STABILITY OF A GENERALIZED MIXED TYPE ADDITIVE, QUADRATIC, CUBIC AND QUARTIC FUNCTIONAL EQUATION

Volume 0 009), Issue 4, Article 4, 9 pp. STABILITY OF A GENERALIZED MIXED TYPE ADDITIVE, QUADRATIC, CUBIC AND QUARTIC FUNCTIONAL EQUATION K. RAVI, J.M. RASSIAS, M. ARUNKUMAR, AND R. KODANDAN DEPARTMENT OF MATHEMATICS SACRED HEART COLLEGE TIRUPATTUR - 35 0 TAMILNADU, INDIA. shckravi@yahoo.co.in PEDAGOGICAL DEPARTMENT E.E. SECTION OF MATHEMATICS AND INFORMATICS NATIONAL AND CAPODISTRIAN UNIVERSITY OF ATHENS 4, AGAMEMNONOS STR. AGHIA PARASKEVI ATHENS 534, GREECE. jrassias@primedu.uoa.gr URL: http://www.primedu.uoa.gr/ jrassias/ DEPARTMENT OF MATHEMATICS SACRED HEART COLLEGE TIRUPATTUR - 35 0 TAMILNADU, INDIA. annarun00@yahoo.co.in DEPARTMENT OF MATHEMATICS SRINIVASA INSTITUTE OF TECHNOLOGY AND MANAGEMENT STUDIES CHITTOOR - 57 7, ANDHRA PRADESH. Rkodandan979@rediff.co.in Received 0 July, 009; accepted 0 November, 009 Communicated by S.S. Dragomir ABSTRACT. In this paper, we obtain the general solution and the generalized Hyers-Ulam- Rassias stability of the generalized mixed type of functional equation f x + ay) + f x ay) = a [f x + y) + f x y)] + a ) f x) a 4 a ) + [f y) + f y) 4f y) 4f y)]. for fixed integers a with a 0, ±. Key words and phrases: Additive function, Quadratic function, Cubic function, Quartic function, Generalized Hyers-Ulam- Rassias stability, Ulam-Gavruta-Rassias stability, J.M. Rassias stability. 000 Mathematics Subject Classification. 39B5, 39B8. 83-09

K. RAVI, J.M. RASSIAS, M. ARUNKUMAR, AND R. KODANDAN. INTRODUCTION S.M. Ulam [3] is the pioneer of the stability problem in functional equations. In 940, while he was delivering a talk before the Mathematics Club of the University of Wisconsin, he discussed a number of unsolved problems. Among them was the following question concerning the stability of homomorphisms: "Let G be group and H be a metric group with metric d, ). Given ɛ > 0 does there exist a δ > 0 such that if a function f : G H satisfies d fxy), fx)fy)) < δ for all x, y G, then there exists a homomorphism a : G H with d fx), ax)) < ε for all x G." In 94, D.H. Hyers [] gave the first affirmative partial answer to the question of Ulam for Banach spaces. He proved the following celebrated theorem. Theorem. []). Let X, Y be Banach spaces and let f : X Y be a mapping satisfying.) f x + y) f x) f y) ε for all x, y X. Then the limit f n x).) ax) = lim n n exists for all x X and a : X Y is the unique additive mapping satisfying.3) f x) a x) ε for all x X. In 950, Aoki [] generalized the Hyers theorem for additive mappings. In 978, Th.M. Rassias [] provided a generalized version of the Hyers theorem which permitted the Cauchy difference to become unbounded. He proved the following: Theorem. []). Let X be a normed vector space and Y be a Banach space. If a function f : X Y satisfies the inequality.4) f x + y) f x) f y) θ x p + y p ) for all x, y X, where θ and p are constants with θ > 0 and p <, then the limit f n x).5) T x) = lim n n exists for all x X and T : X Y is the unique additive mapping which satisfies.) f x) T x) θ p x p for all x X. If p < 0, then inequality.4) holds for x, y 0 and.) for x 0. Also if for each x X the function ftx) is continuous in t R, then T is linear. It was shown by Z. Gajda [9], as well as Th.M. Rassias and P. Semrl [7] that one cannot prove a Th.M. Rassias type theorem when p =. The counter examples of Z. Gajda, as well as of Th.M. Rassias and P. Semrl [7] have stimulated several mathematicians to invent new definitions of approximately additive or approximately linear mappings; P. Gavruta [0] and S.M. Jung [7] among others have studied the Hyers-Ulam-Rassias stability of functional equations. The inequality.4) that was introduced by Th.M. Rassias [] provided much J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

STABILITY OF GENERALIZED MIXED TYPE 3 influence in the development of a generalization of the Hyers-Ulam stability concept. This new concept is known as the Hyers-Ulam-Rassias stability of functions. In 98, J.M. Rassias [4] following the spirit of the approach of Th.M. Rassias [] for the unbounded Cauchy difference proved a similar stability theorem in which he replaced the factor x p + y p by x p y q for p, q R with p + q. Theorem.3 [4]). Let X be a real normed linear space and Y be a real completed normed linear space. Assume that f : X Y is an approximately additive mapping for which there exists constants θ > 0 and p, q R such that r = p + q and f satisfies the inequality.7) f x + y) f x) f y) θ x p y q for all x, y X. Then the limit f n x).8) Lx) = lim n n exists for all x X and L : X Y is the unique additive mapping which satisfies θ.9) fx) Lx) r x r for all x X. If, in addition f : X Y is a mapping such that the transformation t ftx) is continuous in t R for each fixed x X, then L is an R linear mapping. However, the case r = in inequality.9) is singular. A counter example has been given by P. Gavruta []. The above-mentioned stability involving a product of different powers of norms was called Ulam-Gavruta-Rassias stability by M.A. Sibaha et al., [30], as well as by K. Ravi and M. Arunkumar [8]. This stability result was also called the Hyers-Ulam-Rassias stability involving a product of different powers of norms by Park [3]. In 994, a generalization of Th.M. Rassias theorem and J.M. Rassias theorem was obtained by P. Gavruta [0], who replaced the factors θ x p + y p ) and θ x p y p ) by a general control function ϕx, y). In the past few years several mathematicians have published various generalizations and applications of Hyers- Ulam- Rassias stability to a number of functional equations and mappings see [4, 5, 3, 8, 9]). Very recently, J.M. Rassias [9] in the inequality.7) replaced the bound by a mixed one involving the product and sum of powers of norms, that is, θ x p y p + x p + y p )}. The functional equation.0) f x + y) + f x y) = f x) + f y) is said to be a quadratic functional equation because the quadratic function fx) = ax is a solution of the functional equation.0). A quadratic functional equation was used to characterize inner product spaces [, 0]. It is well known that a function f is a solution of.0) if and only if there exists a unique symmetric biadditive function B such that fx) = Bx, x) for all x see [0]). The biadditive function B is given by.) B x, y) = [f x + y) + f x y)]. 4 The functional equation.) f x + y) + f x y) = f x + y) + f x y) + f x) is called a cubic functional equation, because the cubic function fx) = cx 3 is a solution of the equation.). The general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation.) was discussed by K.W. Jun and H.M. Kim [4]. They proved that a function f between real vector spaces X and Y is a solution of.) if and only if there exists J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

