A FIXED POINT APPROACH TO ORTHOGONAL STABILITY OF AN AQ-FUNCTIONAL EQUATIONS IN MODULAR SPACES IZ-IDDINE EL-FASSI
∗
Abstract. In this paper, we present a fixed point method to prove generalized Hyers-Ulam stability of the following orthogonally additive - quadratic functional equation a2 − a [f (x + y) − f (x − y)] 2 where a ∈ N − {0, 1}, in modular spaces. f (x + ay) = f (x) + a2 f (y) −
1. INTRODUCTION
T
he concept of stability for a functional equation arises when one replaces a functional equation by an inequality which acts as a perturbation of the equation. Recall that the problem of stability of functional equations was motivated by a question of Ulam being asked in 1940 (see [39]) and Hyers answer to it was published in [11]. Hyers’s theorem was generalized by Aoki [3] for additive mappings and by Rassias [31] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [31] has provided a lot of influences in the development of the Hyers-Ulam-Rassias stability of functional equations (see [25]). During the last decades several stability problems of functional equations have been investigated by a number of mathematicians in various spaces, such as fuzzy normed spaces, orthogonal normed spaces and random normed spaces; see [4, 6, 12, 13, 23, 34, 42] and reference therein. Recently, Gh. Sadeghi [35] proved the Hyers-Ulam stability of the generalized Jensen functional equation f (rx+sy) = rg(x)+sh(x) in modular space, using the fixed point method. The theory of modulars on linear spaces and the corresponding theory of modular linear spaces were founded by H. Nakano [27] and were intensively developed by his mathematical school: S. Koshi, T. Shimogaki, S. Yamamuro [15, 41] and others. Further and the most complete development of these theories are due to W. Orlicz, S. Mazur, J. Musielak, W. A. Luxemburg, Ph. Turpin [17, 19, 38, 26] and their collaborators. In the present time the theory of modulars and modular spaces is extensively applied, in particular, in the study of various W. Orlicz spaces [28] and interpolation theory [16], which in their turn have broad applications [18, 26]. The importance for applications consists in the richness of the structure of modular spaces, that-besides being Banach spaces (or F −spaces in more general setting)- are equipped with modular equivalent of norm or metric notions. ∗
Corresponding author. 2010 Mathematics Subject Classification. 39B55, 39B52, 39B82, 47H09. Key words and phrases. Hyers-Ulam stability, fixed point, Orthogonality additive-quadratic functional equation, Modular space.
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Assume that E is a real inner product space and f : E → R is a solution of the orthogonal Cauchy functional equation f (x + y) = f (x) + f (y) with < x, y >= 0. 2 By the Pythagorean theorem, f (x) = kxk is a solution of the conditional equation. Of course, this function does not satisfy the additivity equation everywhere. Thus orthogonal Cauchy equation is not equivalent to the classic Cauchy equation on the whole inner product space. Pinsker [30] characterized orthogonally additive functional on an inner product space when the orthogonality is the ordinary one in such spaces. Sundaresan [36] generalized this result to arbitrary Banach spaces equipped with Birkhoff-James orthogonality. The orthogonally Cauchy functional equation f (x + y) = f (x) + f (y), with x⊥y, in which ⊥ is an abstract orthogonality relation, was first investigated by S. Gudder and D. Strawther [10]. R¨atz and Szab´o [33] investigated the problem in a rather more general framework. There are several orthogonality notions on a real normed spaces as BirkhoffJames, semi-inner product, Carlsson, Singer, Roberts, Pythagorean, isosceles and Diminnie (see, e.g., [1]). Let us recall the orthogonality