On stability of the general linear equation

Transcription

1 Aequat. Math. 89 (2015), c The Author(s) This article is pulished with open access at Springerlink.com /15/ pulished online Novemer 14, 2014 DOI /s z Aequationes Mathematicae On staility of the general linear equation Anna Bahyrycz and Jolanta Olko Astract. We prove, using the fixed point approach, some staility results for the general linear functional equation. Namely we otain sufficient conditions for the staility of a wide class of functional equations and control functions. Our results generalize a lot of the well known and recent outcomes concerning staility. In some examples we indicate how our method may e used to check if the particular functional equation is stale and we discuss the optimality of otained ounding constants. Mathematics Suject Classification. Primary 39B82; Secondary 39B52. Keywords. Hyers-Ulam staility, linear functional equation, fixed point theorem. 1. Introduction The theory of staility of functional equations started with Hyers answer to the famous question of Ulam concerning the staility of homomorphisms in metric groups (cf. [9]). Since then many authors have studied this suject dealing with a lot of functional equations, for example: p-wright equation f(px +(1 p)y)+f((1 p)x + py) =f(x)+f(y); linear equation f(ax + y) = Af(x)+Bf(y); quadratic equation f(x + y)+f(x y) =2f(x)+2f(y); Fréchet s equation f(x + y + z)+f(x)+f(y)+f(z) = f(x + y)+f(x + z)+f(y + z), where a,, A, B, p are given (cf., e.g., [1,2,5,7,8,10,11,13 17]). We deal with the general linear equation, in the class of functions mapping a linear space X into a normed space Y (oth over the field F {R, C}), namely

2 1462 A. Bahyrycz, J. Olko AEM m A i f a ij x j + A =0, (1.1) where A, a ij F, A i F \{0}, i {1,...,m}, j {1,...,n}. One of the ways of proving staility is the method of fixed points which is the most popular tool. Analyzing proofs of staility in particular cases of the functional equation (1.1), among others the aove-mentioned (see [3, 4]), we have found sufficient conditions for its staility. What is more interesting, we present some examples how our criterion may e used as a tool to check the staility of particular functional equations of the linear type. Moreover, we discuss the optimality of otained ounding constants in particular cases. Some similar ideas can e found in [6], where the general method for proving staility is descried. Our considerations ased on the fixed point theorem lead to a simpler procedure and sufficient conditions which are easier to check. 2. Statement of the main result and applications In this section we state the main result and we present its applications. The proof of the theorem will e given in the last section. The main theorem of this paper provides a criterion for the staility of the equation (1.1). It implies lots of the well known and recent results concerning particular cases of this functional equation. We descrie a method of determining whether a functional equation of linear type is stale. Our test method for staility of the equation is ased on the use of the fixed point theorem. In proofs of numerous theorems concerning staility an appropriate sustitution is used to otain a contraction operator. The fixed point of the operator is a solution of the equation which is close to a given function. Namely, considering a function g satisfying approximately equation (1.1) we look for a sustitution x j = c j x, c j F, j {1,...,n} such that m A i g a ij c j x + A = g(x) Tg(x), x X, where T is a proper operator. For this purpose, the solution c 1,...,c n F to the system consisting of at least one equation a ij c j =1, i {1,...,m}, should e found [see condition (i)]. In the following theorem additional conditions implying staility are given.

3 Vol. 89 (2015) On staility of the general linear equation 1463 In the sequel R + =[0, + ). A sum of numers over an empty set is defined to e zero. Theorem 2.1. Assume that A =0or (A 0and m A i 0). Let Y e a Banach space, g : X Y, θ : X n R + fulfill m A i g a ij x j + A θ(x 1,...,x n ), x 1,...,x n X. (2.1) Assume that there exist I {1,...,m}, c 1,...,c n F and ω 1,...,ω n [0, + ) such that (i) a ij c j =1,i I, (ii) A i ω i < A i, i/ I i I (iii) θ a ij c j (x 1,...,x n ) ω i θ(x 1,...,x n ) i/ I, x 1,...,x n X. Then there exists a unique solution G : X Y of (1.1) such that θ(c 1 x,...,c n x) i I A i i/ I A, x X. (2.2) i ω i Moreover G is a unique solution of (1.1) such that there exists a constant B (0, ) with Bθ(c 1 x,...,c n x), x X. (2.3) Remark 2.2. Oserve that if (i) holds with I = {1,...,m}, then the conditions (ii), (iii) are satisfied. Consequently the equation is stale. Remark 2.3. Among frequently appearing control functions in (2.1) arethe following 1. θ 1 (x 1,...,x n )=C; 2. θ 2 (x 1,...,x n )=C n c jx j kj ; 3. θ 3 (x 1,...,x n )=Cmax { c j x j kj : j {1,...,n} } ; 4. θ 4 (x 1,...,x n )=C n x j kj ; with some C (0, + ), k j > 0andc j F \{0}. It is easy to see that for each such function θ i there exists ω : F R + fulfilling θ i (βx 1,...,βx n ) ω(β)θ i (x 1,...,x n ) β F, x 1,...,x n X, (for example ω 1forθ 1, ω(c) = max{ c k1,..., c kn }, c F for θ 2 ). Consequently the condition (iii) is satisfied with ω i = ω( n a ijc j ).

