UNIQUENESS THEOREMS ON FUNCTIONAL INEQUALITIES CONCERNING CUBIC QUADRATIC ADDITIVE EQUATION

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1 Journal of Mathematical Inequalities Volume, Number 08, 43 6 doi:0.753/jmi UNIQUENESS THEOREMS ON FUNCTIONAL INEQUALITIES CONCERNING CUBIC QUADRATIC ADDITIVE EQUATION YANG-HI LEE, SOON-MO JUNG AND MICHAEL TH. RASSIAS Communicated by A. Gilányi Abstract. We prove uniqueness theorems concerning the functional inequalities in connection with an n-dimensional cubic-quadratic-additive equation m c i f a i +a i + +a in n = 0 i= by applying the direct method.. Introduction Let V and W be real vector spaces. For a given mapping f : V W,wedefine Af,y := f + y f f y, Qf,y := f + y+ f y f f y, Cf,y := f + y 3 f + y f y+3f 6 f y for all,y V. A mapping f : V W is called an additive mapping, a quadratic mapping, or a cubic mapping provided f satisfies the functional equation Af, y = 0forall,y V, Qf,y =0forall,y V,orCf,y =0forall,y V, respectively. We note that the mappings g,h,k : R R given by g =a, h = a,andk =a 3 are solutions of Ag,y =0, Qh,y =0, and Ck,y =0, respectively. A mapping f : V W is called a cubic-quadratic-additive mapping if and only if f is represented by the sum of an additive mapping, a quadratic mapping, and a cubic mapping. A functional equation is called a cubic-quadratic-additive functional equation provided that each of its solutions is a cubic-quadratic-additive mapping and every cubic-quadratic-additivemapping is also a solution of that equation. The mapping f : R R given by f =a 3 + b + c is a solution of the cubic-quadratic-additive functional equation. For the study of functional inequalities concerning the cubic-quadratic-additive equations and a broad variety of other types of functional inequalities, the reader is referred to [,, 3, 4, 5, 6, 7, 8, 9, 0,,, 3, 5, 6, 7, 8]. Mathematics subject classification 00: 39B8, 39B5. Keywords and phrases: Functional inequality, functional equation, generalized Hyers-Ulam stability, n-dimensional cubic-quadratic-additive functional equation, direct method. c D l,zagreb Paper JMI

2 44 Y.-H. L EE, S.-M.JUNG AND M. TH. RASSIAS Throughout this paper, let V and W be real vector spaces, X a real normed space, Y a real Banach space, and let N 0 denote the set of all nonnegative integers. For a given mapping f : V W,wedefine Df : V n W by Df,,..., n := m i= c i f a i + a i + + a in n. for all,,..., n V,wherem is a positive integer and c i,a ij are real constants. In this paper, we prove uniqueness theorems that can be easily applied to the investigation of functional inequalities concerning a large class of functional equations of the form Df,,..., n =0, which includes the cubic-quadratic-additive functional equation as a special case. This theorem is particularly useful for proving the Hyers-Ulam stability of a variety of functional equations.. Preliminaries For a given mapping f : V W, we use the following abbreviations: f o f o := f f, f e := := a3 f o f o a a 3 a f + f,, f o := af o f o a a 3 a for all V. We will now introduce a lemma that was proved in [4, Corollary ]. LEMMA.. Let k > be a real constant, let φ : V \{0} [0, be a function satisfying either Φ := k i φki <. for all V \{0} or Φ := k 3i φ k i <. for all V \{0}, and let f : V Y be an arbitrarily given mapping. If there eists a mapping F : V Y satisfying for all V \{0} and f F Φ.3 F o k=kf o, F e k=k F e, F o k=k 3 F o.4 for all V, then F is a unique mapping satisfying.3 and.4.