space in the sense of R¨atz; cf. [32]. Suppose E is a real vector space with dim E ≥ 2 and ⊥ is a binary relation on E with the following properties: (O1) totality of ⊥ for zero: x⊥0, 0⊥x for all x ∈ E; (O2) independence: if x, y ∈ E − {0}, x⊥y, then, x, y are linearly independent; (O3) homogeneity: if x, y ∈ E, x⊥y, then αx⊥βy for all α, β ∈ R; (O4) the Thalesian property: if P is a 2-dimensional subspace of E. If x ∈ P and λ ∈ R+ , then there exists y0 ∈ P such that x⊥y0 and x + y0 ⊥λx − y0 . The pair (E, ⊥) is called an orthogonality space. By an orthogonality normed space, we mean an orthogonality space having a normed structure. In 1995, R. Ger and J. Sikorska [9] obtained the orthogonal stability of Cauchy functional equation f (x + y) = f (x) + f (y), with f is a mapping from an orthogonality space E into a real Banach space F . Let (E, ⊥) be an orthogonality space and (G, +) be an Abelian group. A mapping f : E → G is said to be (orthogonally) quadratic if it satisfies f (x + y) + f (x − y) = 2f (x) + 2f (y), x⊥y
(1.1)
for all x, y ∈ E. The orthogonally quadratic functional equation (1.1), was first investigated by Vajzovi´c [40] when E is a Hilbert space, G is equal to C, f is continuous and ⊥ means the Hilbert space orthogonality. Later Drlijevi´c [5], Fochi [8], Szab´ o [37], Moslehian [20, 21] and Moslehian and Th. M. Rassias [22] and Paganoni and R¨ atz [29] have investigated the orthogonal stability of functional equations. Iz. El-Fassi and S. Kabbaj [7] investigated the Hyers-Ulam-Rassias stability of orthogonal quadratic functional equation (1.1) in Modular spaces. M. Almahalebi and S. Kabbaj in [2] proved the Hyers-Ulam stability of the following orthogonally additive- quadratic functional equation (briefly, AQ-functional
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equation) a2 − a [f (x + y) − f (x − y)], x⊥y 2 where a ∈ N − {0, 1}, in orthogonality spaces, using fixed point methode. f (x + ay) = f (x) + a2 f (y) −
(1.2)
In the present paper, we establish the stability of orthogonal additive-quadratic functional equation (1.2) in Modular spaces. Therefore, we generalized the main theorem 2.5 of [2]. 2. Preliminaries In this section, we give the definitions that are important in the following. Definition 2.1. Let X be an arbitrary vector space. (a) A functional ρ : X → [0, ∞] is called a modular if for arbitrary x, y ∈ X, (i) ρ(x) = 0 if and only if x = 0, (ii) ρ(αx) = ρ(x) for every scaler α with |α| = 1, (iii) ρ(αx + βy) ≤ ρ(x) + ρ(y) if and only if α + β = 1 and α, β ≥ 0, (b) if (iii) is replaced by (iii)’ ρ(αx + βy) ≤ αρ(x) + βρ(y) if and only if α + β = 1 and α, β ≥ 0, then we say that ρ is a convex modular. A modular ρ defines a corresponding modular space, i.e, the vector space Xρ given by Xρ = {x ∈ X : ρ(λx) → 0 as λ → 0} . Let ρ be a convex modular, the modular space Xρ can be equipped with a norm called the Luxemburg norm, defined by o n x kxkρ = inf λ > 0 : ρ( ) ≤ 1 . λ A function modular is said to satisfy the ∆2 −condition if there exists κ > 0 such that ρ(2x) ≤ κρ(x) for all x ∈ Xρ . Definition 2.2. Let {xn } and x be in Xρ . Then ρ (i) we say {xn } is ρ−convergent to x and write xn → x if and only if ρ(xn −x) → 0 as n → ∞, (ii) the sequence {xn }, with xn ∈ Xρ , is called ρ−Cauchy if ρ(xn − xm ) → 0 as m, n → ∞, (iii) a subset S of Xρ is called ρ−complete if and only if any ρ−Cauchy sequence is ρ−convergent to an element of S. (iv) a subset S of Xρ is called ρ−bounded if δρ (S) = sup{ρ(x − y); x, y ∈ S} < ∞. The modular ρ has the Fatou property if and only if ρ(x) ≤ limn→∞ inf ρ(xn ) whenever the sequence {xn } is ρ−convergent to x. For further details and proofs, we refer the reader to [26]. Remark 2.3. If x ∈ Xρ then ρ(ax) is a nondecreasing function of a ≥ 0. Suppose that 0 < a < b, then property (iii) of definition 2.1 with y = 0 shows that a ρ(ax) = ρ( bx) ≤ ρ(bx). b
