2 / s f Ž x. q Ž y. . 272 THEMISTOCLES M. RASSIAS For the results of the generalized Hyers᎐Ulam stability of the Jensen’s equation, we may refer to w42, 52x. The Hosszu’s ´ functional equation, f Ž x q y y xy . s f Ž x . q f Ž y . y f Ž xy ., is also a well known variation of the additive equation. The reader can refer to w4, 53, 82x for the stability problems of this equation. In the papers w10, 45, 46, 84x, the results of the generalized Hyers᎐Ulam stability of the homogeneous functional equation, f Ž yx . s y k f Ž x ., were presented. The logarithmic functional equation f Ž xy . s f Ž x . q f Ž y . may be identified with the additive equation if the domain of relevant functions is a semigroup. It is well known that the Hyers’s theorem remains valid if E1 is an abelian semigroup Žsee w14x.. Therefore, we can refer to the previous theorems for the generalized Hyers᎐Ulam stability of the logarithmic functional equation. We will now introduce another logarithmic functional equation. The logarithmic functions f Ž x . s c log x Ž c g ⺢ is a constant. obviously satisfy the functional equation f Ž x y . s yf Ž x . . Hence, this equation is a kind of logarithmic functional equation. The stability of this equation was investigated by S.-M. Jung w38x. 3. EXPONENTIAL FUNCTIONAL EQUATION In 1979, J. Baker et al. w3x observed a strong stability phenomenon while investigating the stability problem of the exponential functional equation f Ž x q y . s f Ž x . f Ž y .. The theorem of Baker et al. was generalized by J. Baker w2x. THEOREM 3.1. Let Ž G, ⭈ . be a semigroup. If a function f : G ª ⺓ satisfies the inequality f Ž x ⭈ y. y f Ž x. f Ž y. F Ž 3.1. for some ) 0 and for all x, y g G, then either < f Ž x .< F Ž1 q '1 q 4 .r2 for all x g G or f Ž x ⭈ y . s f Ž x . f Ž y . for all x, y g G. The multiplicative property of the norm was crucial in the proof of Theorem 3.1. In general, the above proof works for functions f : G ª E, where E is a normed algebra in which the norm is multiplicative, i.e., 5 xy 5 s 5 x 5 5 y 5 for any x, y g E. Examples of such real normed algebras are the quaternions and the Cayley numbers. J. Baker w2x gave the following example which shows that the theorem is false if the algebra does not have the multiplicative norm. STABILITY OF FUNCTIONAL EQUATIONS 273 Gi¨ en ) 0, choose a ␦ with < ␦ y ␦ 2 < s . Let M2 Ž⺓. denote the space of 2 = 2 complex matrices with the usual norm. Let us define f : ⺢ ª M2 Ž⺓. by ex 0 f Ž x. s 0 ␦ ž / for all x g ⺢. Then f is unbounded and it satisfies 5 f Ž x q y . y f Ž x . f Ž y .5 s for all x, y g ⺢. Howe¨ er, f is not exponential. The theorem of Baker et al. was generalized by L. Szekelyhidi in another ´ way. Let Ž G, ⭈ . be a semigroup and let V be a vector space of complex-valued functions on G. Then V is called right in¨ ariant if f g V implies that the function f Ž x ⭈ y . belongs to V for each fixed y g G. Similarly, we may define left in¨ ariant ¨ ector spaces, and we call V in¨ ariant if it is right and left invariant. Following Szekelyhidi, a function m : G ª ⺓ is called exponential if ´ mŽ x ⭈ y . s mŽ x . mŽ y . holds for any x, y g G. The following theorem is due w77x. to Szekelyhidi ´ THEOREM 3.2. Let Ž G, ⭈ . be a semigroup and V be a right in¨ ariant ¨ ector space of complex-¨ alued functions on G. Let f, m : G ª ⺓ be functions such that the function y Ž x . s f Ž x ⭈ y . y f Ž x . mŽ y . belongs to V for each fixed y g G. Then either f belongs to V or m is exponential. During the 31st International Symposium on Functional Equations, Th. M. Rassias w61x raised an open problem concerning the behavior of solutions of the inequality f Ž x q y. y f Ž x. f Ž y. F Ž 5 x5 p q 5 y5 p. Ž 3.2. Žcf. w68x.. In connection with this open problem, S.