4 1464 A. Bahyrycz, J. Olko AEM Our result may e used for proving the staility of different functional equations of the general form (1.1). Below we indicate relevant examples of its applications, to present our method p-wright equation Consider the p-wright equation with p F \{0, 1} f(px +(1 p)y)+f((1 p)x + py) =f(x)+f(y), x,y X, which is a particular case of (1.1), with m =4,n =2,A 1 = A 2 =1,A 3 = A 4 = 1, a 11 a 22 = p, a 12 = a 21 =1 p, a 31 = a 42 =1,a 32 = a 41 =0.We look for solutions c 1,c 2 of susystems of the system of linear equations p 1 p p p [ c1 c 2 ] 1 = which leads to the following two cases pc 1 +(1 p)c 2 =1 { (1 p)c (1) 1 + pc 2 1 c1 = c 1 c 1 1 c 2 = 1 pc 1 p c 2 1 pc 1 +(1 p)c 2 1 { (1 p)c (2) 1 + pc 2 1 c1 =1 c 1 =1 c 2 = c 1. c 2 = c 1 According to Theorem 2.1 applied to these solutions, we otain estimations in the real case for a particular control function. To shorten the paper we omit the second conclusion of Theorem 2.1 (see condition (2.3)) in the statement of the following results., Theorem 2.4. Let F = R, k (0, ), p R \{0, 1}, c R \{1}. Assume that g : X Y satisfies the inequality g(px+(1 p)y)+g((1 p)x+py) g(x) g(y) x k + y k, x,y X. If γ 1 (c) := (1 p)c + p 1 pc 1 p k + c k + 1 pc 1 p k < 1, then there exists a unique p-wright affine function G : X Y such that ( c k + 1 pc 1 p k ) x k, x X. 1 γ 1 (c)

5 Vol. 89 (2015) On staility of the general linear equation 1465 If γ 2 (c) := p +(1 p)c k + (1 p)+pc k + c k < 1, then there exists a unique p-wright function G : X Y such that (1 + c k ) x k, x X. 1 γ 2 (c) An analysis of the conditional minima of the aove otained ounding functions in the case k = 2 ensures the staility of this equation in this case. The detailed verification of the following corollary is left to the reader. Corollary 2.5 (The case k =2). Let p R \{0, 1}, g : X Y satisfy g(px +(1 p)y)+g((1 p)x + py) g(x) g(y) x 2 + y 2, x,y X. Then there exists a unique p-wright affine function G : X Y such that where M x 2, x X, { 1+4p 2 (1 p) 2 +(p 2 +(1 p) 2 ) M = 4p(1 p) if p (0, 1) 1 2p 1 1 if p R \ [0, 1]. Remark 2.6. Applying the aove result for p { 1 2, 1 2 } the optimal constant otained y our method is equal to Oserve that it is smaller than the constant in [12, Th. 1]. In the case p (0, 1) p 2 +(1 p) 2 < 1 our estimation M is smaller 1 than the estimate 1 p 2 (1 p) otained for the equation of the p-wright affine 2 function in [4, Th. 2. 1]. In the same manner we can study other cases in order to get the optimal constants Linear equation Consider the linear equation f(ax + y) =Af(x)+Bf(y), x,y X. (2.4) We look for solutions c 1,c 2 of susystems of the system of linear equations a [ ] c1 = 1. c

