REMARKS ON THE STABILITY OF MONOMIAL FUNCTIONAL EQUATIONS

Fixed Point Theory, Volume 8, o., 007, 01-18 http://www.math.ubbclu.ro/ nodeac/sfptc.html REMARKS O THE STABILITY OF MOOMIAL FUCTIOAL EQUATIOS LIVIU CĂDARIU AD VIOREL RADU Politehnica University of Timişoara, Department of Mathematics Piaţa Victoriei o., 300006 Timişoara, Romania E-mail: liviu.cadariu@mat.upt.ro, lcadariu@yahoo.com West University of Timişoara, Faculty of Mathematics and Computer Science Bv. Pârvan 4, 3003 Timişoara, Romania, E-mail: radu@math.uvt.ro Abstract. The aim of this paper is to prove the generalized stability in the Ulam-Hyers- Bourgin sense of the monomial equations, for functions from -divisible groups into complete β-normed spaces. There is used a fixed point method, previously applied by the authors to some particular functional equations. A special case, the stability of type Aoki-Rassias is emphasized in section 3 and finally we give some counterexamples in order to clarify the role of each of the control conditions. Key Words and Phrases: Monomial functional equation, fixed point, stability. 000 Mathematics Subect Classification: 39B5, 39B6, 39B8, 47H10. 1. Introduction Different methods are known for demonstrating the stability of functional equations. evertheless, almost all proofs use the direct method, conceived by Hyers in [] and revealed by Bourgin in [5] for unbounded differences see also [3], [30], [14] and [18]. For supplementary details, we refer the reader to the expository papers [15] and [31] or to the books [4], [1], [13] and [5]. A slightly different method, by using the fixed point alternative, has been applied in [9], [6]-[11], [6] and [8]. It does essentially insinuate a metrical context and is seen to better clarify the ideas of stability as well as the role of each controlling condition. 01

0 LIVIU CĂDARIU AD VIOREL RADU For an Abelian group X and a vector space Y consider the difference operators defined, for each y X and any mapping f : X Y, in the following manner: 1 yfx := fx + y fx, for all x X, and, inductively, n+1 y = 1 y n y, for all n 1. A mapping f : X Y is called a monomial function of degree if it is a solution of the monomial functional equation y fx!fy = 0, x, y X. 1.1 otice that the monomial equation of degree 1 is exactly the Cauchy equation, while for = the monomial equation has the form fx + y fx + y + fx fy = 0, which is equivalent to the well-known quadratic functional equation: fx + y + fx y = fx + fy, x, y X. 1. In what follows, the positive integer will be fixed. M.H. Albert and A. Baker, in [3], demonstrated the Ulam-Hyers stability of the monomial equation 1.1 see also [0]. Subsequently, A. Gilanyi proved in [1] that the equation is stable in the sense of Aoki-Rassias see also [35] for the asymptotic stability: Proposition 1.1. Let X be a normed linear space, let Y be a Banach space and let p be a non-negative real number. If, for a function f : X Y, there exists a real number η 0 such that y fx!fy η x p + y p, x, y X, 1.3 then there exist a real number c = c, p and a unique monomial function g : X Y of degree with the property fx gx cη x p, x X. On the other hand, by using the fixed point method, we proved in [8] the following generalized stability result for additive Cauchy equations and mappings with values into β-normed spaces: Proposition 1.. Let X, Y be two linear spaces over the same { real or complex field, with Y a complete β-normed space, and set r =, if = 0 1, if = 1.

STABILITY OF THE MOOMIAL EQUATIOS 03 Suppose the mapping f : X Y, with f 0 = 0, verifies the control condition fx + y f x f y β ϕx, y, for all x, y X, where ϕ : X X R + has the following property: ϕ r nx, rn y lim = 0. n r nβ If there exists a positive constant L < 1 such that the mapping x x ψx = ϕ, x satisfies the relation ψx L r β ψ x r, for all x X, then there exists a unique additive mapping a : X Y such that fx ax β L1 ψx, for all x X. 1 L In the present paper, by developing to some extent the above results, we prove a stability property in the Ulam-Hyers-Bourgin sense: For every f : G Y, from a -divisible group G into a complete β-normed space Y, which verifies a slightly perturbed monomial equation i.e. controlled by a mapping ϕ : G G [0, with suitable properties, there exists a unique monomial solution fittingly approximating f see Agarwal, Xu & Zhang [1] for a general definition. A stability theorem via the fixed point alternative We intend to show that Proposition 1.1 and Proposition 1. can really be generalized by the fixed point method, proposed in [9] and already used in [6]- [11] and [8] to different functional equations see also [4] and [16]. Actually, we shall prove a new stability theorem of the Ulam-Hyers-Bourgin type for the monomial functional equation 1.1. As it will be seen, the fixed point alternative is a meaningful device on the road to a better understanding of the stability property, plainly related to some fixed point of a concrete operator. Specifically, our control conditions are perceived to be responsible for three fundamental facts:

