Function Spaces and Applications Volume 203, Article ID 67380, 8 pages http://dx.doi.org/0.55/203/67380 Research Article Solution and Stability of a General Mixed Type Cubic and Quartic Functional Equation Xiaopeng Zhao,,2 Xiuzhong Yang, 3 andchin-tzongpang 4 Department of Mathematics, Zhejiang University, Hangzhou 30027, China 2 Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan 3 College of Mathematics and Information Science, Hebei Normal University, and Hebei Key Laboratory of Computational Mathematics and Applications, Shijiazhuang 050024, China 4 Department of Information Management, Yuan Ze University, Chung-Li 32003, Taiwan Correspondence should be addressed to Chin-Tzong Pang; imctpang@saturn.yzu.edu.tw Received 23 August 203; Accepted 5 September 203 Academic Editor: Jinlu Li Copyright 203 Xiaopeng Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the following mixed type cubic and quartic functional equation λ[f(x + λy) + f(x λy)] = λ 3 [f(x + y) + f(x y)] 2λ 3 (λ + )f(y) 2λ(λ 2 )f(x) + 2(λ + )f(λy), where λ is a fixed integer. We establish the general solution of the functional equation when the integer λ =0, ±, and then, by using the fixed point alternative, we investigate the generalized Hyers-Ulam- Rassias stability for this functional equation when the integer λ 2.. Introduction In 940, Ulam []askedthefundamentalquestionforthestability for the group homomorphisms. Let (G, )be a group, and let (G 2,,d)be a metric group with the metric d(, ).Givenε>0, does there exist a δ(ε) > 0 such that if a mapping h: G G 2 satisfies the inequality d(h(x y),h(x) h(y)) <δ () for all x, y G, then there is a homomorphism H: G G 2 with d (h (x),h(x)) <ε (2) for all x G? In other words, under what conditions, does there exist a homomorphism near an approximately homomorphism? In the next year, Hyers [2] gavethefirstaffirmativeanswerto the question of Ulam for Cauchy equation in the Banach spaces. Then, Rassias [3] generalized Hyers result by considering an unbounded Cauchy difference, and this stability phenomenon is known as generalized Hyers-Ulam-Rassias stability or Hyers-Ulam-Rassias stability. During the last three decades, the stability problem for several functional equations has been extensively investigated by many mathematicians; see, for example, [4 9] and the references therein. We also refer the readers to the books [0 3]. In [4], Jun and Kim introduced the following functional equation f (2x + y) +f(2x y)=2f(x+y)+2f(x y)+2f (x). (3) It is easy to see that the function f(x) = x 3 satisfies the functional equation (3). Thus, it is natural that (3) iscalled a cubic functional equation and every solution of the cubic functional equation is said to be a cubic function. In [4], Jun and Kim established the general solution and the generalized Hyers-Ulam-Rassias stability for (3). They proved that a function f: X Ybetween real vector spaces is a solution of the functional equation (3) if and only if there exists a function G: X X X Ysuch that f(x) = G(x,x,x) for all x Xand G is symmetric for each fixed one variable and is additive for fixed two variables. In [5], Lee et al. considered the following quartic functional equation f (2x + y) + f (2x y) (4) =4[f(x+y)+f(x y)]+24f(x) 6f(y).