-M. Jung w36x generalized the theorem of Baker et al. as follows. Let H : w0, ⬁. = w0, ⬁. ª w0, ⬁. be a monotonically increasing function Žin both variables. for which there exist, for given u, ¨ G 0, an ␣ s ␣ Ž u, ¨ . ) 0 and a w 0 s w 0 Ž u, ¨ . ) 0 such that H Ž u, ¨ q w . F ␣ H Ž w, w . for all w G w 0 . THEOREM 3.3. Let E be a complex normed space and let f : E ª ⺓ be a function for which there exist a z g E Ž z / 0. and a real number  Ž0  - 1. such that ⬁ Ý < f Ž z . <yŽ iq1. H Ž i 5 z 5 , 5 z 5. -  and is1 H Ž n 5 z 5 , n 5 z 5 . s o Ž < f Ž z . < n . as n ª ⬁. 274 THEMISTOCLES M. RASSIAS Furthermore, we assume that f satisfies the inequality f Ž x q y . y f Ž x . f Ž y . F H Ž 5 x 5 , 5 y 5. for all x, y g E. Then f is an exponential function. The group operation in the range of exponential functions is the ‘‘multiplication.’’ R. Ger noticed that the superstability of the functional inequality Ž3.1. is caused by the fact that the natural group structure in the range is disregarded. Hence, the stability problem for the exponential equation can be posed more naturally as f Ž x q y. f Ž x. f Ž y. y 1 F . Ž 3.3. R. Ger w24x investigated the stability problem given by the inequality Ž3.3.. Let Ž G, q. be an amenable semigroup, and let g w0, 1. be a gi¨ en number. If a function f : G ª ⺓ _ 04 satisfies the inequality Ž3.3. for all x, y g G, then there exists an exponential function m : G ª ⺓ _ 04 such that max ½ f Ž x. mŽ x . y1 , mŽ x . f Ž x. y1 5 F 2y 1y for all x g G. The bound Ž2 y .rŽ1 y . in the above inequality does not tend to zero even though does tend to zero. This shortcoming was overcome by R. Ger and P. ˇ Semrl w25x who proved the following theorem. THEOREM 3.4. Let Ž G, q. be a cancellati¨ e abelian semigroup. If a function f : G ª ⺓ _ 04 satisfies the inequality Ž3.3. for a gi¨ en g w0, 1. and for all x, y g G, then there exists a unique exponential function m : G ª ⺓ _ 04 such that max ½ f Ž x. mŽ x . y1 , mŽ x . f Ž x. y1 5 F 1q 1 Ž1 y . 2 y2 ž 1q 1y 1r2 1r2 / for all x g G. We will now introduce some results of the stability of the exponential equation on a restricted domain. Recently, S.-M. Jung w47x proved the following theorem and applied it to the asymptotic properties of exponential functions. THEOREM 3.5. Let E be a real Ž or complex . normed space and d ) 0 be a gi¨ en constant. Suppose a function f : E ª ⺓ satisfies the inequality f Ž x q y. y f Ž x. f Ž y. F 275 STABILITY OF FUNCTIONAL EQUATIONS for some ) 0 and for all x, y g E with 5 x 5 G d or 5 y 5 G d. If there exists a constant C ) 0 such that sup < f Ž x . < F C, 5 x 5Fd then f is either bounded or an exponential function. In the same paper, S.-M. Jung studied the solutions of the inequality Ž3.3. defined on a restricted domain and proved the stability of the exponential equation. THEOREM 3.6. Let E be a real Ž or complex . normed space and d ) 0 be a gi¨ en constant. If a function f : E ª ⺓ _ 04 satisfies the inequality Ž3.3. for some g w0, 1r2. and for all x, y g E with 5 x 5 G d or 5 y 5 G d, then there exists an exponential function m : E ª ⺓ _ 04 such that max ½ f Ž x. mŽ x . y1 , mŽ x . f Ž x. y1 5 F ⌽Ž . for all x g E, where ⌽Ž . s 1 q Ž1 q . 