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Moreover, if ρ is convex modular on X and |α| ≤ 1 then, ρ(αx) ≤ |α|ρ(x) and also ρ(x) ≤ 21 ρ(2x) ≤ κ2 ρ(x) if ρ satisfy the ∆2 − condition for all x ∈ X. 3. Orthogonal Stability of Eq (1.2) in Modular Spaces Throughout this paper, we assume that the convex modular ρ has the Fatou property such that satisfies the ∆2 −condition with 0 < κ ≤ 2. In addition, we assume that (E, ⊥) denotes an orthogonality normed space and Xρ is ρ−complete modular space. For a given function f : E → Xρ , we define a2 − a [f (x + y) − f (x − y)] 2 for all x, y ∈ E with x⊥y, where ⊥ is the orthogonality in the sense of R¨atz. In this section we establish the Hyers-Ulam stability of orthogonal AQ-functional equation (1.2) in modular spaces. Df (x, y) = f (x + ay) − f (x) − a2 f (y) +
Theorem 3.1. Let f : E → Xρ be an even mapping satisfies the condition f (0) = 0 and an inequality of the form ρ(Df (x, y)) ≤ φ(x, y), x⊥y
(3.1)
2
for all x, y ∈ E, where φ : E → [0, ∞) is a given function such that φ(ax, ay) ≤ a2 Lφ(x, y)
(3.2)
for all x, y ∈ E and a constant 0 < L < 1. Then there exist unique quadratic mapping Q : E → Xρ such that ρ(f (x) − Q(x)) ≤
1 φ(0, x) a2 (1 − L)
(3.3)
for all x ∈ E. Proof. Put x = 0 and y = x in (3.1). We can do this because of (O1), we get ρ(f (ax) − a2 f (x)) ≤ φ(0, x) for all x ∈ E. Hence f (ax) 1 1 ρ( 2 − f (x)) ≤ 2 ρ(f (ax) − a2 f (x)) ≤ 2 φ(0, x) a a a for all x ∈ E. We consider the set
(3.4)
(3.5)
Mρe = {g : E → Xρ ; g(0) = 0} and introduce the convex modular ρe on Mρe as follows ρe(g) = inf{K > 0 : ρ(g(x)) ≤ Kφ(0, x)}. It is sufficient to show that ρe satisfies the following condition ρe(αg + βh) ≤ αe ρ(g) + β ρe(h) if α + β = 1 and α, β ≥ 0. Let ε > 0 be given. Then there exist K1 > 0 and K2 > 0 such that K1 ≤ ρe(g) + ε; ρ(g(x)) ≤ K1 φ(0, x) and K2 ≤ ρe(h) + ε; ρ(h(x)) ≤ K2 φ(0, x).
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If α + β = 1 and α, β ≥ 0, then we get ρ(αg(x) + βh(x)) ≤ αρ(g(x)) + βρ(h(x)), whence ρe(αg + βh) ≤ αe ρ(g) + β ρe(h) + (α + β)ε. Hence we have ρe(αg + βh) ≤ αe ρ(g) + β ρe(h). Moreover, ρe satisfies the ∆2 −condition with 0 < κ ≤ 2. Let {gn } be a ρe−Cauchy sequence in Mρe and let ε > 0 be given. Then there exists a positive integer N ∈ N such that ρe(gn − gm ) ≤ ε for all m, n ≥ N. Now by considering the definition of the modular ρe, we see that ρ(gn (x) − gm (x)) ≤ εφ(0, x)
(3.6)
for all x ∈ E and m, n ≥ N. If x is any given point of E, (3.6) implies that {gn (x)} is a ρ−Cauchy sequence in Xρ . Since Xρ is ρ−complete, so {gn (x)} is ρ−convergent in Xρ , for each x ∈ E. Hence, we can define a function g : E → Xρ by g(x) = lim gn (x), n→∞
for any x ∈ E. Let m increase to infinity, then (3.6) implies that ρe(gn − g) ≤ ε for all n > N , since ρ has the Fatou property. Thus {gn } is ρe−convergent sequence in Mρe. Therefore Mρe is ρe−complete. Now, we consider the linear mapping J : Mρe → Mρe defined by Jg(x) :=
1 g(ax) a2
for all g ∈ Mρe and x ∈ E. Let g, h ∈ Mρe and let K ∈ [0, ∞) be an arbitrary constant with ρe(g − h) ≤ K. From the definition of ρe, we have ρ(g(x) − h(x)) ≤ Kφ(0, x) for all x ∈ E. By the assumption and the last inequality, we get ρ(
1 1 g(ax) h(ax) − ) ≤ 2 ρ(g(ax) − h(ax)) ≤ 2 Kφ(0, ax) ≤ K.Lφ(0, x) a2 a2 a a
for all x ∈ E. Hence ρe(Jg − Jh) ≤ Le ρ(g − h) for all g, h ∈ Mρe. That is, J is a ρe−strict contraction. we show that the ρe−strict mapping J satisfies the conditions of theorem 3.4 of [14]. Replacing x by ax in (3.5) we get ρ(
f (a2 x) 1 − f (ax)) ≤ 2 φ(0, ax) a2 a
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(3.7)
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IZ-IDDINE EL-FASSI, JOURNAL OF GLOBAL RESEARCH IN MATHEMATICAL ARCHIVES....