04 LIVIU CĂDARIU AD VIOREL RADU 1 The contraction property of the operator S given by OP mon below. The distance between f and Sf, the first two approximations, is finite. 3 The fixed point function of S is forced to be a monomial function. Let X be a -divisible group 1, let Y be a real or complex complete β- normed space, and assume we are given a function ϕ : X X [0, with the following property: H ϕ lim m r m x, rm y r mβ = 0, x, y X, for r := 1, {0, 1}. Theorem.1. Suppose the mapping f : X Y, with f 0 = 0, verifies the control condition y fx!fy β ϕx, y, x, y X..4 If there exists a positive constant L < 1 such that the mapping x ψx = 1 ix! β ϕ0, x + ϕ i, x, x X, satisfies the inequality H ψr x L r β ψ x, x X, then there exists a unique monomial mapping g : X Y, of degree, with the following fitting property: Est fx gx L1 ψx, x X. 1 L For the proof of our theorem, we need the following two fundamental lemmas. The first one gives a crucial intermediary result and the second one, which is recalled for convenience only, is a celebrated result in fixed point theory. 1 That is to say an Abelian group X, + such that for any x X there exists a unique a X with the property x = a; this unique element a is denoted by x. Usually, a mapping β : Y R +, where β 0, 1], is called a β norm iff it has the properties n I β : y β = 0 y = 0, n II β : λ y β = λ β y β, and n III β : y + z β y β + z β, for all y, z Y, and λ K.

STABILITY OF THE MOOMIAL EQUATIOS 05 Lemma.. Let us consider an Abelian group G, a β normed linear space Y and a mapping ϕ : G G [0,. If the function f : G Y satisfies.4 then, for all x G, fx fx 1 β β! β ϕ0, x + Proof. As in [1], we define the functions F i : G Y by Using.4 we see that and F i x := x fix!fx, x G. ϕix, x. i.5 F i x β ϕix, x, x G.6 F 0 x β ϕ0, x, x G..7 On the other hand, if we consider as in [19], Lemma.. the +1 +1 matrix α 0 0 α 1 0... α 0 α 0 A = 1 α 1 1... α 1...... α 0 α 1... α where α i+k i = 1 k k, if 0 k 0, otherwise for i = 0,...,, k = i,..., i, and the 1 + 1 matrix B = β 0 β 1 β β,.8 with β k 1 k = k, if k, for k = 0,...,,.9 0, if k then, for the positive constants K i =, one has i K 0 A 0 + K 1 A 1 +... + K A = B and K 0 + K 1 +... + K =,.10

06 LIVIU CĂDARIU AD VIOREL RADU where A i, i = 0,..., is the i-th row of the matrix A. Recall the following noteworthy formula for the difference operator: y fx = 1 + fx + y..11 Therefore =0 +i x fix = 1 1 k i fkx k i so that, with the notation.8, hence k=i x fix = 1 F i x = 1 k=0 k=0 α k i fkx, x G, α k i fkx!fx, x G. On the other hand, with the notation.9, xf0 = 1 1 fx = 1 β k fkx, so that =0 k=0 F 0 x = 1 β k fkx!fx, x G. k=0 By.10, we can write: K 0 α k 0 +... + K k=0, α k where K i = i, i = 0, 1,...,. k=0, = β k = B,.1 k=0, If we multiply.1 by fkx and then add for k = 0,, we obtain K 0 x f0 + K 1 x fx +... + K x fx = xf0, x G, hence K 0 F 0 x + K 1 F 1 x +... + K F x +!fx = F 0 x +!fx

