On the probabilistic stability of the monomial functional equation

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1 Available online a J. Nonlinear Sci. Appl. 6 (013), Research Aricle On he probabilisic sabiliy of he monomial funcional equaion Claudia Zaharia Wes Universiy of Timişoara, Deparmen of Mahemaics, Bd. V. Parvan 4, 3003, Timişoara, Romania. This paper is dedicaed o he memory of Professor Viorel Radu Communicaed by Professor D. Miheţ Absrac Using he fixed poin mehod, we esablish a generalized Ulam - Hyers sabiliy resul for he monomial funcional equaion in he seing of complee rom p-normed spaces. As a paricular case, we obain a new sabiliy heorem for monomial funcional equaions in β-normed spaces. Keywords: Rom p-normed space; Hyers - Ulam - Rassias sabiliy; monomial funcional equaion. 010 MSC: Primary 39B8; Secondary 54E Inroducion The problem of Ulam - Hyers sabiliy for funcional equaions concerns deriving condiions under which, given an approximae soluion of a funcional equaion, one may find an exac soluion ha is near i in some sense. The problem was firs saed by Ulam [] in 1940 for he case of group homomorphisms, solved by Hyers [10] in he seing of Banach spaces. Hyers s resul has since seen many significan generalizaions, boh in erms of he conrol condiion used o define he concep of approximae soluion ([], [1], [5]) in erms of he mehods used for he proofs. Radu [0] noed ha he fixed poin alernaive can be used successfully in he sudy of Ulam - Hyers sabiliy, o obain resuls regarding he exisence uniqueness of he exac soluion as a fixed poin of a suiably chosen conracive operaor on a complee generalized meric space. The fixed poin mehod was subsequenly used o obain sabiliy resuls for oher funcional equaions in various seings. The noion of fuzzy sabiliy for funcional equaions was inroduced in he papers [16, 17]. The fixed poin mehod was firs used o sudy he probabilisic sabiliy of funcional equaions in [1, 13, 14]. address: czaharia@mah.uv.ro (Claudia Zaharia) Received

2 C. Zaharia, J. Nonlinear Sci. Appl. 6 (013), Recenly, in [15] [3], he problem of sabiliy was considered in he more general seing of rom p-normed spaces. Following he same approach, we prove a sabiliy resul for he monomial funcional equaion, for mappings aking values in a complee rom p-normed space. As rom p-normed spaces generalize rom normed spaces β-normed spaces, his allows for a uniary framework in which o discuss several sabiliy resuls. Definiion 1.1. Le X Y be linear spaces. A mapping f : X Y is called a monomial funcion of degree N if i is a soluion of he monomial funcional equaion N y f(x) N!f(y) = 0, x, y X. (1.1) Here, denoes he difference operaor, given by y f(x) = f(x + y) f(x), for all x, y X, is ieraes are defined inducively by 1 y = y n+1 y = 1 y n y, for all n 1. I can easily be shown ha N y f(x) = N ( ) N ( 1) f(x + iy). Oher well-known funcional equaions, such as he addiive, quadraic or cubic ones, are paricular cases of equaion (1.1), obained by seing N = 1, or 3 respecively. The (generalized) Ulam - Hyers sabiliy for he monomial funcional equaion was previously sudied in [1], [7], [8] [3]. We also menion he recen papers [18] [19]. We will assume ha he reader is familiar wih he noaions erminology specific o he heory of rom normed spaces. We only recall he definiion of a rom p-normed space, as given in [9]. Definiion 1.. ([9]) Le p (0, 1]. A rom p-normed space is a riple (X, µ, T ) where X is a real vecor space, T is a coninuous -norm, µ is a mapping from X ino D + so ha he following condiions hold: (P1) µ x () = 1 for all > 0 iff x = 0; (P) µ αx () = µ x ( α p ), for all x X, α 0 > 0; (P3) µ x+y ( + s) T (µ x (), µ y (s)), for all x, y X,, s 0. If (X, µ, T ) is a rom p-normed space wih T - a coninuous -norm such ha T T L, hen V = {V (ε, λ) : ε > 0, λ (0, 1)}, V (ε, λ) = {x X : µ x (ε) > 1 λ} is a complee sysem of neighborhoods of he null vecor for a linear opology on X generaed by he p-norm µ ([9]). Definiion 1.3. Le (X, µ, T ) be a rom p-normed space. (i) A sequence {x n } in X is said o be convergen o x in X if for every > 0 ε > 0, here exiss a posiive ineger N such ha µ xn x() > 1 ε whenever n N. (ii) A sequence {x n } in X is said o be Cauchy if, for every > 0 ε > 0, here exiss a posiive ineger N such ha µ xn x m () > 1 ε whenever m, n N. (iii) A rom p-normed space (X, µ, T ) is said o be complee iff every Cauchy sequence in X is convergen o a poin in X.. Main resuls In he following, N is a fixed posiive ineger. Definiion.1. Le X be a linear space, (Y, µ, T M ) be a rom p-normed space Φ be a mapping from X o D +. A mapping f : X Y is said o be probabilisic Φ-approximaely monomial of degree N if µ N y f(x) N!f(y)() Φ x,y (), x, y X, > 0. (.1)