4 K. RAVI, J.M. RASSIAS, M. ARUNKUMAR, AND R. KODANDAN a unique function C : X X X Y such that fx) = Cx, x, x) for all x X and C is symmetric for each fixed one variable and is additive for fixed two variables. The quartic functional equation.3) f x + y) + f x y) f x) = 4 [f x + y) + f x y)] + 4f y) was introduced by J.M. Rassias [5]. Later S.H. Lee et al., [] remodified J.M. Rassias s equation and obtained a new quartic functional equation of the form.4) f x + y) + f x y) = 4 [f x + y) + f x y)] + 4f x) f y) and discussed its general solution. In fact S.H. Lee et al., [] proved that a function f between vector spaces X and Y is a solution of.4) if and only if there exists a unique symmetric multi - additive function Q : X X X X Y such that fx) = Qx, x, x, x) for all x X. It is easy to show that the function fx) = kx 4 is the solution of.3) and.4). A function.5) fx) = Qx, x, x 3, x 4 ) is called symmetric multi additive if Q is additive with respect to each variable x i, i =,, 3, 4 in.5). A function f is defined as fx) = βx) αx) where αx) = fx) fx), βx) = fx) 4fx), further, f satisfies fx) = 4fx) and fx) = fx) is said to be a quadratic - quartic function. K.W. Jun and H.M. Kim [] introduced the following generalized quadratic and additive type functional equation n ).) f x i + n ) i= n f x i ) = i= i<j n f x i + x j ) in the class of functions between real vector spaces. For n = 3, Pl. Kannappan proved that a function f satisfies the functional equation.) if and only if there exists a symmetric biadditive function A and an additive function B such that fx) = Bx, x) + Ax) for all x see [0]). The Hyers-Ulam stability for the equation.) when n = 3 was proved by S.M. Jung [8]. The Hyers-Ulam-Rassias stability for the equation.) when n = 4 was also investigated by I.S. Chang et al., [3]. The general solution and the generalized Hyers-Ulam stability for the quadratic and additive type functional equation.7) f x + ay) + af x y) = f x ay) + af x + y) for any positive integer a with a, 0, was discussed by K.W. Jun and H.M. Kim [5]. Recently A. Najati and M.B. Moghimi [] investigated the generalized Hyers-Ulam-Rassias stability for a quadratic and additive type functional equation of the form.8) f x + y) + f x y) = f x + y) + f x y) + f x) 4f x) Very recently, the authors [, 7] investigated a mixed type functional equation of cubic and quartic type and obtained its general solution. The stability of generalized mixed type functional equations of the form.9) f x + ky) + f x ky) = k [f x + y) + f x y)] + k ) f x) J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

STABILITY OF GENERALIZED MIXED TYPE 5 for fixed integers k with k 0, ± in quasi -Banach spaces was investigated by M. Eshaghi Gordji and H. Khodaie [8]. The mixed type functional equation.9) is additive, quadratic and cubic. In this paper, the authors introduce a mixed type functional equation of the form.0) f x + ay) + f x ay) = a [f x + y) + f x y)] + a ) f x) + a4 a [f y) + f y) 4f y) 4f y)] which is additive, quadratic, cubic and quartic and obtain its general solution and generalized Hyers-Ulam-Rassias stability for fixed integers a with a 0, ±.. GENERAL SOLUTION In this section, we present the general solution of the functional equation.0). Throughout this section let E and E be real vector spaces. Theorem.. Let f : E E be a function satisfying.0) for all x, y E. If f is even then f is quadratic - quartic. Proof. Let f be an even function, i.e., f x) = fx). Then equation.0) becomes.) f x + ay) + f x ay) = a [f x + y) + f x y)] + a ) f x) + a4 a for all x, y E. Interchanging x and y in.) and using the evenness of f, we get.) f ax + y) + f ax y) = a [f x + y) + f x y)] + a ) f y) + a4 a [f y) 4f y)] [f x) 4f x)] for all x, y E. Setting x, y) as 0, 0) in.), we obtain f0) = 0. Replacing y by x + y in.) and using the evenness of f, we have.3) f a + ) x + y) + f a ) x y) = a [f x + y) + f y)] + a ) f x + y) + a4 a [f x) 4f x)] for all x, y E. Replacing y by x y in.), we obtain.4) f a + ) x y) + f a ) x + y) = a [f x y) + f y)] + a ) f x y) + a4 a [f x) 4f x)] for all x, y E. Adding.3) and.4), we get.5) f a + ) x + y) + f a ) x y) + f a + ) x y) + f a ) x + y) = a [f x + y) + f x y) + f y)] + a ) [f x + y) + f x y)] + a4 a [f x) 8f x)] J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