STABILITY OF THE MOOMIAL EQUATIOS 07 for all x G. Using.6 and.7 we get, for all x G, fx fx 1 β β! β ϕ0, x + ϕix, x. i Remark.3. It easy to see e.g. by taking ϕ 0 in Lemma. that gx = gx and g m x = m gx, x G, m, for any monomial function g of degree. Remark.4. Let G be a - divisible group and formally replace x with x in.5. Then one has the following result: Let Y be a β normed linear space and ϕ : G G [0, a given mapping. If f : G Y satisfies y fx!fy β ϕx, y, x, y G, then x fx β f 1! β ϕ0, x + ix ϕ i, x, x G. ow we recall the fixed point alternative see, e.g., [7, 3]: Lemma.5. Suppose we are given a complete generalized metric space E, d i.e. one for which d may assume infinite values, and a strictly contractive mapping S : E E, with the Lipschitz constant L < 1. Then, for each of the elements x E, exactly one of the following assertions is true: A 1 ds n x, S n+1 x = +, for all n 0, A There exists k such that ds n x, S n+1 x < +, for each n k. Actually, if A holds, then see [34] A 1 The sequence S n x is convergent to a fixed point z of S; A z is the unique fixed point of S in the set { } Z := z E, d S k x, z < + ; A 3 The following estimation holds: d z, z 1 d z, Sz, z Z. 1 L

08 LIVIU CĂDARIU AD VIOREL RADU The proof of Theorem.1. We consider the set E := {g : X Y, g0 = 0} and introduce a generalized metric d = d ψ on E, where { } GM ψ d ψ g, h = inf K R +, g x h x β Kψx, x X. It is easy to see that E, d is complete. ow, consider the mapping OP mon S : E E, Sg x := g r x r. and notice that r = 1, for fixed in {0, 1}. Step I. Using the hypothesis H, one can see that S is strictly contractive on E. amely, we can write, for any g, h E : Therefore dg, h < K = g x h x β Kψx, x X = 1 r g r x 1 r h r x 1 β r β Kψr x, x X = 1 g r x 1 h r x LKψx, x X = β r r β d Sg, Sh LK. CC L d Sg, Sh Ld g, h, g, h E, that is S is a strictly contractive self-mapping of E relative to d, with the Lipschitz constant L < 1. Step II. We show that d f, Sf <. For =0, by Lemma., we have fx fx β 1 β! β ϕ0, x + i Therefore, using H 0, f x fx ψx β β Lψx, x X, that is d f, Sf L <. ϕix, x, x X.

STABILITY OF THE MOOMIAL EQUATIOS 09 For =1, by Remark.4, we see that x fx β f 1! β ϕ0, x + ix ϕ i, x = ψx, x X. Therefore d f, Sf 1 = L 0 <. Step III. In both cases we can apply the fixed point alternative see Lemma.5, which tells us that there is a mapping g : X Y such that: g is a fixed point of S, that is g x = g x, x X..13 The mapping g is the unique fixed point of S in the set F = {h E, d f, h < }. This says that g is the unique mapping with both the properties.13 and.14, where K < such that f x g x Kψx, x X..14 ds m f, g 0, for m, which implies the equality f r mx lim = lim g mx = gx, x X. m m r m df, g 1 d f, Sf, which implies the inequality 1 L df, g L1 1 L, from which Est is seen to be true. Step IV. We show that g is a monomial function of degree. To this end, we replace x by r mx and y by rm y in relation.4, then divide the obtained relation by r m and we obtain r m yfrm x r m! frm y r m ϕrm x, rm y, x, y X. β r mβ

10 LIVIU CĂDARIU AD VIOREL RADU On the other hand, by.11, r m yfrm x r m = = 1 k k=0 1 k k=0 k k fr m x + kr m y = y g m x, x, y X. r m g m x + ky = = And we get y g m x! g m y β ϕrm x, rm y r mβ By letting m in the above relation and using, x, y X. H, we obtain y gx! gy = 0, x, y G. Remark.6. For = 1 in the above theorem, we obtain a generalized stability result for the additive Cauchy equation for functions with values in complete β-normed spaces, with ψx = ϕ 0, x + ϕ 0, x x + ϕ, x, x X. compare with Proposition 1.. If = in Theorem.1 it results as in [7] for β = 1 that the quadratic functional equation again for functions with values in complete β-normed spaces is stable in the Ulam-Hyers-Bourgin sense, with ψx = 1 β ϕ 0, x + ϕ 0, x x + ϕ, x + ϕ x, x, x X. It is worth noting that the estimations obtained directly for particular values of as in [6], [7], [8] or [11] are generally better than those resulting from Est, which in its turn is applicable for all. 3. Stability of the Aoki-Rassias type Let α R +. A mapping. α : X X R + is called an sh-functional of order α iff it is α sub-homogeneous: h α : λ z α λ α z α, λ K, z X X.