3 C. Zaharia, J. Nonlinear Sci. Appl. 6 (013), We will prove ha, under suiable condiions on he funcion Φ, every probabilisic Φ-approximaely monomial mapping can be approximaed, in a probabilisic sense, by a monomial mapping of he same degree. In doing so, we will need he following lemmas: Lemma.. ([11]) Le (X, d) be a complee generalized meric space A : X X be a sric conracion wih he Lipschiz consan L (0, 1), such ha d(x 0, A(x 0 )) < + for some x 0 X. Then A has a unique fixed poin in he se Y := {y X, d(x 0, y) < } he sequence (A n (x)) n N converges o he fixed poin x for every x Y. Moreover, d(x 0, A(x 0 )) δ implies d(x, x 0 ) Lemma.3. ([6])Le n, λ be inegers, A = α (0) 0 α (λn) α (0) (λ 1)n α (λn) (λ 1)n δ 1 L. where for i = 0,..., (λ 1)n k = i,..., λn i ( ) n α (i+k) ( 1) k, if 0 k n, i = n k 0, oherwise. Le a i denoe he i h row in A, i = 0,..., (λ 1)n, b = (β (0) β (λn) ), where ( ) β (k) ( 1) k n λ = n k, if λ k, λ 0, if λ k, for k = 0,..., λn. Then here exis posiive inegers K 0,..., K (λ 1)n so ha K 0 + K (λ 1)n = λ n K 0 a K (λ 1)n a (λ 1)n = b. Remark.4. In he case of λ =, K i = ( n n i), for all i = 0, N (see [6]). Nex, given linear spaces X Y a mapping f : X Y, using he noaions of he previous lemma for λ =, one can wrie N N x f(ix) = ( 1) N α (k) i f(kx), i = 0, N, By Lemma.3, N K i α (k) i k=0 N N xf(0) = ( 1) N β (k) f(kx). k=0 = β (k) for all k = 0, N, wih K i = ). Therefore we have shown ha N ( ) N N x f(ix) = N xf(0), x X. (.)