2 Ž1 y . 4 y2 1r2 1q Ž1 y . 2 ž4Ž 1 y 2 3r2 . 2 1r2 y 3Ž 1 y . / . Moreo¨ er, if - sinŽr9., the exponential function m is uniquely determined. The exponential functions f Ž x . s a x are solutions of the functional equation y f Ž xy . s f Ž x . . Therefore, it may be considered as another exponential functional equation. S.-M. Jung w44x investigated the stability problems of that equation in a similar setting as Ž3.3.. If the domain of related functions is a semigroup, we may identify the multiplicati¨ e functional equation f Ž xy . s f Ž x . f Ž y . with the exponential equation. Hence, for the results of the stability of the multiplicative equation, we can refer to the theorems presented in the first part of this section. Let E1 and E2 be Banach algebras. Let us denote by LŽ E1 , E2 . the space of bounded linear mappings from E1 into E2 . A pair Ž E1 , E2 . is called almost multiplicati¨ e near multiplicati¨ e Žor AMNM for short. if for 276 THEMISTOCLES M. RASSIAS given ) 0 and K ) 0 there exists ) 0 such that if T g LŽ E1 , E2 . with 5 T 5 F K satisfies T Ž xy . y T Ž x . T Ž y . F 5 x 5 5 y 5 for all x, y g E1 , then there is a multiplicative function M g LŽ E1 , E2 . with 5 T Ž x . y M Ž x .5 F 5 x 5 for any x g E1. We refer the reader interested in this subject to the papers w32᎐34x. The following functional equation f Ž x y. s f Ž x. y may be regarded as a variation of the multiplicative functional equation. S.-M. Jung investigated the stability problems of the above functional equation Žsee w48x.. 4. THE QUADRATIC FUNCTIONAL EQUATION In this section, we will introduce the results concerning the generalized Hyers᎐Ulam stability of the quadratic functional equation f Ž x q y. q f Ž x y y. s 2 f Ž x. q 2 f Ž y. . By a quadratic function we mean a solution of the quadratic functional equation. The Hyers᎐Ulam stability of the quadratic equation was first proved by F. Skof w74x for the functions f : E1 ª E2 where E1 is a normed space and E2 is a Banach space. P. W. Cholewa w8x demonstrated that Skof’s theorem is also valid if E1 is replaced by an abelian group G Žcf. w11x.. THEOREM 4.1. Let G and E be an abelian group and a Banach space, respecti¨ ely. If a function f : G ª E satisfies the inequality f Ž x q y. q f Ž x y y. y 2 f Ž x. y 2 f Ž y. F Ž 4.1 . for some G 0 and for all x, y g G, then there exists a unique quadratic function q : G ª E such that f Ž x . y q Ž x . F r2 for all x g G. The theorem of Skof was later extended by S. Czerwik w9x to the generalized Hyers᎐Ulam stability of that equation. We will introduce the theorem of Czerwik for the cases p - 2 and p ) 2 separately. First, we deal with the case p - 2. STABILITY OF FUNCTIONAL EQUATIONS 277 THEOREM 4.2. Let E1 and E2 be a normed space and a Banach space, respecti¨ ely. If a function f : E1 ª E2 satisfies the inequality f Ž x q y . q f Ž x y y . y 2 f Ž x . y 2 f Ž y . F q Ž 5 x 5 p q 5 y 5 p . Ž 4.2. for some , G 0, p - 2 and for all x, y g E1 _ 04 , then there exists a unique quadratic function q : E1 ª E2 such that f Ž x. y qŽ x. F q 5 f Ž 0. 5 3 q 2 4y2p 5 x5 p for any x g E1 _ 04 . S. Czerwik also proved the generalized Hyers᎐Ulam stability of the quadratic equation for p ) 2. THEOREM 4.3. Let E1 be a normed space and let E2 be a Banach space. If a function f : E1 ª E2 satisfies the inequality 5 f Ž x q y . q f Ž x y y . y 2 f Ž x . y 2 f Ž y . 