2 Function Spaces and Applications Since the function f(x) = x 4 satisfies the functional equation (4), the functional equation (4)iscalledaquarticfunctional equation and every solution of the quartic functional equation is said to be a quartic function. In [5], the authors solved the functional equation (4) and proved the stability forit.actually,theyobtainedthatafunctionf: X Y between real vector spaces satisfies the functional equation (4) if and only if there exists a symmetric biquadratic function B: X X Ysuch that f(x) = B(x, x) for all x X.A function f: X Ybetween real vector spaces is said to be quadratic if it satisfies the following functional equation f (x+y) +f(x y) =2f(x) +2f(y) (5) for all x, y X,andafunctionB:X X Yis said to be bi-quadratic if B is quadratic for each fixed one variable (see [8]).Thefollowingmixedtypecubicandquarticfunctional equation was introduced by Eshaghi Gordji et al. [6] f(x+2y)+f(x 2y) =4[f(x+y)+f(x y)] 24f (y) 6f (x) +3f(2y). It can be verified that the function f(x) = x 3 +x 4 satisfies the functional equation (6). In [6], the authors obtained thegeneralsolutionandthegeneralizedhyers-ulam-rassias stability of (6) in quasi-banach space. The literature on the stability of the mixed type functional equations is very rich; see [7 22]. In the present paper, we extend (6) and consider the following functional equation λ[f(x+λy)+f(x λy)] =λ 3 [f(x+y)+f(x y)] 2λ 3 (λ+) f(y) 2λ(λ 2 )f(x) +2(λ+) f(λy), where λ is a fixed integer. One can see that the functional equation (6)isaspecialcaseof(7) when we take the integer λ=2. In 2003, Radu [23] noticed that the fixed point theorem, whichwasestablishedbydiazandmargolis[24], plays an important part in solving the stability problem of functional equations. Subsequently, this method has been successfully used by many mathematicians to investigate the stability of several functional equations; see, for example, [2, 25, 26]and the references therein. In this paper, we first establish the general solution of functional equation (7) when the integer λ =0,± and f is a mapping between vector spaces. Then, by using the fixed point method, we prove the generalized Hyers-Ulam-Rassias stability of the functional equation (7) when the integer λ 2 and f is a mapping from the normed space to the Banach space. 2. Solution of the Functional Equation (7) Recall form [4, 5] that every solution of the cubic functional equation (3) and the quartic functional equation (4)is said to (6) (7) be a cubic function and a quartic function, respectively. In this section, we investigate the general solution of the mixed type cubic and quartic functional equation (7). Throughout this section, let X and Y be two real vector spaces, and we always assume that the integer λ in the functional equation (7) is different from 0,, and. Before proving our main theorem, we first give the following two lemmas. Lemma. If an odd mapping f: X Ysatisfies (7),thenf is cubic. Proof. Note that, in view of the oddness of f,wehavef(0) = 0 and f( x) = f(x) for all x X.Lettingx=0in (7), we