3 UNIQUENESS THEOREMS ON FUNCTIONAL INEQUALITIES 45 We introduce lemmas that were proved in [4, Corollary 3]. LEMMA.. Let k > be a real number, let φ,ψ : V \{0} [0, be functions satisfying each of the following conditions k i ψ k i <, k i φ k i <, Φ := k i φ k i <, Ψ := k i ψ k i < for all V \{0}, and let f : V Y be an arbitrarily given mapping. If there eists a mapping F : V Y satisfying the inequality f F Φ+ Ψ.5 for all V \{0} and the conditions in.4 for all V, then F is a unique mapping satisfying the conditions.4 for all V and the inequality.5 for all V \{0}. LEMMA.3. Let k > be a real number, let φ,ψ : V \{0} [0, be functions satisfying each of the following conditions k i ψ k i <, k 3i φ k i <, Φ := k i φ k i <, Ψ := k 3i ψ k i < for all V \{0}, and let f : V Y be an arbitrarily given mapping. If there eists a mapping F : V Y satisfying the inequality f F Φ+ Ψ.6 for all V \{0} and the conditions in.4 for all V, then F is a unique mapping satisfying the conditions.4 for all V and the inequality.6 for all V \{0}. 3. Main results In the following four theorems, we prove that there eists only one eact solution near every approimate solution to Df,,..., n =0. THEOREM 3.. Let a be a real constant with a {,0,}, letnbeafied integer greater than,let μ,ν : V \{0} [0, be functions satisfying the conditions μa i 3i < and μa i i < and νa i 3i < when <, νa i i < when > 3.

4 46 Y.-H. L EE, S.-M.JUNG AND M. TH. RASSIAS for all V \{0}, and let ϕ : V \{0} n [0, be a function satisfying the condition ϕa i,a i,...,a i n 3i < when <, 3. ϕa i,a i,...,a i n i < when > for all,,..., n V \{0}. If apping f : V Y satisfies f 0=0, f e a a f e μ and f o a a + a 3 f o a+a 4 f o ν 3.3 for all V \{0}, and if f moreover satisfies Df,,..., n ϕ,,..., n 3.4 for all,,..., n V \{0}, then there eists a unique mapping F : V Y such that for all,,..., n V \{0}, and DF,,..., n =0 3.5 F e a=a F e, F o a=af o, F o a=a 3 F o 3.6 for all V, and such that for all V \{0}. f F μa i a i+ + ai+ νa i 3i Proof. First, we define A := { f : V Y f 0=0} and apping J m : A A by J m f := f o a 3m + f e a m + f o for V and m N 0. It follows from 3.3that J m f J m+l f J i f J i+ f = f o af o a i a 3 aa 3i f oa i+ af o a 3 aa 3i+3 + f ea i a i f e a i+ f o a 3 f o a i a 3 aa i + f oa i+ a 3 f o a 3 a 3.8

5 UNIQUENESS THEOREMS ON FUNCTIONAL INEQUALITIES 47 f e a f e a i a i+ + f oa a i a + a 3 f o +a 4 f o a i a 3 aa 3i+3 + f oa a i a + a 3 f o +a 4 f o a i a 3 a μa i a i+ + ai+ νa i 3i+3 for all V \{0}. In view of 3. and3.8, the sequence {J m f } is a Cauchy sequence for all V \{0}. SinceY is complete and f 0=0, the sequence {J m f } converges for all V. Hence, we can define apping F : V Y by F := lim J f o m f = lim m m a 3m + f e a m + f o for all V. We easily obtain from the definition of F that F o a = F oa a 3 F o a a 3 a fo +3 af o + = lim m a 3m a 3 a f o+3 a 3 f o + a 3 a fo ++ af o ++ + lim m+ a3 a 3m+ a 3 a f o++ a 3 f o ++ + a 3 a fo +3 a 3 f o + = lim m + a 3 a fo + a a 3 f o + a = lim m + a 3 a = lim m+ a = af o, fo ++ a 3 f o a 3 a F e a = Fa+F a = lim m f + + f + a m

6 48 Y.-H. L EE, S.-M.JUNG AND M. TH. RASSIAS = a f + + f + lim m+ a m+ = a F e, F o a = Fa afa a 3 a fo +3 af o + = lim m a 3m a 3 a f o+3 a 3 f o + a 3 a lim a fo ++ af o ++ m+ a 3m+ a 3 a f o++ a 3 f o ++ + a 3 a fo + a af o + a = lim m a 3m+3 a 3 a = a 3 lim m+ = a 3 F o for all V, and by. and3., we get fo ++ af o ++ a 3m++3 a 3 a DF,,..., n Df o +,...,+ n ad fo,..., n = lim m a 3m a 3 a + Df e,..., n a m Df o +,...,+ n a 3 Df o,..., n a 3 a lim m m + m+3 + a 3m ϕ e,..., n + m + 3m ϕ e +,...,+ n = 0 for all,,..., n V \{0},whereϕ e,..., n := ϕ,..., n +ϕ,..., n ; i.e., DF,,..., n =0 for all,,..., n V \{0}. Moreover, if we put m = 0andletl in 3.8, then we obtain the inequality 3.7.