K. RAVI, J.M. RASSIAS, M. ARUNKUMAR, AND R. KODANDAN for all x, y E. Replacing y by ax + y in.), we obtain.) f ax + y) + f y) = a [f a + ) x + y) + f a) x y)] + a ) f ax + y) + a4 a [f x) 4f x)] for all x, y E. Replacing y by ax y in.3), we get.7) f ax y) + f y) = a [f a + ) x y) + f a) x + y)] + a ) f ax y) + a4 a [f x) 4f x)] for all x, y E. Adding.) and.7), we obtain.8) f ax + y) + f ax y) + f y) = a [f a + ) x + y) + f a + ) x y) + f a ) x + y) + f a ) x y)] + a ) [f ax + y) + f ax y)] + a4 a [f x) 4f x)] 3 for all x, y E. Using.5) in.8), we arrive at.9) f ax + y) + f ax y) + f y) = a 4 [f x + y) + f x y)] + a 4 f y) + a a ) [f x + y) + f x y)] + a a 4 a ) 3 for all x, y E. Replacing x by x in.), we get [f x) 4f x)] + a ) [f ax + y) + f ax y)] + a4 a 3 [f x) f x)].0) f ax + y) + f ax y) = a [f x + y) + f x y)] + a ) f y) + a4 a [f 4x) 4f x)] for all x, y E. Using.0) in.9), we obtain.) a [f x+y)+f x y)]+ a ) f y)+ a4 a [f 4x) 4f x)]+f y) = a 4 [f x + y) + f x y)] + a a ) [f x + y) + f x y)] + a a 4 a ) 3 [f x) 4f x)] + a ) [f ax + y) + f ax y)] + a4 a 3 [f x) 4f x)] + a 4 f y) J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

STABILITY OF GENERALIZED MIXED TYPE 7 for all x, y E. Using.) in.), we get.) a [f x + y) + f x y)] + a ) f y) + a4 a [f 4x) 4f x)] + f y) = a 4 [f x + y) + f x y)] + a a ) [f x + y) + f x y)] + a a 4 a ) [f x) 4f x)] + a4 a [f x) 4f x)] + a 4 f y) 3 3 + a ) [ a f x + y) + f x y)) + a ) ] f y) + a4 a [f x) 4f x)] for all x, y E. Letting y = 0 in.), we obtain.3) f ax) = a f x) + a4 a [f x) 4f x)] for all x, y E. Replacing y by x in.), we get.4) f a + ) x)+f a ) x) = a f x)+ a ) f x)+ a4 a [f x) 4f x)] for all x E. Replacing y by ax in.), we obtain.5) f ax) = a [f + a) x) + f a) x)] + a ) f ax) + a4 a [f x) 4f x)] for all x E. Letting y = 0 in.0), we get.) f ax) = a f x) + a4 a [f 4x) 4f x)] for all x E. From.5) and.), we arrive at.7) a f x) + a4 a [f 4x) 4f x)] = a [f + a) x) + f a) x)] + a ) f ax) + a4 a [f x) 4f x)] for all x E. Using.3) and.4) in.7), we obtain.8) a f x) + a4 a [f 4x) 4f x)] = a [ a f x) + a ) f x) + a4 a ] [f x) 4f x)] ] + a ) [ a f x) + a4 a [f x) 4f x)] for all x E. Comparing.) and.8), we arrive at + a4 a [f x) 4f x)].9) f x + y) + f x y) = 4 [f x + y) + f x y)] 8f x) + f x) f y) for all x, y E. Replacing y by y in.9), we get.0) f x + y)+f x y) = 4 [f x + y) + f x y)] 8f x)+f x) f y) J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

8 K. RAVI, J.M. RASSIAS, M. ARUNKUMAR, AND R. KODANDAN for all x, y E. Interchanging x and y in.9) and using the evenness of f, we obtain.) f x + y) + f x y) = 4 [f x + y) + f x y)] 8f y) + f y) f x) for all x, y E. Using.) in.0), we get.) f x + y) + f x y) = [f x + y) + f x y)] + f y) 3f y) + f x) 3f x) for all x, y E. Rearranging.), we have.3) f x + y) f x + y)} + f x y) f x y)} for all x, y E. Let α : E E defined by.4) α x) = f x) f x), x E. Applying.4) in.3), we arrive at = f x) f x)} + f y) f y)}.5) α x + y) + α x y) = α x) + α y) x E. Hence α : E E is quadratic mapping. Since α is quadratic, we have α x) = 4α x) for all x E. Then.) f 4x) = 0f x) 4f x) for all x E. Replacing x, y) by x, y) in.9), we get.7) f x + y)) + f x y)) for all x, y E. Using.) in.7), we obtain.8) f x + y)) + f x y)) = 4 [f x + y)) + f x y))] 8f x) + f 4x) f y) = 4 [f x + y)) + f x y))] + 3 f x) 4f x)} f y) for all x, y E. Multiplying.9) by 4, we arrive at.9) 4f x + y) + 4f x y) for all x, y E. Subtracting.9) from.8), we get = [f x + y) + f x y)] + 8 f x) 4f x)} 4f y).30) f x + y)) 4f x + y)} + f x y)) 4f x y)} = 4 f x + y)) 4f x + y)} + 4 f x y)) 4f x y)} for all x, y E. Let β : E E be defined by.3) β x) = f x) 4f x), x E. Applying.30) in.3), we arrive at + 4 f x) 4f x)} f y) 4f y)}.3) β x + y) + β x y) = 4 [β x + y) + β x y)] + 4β x) β y) for all x, y E. Hence β : E E is quartic mapping. On the other hand, we have β x) α x).33) f x) = x E. J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

STABILITY OF GENERALIZED MIXED TYPE 9 This means that f is quadratic-quartic function. This completes the proof of the theorem. Theorem.. Let f : E E be a function satisfying.0) for all x, y E. If f is odd then f is additive - cubic. Proof. Let f be an odd function i.e., f x) = f x)). Then equation.0) becomes.34) f x + ay) + f x ay) = a [f x + y) + f x y)] + a ) f x) for all x, y E. By Lemma. of [3], f is additive-cubic. Theorem.3. Let f : E E be a function satisfying.0) for all x, y E if and only if there exists functions A : E E, B : E E E, C : E E E E and D : E E E E E such that.35) f x) = A x) + B x, x) + C x, x, x) + D x, x, x, x) for all x E, where A is additive, B is symmetric bi-additive, C is symmetric for each fixed one variable and is additive for fixed two variables and D is symmetric multi-additive. Proof. Let f : E E be a function satisfying.0). We decompose f into even and odd parts by setting f e x) = f x) + f x)}, f o x) = f x) f x)} for all x E. It is clear that f x) = f e x) + f o x) for all x E. It is easy to show that the functions f e and f o satisfy.0). Hence by Theorem. and., we see that the function f e is quadratic-quartic and f o is additive-cubic, respectively. Thus there exist a symmetric bi-additive function B : E E E and a symmetric multi-additive function D : E E E E E such that f e x) = B x, x) + D x, x, x, x) for all x E, and the function A : E E is additive and C : E E E E such that f o x) = A x) + C x, x, x), where C is symmetric for each fixed one variable and is additive for fixed two variables. Hence we get.35) for all x E. Conversely let f x) = A x) + B x, x) + C x, x, x) + D x, x, x, x) for all x E, where A is additive, B is symmetric bi-additive, C is symmetric for each fixed one variable and is additive for fixed two variables and D is symmetric multi-additive. Then it is easy to show that f satisfies.0). 3. STABILITY OF THE FUNCTIONAL EQUATION.0) In this section, we investigate the generalized Hyers-Ulam-Rassias stability problem for the functional equation.0). Throughout this section, let E be a real normed space and E be a Banach space. Define a difference operator Df : E E E by Df x, y) = f x + ay) + f x ay) a [f x + y) + f x y)] a ) f x) for all x, y E. a4 a [f y) + f y) 4f y) 4f y)] Theorem 3.. Let φ b : E E [0, ) be a function such that φ b n x, n y) φ b n x, n y) 3.) converges and lim = 0 4 n n 4 n n=0 for all x, y E and let f : E E be an even function which satisfies the inequality 3.) Df x, y) φ b x, y) J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