for all x ∈ E. Hence ρ(
f (a2 x) 1 f (a2 x) − f (x)) = ρ( 2 ( − a2 f (x))) 2 2 (a ) a a2 1 f (a2 x) ≤ 2 ρ( − f (ax) + f (ax) − a2 f (x)) a a2 f (a2 x) κ − f (ax)) + ρ(f (ax) − a2 f (x))] ≤ 2 [ρ( 2a a2 κ κ φ(0, ax) + 2 φ(0, x) ≤ 2(a2 )2 2a κ 1 ≤ φ(0, ax) + 2 φ(0, x) 2 2 2(a ) a
(3.8)
for all x ∈ E. By using (3.5), (3.8) and the principle of mathematical induction, we can easily see that n
ρ(
X κi−1 f (an x) − f (x)) ≤ φ(0, ai−1 x) i−1 a2i a2n 2 i=1
(3.9)
for all x ∈ E and n ∈ N. Indeed, for n = 1 the relation (3.9) is true. Assume that the relation (3.9) is true for n, and show this relation rest true for n + 1. Thus we have ρ(
f (an+1 x) 1 f (an+1 x) 1 f (an+1 x) 2 − f (x)) = ρ( ( − a f (x)) ≤ ρ( − a2 f (x)) a2 a2n a2 a2n a2(n+1) κ f (an+1 x) ≤ 2 [ρ( − f (ax)) + ρ(f (ax) − a2 f (x))] 2a a2n n κ X κi−1 ≤ 2[ φ(0, ai x) + φ(0, x)] 2a i=1 2i−1 a2i ≤
n X i=0
=
n+1 X i=1
κi 2i a2(i+1) κi−1 2i−1 a2i
φ(0, ai x)
φ(0, ai−1 x)
for all x ∈ E. Hence the relation (3.9) is true for all x ∈ E and n ∈ N∗ . Thus ρ(
n n X f (an x) 1 X i−1 κi−1 i−1 L φ(0, x) − f (x)) ≤ φ(0, a x) ≤ a2n 2i−1 a2i a2 i=1 i=1
≤
(1 − Ln )φ(0, x) φ(0, x) ≤ 2 2 a (1 − L) a (1 − L)
(3.10)
for all x ∈ E. Next, we assert that δρe(f ) = sup{e ρ(J n f − J m f ); m, n ∈ N} < ∞.