7 UNIQUENESS THEOREMS ON FUNCTIONAL INEQUALITIES 49 Notice that the equalities F o =F o, F o F e = F e, F e = F o, = F e, 3.9 F o = 3 F o, F o = F o 3 are true in view of 3.6. Therefore, the equalities in.4 hold, for all V, with k = if > ork = if <. When >, in view of Lemma., there eists a unique mapping F : V Y satisfying the equalities in 3.6 and the inequality 3.7, since the inequality f F μa i a i+ + ai+ νa i 3i+3 μa i i + νai i φk i k i holds for all V \{0},wherewesetk:= and φ := μ+μ + ν+ν. a 3 a When <, in view of Lemma., there eists a unique mapping F : V Y satisfying the equalities in 3.6 and the inequality 3.7, since the inequality f F μa i a i+ + ai+ νa i 3i+3 μa i 3i+3 + νa i 3i+3 k 3i φ holds for all V \{0},wherek := and φ := μ+μ + ν+ν. 3 a 3 a 3 In the following theorem, we assume that μ, ν and ϕ satisfy other conditions than those of Theorem 3. and we prove that there eists a unique eact solution near every approimate solution to Df,,..., n =0. THEOREM 3.. Assume that a is a real constant with a {,0,}. Letnbe a fied integer greater than, letμ,ν : V \{0} [0, be functions satisfying the k i

8 50 Y.-H. L EE, S.-M.JUNG AND M. TH. RASSIAS conditions i μ a i < and 3i μ a i < and i ν a i < when <, 3i ν a i < when > 3.0 for all V \{0}, and let ϕ : V \{0} n [0, be a function satisfying the condition i ϕ a i, a i,..., n a i < when <, 3i ϕ a i, 3. a i,..., n a i < when > for all,,..., n V \{0}. If apping f : V Y satisfies f 0=0, the inequalities in 3.3 for all V \{0}, and 3.4 for all,,..., n V \{0}, then there eists a unique mapping F : V Y satisfying 3.5 for all,,..., n V \{0} and 3.6 for all V, and such that f F for all V \{0}. a i μ + a3i+3 a 3 ν a a i+ Proof. First, we define the mappings J m f : V Y by J m f := a 3m+3 f o + + a m f e + + f o + for all V and m N 0. It follows from 3.3 that J m f J m+l f a i f e a + a3i+3 a f e f o a a i+ f o a a + a 3 f o a a i+ a i μ + a3i+3 a 3 ν a for all V \{0}. a i+ a + a 3 f a i+ + a 4 f a a i+ a i a 4 f o a i+

9 UNIQUENESS THEOREMS ON FUNCTIONAL INEQUALITIES 5 On account of 3.0and3.3, the sequence {J m f } is a Cauchy sequence for all V \{0}. SinceY is complete and f 0=0, the sequence {J m f } converges for all V. Hence, we can define apping F : V Y by F := lim a 3m+3 f o m + + a m f e + + f o + for all V. Moreover, if we put m = 0andletl in 3.3, we obtain the inequality 3.. In view of the definition of F,3.4, 3., and DF,,..., n = lim m a m Df e,..., n + am+4 a 3m+4 a 3 a + a3m+3 + a 3 Df o a,..., n n Df o,..., am+ + for all,,..., n V \{0}, we get the equalities in 3.6forall V and we further obtain DF,,..., n =0forall,,..., n V \{0}. We notice that the equalities in 3.9 hold in view of 3.6. Therefore, the equalities in.4 hold, for all V, with k = if > ork = if <. When >, according to Lemma., there eists a unique mapping F : V Y satisfying the equalities in 3.6 and the inequality 3., since the inequality f F a i μ = a μ i k 3i φ k i + a3i+3 + μ + ν a i+ a 3i+3 ν a i+ + ν a i+ holds for all V \{0}, wherek := and φ := μ a + μ a + a 3 a 3 a ν a + a 3 a 3 a ν a. When <, according to Lemma., there eists a unique mapping F : V Y satisfying the equalities in 3.6 and the inequality 3., since the inequality f F a i μ + a3i+3 a 3 ν a a i+