4 C. Zaharia, J. Nonlinear Sci. Appl. 6 (013), Theorem.5. Le X be a real linear space, (Y, µ, T M ) be a complee rom p-normed space, Φ : X D + be a mapping such ha, for some α (0, Np ), he following relaions hold: min {Φ ix,x (α), Φ 0,4x (α)} min {Φ ix,x (), Φ 0,x ()}, x X, > 0, (.3),N,N lim Φ n n x, n y( nnp ) = 1, x, y X, > 0. (.4) If f : X Y is a probabilisic Φ-approximaely monomial mapping of degree N wih f(0) = 0, hen here exiss a unique monomial mapping of degree N, M : X Y, so ha In addiion, µ f(x) M(x) () min,n {Φ ix,x (N!) p ( Np α) 1 + N ) p, Φ (N!) p ( Np α) 0,x 1 + N ) p }, M(x) = lim n x X, > 0. (.5) f( n x), x X. (.6) nn Proof. We will follow an idea of Gilányi (see [8]) o obain an esimae of µ f(x) f( n x) nn (). For i = 0, N, subsiue (x, y) wih (ix, x) in (.1) o ge which implies By using (P3), we obain µ N or equivalenly, via (.), µ N x f(ix) N!f(x)() Φ ix,x (), x X, > 0, (.7) µ ( N ) N x f(ix) )N!f(x) ) N x f(ix) N N!f(x) µ N x f(0) N N!f(x) Also, by seing i = 0 replacing x wih x in (.7), we ge (( ) N p ) Φ ix,x (), x X, > 0. ( ) N p ) min {Φ ix,x ()}, x X, > 0,,N ( ) N p ) min {Φ ix,x ()}, x X, > 0.,N µ N x f(0) N!f(x) () Φ 0,x(), x X, > 0. Consequenly, µ N! N f(x) N!f(x) (( 1 + N ( ) ) ) N p min {Φ ix,x (), Φ 0,x ()},,N x X, > 0,

5 C. Zaharia, J. Nonlinear Sci. Appl. 6 (013), or µ f(x) f(x) N 1 + N ) p Np (N!) p min {Φ ix,x (), Φ 0,x ()}, x X, > 0. (.8),N Now, le G(x, ) := min {Φ ix,x (), Φ 0,x ()}. Noe ha, by (.3), G has he propery G(x, α) G(x, ),,N for all x X all > 0. We denoe by E he space of all mappings g : X Y wih g(0) = 0, define he mapping d G : E E [0, ] as d G (g, h) = inf{a R : µ g(x) h(x) (a) G(x, ), x X, > 0}. Following he same reasoning as in [13], i can be shown ha (E, d G ) is a complee generalized meric space. We claim ha J : E E, Jg(x) = g(x) α, is a sric conracion, wih he Lipschiz consan. Indeed, N Np le g, h E be so ha d G (g, h) < ε. This implies µ g(x) h(x) (ε) G(x, ), x X, > 0. Then ( α ) µ Jg(x) Jh(x) Np ε = µ g(x) h(x) (αε) G(x, α) G(x, ), x X, > 0, so d G (Jg, Jh) α ε. Therefore d Np G (Jg, Jh) α d Np G (g, h), our claim is proved. Moreover, from (.8), 1 + N ) p d G (f, Jf) Np (N!) p. By Lemma., J has a fixed poin M : X Y wih he following properies: f( n x) (i) d G (J n f, M) 0 when n, so lim = M(x), for all x X. n nn (ii) d G (f, M) 1 α d G (f, Jf), so he esimaion (.5) holds. 1 N p (iii) M is he unique fixed poin of J in he se {g E : d G (f, g) < }. Finally, we mus show ha M is a monomial mapping of degree N. Subsiuing x y by n x n y in (.1), we obain µ N n y f( n x) N!f( n y) () Φ n x, n y() or for all x X all > 0, so Now, µ N ( 1) ( )f( N n (x+iy)) N!f( n y) µ N ( 1) ( ) N f(n (x+iy)) nn N! f(n y) µ N y M(x) N!M(y)() = µ N min{µ N () Φ n x, n y() nn () Φ n x, n y( nnp ), x X, > 0. (.9) ( 1) ( )M(x+iy) N!M(y) N () ( 1) ( )(M(x+iy) N f(n (x+iy)) nn ) N!(M(y) f(n y) nn ) µ N ( 1) ( ) N f(n (x+iy)) nn N! f(n y) nn ( ( ), ) }, x X, > 0. Boh expressions on he righ h side of he inequaliy above end o 1 as n ends o infiniy, he laer due o (.4) (.9). Thus, we have shown ha N y M(x) N!M(y) = 0, which concludes he proof.