0 K. RAVI, J.M. RASSIAS, M. ARUNKUMAR, AND R. KODANDAN for all x, y E. Then there exists a unique quadratic function B : E E such that 3.3) f x) f x) B x) Φ b k x ) 4 4 k for all x E, where the mapping Bx) and Φ b k x) are defined by 3.4) B x) = lim f n+ x ) f n x) } n 4 n 3.5) Φ b k x ) = for all x E. [ a ) φ a 4 a b 0, k x ) + a φ b k x, k x ) + φ b 0, k+ x ) + φ b k ax, k x ) ] Proof. Using the evenness of f, from 3.) we get 3.) f x + ay) + f x ay) a [f x + y) + f x y)] a ) f x) a4 a ) [f y) 8f y)] φ b x, y) for all x, y E. Interchanging x and y in 3.), we obtain 3.7) f ax + y) + f ax y) a [f x + y) + f x y)] a ) f y) a4 a ) [f x) 8f x)] φ b y, x) for all x, y E. Letting y = 0 in 3.7), we get 3.8) f ax) a f x) a4 a ) [f x) 8f x)] φ b 0, x) for all x E. Putting y = x in 3.7), we obtain 3.9) f a + ) x) + f a ) x) a f x) a ) f x) a4 a ) [f x) 8f x)] φ b x, x) for all x E. Replacing x by x in 3.8), we get 3.0) f ax) a f x) a4 a ) [f 4x) 8f x)] φ b 0, x) for all x E. Setting y by ax in 3.7), we obtain 3.) f ax) a [f + a) x) + f a) x)] a ) f ax) a4 a ) [f x) 8f x)] φ b ax, x) J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

STABILITY OF GENERALIZED MIXED TYPE for all x E. Multiplying 3.8), 3.9), 3.0) and 3.) by a ), a, and respectively, we have a 4 a ) f 4x) 0f x) + 4f x) = 4 a ) f ax) 4a a ) f x) } a ) a 4 a ) [f x) 8f x)] + a f a + ) x) + a f a ) x) a 4 f x) 4a a ) } f x) a a 4 a ) [f x) 8f x)] + f ax) +a f x) + } a4 a ) [f 4x) 8f x)] + f ax) a [f + a) x) + f a) x)] 4 a ) f ax) a4 a ) [f x) 8f x)]} a ) φ b 0, x) + a φ b x, x) + φ b 0, x) + φ b ax, x) for all x E. Hence from the above inequality, we get 3.) f 4x) 0f x) + 4f x) [ ) a φ a 4 a b 0, x) + a φ b x, x) + φ b 0, x) + φ b ax, x) ] ) for all x E. From 3.), we arrive at 3.3) f 4x) 0f x) + 4f x) Φ b x), where Φ b x) = [ ) a φ a 4 a b 0, x) + a φ b x, x) + φ b 0, x) + φ b ax, x) ] for all x E. It is easy to see from 3.3) that 3.4) f 4x) f x) 4 f x) f x)} Φ b x) for all x E. Using.4) in 3.4), we obtain 3.5) α x) 4α x) Φ b x) for all x E. From 3.5), we have 3.) α x) α x) 4 Φ b x) 4 for all x E. Now replacing x by x and dividing by 4 in 3.), we obtain 3.7) α x) α x) 4 4 Φ b x) 4 J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

K. RAVI, J.M. RASSIAS, M. ARUNKUMAR, AND R. KODANDAN for all x E. From 3.) and 3.7), we arrive at α x) α x) 4 α x) α x) 4 4 + α x) 3.8) α x) 4 [ Φ b x) + Φ ] b x) 4 4 for all x E. In general for any positive integer n, we get α n x) α x) 4 n n Φ b k x ) 3.9) 4 4 k Φ b k x ) 4 4 k for all x E. In order to prove the convergence of the sequence α n x) }, replace x by m x 4 n and divide by 4 m in 3.9). For any m, n > 0, we have α n+m x) α m x) 4 n+m 4 m = α n m x) 4 m α m x) 4 n n Φ b k+m x ) 4 4 k+m Φ b k+m x ) 4 4 k+m } α n x) 4 n 0 as m for all x E. Hence the sequence is a Cauchy sequence. Since E is complete, there exists a quadratic mapping B : E E such that B x) = lim n α n x) 4 n x E. Letting n in 3.9) and using.4), we see that 3.3) holds for all x E. To prove that B satisfies.0), replace x, y) by n x, n y) and divide by 4 n in 3.). We obtain 4 n f n x + ay)) + f n x ay)) a [f n x + y)) + f n x y))] a ) f n x) a4 a ) [f n y)) + f n y))] a4 a ) [ 4f n y) 4f n y))] φ n x, n y) 4 n for all x, y E. Letting n in the above inequality, we see that B x + ay) + B x ay) a [B x + y) + B x y)] a ) B x) a4 a ) [B y) + B y) 4B y) 4Bf y)] 0, J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