get f(λy)=λ 3 f(y) (8) for all y X. Applying (8)to(7), we obtain f(x+λy)+f(x λy) =λ 2 [f(x+y)+f(x y)] 2(λ 2 )f(x) for all x, y X.Replacingy by x+yin (9), we get f((λ+) x+λy) f((λ ) x+λy) =λ 2 [f (2x + y) f (y)] 2 (λ 2 )f(x) (9) (0) for all x, y X.Now,ifwereplacex by λx + y in (9)anduse (8), we see that f (λx + (λ+) y)+f(λx (λ ) y) =λ 2 [f (λx + 2y) + λ 3 f (x)] 2(λ 2 )f(λx+y) () for all x, y X.Interchangingx with y in ()andusingthe oddness of f,wegettherelation f((λ+) x+λy) f((λ ) x λy) =λ 2 [f (2x + λy) + λ 3 f(y)] 2(λ 2 )f(x+λy) (2) for all x, y X. Then, subtracting (2)from(0), one has f((λ ) x λy) f((λ ) x+λy) =λ 2 [f(2x+y) f(2x+λy)] (λ 2 +λ 5 )f(y)+2(λ 2 )(f(x+λy) f(x)) (3) for all x, y X. Now,replacingy by y in (3) and noting that f is odd, we have f((λ ) x+λy) f((λ ) x λy) =λ 2 [f(2x y) f(2x λy)]+(λ 2 +λ 5 )f(y) +2(λ 2 )(f(x λy) f(x)) (4)
Function Spaces and Applications 3 for all x, y X. Adding (3)to(4)gives λ 2 [f(2x+y)+f(2x y)] +2(λ 2 )[f(x+λy)+f(x λy)] =λ 2 [f(2x+λy)+f(2x λy)]+4(λ 2 )f(x) (5) for all x, y X.Replacingx by 2x in (9)andusing(8), we get f(2x+λy)+f(2x λy) =λ 2 [f(2x+y)+f(2x y)] 6(λ 2 )f(x). (6) Applying (9)and(6)to(5)yields that f(2x+y)+f(2x y) =2[f(x+y)+f(x y)]+2f(x) (7) for all x, y X.Thus,themappingf: X Yis cubic. This completes the proof. Lemma 2. If an even mapping f: X Ysatisfies (7) for all x, y X,thenf is quartic. Proof. In view of the evenness of f, wehavef( x) = f(x) for all x X.Puttingx=y=0in (7), we get f(0) = 0.Then, let x=0in (7), we obtain f(λy)=λ 4 f(y) (8) for all y X.Combing(7) and(8) implies the following equation f(x+λy)+f(x λy) =λ 2 [f (x + y) + f (x y)] 2(λ 2 )f(x) +2λ 2 (λ 2 )f(y) (9) for all x, y X.Replacingy by x+yin (9) and note that f is even, we get f((λ+) x+λy)+f((λ ) x+λy) =λ 2 [f (2x + y) + f (y)] 2 (λ 2 )f(x) +2λ 2 (λ 2 )f(x+y) (20) for all x, y X.Replacingx by λx+yin (9)andusing(8), we get f (λx + (λ+) y)+f(λx (λ ) y) =λ 2 [f (λx + 2y) + λ 4 f (x)] 2(λ 2 )f(λx+y)+2λ 2 (λ 2 )f(y) (2) for all x, y X. Interchanging the roles of x and y in (2), we obtain f((λ+) x+λy)+f((λ ) x λy) =λ 2 [f (2x + λy) + λ 4 f(y)] 2(λ 2 )f(x+λy)+2λ 2 (λ 2 )f(x) for all x, y X.Ifwesubtract(22)from(20), we obtain f((λ ) x+λy) f((λ ) x λy) =λ 2 [f (2x + y) f (2x + λy)] +2(λ 2 )f(x+λy)+(λ 2 λ 6 )f(y) (22) +2λ 2 (λ 2 )f(x+y) 2(λ 2 +)(λ 2 )f(x) (23) for all x, y X.Replacingy by y in (23), we get f((λ ) x λy) f((λ ) x+λy) =λ 2 [f (2x y) f (2x λy)] +2(λ 2 )f(x λy)+(λ 2 λ 6 )f(y) +2λ 2 (λ 2 )f(x y) 2(λ 2 +)(λ 2 )f(x) (24) for all x, y X.Ifweadd(23)to(24), we have λ 2 [f(2x+y)+f(2x y)] λ 2 [f (2x + λy) + f (2x λy)] +2(λ 2 )[f(x+λy)+f(x λy)] +2λ 2 (λ 2 )[f(x+y)+f(x y)] =4(λ 2 +)(λ 2 )f(x) +2(λ 6 λ 2 )f(y) (25) for all x, y X.Replacingx by 2x in (9)andusing(8), we get f(2x+λy)+f(2x λy) =λ 2 [f (2x + y) + f (2x y)] 32(λ 2 )f(x) +2λ 2 (λ 2 )f(y) (26) for all x, y X. Applying (9)and(26)to(25), we obtain that f (2x + y) + f (2x y) =4[f(x+y)+f(x y)]+24f(x) 6f(y) (27) for all x, y X. Therefore, the mapping f: X Yis quartic and the proof is complete. Now,wearereadytofindoutthegeneralsolutionof(7).