6 1466 A. Bahyrycz, J. Olko AEM Thus, we have the following possiilities: (1) c 1 = c 2 =1,a+ =1 (2) c 1 = c 2 =1,a+ 1 (3) c 1 =1,c 2 = 1 a 1, 0 (3 ) c 2 =1,c 1 = 1 a 1,a 0 (4) c 1 = c 1,c 2 = 1 ac 1, 0 (4 ) c 2 = c 1,c 1 = 1 c a 1,a 0 (5) c 1 =1,c 2 = c 1,a+ c 1 (5 ) c 2 =1,c 1 = c 1,ac+ 1. Therefore we otain the results corresponding to each of the cases (numer of each case is indicated in parentheses). Theorem 2.7 (1). Let Y e a Banach space, a+ =1, A+B 1, θ : X 2 R +. If g : X Y, satisfies g(ax + y) Ag(x) Bg(y) θ(x, y), x,y X, (2.5) then there exists a unique solution G : X Y of (2.4) such that θ(x, x) 1 A B, x X. Moreover G is a unique solution of (2.4) such that there exists a constant K (0, ) with Kθ(x, x), x X. Theorem 2.8 (2). Let Y e a Banach space, a+ 1, g : X Y, θ : X 2 R + satisfy (2.5). Assume that there exists ω R + such that ω< A + B and θ((a + )x, (a + )y) ωθ(x, y), x,y X. Then there exists a unique solution G : X Y of (2.4) such that θ(x, x) A + B ω, x X. Proof. Applying Theorem 2.1 for c 1 = c 2 = 1, I = {2, 3} we otain our claim. Theorem 2.9 (3). Let Y e a Banach space, a + 1, 0, g : X Y, θ : X 2 R + satisfy (2.5). Assume that there exists ω R + such that B ω < 1 A and ( 1 a θ x, 1 a ) y ωθ(x, y). Then there exists a unique solution G : X Y of (2.4) such that θ(x, 1 a x) 1 A B ω, x X.

7 Vol. 89 (2015) On staility of the general linear equation 1467 Theorem 2.10 (4). Let Y e a Banach space, ac + 1, 0, c 1, g : X Y, θ : X 2 R + satisfy (2.5). Assume that there exist ω(c),ω( 1 ac ) R + such that A ω(c)+ B ω( 1 ac ) < 1 and { θ(βx,βy) ω(β)θ(x, y), β c, 1 ac }. Then there exists a unique solution G : X Y of (2.4) such that θ(cx, 1 ac x) 1 A ω(c) B ω( 1 ac ), x X. Theorem 2.11 (5). Let Y e a Banach space, a + c 1, c 1, g : X Y, θ : X 2 R + satisfy (2.5). Assume that there exist ω(a + c),ω(c) R + such that ω(a + c)+ B ω(c) < A and θ(βx,βy) ω(β)θ(x, y), β {c, a + c}. Then there exists a unique solution G : X Y of (2.4) such that θ(x, cx) A ω(a + c) B ω(c), x X. Note that interchanging a and, A and B in the aove three theorems, we otain results for the cases (3 ) (5 ). According to Remark 2.3, in the case θ(x, y) :=ε we can take ω 1. Consequently, comining all the aove results in this case we deduce that the linear equation is stale and the ounding constant is the smallest one otained y our method. Theorem Let F = R, a + =1, A + B 1.Ifg : X Y satisfies g(ax + y) Ag(x) Bg(y) ε, x, y X, then the linear equation (2.4) is stale. Namely, there exists its unique solution G : X Y such that ε A + B 1, x X. Proof. Take a,, A, B R such that a + =1andA + B 1. Then at least the condition (1) holds, the conditions (2), (3), (3 ) are not satisfied. Define constants M 1 = A + B 1, M 4 = M 4 =1 A B, M 5 = A 1 B, M 5 = B 1 A. If M i > 0 for some i {1, 4, 5, 5 }, then there exists a unique soluton G i : X Y of (2.4) such that g(x) G i (x) ε M i, x X, see Theorems 2.7, 2.10, 2.11 and analogons of the last two ones.