STABILITY OF THE MOOMIAL EQUATIOS 11 As usual, X is identified with X {0} in X X, so that x α = x, 0 α defines an sh-functional of order α on X. As a consequence of Theorem.1, we have the following result, which directly extends many stability theorems of the Aoki-Rassias type: Proposition 3.1. Let X, Y be two linear spaces over the same real or complex field. Suppose we are given a complete β norm on Y and an shfunctional of order α on X X, with α β. Under these conditions we have the following stability property: For each ε > 0, there exists δε > 0 such that, for every mapping f : X Y which satisfies C αβ y fx!fy β δε x, y α, x, y X, there exists a unique monomial mapping g : X Y, of degree, such that, x X, Est αβ fx gx β ε! β 0, x α + ix i, x. Proof. Having in mind Theorem.1, we take the following control function, appearing in the hypothesis C αβ : ϕ x, y := δε x, y α, for all x, y X. We shall consider two cases. Case 1. For α β < 0, we work with = 0, that is r 0 =. We then have: and ϕ m x, m y mβ δε mα β x, y α 0, x, y X ψx = δε! β 0, x α + ix, x i α α ψ x = L β ψ x, x X, with L = α β < 1 and so H 0 and H 0 hold. Case. For α β > 0, we take = 1, that is r 1 = 1. We then have: x βm ϕ m, y m δε mβ α x, y α 0, x, y X α

1 LIVIU CĂDARIU AD VIOREL RADU and ψx 1 α ψx = L 1 β ψx, x X, with L = β α < 1, so that H 1 and H 1 are verified in this case. Theorem.1 tells us that there is a unique monomial mapping g : X Y such that either fx gx β L ψx, for all x X, 1 L holds, with L = α β, or fx gx β 1 ψx, for all x X, 1 L holds, with L = β α. Thus, the inequality Est αβ holds true for δε = ε β α α. Remark 3.. As a direct consequence of our proposition, we obtain the result of Gilanyi, [1] formulated in Proposition 1.1 above. Indeed, we apply Proposition 3.1 for the complete 1 normed space Y and for the sh-functional of order p on X X, given by x, y p = x p + y p, with p 0, p. In this case the generalized metric is of the form { } d p g, h = inf K R +, g x h x β K γ, p x p, x X, and we obtain c = c, p = p + i p + 1 i p.! Remark 3.3. Obviously, Proposition 3.1 can be proved by using directly the alternative of fixed point. 4. Comments, examples and counterexamples As we shall see by examples, our hypotheses in Theorem.1 are essential for the stability result. otice that in the proof we used the hypothesis H to show that the operator S is contractive and, subsequently, to obtain the estimation Est, while the hypothesis H was fundamentally used to show that the fixed point function g does satisfy the monomial equation 1.1. In what follows we shall use the notations in Theorem.1.

STABILITY OF THE MOOMIAL EQUATIOS 13 Assertion 4.1. There exist f : R R and ϕ : R [0, such that a the relation.4 holds, b none of the conditions H and H is satisfied and c there exist infinitely many monomial functions g which satisfy the relation Est. Proof. Indeed, if we take X = Y = R, β =, f x := x and ϕx, y = x + y, then the inequality.4 clearly holds. Obviously, ψx = x 1 + 1! i + 1 = γ x, i and we easily see that none of the conditions H and H is satisfied. However, there exist infinitely many fixed points of S, gx = λx, which satisfy fx gx = x λx 1 + λ x = 1 + λ γ ψx Actually, every mapping g with gx = gx and gx k x is seen to satisfy fx gx 1 + k x. Assertion 4.. Generally, the monomial functional equation 1.1 is not stable for α = β. More exactly, there exists f : R R for which 1.3 holds for p = and there exists no monomial mapping g of degree which could verify the relation Est. Proof. In [1] see also [17], [5]-pp. 3-4, [4]-pp. 59-60 the following example is given: Let be a positive integer, ε be a positive real number and let ε = We consider the mapping ϕ : R R ε, ϕx = ε x, 1 ε, ε +!. if x if < x < if x and we define a function f: R R by ϕ m x fx =, for all x R. m m=0