4 Function Spaces and Applications Theorem 3. Amappingf: X Ysatisfies (7) for all x, y X if and only if there exist a symmetric multiadditive mapping G:X X X Yand a symmetric bi-quadratic mapping B: X X Ysuch that f(x) = G(x, x, x) + B(x, x) for all x X. Proof. First, we assume that there exist a symmetric multiadditive mapping G: X X X Yand a symmetric bi-quadratic mapping B: X X Ysuch that f(x) = G(x, x, x) + B(x, x) for all x X. We need to show that the function f satisfies (7). By a simple computation, one can obtain that the function x G(x,x,x)satisfies (7). Thus, to show that the function f satisfies (7), we only need to show that the function x B(x, x) also satisfies (7); namely, λ[b(x+λy,x+λy)+b(x λy,x λy)] =λ 3 [B(x+y,x+y)+B(x y,x y)] 2λ 3 (λ+) B (y, y) 2λ(λ 2 )B(x, x) +2(λ+) B(λy,λy) (28) for all x, y X. SinceB: X X Yis a bi-quadratic mapping,itcanbeverifiedthat B(λx,λy)=λ 4 B(x,y) (29) for all integer λ and all x, y X.Then,(28)becomes B(x+λy,x+λy)+B(x λy,x λy) =λ 2 [B(x+y,x+y)+B(x y,x y)] 2(λ 2 )B(x, x) +2λ 2 (λ 2 )B(y,y). (30) To establish (28), it suffices to show (30). Note that if (30) holds for some integer λ, thensodoes λ. Thus,inthe following,we will show that (30) holds for all positive integers λ with λ 2and all x, y X.Todothis,weuseinductionon λ.fixanyx, y X.Inthecasewhenλ=2,wehave B(x+2y,x+2y)+B(x 2y,x 2y) =[B(x+y+y,x+2y)+B(x+y y,x+2y)] +[B(x y+y,x 2y)+B(x y y,x 2y)] [B(x,x+2y)+B(x,x 2y)] =2[B(x+y,x+2y)+B(y,x+2y)] +2[B(x y,x 2y)+B(y,x 2y)] [2B(x, x) +2B(x,2y)] =2[B(x+y,x+y+y)+B(x+y,x+y y)] +2[B(x y,x y y)+b(x y,x y+y)] 2[B(x+y,x)+B(x y,x)] =4[B(x+y,x+y)+B(x+y,y)] +4[B(x y,x y)+b(x y,y)] 4[B(x, x) +B(y,x)]+4[B(y,x)+B(y,2y)] [2B(x, x) +8B(x,y)] =4[B(x+y,x+y)+B(x y,x y)] +4[B(x+y,y)+B(x y,y)] 6B(x, x) 8B (x, y) + 6B (y, y) =4[B(x+y,x+y)+B(x y,x y)] + 8 [B (x, y) + B (y, y)] 6B(x, x) 8B (x, y) + 6B (y, y) =4[B(x+y,x+y)+B(x y,x y)] 6B(x, x) + 24B (y, y). (3) So (30)istrueforλ=2. Here, we have used the bi-quadratic property of the function B and (29). Now, assume that (30)is true for all positive integers that are less than or equal to some integer λ(> 2).Then, B(x+(λ+) y, x + (λ+) y) +B(x (λ+) y, x (λ+) y) =[B(x+y+λy,x+y+λy) +B(x+y λy,x+y λy)] +[B(x y+λy,x y+λy) +B(x y λy,x y λy)] [B(x+(λ ) y, x + (λ ) y) +B (x (λ ) y, x (λ ) y)] =λ 2 [B(x+2y,x+2y)+B(x, x)] 2(λ 2 )B(x+y,x+y)+2λ 2 (λ 2 )B(y,y) +λ 2 [B (x, x) +B(x 2y,x 2y)] 2(n 2 )B(x y,x y)+2λ 2 (λ 2 )B(y,y) {(λ ) 2 [B(x+y,x+y)+B(x y,x y)] 2((λ ) 2 )B(x, x) +2[B(y,x+2y)+B(y,x 2y)] [2B(x, x) +8B(x,y)] +2(λ ) 2 ((λ ) 2 )B(y,y)}. (32)