8 1468 A. Bahyrycz, J. Olko AEM It is easy to verify that M 1 = max{m 1,M 4,M 5,M 5 }, therefore 1 M 1 is the smallest ound otained using linear sustitutions. Moreover, according to the second assertion of Theorem 2.7 the solution G 1 is a unique one satisfying the inequality Kε, x X with any constant K R, which completes the proof. The same, ut more complicated reasoning, can e drawn for the case a + 1. To state our result in this case, define sets P (3) := {(A, B) R 2 : (B 1 1+A B<1 A) ( 1 B 0 A 1 <B A) (A + B<1 A 0 B 0)}, P (3 ):={(A, B) R 2 : (A 1 1+B A<1 B) ( 1 A 0 B 1 <A B) (A + B<1 A 0 B 0)}, P := {(A, B) R 2 : A + B =1}. Theorem 2.13 (The case θ ε). Let F = R, a+ 1, A+B 1.Ifg : X Y satisfies g(ax + y) Ag(x) Bg(y) ε, x, y X, then the linear equation is stale. Namely, there exists its unique solution G : X Y such that ε M(A, B), x X, where 1 A B if (A, B) P (3) M(A, B) = 1 B A if (A, B) P (3 ). A + B 1 if (A, B) R 2 \ (P (3) P (3 ) P ) In the case A + B = 1 our criterion does not determine the staility of the equation (2.4) for the constant control function. Remark Among a lot of consequences of Theorems , we have the following results for Cauchy s equation (the case a = = A = B = 1). Applying Theorem 2.13 for θ(x, y) :=ε and ω = 1 we have the well known staility result proved y Hyers in [9]. Oserve that the constant M(A, B) = A + B 1 otained y our method is optimal. Theorem 2.8 used for the case θ(x, y) :=C( x p + y p ), ω = a + p leads to Aoki s outcome from [1]. Setting in Theorem 2.10 c = 1 2 and θ(x, y) :=C( x p + y p ), ω(c) := c p, where C 0, p>1 we get the staility result for Cauchy s equation proved y Gajda (see [7]).

9 Vol. 89 (2015) On staility of the general linear equation 1469 Itwasprovedin[2] that the estimation and p 1 in the general case. C x p 2 p 1 1 is the optimum for p Quadratic equation Consider the quadratic equation f(x + y)+f(x y) =2f(x)+2f(y), x,y X, (2.6) which is a particular case of (1.1), with m =4,n =2,A 1 = A 2 =1,A 3 = A 4 = 2, a 12 = a 42 = a i1 =1fori =1, 2, 3, a 32 = a 41 =0,a 22 = 1. Applying Theorem 2.1 for c 1 = c 2 =1,I = {3, 4} we otain the following staility result for this equation. Theorem Let Y e a Banach space, g : X Y, θ : X 2 R + satisfy g(x + y)+g(x y) 2g(x) 2g(y) θ(x, y), x,y X. Assume that there exist ω(2),ω(0) R + such that ω(2) + ω(0) < 4 and θ(cx, cy) ω(c)θ(x, y), c {2, 0}. Then there exists a unique solution G : X Y of (2.6) such that θ(x, x) 4 ω(2) ω(0), x X. Moreover G is a unique solution of (3.2) such that there exists a constant K (0, ) with Kθ(x, x), x X. Remark Applying the aove theorem for θ(x, y) := ε, ω(c) := 1 we otain the known result of Skof. For the functions θ(x, y) :=C( x p + y p ), ω(c) = c p, p<2 we otain the estimation C x p 4 2 p. 3. Proof of the main result In the sequel, T and Λ stand for maps of the forms (T ξ)(x) := α i ξ(β i x), ξ Y X,x X, (3.1) (Λδ)(x) := α i δ(β i x), δ R X +,x X, (3.2) for some α 1,...,α k,β 1,...,β k F, k N. Moreover we have the following lemmas, which will e needed to otain our main result.