10 5 Y.-H. L EE, S.-M.JUNG AND M. TH. RASSIAS a φk i k i μ + μ + ai ν a i+ + ν a i+ holds for all V \{0}, wherek := and φ := μ a + μ a + a 3 a ν a + a 3 a ν a. We assume that μ, ν and ϕ satisfy different conditions from those of Theorems 3. and 3. and prove that there eists a unique eact solution near every approimate solution to Df,,..., n =0. THEOREM 3.3. Let a be a real constant with a {,0,}, letnbeafied integer greater than, letμ : V \{0} [0, be a function satisfying the conditions μa i i < when >, i μ a i < when < 3.4 for all V \{0},let ν : V \{0} [0, be a function satisfying the conditions νa i i < and i ν a i < when >, νa i 3.5 i < and i ν a i < when < for all V \{0}, and let ϕ : V \{0} n [0, be a function satisfying the conditions ϕa i,a i,...,a i n i < and i ϕ a i, a i,..., n a i < when >, ϕa i,a i,...,a i n i < and when < i ϕ a i, a i,..., n a i < 3.6 for all,,..., n V \{0}. If apping f : V Y satisfies f 0 =0 and the inequalities in 3.3 for all V \{0} and 3.4 for all,,..., n V \{0}, then there eists a unique mapping F : V Y satisfying the equality 3.5 for all

11 UNIQUENESS THEOREMS ON FUNCTIONAL INEQUALITIES 53,,..., n V \{0}, the equalities in 3.6 for all V, and f F μa i a i+ + when >, i μ + when < νa i 3i+3 + i ν νa i i+ + 3i ν 3.7 for all V \{0}. Proof. We will divide the proof of this theorem into two cases, the case for > and the other case for <. Case. Assume that >. We define a set A := { f : V Y f 0=0} and a mapping J m : A A by J m f := f o a 3m + f e a m + f o for all V and m N 0. It follows from 3.3 that J m f J m+l f f e a f e a i a i+ + f oa a i a + a 3 f o +a 4 f a i a 3i a a i a f o a + a 3 f o + a 4 f o μa i a i+ + νa i 3i+3 + i ν for all V \{0}. In view of 3.4, 3.5, and 3.8, the sequence {J m f } is a Cauchy sequence for all V \{0}. SinceY is complete and f 0 =0, the sequence {J m f } converges for all V. Hence, we can define apping F : V Y by F := lim m f o a 3m + f e a m + f o for all V. Moreover, if we put m = 0andletl in 3.8, we obtain the first inequality of 3.7.

12 54 Y.-H. L EE, S.-M.JUNG AND M. TH. RASSIAS Using the definition of F,3.4, 3.6, and DF,,..., n Dfo +,...,+ n ad fo,..., n = lim m a 3m a 3 a + Df e,..., n am a 3 a a m a Df o,...,a n a 3 Df o,..., n for all,,..., n V \{0}, we get the equalities in 3.6forall V and we further get DF,,..., n =0forall,,..., n V \{0}. We notice that the equalities in.4 are true in view of 3.6, where k =. Using Lemma., we conclude that there eists a unique mapping F : V Y satisfying the equalities in 3.6 andthefirst inequality in 3.7, since the inequality f F μa i a i+ + νa i 3i+3 + i ν μa i a i + νa i i + i ν ψk i + k i φ holds for all V \{0}, wherek :=, φ := ν a μ + ν+ν a 3 a 3. k i k i +ν a a 3 a,andψ := μ+ Case. We now consider the case of < and define apping J m : A A by J m f := a 3m f o + a m f e + f o for all V and n N 0. It follows from 3.3that J m f J m+l f a i f e a + a3i a f e f o a i μ + a + a 3 f o a + a 4 f o + f oa a i a + a 3 f o +a 4 f o a i νa i i+ + 3i ν 3.9