Function Spaces and Applications 5 Applying (3)to(32), onecan obtainthat B(x+(λ+) y, x + (λ+) y) +B(x (λ+) y, x (λ+) y) = (λ+) 2 [B(x+y,x+y)+B(x y,x y)] 2((λ+) 2 )B(x, x) +2(λ+) 2 ((λ+) 2 )B(y,y). (33) This means that (30)istrueforλ+,andwehaveshowedthat the function x B(x, x) satisfies (7). Therefore, the mapping f: X Ysatisfies (7). Conversely, we decompose f into the odd part and the even part by putting f (x) f( x) f (x) +f( x) f o (x) =, f 2 e (x) = 2 (34) for all x X.Then,f(x) = f o (x) + f e (x) for all x X.Itis easy to show that the mappings f o and f e satisfy (7). Hence, it follows from Lemmas and 2 that the function f o is cubic and f e is quartic, respectively. Therefore, there exist a symmetric multi-additive mapping G: X X X Ysuch that f o (x) = G(x, x, x) for all x X(see [4,Theorem2.])anda symmetric bi-quadratic mapping B: X X Ysuch that f e (x) = B(x, x) for all x X(see [5, Theorem2.]).Hence, we get f(x) = G(x, x, x) + B(x, x) for all x X.Theproofis complete. 3. Generalized Hyers-Ulam-Rassias Stability of the Functional Equation (7) In this section, we will investigate the stability of the functional equation (7)byusingthefixedpointalternative.Throughout this section, let X be a real normed space and Y be a real Banachspace,andwealwaysassumethattheintegerλused in thesectionisgreaterthanorequalto2.forconvenience,we use the following abbreviation for a given function f: X Y: Df(x,y):=λ[f(x+λy)+f(x λy)] λ 3 [f(x+y)+f(x y)] +2λ 3 (λ+) f(y)+2λ(λ 2 )f(x) 2(λ+) f(λy) for all x, y X. LetusrecallthefollowingresultbyDiazandMargolis. (35) Proposition 4 (see [24]). Let (E, d) be a complete generalized metricspace(i.e.,oneforwhichd may assume infinite value), and let J: E Ebe a strictly contractive mapping with Lipschitz constant L<;thatis, d (Jx, Jy) Ld(x, y) x, y X. (36) Then, for each fixed element x 0 E,either d(j n x 0,J n+ x 0 )=+ (37) for all nonnegative integers n or there exists a non-negative integer n 0 such that (a) d(j n x 0,J n+ x 0 )<+ for all n n 0 ; (b) the sequence {J n x 0 } converges to a fixed point x of J; (c) x istheuniquefixedpointofj in the set F:= {x E d(j n 0 x 0,x)<+ }; (d) d(x, x ) (/( L))d(x, Jx) for all x F. Lemma 5. Let f: X Ybe an odd function for which there exists a function φ:x X [0,+ )such that Df (x, y) φ(x, y) (38) φ(λx,λy) λ 3 Lφ (x, y) (39) for all x, y X, then there exists a unique cubic mapping C: X Ysuch that f (x) C(x) L φ (x) (40) L for all x X,whereφ(x):= (/2(λ + ))φ(0, x/λ). Proof. It follows from (39)that φ(λ r x, λ r y) lim r λ 3r =0 (4) for all x, y X. Lettingx=0in (38) and replacing y by x, we have λ3 f (x) f(λx) φ(λx) (42) for all x X.By(39), we have φ(λx) λ 3 Lφ(x) for all x X. This,together with (42), implies that f (x) λ 3 f (λx) Lφ(x) (43) for all x X.LetΩbetheset of all odd mappings g: X Y. We introduce the generalized metric on Ω: d(g,h) := inf {K [0, ) g (x) h(x) Kφ(x), x X}. (44) It is easy to show that (Ω, d) is complete. Now, we define a function T: Ω Ωby Tg (x):= g (λx) (45) λ3