10 1470 A. Bahyrycz, J. Olko AEM Lemma 3.1. Let X e a linear space over F, ε R X + and let Λ:R X X + R + e given y (3.2). If there exist ω 1,...,ω k R + such that ε(β i x) ω i ε(x), i {1,...,k}, x X, (3.3) then (Λ n ( k ) n. ε)(x) ε(x) α i ω i n=0 n=0 Moreover if γ := k α i ω i < 1, then n=0 (Λn ε)(x) ε(x) 1 γ. Proof. Denote γ := k α i ω i. We prove that for every x X and l N 0 (nonnegative integers) Λ l ε(x) ε(x)γ l. (3.4) Oviously, the aove inequality is fulfilled for l =0.Takex X and l N 0 and assume (3.4). Thus (Λ l+1 ε)(x) =Λ(Λ l ε)(x) = α i (Λ l ε)(β i x) α i ε(β i x)γ l ( ) γ l α i ω i ε(x) =ε(x)γ l+1 and y induction the proof of the first assertion is completed. The second one is a consequence of the convergence of the power series. Oserve that operators (3.1) and (3.2) satisfy the assumptions of Theorem 1 in[3], therefore applying this version of the fixed point theorem and the aove lemma we have the following result. Lemma 3.2. Let Y e a Banach space, ε R X + and let T : Y X Y X e given y (3.1). Assume that there exist ω 1,...,ω k R + such that γ := k α i ω i < 1 and the condition (3.3) holds. If g : X Y satisfies the inequality (T g)(x) g(x) ε(x), x X, (3.5) then there exists a unique fixed point G of T with ε(x) 1 γ, x X. Moreover G(x) := lim n (T n g)(x), x X. Lemma 3.3. Assume that Y is a Banach space, θ : X n R + and let T : Y X Y X e given y (3.1). Assume that there exist ω 1,...ω k R + such that k α i ω i < 1, and θ(β i x 1,...,β i x n ) ω i θ(x 1,...,x n ) i {1,...,k}, x 1,...,x n X. (3.6)

11 Vol. 89 (2015) On staility of the general linear equation 1471 If g : X Y satisfies the inequality m A i g a ij x j θ(x 1,...,x n ), x 1,...,x n X, (3.7) and for every x X there exists G(x) := lim n T n g(x), then G : X Y is asolutionto(1.1) (with A =0). Proof. Denote m Φ(ξ)(x 1,...,x n )= A i ξ a ij x j, ξ Y X,x 1,...,x n X. Let γ := k α j ω j. We prove that for every x 1,...,x n X and l N 0 Φ(T l g)(x 1,...,x n ) γ l θ(x 1,...,x n ). (3.8) Clearly, the case l =0isjust(3.7). Next, fix l N 0 and assume that (3.8) holds for every x 1,...,x n X. Then for every x 1,...,x n X m m Φ(T l+1 g)(x 1,...,x n )= A i (T l+1 g) a ij x j = A i T (T l g) a ij x j = = m A i p=1 α p (T l g) β p n α p Φ(T l g)(β p x 1,...,β p x n ). p=1 a ij x j = p=1 α p m A i (T l g) a ij (β p x j ) Consequently, applying the inductive assumption and (3.6) Φ(T l+1 g)(x 1,...,x n ) α p Φ(T l g)(β p x 1,...,β p x n ) p=1 α p γ l θ(β p x 1,...,β p x n ) p=1 α p γ l ω p θ(x 1,...,x n ) = γ l+1 θ(x 1,...,x n ). Thus, y induction we have shown that (3.8) holds for x 1,...,x n X, l N 0. Oserve that lim Φ(T l g)(x 1,...,x n ) = Φ( lim T l g)(x 1,...,x n ). l l Since y our assumptions γ = k α i ω i < 1, letting l in (3.8), we otain that p=1

12 1472 A. Bahyrycz, J. Olko AEM Φ(G)(x 1,...,x n )=0, x 1,...,x n X. Now we are in a position to prove Theorem 2.1 Proof. Assume that I {1,...,m} and c 1,...,c n F, ω 1,...,ω n F such that assumptions (i) (iii) hold. Note that A I := i I A i 0 y (ii). The proof will e divided into 2 steps. First assume that A = 0. Sustituting x j = c j x, j {1,...,n} in (2.1) we have g(x) Tg(x) θ(c 1x,...,c n x), x X, A I where for every ξ Y X, x X { A i T ξ(x) := i/ I A I ξ( n a ijc j x) if I {1,...,m} 0 if I = {1,...,m}. Define moreover ε(x) := θ(c1x,...,cnx) A I for x X and put γ := i/ I Ai ωi A I, according to our convention that the value of an empty sum is zero. Oserve that all the assumptions of Lemma 3.2 are fulfilled. Hence there exists a unique fixed point G : X Y of T such that ε(x) 1 γ = θ(c 1x,...,c n x) A I i/ I A, x X, (3.9) i ω i and G(x) = lim n (T n g)(x) forx X. By Lemma 3.3, it is a solution of the equation (1.1) (with A = 0). Suppose now that there exist B>0andH : X Y a solution of (1.1) such that (2.3) is satisfied. By the triangle inequality and (3.9) G(x) H(x) ε(x) 1 γ + Bθ(c 1x,...,c n x)= ε(x) 1 γ + B A I ε(x) 1 =(1+B A I (1 γ))ε(x) 1 γ =(1+BA I (1 γ))ε(x) γ p, x X. (3.10) Let C := 1 + B A I (1 γ). We show that for all l N 0 and x X G(x) H(x) Cε(x) γ p. (3.11) The case l = 0 is exactly (3.10). So fix x X and assume that (3.11) holds for l N 0, and we will prove it for l + 1. Oserve that for every x X the p=0 p=l