6 C. Zaharia, J. Nonlinear Sci. Appl. 6 (013), Similarly, one can obain he following resul for α > Np. Theorem.6. Le X be a real linear space, (Y, µ, T M ) be a complee rom p-normed space, Φ : X D + be a mapping such ha, for some α > Np, min {Φ ix,x (), Φ 0,x ()} min {Φ ix,x (α), Φ 0,4x (α)}, x X, > 0, (.10),N,N ( ) lim Φ n n x, n y nnp = 1, x, y X, > 0. (.11) If f : X Y is a probabilisic Φ-approximaely monomial mapping of degree N wih f(0) = 0, hen here exiss a unique monomial mapping of degree N, M : X Y, so ha µ f(x) M(x) () min {Φ ix,n, x Moreover, M(x) = lim n nn f ( x n ), for all x X. (N!) p (α Np ) α(1 + N ) p), Φ (N!) p (α Np ) 0,x α(1 + N ) p) }, Proof. Relaion (.8) implies 1 + N ) p µ N f( x ) f(x) (N!) p min,n {Φ ix, x (), Φ 0,x ()}, x X, > 0. x X, > 0. (.1) Se G(x, ) := min {Φ ix,n, x (), Φ 0,x ()} noe ha, by (.10), i has he propery G ( x, α) G(x, ). We define d G (g, h) = inf{a R + : µ g(x) h(x) (a) G(x, ), x X, > 0} on he space E = {g : X Y : g(0) = 0} noe ha (E, d G ) is a complee generalized meric space. As in he proof of Theorem.5, we can show ha J : E E, Jg(x) = N g ( ) x, is a sric conracion, wih he Lipschiz consan Np α, is only fixed poin M : X Y so ha d G(f, M) < is he unique monomial mapping wih he required properies. Remark.7. Noe ha, by (.8), min {Φ ix,n, x min,n (N!) p (α Np ) α(1 + N ) p), Φ (N!) p (α Np ) 0,x α(1 + N ) p) } {Φ ix,x (N!) p (α Np ) 1 + N ) p, Φ (N!) p (α Np ) 0,x 1 + N ) p }, so ha he esimaion (.1) is comparable o ha in Theorem.5.

7 C. Zaharia, J. Nonlinear Sci. Appl. 6 (013), Remark.8. Insead of he hypohesis (.3) + (.4), one can consider a condiion wih a simpler formulaion, namely Φ x,y (α) Φ x,y (), x, y X, > 0. (.13) However, we noe ha he wo are no equivalen. I is immediae ha (.13) implies (.3) (.4). The following example shows ha he converse does no hold: Example.9. Le (X, ) be a normed space. The mapping Φ : X X D + defined by { 1, if here exiss a R so ha y = ax, Φ x,y () = oherwise, + x y Np+1, saisfies he condiions (.3) (.4), bu, for all linearly independen x, y X, Φ x,y (α) < Φ x,y (). Similarly, he condiion Φ x,y (α) Φ x,y (), x, y X, > 0 can be considered insead of he hypohesis (.10) + (.11) in Theorem Applicaions As consequences of Theorem.5, we will obain generalized Ulam - Hyers sabiliy resuls for he case of rom normed spaces β-normed spaces compare hem wih hose already exising in he lieraure. Resuls regarding he case α > Np can be derived in an idenical manner from Theorem.6. In he seing of rom normed spaces, our heorem reads as follows: Theorem 3.1. (compare wih [4, Theorem 4.1]) Le X be a real linear space (Y, µ, T M ) be a complee rom normed space. Suppose ha he mapping f : X Y wih f(0) = 0 saisfies µ N y f(x) N!f(y)() Φ x,y (), x, y X, > 0, where Φ : X D + is a given funcion. If here exiss α (0, N ) such ha min {Φ ix,x (α), Φ 0,4x (α)} min {Φ ix,x (), Φ 0,x ()}, x X, > 0,N,N lim Φ n n x, n y( nn ) = 1, x X, > 0, hen here exiss a unique monomial mapping of degree N, M : X Y, which saisfies he inequaliy!( N ) ( α) N!( N ) α) µ f(x) M(x) () min {Φ ix,x,n N, Φ 0,x + 1 N }, + 1 Proof. Se p = 1 in Theorem.5. x X, > 0. In view of Remark.8, our wo hypoheses on Φ could have been replaced wih Φ x,y (α) Φ x,y (), which is he condiion ha appears in [4]. Recall ha a β-normed space (0 < β 1) is a pair (Y, β ), where Y is a real linear space β is a real valued funcion on Y (called a β-norm) saisfying he following condiions: (i) x β 0 for all x Y x β = 0 if only if x = 0; (ii) λx β = λ β x β for all x Y λ R; (iii) x + y β x β + y β for all x, y Y.