6 Function Spaces and Applications for all g Ωand all x X.Notethat,forallg, h Ω, d(g,h) K g (x) h(x) Kφ(x), x X λ 3 g (λx) λ 3 h (λx) Kφ (λx), λ3 λ 3 g (λx) λ 3 h (λx) LKφ(x), d(tg,th) LK. Hence, we obtain that x X x X (46) d (Tg, Th) Ld(g, h) (47) for all g, h Ω;thatis,Tisastrictly contractive mapping of Ω with Lipschitz constant L.Itfollowsfrom(43)thatd(Tf, f) L. Therefore, according to Proposition 4,thesequence{T r f} converges to a fixed point C of T;thatis, f(λ r x) C: X Y, C(x) = lim r (Tr f) (x) = lim r λ 3r (48) and C(λx) = λ 3 C(x) for all x X.AlsoC is the unique fixed point of T in the set Δ = {g Ω d(f, g) < } and d(f,c) L d(tf,f) L L, (49) which yields the inequality (40). It follows from the definition of C,(38), and (4)that DC (x, y) = lim Df (λr x, λ r y) φ(λ r x, λ r y) r λ 3r lim r λ 3r =0 (50) for all x, y X; that is,the mapping C: X Ysatisfies (7). Since f is odd, C is odd. Therefore, Lemma guarantees that C is cubic. Finally, it remains to prove the uniqueness of C. LetS: X Ybe another cubic function satisfying (40). Since d(f, S) L/( L) and S is cubic, we get S Δand TS(x) = (/λ 3 )S(λx) = S(x) for all x X;thatis,S is a fixed point of T.SinceC is the unique fixed point of T in Δ,it follows that S=C. Lemma 6. Let f: X Ybe an odd function for which there exists a function φ: X X [0,+ )such that Df (x, y) φ(x, y) (5) λ 3 φ (x, y) Lφ(λx, λy) (52) for all x, y X, then there exists a unique cubic mapping C: X Ysuch that f (x) C(x) ψ (x) (53) L for all x X,whereψ(x):= (/2(λ + ))φ(0, x/λ). Proof. It follows from (52)that lim r λ3r φ( x λ r, y )=0 (54) λr for all x, y X.Lettingx=0in (5) and replacing y by x,we have λ3 f( x λ ) f(x) ψ(x) (55) for all x X.WeintroducethesamedefinitionsforΩand d as in the proof of Lemma 5 (by replacing φ by ψ)suchthat(ω, d) becomes a generalized complete metric space. Let T: Ω Ω bethemappingdefinedby Tg (x):= λ 3 g( x λ ) (56) for all g Ωand all x X.Onecanshowthatd(Tg, Th) Ld(g, h) for all g, h Ω. It follows from (55)thatd(Tf, f). Due to Proposition 4,thesequence{T r f} converges to a fixed point C of T;thatis, C: X Y, C(x) = lim r (Tr f) (x) = lim r λ3r f( x λ r ) (57) and C(λx) = λ 3 C(x) for all x X.Also, d (f, C) L d (Tf, f) L, (58) which yields the inequality (53). The rest of the proof is similar to the proof of Lemma 5, and we omit the details. Similarly, we can prove the following two lemmas on even functions. Lemma 7. Let f: X Ybe an even function with f(0) = 0 for which there exists a function φ: X X [0,+ )such that Df (x, y) φ(x, y) (59) φ(λx,λy) λ 4 Lφ (x, y) (60) for all x, y X, then there exists a unique quartic mapping Q: X Ysuch that f (x) Q(x) L φ (x) (6) L for all x X,whereφ(x):= (/2)φ(0, x/λ). Lemma 8. Let f: X Ybe an even function with f(0) = 0 for which there exists a function φ: X X [0,+ )such that Df (x, y) φ(x, y) (62)