5 F Ž 5 x 5 p q 5 y 5 p . Ž 4.3. for some G 0, p ) 2 and for all x, y g E1 , then there exists a unique quadratic function q : E1 ª E2 such that 5 f Ž x. y qŽ x. 5 F 2 2 y4 p 5 x5 p for all x g E1. S. Czerwik w9x indicated that the quadratic equation is not stable in the sense of Hyers, Ulam, and Rassias if p s 2 is assumed in the inequality Ž4.2. or Ž4.3.. Indeed, he slightly modified Gajda’s construction Žsee Theorem 2.2. to prove the following theorem: THEOREM 4.4. Suppose the function f : ⺢ ª ⺢ is defined by f Ž x. s ⬁ Ý 4yn Ž 2 n x . , ns0 where the function : ⺢ ª ⺢ is gi¨ en by Ž x. s ½ x 2 for < x < G 1, for < x < - 1 with a constant ) 0. Then the function f satisfies the inequality f Ž x q y . q f Ž x y y . y 2 f Ž x . y 2 f Ž y . F 32 Ž x 2 q y 2 . 278 THEMISTOCLES M. RASSIAS for all x, y g ⺢. Moreo¨ er, there exists no quadratic function q : ⺢ ª ⺢ such that the image set of < f Ž x . y q Ž x .<rx 2 Ž x / 0. is bounded. Theorem 4.3 is an immediate consequence of the following stability theorem, which was presented by C. Borelli and G. L. Forti w5x, for a wide class of functional equations which contains the quadratic equation as a special case: THEOREM 4.5. Let G be an abelian group, E a Banach space, and let f : G ª E be a function with f Ž0. s 0 and satisfying the inequality f Ž x q y . q f Ž x y y . y 2 f Ž x . y 2 f Ž y . F Ž x, y . for all x, y g G. Assume that one of the series ⬁ Ý 2y2 i Ž 2 iy1 x, 2 iy1 x . ⬁ Ý 2 2Ž iy1. Ž 2yi x, 2yi x . and is1 is1 con¨ erges for each x g G and denote by ⌽ Ž x . its sum. If 2y2 i Ž 2 iy1 x, 2 iy1 y . ª 0 or 2 2Ž iy1. Ž 2yi x, 2yi y . ª 0, as i ª ⬁, then there exists a unique quadratic function q : G ª E such that f Ž x. y qŽ x. F ⌽Ž x. for all x g G. As F. Skof w73x did for the additive equation, F. Skof and S. Terracini w76x proved the Hyers᎐Ulam stability of the quadratic equation on a finite interval in ⺢ Žcf. w75x.. THEOREM 4.6. Let E be a Banach space and let c, ) 0 be gi¨ en. If a function f : w0, c . ª E satisfies the inequality Ž4.1. for all x G y G 0 with x q y - c, then there exists a quadratic function q : ⺢ ª E such that f Ž x. y qŽ x. F 79 2 for any x g w0, c .. With the help of Theorem 4.6, F. Skof and S. Terracini w76x proved the following theorem. THEOREM 4.7. Let E be a Banach space and let c, ) 0 be gi¨ en. If a function f : Žyc, c . ª E satisfies the inequality Ž4.1. for all x, y g ⺢ with < x q y < - c and < x y y < - c, then there exists a quadratic function q : ⺢ ª E such that 81 f Ž x. y qŽ x. F 2 for any x g Žyc, c .. STABILITY OF FUNCTIONAL EQUATIONS 279 The theorem of Skof and Terracini was recently generalized by S.-M. Jung and B. Kim w51x as follows: THEOREM 4.8. Let E be a Banach space and n g ⺞ be a gi¨ en integer. For a gi¨ en c ) 0, assume that a function f : wyc, c x n ª E satisfies the inequality Ž4.1. for some ) 0 and for all x, y g wyc, c x n with x q y, x y y g wyc, c x n. Then there exists a quadratic function q : ⺢ n ª E such that f Ž x . y q Ž x . - Ž 2912 n2 q 1872 n q 334 . for any x g wyc, c x n. On the other hand, S.-M. Jung w43x proved the Hyers᎐Ulam stability of the quadratic equation on a restricted Žunbounded. domain and applied this result to a study of an asymptotic behavior of quadratic functions. THEOREM 4.9. Let E1 and E2 be a real normed space and a real Banach space, respecti¨ ely. Suppose d ) 0 and G 0 are gi¨ en. If a function f : E1 ª E2 satisfies the inequality Ž4.1. for all x, y g E1 with 5 x 5 q 5 y 5 G d, then there exists a unique quadratic function q : E1 ª E2 such that f Ž x. y qŽ x. F 7 2 Ž 4.4. for all x g E1. If, moreo¨ er, f is measurable or f Ž tx . is continuous in t for each fixed x g E1 , then q Ž tx . s t 2 q Ž x . for all x g E1 and all t g ⺢. There is a survey paper w63x which provides a kind introduction to the theory of stability of the quadratic functional equation. 