STABILITY OF GENERALIZED MIXED TYPE 3 which gives B x + ay) + B x ay) = a [B x + y) + B x y)] + a ) B x) + a4 a ) [B y) + B y) 4B y) 4Bf y)] for all x, y E. Hence B satisfies.0). To prove that B is unique, let B be another quadratic function satisfying.0) and 3.3). We have B x) B x) = 4 n B n x) B n x) 4 B n n x) α n x) + α n x) B n x) } Φ b k x ) 4 n 4 k 0 as n for all x E. Hence B is unique. This completes the proof of the theorem. The following corollary is an immediate consequence of Theorem 3. concerning the stability of.0). Corollary 3.. Let ε, p be nonnegative real numbers. Suppose that an even function f : E E satisfies the inequality ε x p + y p ), 0 p < ; ε, 0 p < ; 3.0) Df x, y) ε x p y p, 0 p < ; ε x p y p + x p + y p}), for all x, y E. Then there exists a unique quadratic function B : E E such that λ x p, 4 p 0λ, 3.) f x) f x) B x) where λ = ε 4 + a + a p ) + p )} a 4 a, λ = λ 3 x p 4 p, λ 4 x p 4 p, ε a 4 a, λ 3 = ε a + a p } and λ a 4 a 4 = ε 4 + 4a + a p ) + a p ) + p )} a 4 a for all x E. Theorem 3.3. Let φ d : E E [0, ) be a function such that φ d n x, n y) φ d n x, n y) 3.) converges and lim = 0 n n n n=0 for all x, y E and let f : E E be an even function which satisfies the inequality 3.3) Df x, y) φ d x, y) J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

4 K. RAVI, J.M. RASSIAS, M. ARUNKUMAR, AND R. KODANDAN for all x, y E. Then there exists a unique quartic function D : E E such that 3.4) f x) 4f x) D x) Φ d k x ) k for all x E, where the mapping Dx) and Φ d k x) are defined by 3.5) D x) = lim f n+ x ) 4f n x) }, n n 3.) Φ d k x ) = for all x E. [ a ) φ a 4 a d 0, k x ) + a φ d k x, k x ) Proof. Along similar lines to those in the proof of Theorem 3., we have 3.7) f 4x) 0f x) + 4f x) Φ d x), where Φ d x) = + φ d 0, k+ x ) + φ d k ax, k x ) ] [ ) a φ a 4 a d 0, x) + a φ d x, x) + φ d 0, x) + φ d ax, x) ] for all x E. It is easy to see from 3.7) that 3.8) f 4x) 4f x) f x) 4f x)} Φ d x) for all x E. Using.3) in 3.8), we obtain 3.9) β x) β x) Φ d x) for all x E. From 3.9), we have 3.30) β x) β x) Φ d x) for all x E. Now replacing x by x and dividing by in 3.30), we obtain 3.3) β x) β x) Φ d x) for all x E. From 3.30) and 3.3), we arrive at β x) β x) β x) β x) + β x) 3.3) β x) [ Φ d x) + Φ ] d x) for all x E. In general for any positive integer n, we get β n x) β x) n n Φ d k x ) 3.33) k Φ d k x ) k J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

STABILITY OF GENERALIZED MIXED TYPE 5 for all x E. In order to prove the convergence of the sequence β n x) }, replace x by m x n and divide by m in 3.33). For any m, n > 0, we then have β n+m x) β m x) n+m m = β n m x) m β m x) n n Φ d k+m x ) k+m Φ d k+m x ) k+m } β n x) n 0 as m for all x E. Hence the sequence is a Cauchy sequence. Since E is complete, there exists a quartic mapping D : E E such that β n x) D x) = lim x E n n. Letting n in 3.33) and using.3) we see that 3.4) holds for all x E. The proof that D satisfies.0) and is unique is similar to that for Theorem 3.. The following corollary is an immediate consequence of Theorem 3.3 concerning the stability of.0). Corollary 3.4. Let ε, p be nonnegative real numbers. Suppose that an even function f : E E satisfies the inequality ε x p + y p ), 0 p < 4; ε, 3.34) Df x, y) ε x p y p, 0 p < ; ε x p y p + x p + y p}), 0 p < for all x, y E. Then there exists a unique quartic function D : E E such that λ x p p, λ, 3.35) f x) 4f x) D x) λ 3 x p p, λ 4 x p p, for all x E, where λ i i =,, 3, 4) are given in Corollary 3.. Theorem 3.5. Let φ : E E [0, ) be a function such that φ b n x, n y) φ d n x, n y) 3.3), converges 4 n n and n=0 φ b n x, n y) φ d n x, n y) 3.37) lim = 0 = lim n 4 n n n n=0 J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

K. RAVI, J.M. RASSIAS, M. ARUNKUMAR, AND R. KODANDAN for all x, y E. Suppose that an even function f : E E satisfies the inequalities 3.) and 3.3) for all x, y E. Then there exists a unique quadratic function B : E E and a unique quartic function D : E E such that 3.38) f x) B x) D x) Φ b k x ) + Φ d k x ) } 4 4 k k for all x E, where Φ b k x ) and Φ d k x ) are defined in 3.5) and 3.), respectively for all x E. Proof. By Theorems 3. and 3.3, there exists a unique quadratic function B : E E and a unique quartic function D : E E such that 3.39) f x) f x) B x) 4 Φ b k x ) 4 k and 3.40) f x) 4f x) D x) Φ d k x ) k for all x E. Now from 3.39) and 3.40), one can see that f x) + B x) D x) = f x) f x) + + B } x) f x) 4f x) + D } x) f x) f x) B x) + f x) 4f x) D x) } Φ b k x ) + Φ d k x ) } 4 4 k k for all x E. Thus we obtain 3.38) by defining B x) = B x) and D x) = D x), where Φ b k x ) and Φ d k x ) are defined in 3.5) and 3.), respectively for all X E. The following corollary is the immediate consequence of Theorem 3.5 concerning the stability of.0). Corollary 3.. Let ɛ, p be nonnegative real numbers. Suppose an even function f : E E satisfies the inequality ε x p + y p ), 0 p < ; 3.4) Df x, y) ε, ε x p y p, 0 p < ; ε x p y p + x p + y p}), 0 p < J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