Function Spaces and Applications 7 λ 4 φ (x, y) Lφ (λx, λy) (63) for all x, y X, then there exists a unique quartic mapping Q: X Ysuch that f (x) Q(x) ψ (x) (64) L for all x X,whereψ(x):= (/2)φ(0, x/λ). Now, we are ready to give our main theorems in this section. Theorem 9. Let f: X Ybe a function with f(0) = 0 for which there exists a function φ: X X [0,+ )such that Df (x, y) φ(x, y) (65) φ(λx,λy) λ 3 Lφ (x, y) (66) for all x, y X, thenthereexistauniquecubicmappingc: X Yand a unique quartic mapping Q: X Ysuch that f (x) C(x) Q(x) for all x X,where φ o (x):= L L [φ o (x) +φ e (x)] (67) 2 (λ+) Φ(0,x λ ), φ e (x):= 2 Φ(0,x λ ), Φ(x,y):= 2 [φ (x, y) + φ ( x, y)]. (68) Proof. Let f o and f e denotetheoddandtheevenpartoff, respectively. Then, it can be verified from (65) that Df o (x, y) Φ(x,y), Df e (x, y) Φ(x,y) (69) for all x, y X.Moreover,by(66), it is easy to compute that Φ (λx, λy) λ 3 LΦ (x, y) (70) for all x, y X. Thus, by applying Lemmas 5 and 7, onecan obtain that there exist a unique cubic mapping C: X Y andauniquequarticmappingq: X Ysuch that f o (x) C(x) L L φ o (x), (7) f e (x) Q(x) L L φ e (x) (72) for all x X,whereφ o (x) := (/2(λ + ))Φ(0, x/λ) and φ e (x) := (/2)Φ(0, x/λ). Moreover, combining (7) and(72) yields the inequality (67). The proof is complete. Theorem 0. Let f: X Ybe a function with f(0) = 0 for which there exists a function φ: X X [0,+ )such that Df (x, y) φ(x, y) (73) λ 4 φ (x, y) Lφ(λx, λy) (74) for all x, y X, then there exist a unique cubic mapping C: X Yand a unique quartic mapping Q: X Ysuch that f (x) C(x) Q(x) L [ψ o (x) +ψ e (x)] (75) for all x X,where ψ o (x):= 2 (λ+) Ψ(0,x λ ), ψ e (x):= 2 Ψ(0,x λ ), Ψ(x,y):= 2 [φ (x, y) + φ ( x, y)]. (76) Proof. Similar to the proof of Theorem 9, theresultfollows from Lemmas 6 and 8. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References [] S. M. Ulam, Problems in Modern Mathematics, chapter6,john Wiley & Sons, New York, NY, USA, 964. [2] D. H. Hyers, On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America,vol.27,pp.222 224,94. [3] Th.M.Rassias, OnthestabilityofthelinearmappinginBanach spaces, Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297 300, 978. [4] S.-M. Jung, Hyers-Ulam-Rassias stability of Jensen s equation and its application, Proceedings of the American Mathematical Society,vol.26,no.,pp.337 343,998. [5] K.-W. Jun, H.-M. Kim, and I.-S. Chang, On the Hyers-Ulam stability of an Euler-Lagrange type cubic functional equation, Computational Analysis and Applications,vol.7,no., pp.2 33,2005. [6] K.W.Jun,S.B.Lee,andW.G.Park, Solutionandstabilityof a cubic functional equation, Acta Mathematica Sinica, vol.26, no.7,pp.255 262,200. [7] G. H. Kim, A stability of the generalized sine functional equations, Mathematical Analysis and Applications, vol. 33, no. 2, pp. 886 894, 2007. [8] W.-G. Park and J.-H. Bae, On a bi-quadratic functional equation and its stability, Nonlinear Analysis. Theory, Methods &Applications,vol.62,no.4,pp.643 654,2005. [9] J. Sikorska, Generalized stability of the Cauchy and the Jensen functional equations on spheres, Mathematical Analysis and Applications,vol.345,no.2,pp.650 660,2008.
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