8 C. Zaharia, J. Nonlinear Sci. Appl. 6 (013), In [3], Cădariu Radu used he fixed poin mehod o obain he following generalized Ulam - Hyers sabiliy resul for he monomial funcional equaion in β-normed spaces: Theorem 3.. ([3, Theorem.1]) Le X be a linear space, Y be a complee β-normed space, assume we are given a funcion ϕ : X X [0, ) wih he following propery: ϕ( n x, n y) lim n nnβ = 0, x, y X. (3.1) Suppose ha he mapping f : X Y wih f(0) = 0 verifies he conrol condiion If here exiss a posiive consan L < 1 such ha he mapping saisfies he inequaliy N y f(x) N!f(y) β ϕ(x, y), x, y X. (3.) ( x ψ(x) = 1 (N!) β ϕ(0, x) + N ( ) ix ϕ, x ) ), x X, ψ(x) Nβ Lψ(x), x X, (3.3) hen here exiss a unique monomial mapping of degree N, M : X Y, wih he following propery: f(x) M(x) β L ψ(x), x X. (3.4) 1 L By noing ha every β-normed space (Y, β ) induces a rom p-normed space (Y, µ, T M ) wih β = p µ x () = + x β, from Theorem.5 we obain he following new sabiliy resul. Theorem 3.3. Le X be a real linear space, (Y, β ) be a complee β-normed space, ϕ : X [0, ) be a mapping so ha (3.1) holds, for some α (0, Nβ ), max,n {ϕ(ix, x), ϕ(0, 4x)} α max {ϕ(ix, x), ϕ(0, x)}, x X. (3.5),N Suppose ha f : X Y wih f(0) = 0 verifies he conrol condiion (3.). Then here exiss a unique monomial mapping of degree N, M : X Y, wih he following propery: f(x) M(x) β 1 + N ) β (N!) β ( Nβ α) max {ϕ(ix, x), ϕ(0, x)}, x X. (3.6),N Proof. Consider he induced rom p-normed space (Y, µ, T M ) apply Theorem.5 wih Φ x,y () = +ϕ(x,y). Remark 3.4. Theorem 3.3 provides an alernaive version for he sabiliy resul obained in [18] in he paricular case of quasi-p-normed spaces. References [1] M. Alber J. A. Baker, Funcions wih bounded n-h differences, Ann. Polonici Mah. 43 (1983), [] T. Aoki, On he sabiliy of he linear ransformaion in Banach spaces, J. Mah. Soc. Japan (1950), [3] L. Cădariu V. Radu, Remarks on he sabiliy of monomial funcional equaions, Fixed Poin Theory 8, No. (007), , 3, 3. [4] L. Cădariu V. Radu, Fixed poins generalized sabiliy for funcional equaions in absrac spaces, J. Mah. Inequal. 3, No. 3 (009), , 3

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