STABILITY OF GENERALIZED MIXED TYPE 7 for all x, y E. Then there exists a unique quadratic function B : E E and a unique quartic function D : E E such that λ x p } + 4 p, p 3.4) f x) B x) D x) λ λ 3 x p λ 4 x p 4 p + p }, 4 p + p }, for all x E, where λ i i =,, 3, 4) are given in Corollary 3.. Theorem 3.7. Let φ a : E E [0, ) be a function such that φ a n x, n y) φ a n x, n y) 3.43) converges and lim = 0 n n n n=0 for all x, y E and let f : E E be an odd function with f0) = 0 which satisfies the inequality 3.44) Df x, y) φ a x, y) for all x, y E. Then there exists a unique additive function A : E E such that 3.45) f x) 8f x) A x) Φ a k x ) k for all x E, where the mapping Ax) and Φ a k x) are defined by 3.4) A x) = lim f n+ x ) 8f n x) } n n 3.47) Φ a k x ) = for all x E. [ ) 5 4a φ a 4 a a k x, k x ) + a φ a k+ x, k+ x ) + a φ a k+ x, k x ) + 4 a ) φ a k x, k+ x ) + φ a k x, k 3x ) + φ a k + a) x, k x ) +φ a k a) x, k x ) + φ a k + a) x, k x ) + φ a k a) x, k x )] Proof. Using the oddness of f and from 3.44), we get 3.48) f x + ay) + f x ay) a [f x + y) + f x y)] a ) f x) φ a x, y) for all x E. Replacing y by x in 3.48), we obtain 3.49) f + a) x) + f a) x) a f x) a ) f x) φa x, x) for all x E. Replacing x by x in 3.49), we get 3.50) f + a) x) + f a) x) a f 4x) a ) f x) φa x, x) for all x E. Again replacing x, y) by x, x) in 3.48), we obtain 3.5) f + a) x) + f a) x) a f 3x) a f x) a ) f x) φ a x, x) J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

8 K. RAVI, J.M. RASSIAS, M. ARUNKUMAR, AND R. KODANDAN for all x E. Replacing y by x in 3.48), we get 3.5) f + a) x) + f a) x) a f 3x) + a f x) a ) f x) φ a x, x) for all x E. Replacing y by 3x in 3.48), we obtain 3.53) f + 3a) x) + f 3a) x) a f 4x) + a f x) a ) f x) φ a x, 3x) for all x E. Replacing x, y) by + a)x, x) in 3.48), we get 3.54) f + a) x) + f x) a f + a) x) a f ax) a ) f + a) x) φ a + a) x, x) for all x E. Again replacing x, y) by a)x, x) in 3.48), we obtain 3.55) f a) x) + f x) a f a) x) + a f ax) a ) f a) x) φ a a) x, x) for all x E. Adding 3.54) and 3.55), we arrive at 3.5) f + a) x) + f a) x) + f x) a f + a) x) a f a) x) a ) f + a) x) a ) f a) x) for all x E. Replacing x, y) by + a)x, x) in 3.48), we get φ a + a) x, x) + φ a a) x, x) 3.57) f + 3a) x) + f + a) x) a f + a) x) a f ax) a ) f + a) x) φa + a) x, x) for all x E. Again replacing x, y) by a)x, x) in 3.48), we obtain 3.58) f 3a) x) + f a) x) a f a) x) + a f ax) a ) f a) x) φa a) x, x) for all x E. Adding 3.57) and 3.58), we arrive at 3.59) f + 3a) x) + f 3a) x) + f + a) x) + f a) x) a f + a) x) a f a) x) a ) f + a) x) a ) f a) x) φ a + a) x, x) + φ a a) x, x) J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

STABILITY OF GENERALIZED MIXED TYPE 9 for all x E. Now multiplying 3.49) by a ), 3.5) by a and adding 3.5) and 3.5), we have a 4 a ) f 3x) 4f x) + 5f x) = a ) f + a) x) + a ) f a) x) a a ) f x) 4 a ) f x) } + a f + a) x) + a f a) x) a 4 f 3x) a 4 f x) a a ) f x) } + f + a) x) f a) x) + a f 3x) a f x) + a ) f x) } + f + a) x) + f a) x) + f x) a f + a) x) a f a) x) a ) f + a) x) a ) f a) x)} a ) φ a x, x) + a φ a x, x) + φ a x, x) + φ a + a) x, x) + φ a a) x, x) for all x E. Hence from the above inequality, we get 3.0) f 3x) 4f x) + 5f x) [ ) a φ a 4 a a x, x) + a φ a x, x) ) +φ a x, x) + φ a + a) x, x) + φ a a) x, x)] for all x E. Now multiplying 3.50) by a, 3.5) by a ) and adding 3.49), 3.53) and 3.59), we have a 4 a ) f 4x) f 3x) f x) + f x) = f + a) x) f a) x) + a f x) + a ) f x)} + a f + a) x) + a f a) x) a 4 f 4x) a a ) f x) } + a ) f + a) x) + a ) f a) x) a a ) f 3x) + a a ) f x) 4 a ) f x) } + f + 3a) x) f 3a) x) + a f 4x) a f x) + a ) f x) } + f + 3a) x) + f 3a) x) + f + a) x) + f a) x) a f + a) x) a f a) x) a ) f + a) x) a ) f a) x) } φ a x, x) + a φ a x, x) + a ) φ a x, x) + φ a x, 3x) + φ a + a) x, x) + φ a a) x, x) for all x E. Hence from the above inequality, we get 3.) f 4x) f 3x) f x) + f x) [ φa x, x) + a φ a 4 a a x, x) + a ) φ a x, x) + φ a x, 3x) ) +φ a + a) x, x) + φ a a) x, x)] J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

0 K. RAVI, J.M. RASSIAS, M. ARUNKUMAR, AND R. KODANDAN for all x E. From 3.0) and 3.), we arrive at 3.) f 4x) 0f x) + f x) = f 3x) 8f x) + 0f x) + f 4x) f 3x) f x) + f x) f 3x) 4f x) + 5f x) + f 4x) f 3x) f x) + f x) [ ) 5 4a φ a 4 a a x, x) + a φ a x, x) + a φ a x, x) ) for all x E. From 3.), we have + 4 a ) φ a x, x) + φ a x, 3x) + φ a + a) x, x) +φ a a) x, x) +φ a + a) x, x) + φ a a) x, x)] 3.3) f 4x) 0f x) + f x) Φ a x), where Φ a x) = [ ) 5 4a φ a 4 a a x, x) + a φ a x, x) + a φ a x, x) ) + 4 a ) φ a x, x) + φ a x, 3x) + φ a + a) x, x) for all x E. It is easy to see from 3.3) +φ a a) x, x) +φ a + a) x, x) + φ a a) x, x)] 3.4) f 4x) 8f x) f x) 8f x)} Φ a x) for all x E. Define a mapping γ : E E by 3.5) γ x) = f x) 8f x) for all x E. Using 3.5) in 3.4), we obtain 3.) γ x) γ x) Φ a x) for all x E. From 3.), we have 3.7) γ x) γ x) Φ a x) for all x E. Now replacing x by x and dividing by in 3.7), we obtain 3.8) γ x) γ x) Φ a x) for all x E. From 3.7) and 3.8), we arrive at γ x) γ x) γ x) γ x) + γ x) 3.9) γ x) [ Φ a x) + Φ ] a x) for all x E. In general for any positive integer n, we get γ n x) γ x) n n Φ a k x ) 3.70) k Φ a k x ) k J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