14 LIVIU CĂDARIU AD VIOREL RADU As it is proven in [1], i f verifies the following inequality: y fx!fy ε x + y, x, y R; 4.15 ii There is no real number c for which there could exist a monomial function g : R R of degree such that fx gx cε x, for all x R. From 4.15 it is clear that, for β = 1, ϕx, y = ε x + y, hence ψx = ε x! 1 + 1 i i + 1 and none of the conditions H and H is satisfied. Indeed, it is easy to see that = ε γ x, ψx ϕ r m x, r m y = ψx and lim = ϕ x, y 0. m r m On the other hand, by ii, there exists no monomial mapping g satisfying a relation of the form fx gx k ψx, x R, of type Est. Assertion 4.3. There exist f : R R and ϕ : R [0, such that: a the relation.4 holds, b the conditions H 0 and H 1 are verified, c the mapping ϕ does not satisfy the condition H and d no monomial mapping can satisfy the relation Est. Proof. Indeed, if we take X = Y = R, β =, { 0, if x Q fx = x, if x R \ Q and ϕx, y := y fx!fy, then the inequality.4 is verified. Moreover, ψx = 0. Both conditions H 0 and H 1 hold for every L 0, 1, while the mapping ϕ does not satisfy the condition H. Since f r m x r m = fx, it is clear that the unique fixed point of the operator S in the set F = {g E, d f, g < + } is g = f, which clearly is not monomial. Assertion 4.4. There exist f : R R and ϕ : R [0, such that a the relation.4 holds, b none of the conditions H 0 and H 1 is satisfied, c the relation H holds

STABILITY OF THE MOOMIAL EQUATIOS 15 and d no monomial mapping g can satisfy the relation Est. Proof. Let us take X = Y = R, β =, f x := sin x and ϕx, y := y fx!fy. Then the inequality.4 is trivially verified. Obviously ψx = 1! 1 sin x! sin x + =0 =0 + i 1 i + x sin! sin x, for all x R. For x 0 we obtain ψx ψx ψx 4, and for x π we have that ψx 0. Therefore neither H 0 nor H 1 is satisfied. Clearly the mapping ϕ satisfies the condition H. Let us suppose, for a contradiction, that a monomial mapping g satisfies Est fx gx kψx, for all x R, k a real constant. Then g is a bounded mapping on R. Therefore see [33], gx = η x, for x R, where η is a constant, hence gx = 0 for x R. ow, the inequality Est implies sin x k! 1 =0 =0 sin x! sin x + + i 1 i + x sin! sin x, for all x R, which is impossible. Therefore, there exists no monomial mapping g which satisfies the estimation relation Est. Remark 4.1. In Forti, [16] a stability result for equations of the form SF = F is presented, with S a functional operator of the Schröder type: SF x =

16 LIVIU CĂDARIU AD VIOREL RADU H F Gx. The hypotheses as well as the proofs insinuate a Banach- Caccioppoli type condition: dists n f, S n+1 f <. n=0 We only remark that such a condition and the continuity of S together ensure the existence of a fixed point function for S, which is the limit of S n f. It is worth noting that in Baker, [4] the Banach contraction principle is applied to obtain the Hyers-Ulam stability for nonlinear functional equations of a similar form: ϕx = F x, ϕfx see also [1] for other details and examples. Remark 4.. In [10] we used the direct method to prove, among others, the following Ulam-Hyers-Bourgin stability property for monomial equations: Let there be given a complete β normed space Y, an Abelian group G and a controlling mapping ϕ : G G [0, such that Φ i x := k=0 ϕ k ix, k x β k <, x G, for i = 0, 1,...,, and ϕ m x, m y lim m β m = 0, x, y G. Then, for every mapping f : G Y which verifies y fx!fy β ϕx, y, x, y G, there exists a unique monomial function g : G Y of degree such that, for all x G, 1 fx gx β β! β Φ 0 x + Φ i x. i References [1] R.P. Agarwal, B. Xu, B. and W. Zhang, Stability of functional equations in single variable, J. Math. Anal. Appl., 88 003, 85-869. [] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 1950, 64-66. [3] M.H. Albert and J. A. Baker, Functions with bounded n-th differences, Ann. Polonici Math., 431983, 93-103. [4] J.A. Baker, The stability of certain functional equations, Proc. Amer. Math. Soc., 31991, 79-73.

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