13 UNIQUENESS THEOREMS ON FUNCTIONAL INEQUALITIES 55 for all V \{0}. On account of 3.4, 3.5, and 3.9, the sequence {J m f } is a Cauchy sequence for all V \{0}. SinceY is complete and f 0=0, the sequence {J m f } converges for all V. Hence, we can define apping F : V Y by F := lim m a 3m f o + a m f e + f o for all V. Moreover, if we put m = 0andletl in 3.9, we obtain the second inequality in 3.7. By the definition of F,3.4, 3.6, and DF,,..., n = lim a m Df e m,..., n + a3m a 3 a Df n o,..., am a3m+ a 3 a Df o,..., n Df o +,...,+ n a 3 Df o,..., n a 3 a for all,,..., n V \{0}, we get the equalities in 3.6forall V and we moreover obtain DF,,..., n =0forall,,..., n V \{0}. We remark that the equalities in.4 hold by considering 3.6 with k =. Using Lemma., we conclude that there eists a unique mapping F : V Y satisfying the equalities in 3.6 and the second inequality in 3.7, since the inequality f F i μ + νa i = νa i +ν a i i+ + i + i a 3 ν a ψk i + k i φ k i k i i+ + 3i ν μ + μ + ν holds for all V \{0},wherek:=, φ := a a ν 3 a + ν a + μ a + μ a. a 3 a ν+ν,andψ := Suppose μ, ν and ϕ satisfy other conditions from those of preceding three theorems. In the following theorem, we prove that there eists a unique eact solution near every approimate solution to Df,,..., n =0.

14 56 Y.-H. L EE, S.-M.JUNG AND M. TH. RASSIAS THEOREM 3.4. Suppose a is a real constant with a {,0,}. Letnbeafied integer greater than, letμ : V \{0} [0, be a function satisfying the conditions i μ a i < when >, 3.0 μa i i < when < for all V \{0},let ν : V \{0} [0, be a function satisfying the conditions νa i 3i < and i ν a i < when >, νa i 3. i < and 3i ν a i < when < for all V \{0}, and let ϕ : V \{0} n [0, be a function satisfying the conditions ϕa i,a i,...,a i n 3i < and i ϕ a i, a i,..., n a i < when >, ϕa i,a i,...,a i n i < and when < 3i ϕ a i, a i,..., n a i < 3. for all,,..., n V \{0}. If apping f : V Y satisfies f 0 =0 and the inequalities in 3.3 for all V \{0} and 3.4 for all,,..., n V \{0}, then there eists a unique mapping F : V Y satisfying the equality 3.5 for all,,..., n V \{0}, the equalities in 3.6 for all V, and f F i μ + μa i i+ + νa i 3i+3 + i ν νa i i+ + 3i ν when >, when < 3.3 for all V \{0}. Proof. We will divide the proof of this theorem into two cases; namely, the case for > and the other case for <. Case. Assume that >. We define a set A := { f : V Y f 0=0} and a mapping J m : A A by J m f := f o a 3m + a m f e + f o

15 UNIQUENESS THEOREMS ON FUNCTIONAL INEQUALITIES 57 for all V and m N 0. It follows from 3.3 that J m f J m+l f a ai f e + a f e f oa a i a + a 3 f o +a 4 f a i a 3i a a i a f o a + a 3 f o + a 4 f i μ + νa i 3i+3 + i ν for all V \{0}. In view of 3.0, 3., and 3.4, the sequence {J m f } is a Cauchy sequence for all V \{0}. SinceY is complete and f 0 =0, the sequence {J m f } converges for all V. Hence, we can define apping F : V Y by F := lim m f o a 3m + a m f e + f o for all V. Moreover, if we put m = 0andletl in 3.4, we obtain the first inequality of 3.3. Using the definition of F,3.4, 3., and DF,,..., n Dfo +,...,+ n ad fo,..., n = lim m a 3m a 3 a + am n a 3 Df o,..., a am a 3 Df o,..., n + a m Df e,..., n for all,,..., n V \{0}, we obtain the equalities in 3.6 forall V and we further get DF,,..., n =0forall,,..., n V \{0}. We notice that the equalities in.4 are true in view of 3.6, where k =. Using Lemma.3, we conclude that there eists a unique mapping F : V Y satisfying the equalities in 3.6 andthefirst inequality in 3.3, since the inequality f F i μ + νa i 3i+3 + i ν

16 58 Y.-H. L EE, S.-M.JUNG AND M. TH. RASSIAS i μ + i μ + i + νa i 3i+3 + ν ai 3i+3 ψk i + k i φ k 3i k i ν + ν holds for all V \{0}, wherek :=, φ := μ a + μ ν a + a ψ := ν+ν. a 3 a 3 +ν a a 3 a,and Case. We now consider the case of < and define apping J m : A A by J m f := a 3m f o + f e a m + f o for all V and n N 0. It follows from 3.3that J m f J m+l f a f e a i f e a i a a i+ + a3i μa i i+ + νa i i+ + 3i ν a a f o a + a 3 f o + a 4 f o + f oa a i a + a 3 f o +a 4 f o a i 3.5 for all V \{0}. On account of 3.0, 3. and3.5, the sequence {J m f } is a Cauchy sequence for all V \{0}. SinceY is complete and f 0=0, the sequence {J m f } converges for all V. Hence, we can define apping F : V Y by F := lim m a3m f o + f e a m + f o for all V. Moreover, if we put m = 0andletl in 3.5, we obtain the second inequality in 3.3.