STABILITY OF GENERALIZED MIXED TYPE for all x E. In order to prove the convergence of the sequence γ n x) }, replace x by m x n and divide by m in 3.70). Then for any m, n > 0, we have γ n+m x) γ m x) n+m m = γ n m x) m γ m x) n n Φ a k+m x ) k+m Φ a k+m x ) k+m } γ n x) n 0 as m for all x E. Hence the sequence is a Cauchy sequence. Since E is complete, there exists a additive mapping A : E E such that γ n x) A x) = lim x E n n. Letting n in 3.70) and using 3.5) we see that 3.45) holds for all x E. The proof that A satisfies.0) and is unique is similar to that of Theorem 3.. The following corollary is the immediate consequence of Theorem 3.7 concerning the stability of.0). Corollary 3.8. Let ε, p be nonnegative real numbers. Suppose that an odd function f : E E with f0) = 0 satisfies the inequality ε x p + y p ), 0 p < ; ε, 3.7) Df x, y) ε x p y p, 0 p < ; ε x p y p + x p + y p}), 0 p < for all x, y E. Then there exists a unique additive function A : E E such that λ, 3.7) f x) 8f x) A x) where λ 5 = λ 5 x p p, λ 7 x p p, λ 8 x p p, ε a 4 a 8a + p a + 4 ) + 3 p + + a) p + a) p + + a) p + a) p }, λ = ε 3a ), a 4 a ε λ 7 = 5 4a + p a + 4 p ) + 3 p + + a) p a 4 a + a) p + + a) p + a) p } J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

K. RAVI, J.M. RASSIAS, M. ARUNKUMAR, AND R. KODANDAN and λ 8 = for all x E. ε a 4 a a + p 3a + 4 ) + 3 p + + a) p + a) p + + a) p + a) p + 4 p ) +3 p + + a) p + a) p + + a) p + a) p } Theorem 3.9. Let φ c : E E [0, ) be a function such that φ c n x, n y) φ c n x, n y) 3.73) converges and lim = 0 8 n n 8 n n=0 for all x, y E and let f : E E be an odd function with f0) = 0 that satisfies the inequality 3.74) Df x, y) φ c x, y) for all x, y E. Then there exists a unique cubic function C : E E such that 3.75) f x) f x) C x) Φ c k x ) 8 8 k for all x E, where the mapping Cx) and Φ c k x) are defined by 3.7) C x) = lim f n+ x ) f n x) } n 8 n 3.77) Φ c k x ) [ = ) 5 4a φ a 4 a c k x, k x ) + a φ c k+ x, k+ x ) +a φ c k+ x, k x ) + 4 a ) φ c k x, k+ x ) + φ c k x, k 3x ) for all x E. + φ c k + a) x, k x ) +φ c k a) x, k x ) +φ c k + a) x, k x ) + φ c k a) x, k x )] Proof. Following along similar lines to those in the proof of Theorem 3.7, we have 3.78) f 4x) 0f x) + f x) Φ c x), where Φ c x) = [ ) 5 4a φ a 4 a c x, x) + a φ c x, x) + a φ c x, x) ) + 4 a ) φ c x, x) + φ c x, 3x) + φ c + a) x, x) for all x E. It is easy to see from 3.78) that + φ c a) x, x) +φ c + a) x, x) + φ c a) x, x)] 3.79) f 4x) f x) 8 f x) f x)} Φ c x) for all x E. Define a mapping δ : E E by 3.80) δ x) = f x) f x) for all x E. Using 3.80) in 3.79), we obtain 3.8) δ x) 8δ x) Φ c x) J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

STABILITY OF GENERALIZED MIXED TYPE 3 for all x E. From 3.8), we have 3.8) δ x) δ x) 8 Φ c x) 8 for all x E. Now replacing x by x and dividing by 8 in 3.8), we obtain 3.83) δ x) δ x) 8 8 Φ c x) 8 for all x E. From 3.8) and 3.83), we arrive at δ x) δ x) 8 δ x) δ x) 8 8 + δ x) 3.84) δ x) 8 [ Φ c x) + Φ ] c x) 8 8 for all x E. In general for any positive integer n, we get 3.85) δ n x) δ x) 8 n 8 8 n Φ c k x ) 8 k Φ c k x ) for all x E. In order to prove the convergence of the sequence δ n x) }, replace x by m x 8 n and divide by 8 m in 3.85). Then for any m, n > 0, we have δ n+m x) δ m x) 8 n+m 8 m = δ n m x) 8 m δ m x) 8 n n Φ c k+m x ) 8 8 k+m Φ c k+m x ) 8 8 k+m } δ n x) 8 n 8 k 0 as m for all x E. Hence the sequence is a Cauchy sequence. Since E is complete, there exists a cubic mapping C : E E such that C x) = lim n δ n x) 8 n x E. Letting n in 3.84) and using 3.80) we see that 3.75) holds for all x E. The rest of the proof, which proves that C satisfies.0) and is unique, is similar to that of Theorem 3.. The following corollary is an immediate consequence of Theorem 3.9 concerning the stability of.0). J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