13 Vol. 89 (2015) On staility of the general linear equation 1473 equation (1.1) is satisfied for c 1 x,...,c n x, therefore T H(x) = A i H a ij c j x = A i H a ij c j x = H(x) A I A i/ I I i I and H is a fixed point of T. Consequently G(x) H(x) = T G(x) TH(x) = A i G a ij c j x A i H a ij c j x A I A i/ I I i/ I A i A I i/ I G a ij c j x H( a ij c j x) A i Cε a ij c j x γ p C A i ωi ε(x) γ p A I A i/ I I p=l i/ I p=l = Cε(x) γ p. p=l+1 Letting l in (3.11) wegetg = H and the proof in the first case is complete. If A 0and m A i 0 define f(x) :=g(x)+ A m, x X. By(2.1) Ai m A i f a ij x j θ(x 1,...,x n ), x 1,...,x n X and consequently, according to our previous considerations, there exists a unique function F : X Y such that f(x) F (x) θ(c 1x,...,c n x) A I i/ I A, x X. i ω i Oviously G(x) :=F (x) A m, x X is the desired function. Ai Open Access. This article is distriuted under the terms of the Creative Commons Attriution License which permits any use, distriution, and reproduction in any medium, provided the original author(s) and the source are credited. References [1] Aoki, T.: On the staility of the linear transformation in Banach spaces. J. Math. Soc. Japan 2, (1950)

14 1474 A. Bahyrycz, J. Olko AEM [2] Brzdęk, J. : A note on staility of additive mappings. In: Rassias, Th.M., Taor, J. (eds.) Staility of Mappings of Hyers- Ulam Type, pp Hadronic Press, Inc., Florida (1994) [3] Brzdęk, J., Chudziak, J., Páles, Zs.: A fixed point approach to staility of functional equations. Nonlinear Anal. 74, (2011) [4] Brzdęk, J.: Staility of the equation of the p-wright affine functions. Aequationes Math. 85, (2013) [5] Brzdęk J.: Ciepliński, hyperstaility and superstaility. Astr. Appl. Anal. (2013). doi: /2013/ [6] Forti, G.-L.: Elementary remarks on Ulam-Hyers staility of linear functional equations. J. Math. Anal. Appl. 328, (2007) [7] Gajda, Z.: On staility of additive mappings. Int. J. Math. Sci. 14, (1991) [8] Gilányi, A., Páles, Zs.: On Dinghas-type derivatives and convex functions of higher order. Real Anal. Exchange, 27, (2001/2002) [9] Hyers, D.H.: On the staility of the linear functional equation. Proc. Nat. Acad. Sci. USA 27, (1941) [10] Hyers, D.H., Isac, G., Rassias, Th. M.: Staility of Functional Equations in Several Variales. Birkhäuser, Boston (1998) [11] Jung S.-M.: Hyers-Ulam-Rassias Staility of Functional Equations in Nonlinear Analysis. Springer Optimization and Its Applications, vol. 48, Springer, New York-Dordrecht- Heidelerg-London (2011) [12] Jung, S.-M.: Hyers-Ulam-Rassias staility of Jensen s equation and its application. Proc. Am. Math. Soc. 126(11), (1998) [13] Lajkó, K.: On a functional equation of Alsina and García-Roig. Pul. Math. Derecen 52, (1998) [14] Maksa, Gy., Nikodem, K., Páles, Zs.: Results on t-wrightconvexity.c.r.math.rep. Acad. Sci. Canada 13, (1991) [15] Paneah, B.: A new approach to the staility of linear functional operators. Aequationes Math. 78, (2009) [16] Rassias, M.: On a modified Hyers-Ulam sequence. J. Math. Anal. Appl. 158, (1991) [17] Skof, F.: Proprieta locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano 53, (1983) Anna Bahyrycz and Jolanta Olko Institute of Mathematics Pedagogical University Podchorążych Kraków Poland ah@up.krakow.pl jolko@up.krakow.pl Received: July 29, 2014