17 UNIQUENESS THEOREMS ON FUNCTIONAL INEQUALITIES 59 By the definition of F,3.4, 3., and DF,,..., n = lim m a 3m a 3 a Df o n a3m+ a 3 a Df o,..., n,..., am + Df e,..., n a m Df o +,...,+ n a 3 Df o,..., n a 3 a for all,,..., n V \{0}, we get the equalities in 3.6forall V and we moreover have DF,,..., n =0forall,,..., n V \{0}. We remark that the equalities in.4 hold by considering 3.6 with k =. Using Lemma.3, we conclude that there eists a unique mapping F : V Y satisfying the equalities in 3.6 and the second inequality in 3.3, since the inequality f F μa i i+ + μa i +μ a i = νa i i+ + 3i ν i+ + νai +ν a i a 3 a i+ + 3i a 3 a ψk i + k i φ k 3i k i ν +ν holds for all V \{0},wherek:=, φ := μ+μ + ν,andψ:= a 3 a μ a + μ a. a 3 a ν+ By using Theorems 3., 3., 3.3, and3.4, we can prove the following corollary. COROLLARY 3.5. Let X be a normed space and let p, ε, θ, ξ be real constants such that p {,,3}, a {,0,}, ξ > 0, and θ > 0. If apping f : X Y satisfies f 0=0, fe a a f e ε p, 3.6 and f o a a + a 3 f o a+a 4 f o θ p 3.7 for all X \{0},as well as if f satisfies the inequality Df,,..., n ξ p + + n p 3.8

18 60 Y.-H. L EE, S.-M.JUNG AND M. TH. RASSIAS for all,,..., n X \{0}, then there eists a unique mapping F : X Y satisfying 3.5 for all,,..., n X \{0}, 3.6 for all X, as well as p p θ p p + ε p a p when either and p < or > and p > 3, f F 3.9 p + θ p 3 p + ε p a p for the other cases for all X \{0}. Proof. Let us put μ := ε p, ν := θ p,andϕ,,..., n := θ p + + n p for all,,..., n X \{0}. Then ϕ, μ, ν satisfy 3. and3. when either > andp < orwhen < andp > 3. If either > andp > 3 or if < andp <, then ϕ, μ, ν satisfy 3.0 and3.. Moreover, ϕ, μ, ν satisfy 3.4, 3.5, and 3.6 when< p < andϕ, μ, ν satisfy 3.0, 3., and 3.when< p < 3. Therefore, by Theorems 3., 3., 3.3,and3.4, there eists a unique mapping F : X Y such that 3.5 holds for all,,..., n X \{0}, and 3.6 holds for all X, and such that 3.9 holds for all X \{0}. ConflictofInterests. The authors declare that there is no conflict of interests regarding the publication of this article. Authors contribution. All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Acknowledgement. Soon-Mo Jung was supported by Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education No. 06RDAB REFERENCES [] S. CZERWIK, Functional Equations and Inequalities in Several Variables, World Sci. Publ., Singapore, 00. [] P. GĂVRUŢA, A generalization of the Hyers-Ulam-Rassias stability of approimately additive mappings, J. Math. Anal. Appl , [3] L. GĂVRUŢA AND P. GĂVRUŢA, On a problem of John M. Rassias concerning the stability in Ulam sense of Euler-Lagrange equation, in: Functional Equations, Diferrence Inequalities and Ulam Stability Notions J. M. Rassias ed., Nova Science Publishers, 00 pp [4] A. GILANYI, Hyers-Ulam stability of monomial functional equations on a general domain, Proc. Natl. Sci. USA , [5] M. E. GORDJI, S. K. GHARETAPEH, J. M. RASSIAS AND S. ZOLFAGHARI, Solution and stability of ied type additive, quadratic, and cubic functional equation, Adv. Difference Equ , Article ID 8630, 7 pages.

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