5. OTHER FUNCTIONAL EQUATIONS The addition and subtraction rules for trigonometric functions can be represented by using functional equations. In this section, we survey these equations and the stability problem of them. The addition rule cosŽ x q y . q cosŽ x y y . s 2 cos x cos y for the cosine function may be represented by the functional equation f Ž x q y. q f Ž x y y. s 2 f Ž x. f Ž y. , Ž 5.1. which is called the cosine functional equation or d’ Alembert equation. The superstability of this equation was first proved by J. Baker w2x. Later w20x presented a simpler proof of it. P. Gavruta ˇ THEOREM 5.1. Let ) 0 and let Ž G, q. be an abelian group. If a function f : G ª ⺓ satisfies the inequality f Ž x q y. q f Ž x y y. y 2 f Ž x. f Ž y. F Ž 5.2. 280 THEMISTOCLES M. RASSIAS for all x, y g G, then either < f Ž x .< F Ž1 q '1 q 2 .r2 for any x g G or f satisfies the cosine equation Ž5.1. for all x, y g G. The sine functional equation 2 f Ž x q y. f Ž x y y. s f Ž x. y f Ž y. 2 Ž 5.3. reminds us of a trigonometric formula, sinŽ x q y . sinŽ x y y . s sin 2 x y sin 2 y. The superstability of the sine equation Ž5.3. was proved by P. W. Cholewa w7x. THEOREM 5.2. Let Ž G, q. be an abelian group in which di¨ ision by 2 is uniquely performable. E¨ ery unbounded function f : G ª ⺓ satisfying the inequality 2 f Ž x q y. f Ž x y y. y f Ž x. q f Ž y. 2 F , Ž 5.4. for some ) 0 and for all x, y g G, is a solution of the sine equation Ž5.3.. w20x proved in Theorem 5.1 that, concerning J. Baker w2x and P. Gavruta ˇ the superstability of the cosine equation, a function f satisfying the inequality Ž5.2. is either a solution of the cosine equation Ž5.1. or it is bounded by a constant depending on only. It is not the case for the sine equation. Indeed, the bounded functions f n Ž x . s n sin x q 1rn satisfy the inequality Ž5.4. with s 3, for all x, y g ⺢ and for all n g ⺞. Nevertheless, for each ␦ ) 0, the inequality < f nŽ x .< F ␦ does not hold for certain x and n. Obviously, the sine and cosine functions satisfy the following functional equations as well, f Ž xy . s f Ž x . f Ž y . y g Ž x . g Ž y . Ž 5.5. f Ž xy . s f Ž x . g Ž y . q f Ž y . g Ž x . Ž 5.6. and which were introduced by L. Szekelyhidi, who proved w79x the Hyers᎐Ulam ´ stability of Eqs. Ž5.5. and Ž5.6. separately. R. Ger w21x proved the generalized Hyers᎐Ulam stability of the system consisting of both equations. Indeed, R. Ger considered the system ½ f Ž xy . s f Ž x . g Ž y . q f Ž y . g Ž x . , g Ž xy . s g Ž x . g Ž y . y f Ž x . f Ž y . STABILITY OF FUNCTIONAL EQUATIONS 281 and proved that the system is not superstable, but that it is stable in the sense of Hyers, Ulam, and Rassias. One is referred to w12, 13x for the stability of generalized trigonometric functional equations. Some results of the generalized Hyers᎐Ulam stability of the gamma functional equation, f Ž x q 1. s xf Ž x ., were presented in the papers w1, 39᎐41x. P. ˇ Semrl w71x introduced a functional equation, f Ž xy . s xf Ž y . q f Ž x . y, which defines multiplicative derivations in algebras and he proved the superstability of this equation. 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