4 K. RAVI, J.M. RASSIAS, M. ARUNKUMAR, AND R. KODANDAN Corollary 3.0. Let ε, p be nonnegative real numbers. Suppose that an odd function f : E E with f0) = 0 satisfies the inequality ε x p + y p ), 0 p < 3; 3.8) Df x, y) ε, ε x p y p, 0 p < 3 ; ε x p y p + x p + y p}), 0 p < 3 for all x, y E. Then there exists a unique cubic function C : E E such that 3.87) f x) 8f x) A x) λ 5 x p 8 p, λ 7, λ 7 x p 8 p, λ 8 x p 8 p, for all x E, where λ i i = 5,, 7, 8) are given in Corollary 3.8. Theorem 3.. Let φ : E E [0, ) be a function such that 3.88) and n=0 φ a n x, n y) n, n=0 φ c n x, n y) 8 n converges φ a n x, n y) φ c n x, n y) 3.89) lim = 0 = lim n n n 8 n for all x, y E. Suppose that an odd function f : E E with f0) = 0 satisfies the inequalities 3.44) and 3.74) for all x, y E. Then there exists a unique additive function A : E E and a unique cubic function C : E E such that 3.90) f x) A x) C x) Φ a k x ) + Φ c k x ) } k 8 8 k for all x E, where Φ a k x ) and Φ c k x ) are defined by 3.47) and 3.77), respectively for all x E. Proof. By Theorems 3.7 and 3.9, there exists a unique additive function A : E E and a unique cubic function C : E E such that 3.9) f x) 8f x) A x) Φ a k x ) k and 3.9) f x) f x) C x) 8 Φ c k x ) 8 k J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

STABILITY OF GENERALIZED MIXED TYPE 5 for all x E. Now from 3.9) and 3.9), one can see that f x) + A x) C x) = f x) 8f x) + + A } x) f x) f x) + C } x) f x) 8f x) A x) + f x) f x) C x) } Φ a k x ) + Φ c k x ) } k 8 8 k for all x E. Thus we obtain 3.90) by defining A x) = A x) and C x) = C x), where Φ a k x ) and Φ c k x ) are defined in 3.47) and 3.77), respectively for all x E. The following corollary is an immediate consequence of Theorem 3. concerning the stability of.0). Corollary 3.. Let ε, p be nonnegative real numbers. Suppose that an odd function f : E E with f0) = 0 satisfies the inequality ε x p + y p ), 0 p < ; ε, 3.93) Df x, y) ε x p y p, 0 p < ; ε x p y p + x p + y p}), 0 p < for all x, y E. Then there exists a unique additive function A : E E and a unique cubic function C : E E such that λ 5 x p } + p 8, p 3.94) f x) A x) C x) 4λ, λ 7 x p λ 8 x p p + 8 p }, p + 8 p }, for all x E, where λ i i = 5,, 7, 8) are given in Corollary 3.8. Theorem 3.3. Let φ : E E [0, ) be a function that satisfies 3.3), 3.37), 3.88) and 3.89) for all x, y E. Suppose that a function f : E E with f0) = 0 satisfies the inequalities 3.), 3.3), 3.44) and 3.74) for all x, y E. Then there exists a unique additive function A : E E, a unique quadratic function B : E E, a unique cubic function C : E E and a unique quartic function D : E E such that 3.95) f x) A x) B x) C x) D x) Φa x) + Φ b x) + Φ c x) + Φ } d x) for all x E, where Φ a x), Φ b x), Φ c x) and Φ d x) are defined by 3.9) Φa x) = Φ a k x ) + Φ a k x ) }, k k J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

K. RAVI, J.M. RASSIAS, M. ARUNKUMAR, AND R. KODANDAN 3.97) Φb x) = 3.98) Φc x) = 4 8 Φ b k x ) 4 k + 4 Φ c k x ) 8 k + 8 Φ b k x ) }, 4 k Φ c k x ) }, 8 k 3.99) Φd x) = Φ d k x ) + k Φ d k x ) }, k respectively for all x E. Proof. Let f e x) = f x) + f x)} for all x E. Then f e 0) = 0, f e x) = f e x). Hence Df e x, y) = Df x, y) + Df x, y) } Df x, y) + Df x, y) } φ x, y) + φ x, y)} for all x E. Hence from Theorem 3.5, there exists a unique quadratic function B : E E and a unique quartic function D : E E such that 3.00) f x) B x) D x) [ Φ b k x ) + Φ d k x ) ] 4 4 k k [ + Φ b k x ) + Φ d k x ) ]} 4 4 k k Φb x) + Φ } d x), where Φ b x) and Φ d x) are given in 3.97) and 3.99) for all x E. Again f o x) = f x) f x)} for all x E. Then f o 0) = 0, f o x) = f o x). Hence Df o x, y) = Df x, y) Df x, y) } Df x, y) + Df x, y) } φ x, y) + φ x, y)} J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

STABILITY OF GENERALIZED MIXED TYPE 7 for all x E. Hence from Theorem 3., there exists a unique additive function A : E E and a unique cubic function C : E E such that 3.0) f x) A x) C x) [ Φ a k x ) + Φ c k x ) ] k 8 8 k [ + Φ a k x ) + Φ c k x ) ]} k 8 8 k Φa x) + Φ } c x), where Φ a x) and Φ c x) are given in 3.9) and 3.98) for all x E. Since f x) = f e x) + f o x), then it follows from 3.00) and 3.0) that f x) A x) B x) C x) D x) = f e x) B x) D x)} + f o x) C x) D x)} f e x) B x) D x) + f o x) C x) D x) Φa x) + Φ b x) + Φ c x) + Φ } d x) for all x E. Hence the proof of the theorem is complete. The following corollary is an immediate consequence of Theorem 3.3 concerning the stability of.0). Corollary 3.4. Let ε, p be nonnegative real numbers. Suppose a function f : E E with f0) = 0 satisfies the inequality ε x p + y p ), 0 p < ; ε, 3.0) D f x, y) ε x p y p, 0 p < ; ε x p y p + x p + y p}), 0 p < for all x, y E. Then there exists a unique additive function A : E E, a unique quadratic function B : E E, a unique cubic function C : E E and a unique quartic function D : E E such that 3.03) f x) A x) B x) C x) D x) λ } + 4 p + λ 5 p 3 + p } ; λ + 4λ λ3 4 p + p } + λ 7 3 p + 8 p }} x p ; 8 p }} x p ; λ4 4 p + p } + λ 8 3 p + 8 p }} x p for all x E, where λ i i =,..., 8) are respectively, given in Corollaries 3. and 3.. J. Inequal. Pure and Appl. Math., 04) 009), Art. 4, 9 pp. http://jipam.vu.edu.au/

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