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It follow from (3.10) that ρ(
f (an x) f (am x) f (an x) f (am x) − ) = ρ( − f (x) + f (x) − ) a2n a2m a2n a2m n κ f (a x) f (am x) ≤ [ρ( 2n − f (x)) + ρ(f (x) − )] 2 a a2m κ ≤ 2 φ(0, x) a (1 − L)
for all x ∈ E. Hence
κ − L) for all m, n ∈ N. By the definition of δρe(f ), we have δρe(f ) < ∞. Lemme 3.3 of [14] schows that {J n f } is ρe− converges to Q ∈ Mρe. Since ρ has Fatou property inequality (3.10) give ρe(JQ − f ) < ∞. Replacing x by an x in (3.5) we obtain ρe(J n f − J m f ) ≤
ρ(
a2 (1
f (an+1 x) 1 − f (an x)) ≤ 2 φ(0, an x) 2 a a
for all x ∈ E. Thus f (an+1 x) f (an x) 1 f (an+1 x) 1 f (an+1 x) ρ( 2n+2 − ) = ρ( 2n ( − f (an x))) ≤ 2n ρ( − f (an x)) 2n 2 a a a a a a2 1 Ln ≤ 2n+2 φ(0, an x) ≤ 2 φ(0, x) ≤ φ(0, x) a a for all x ∈ E. Therefore ρe(JQ − Q) < ∞. It follow from [[14], Theorem 3.4] that, the ρe−limit Q ∈ Mρe of {J n f } is a fixed point of J. If we replace x by an x and y by an y in (3.1) and use (O3) we get 1 1 Df (an x, an y) ) ≤ 2n ρ(Df (an x, an y) ≤ 2n φ(an x, an y) ≤ Ln φ(x, y) 2n a a a for all x, y ∈ E. Taking the limit and since 0 < L < 1, we deduce that DQ(x, y) = 0 for all x, y ∈ E. It follow from (3.10) that 1 . ρe(Q − f ) ≤ 2 a (1 − L) ρ(
If Q0 is another fixed point of J, then κ κ ρe(Q − Q0 ) = ρe(JQ − JQ0 ) ≤ [e ρ(JQ − f ) + ρe(JQ0 − f )] ≤ 2 < ∞. 2 a (1 − L) Since J is ρe−strict contraction, we get ρe(Q − Q0 ) = ρe(JQ − JQ0 ) ≤ Le ρ(Q − Q0 ), which implies that ρe(Q − Q0 ) = 0 or Q = Q0 . This completes the proof of our theorem. Corollary 3.2. [2] Assume that (X, k.k) is Banach space. Let f : E → X be an even mapping satisfies the condition f (0) = 0 and an inequality of the form kDf (x, y)k ≤ φ(x, y), x⊥y
(3.11)
2
for all x, y ∈ E, where φ : E → [0, ∞) is a given function such that φ(ax, ay) ≤ a2 Lφ(x, y)
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for all x, y ∈ E and a constant 0 < L < 1. Then there exist unique quadratic mapping Q : E → X such that 1 kf (x) − Q(x)k ≤ 2 φ(0, x) a (1 − L) for all x ∈ E. Proof. It is well known the every normed space in a modular space with the modular ρ(x) = kxk. Theorem 3.3. Let f : E → Xρ be an odd mapping satisfies the condition f (0) = 0 and an inequality of the form ρ(Df (x, y)) ≤ φ(x, y), x⊥y
(3.12)
for all x, y ∈ E, where φ : E 2 → [0, ∞) is a given function such that φ(ax, ay) ≤ aLφ(x, y)
(3.13)
for all x, y ∈ E and a constant 0 < L < 1. Then there exist unique additive mapping T : E → Xρ such that 1 φ(0, x) (3.14) ρ(f (x) − T (x)) ≤ a(1 − L) for all x ∈ E. Proof. Letting x = 0 and y = x in (3.12). We can do this because of (O1), we get ρ(f (ax) − af (x)) ≤ φ(0, x) for all x ∈ E. Hence f (ax) 1 1 ρ( − f (x)) ≤ ρ(f (ax) − af (x)) ≤ φ(0, x) a a a for all x ∈ E. We consider the set
(3.15)
(3.16)
Mρe = {g : E → Xρ ; g(0) = 0} and introduce the convex modular ρe on Mρe as follows ρe(g) = inf{K > 0 : ρ(g(x)) ≤ Kφ(0, x)}. Similarly to the proof of Theorem 3.1 we obtain ρe satisfies the following conditions: (1) ρe(αg + βh) ≤ αe ρ(g) + β ρe(h) if α + β = 1 and α, β ≥ 0. (2) ρe satisfies the ∆2 −condition with 0 < κ ≤ 0. Moreover Mρe is ρe−complete. Now, we consider the function J : Mρe → Mρe defined by 1 Jg(x) := g(ax) a for all g ∈ Mρe and x ∈ E. Let g, h ∈ Mρe and let K ∈ [0, ∞) be an arbitrary constant with ρe(g − h) ≤ K. Then, we have ρ(g(x) − h(x)) ≤ Kφ(0, x) for all x ∈ E. By the assumption and the last inequality, we get g(ax) h(ax) 1 1 ρ( − ) ≤ ρ(g(ax) − h(ax)) ≤ Kφ(0, ax) ≤ K.Lφ(0, x) a a a a
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for all x ∈ E. Hence ρe(Jg − Jh) ≤ Le ρ(g − h) for all g, h ∈ Mρe. That is, J is a ρe−strict contraction. we show that the ρe−strict mapping J satisfies the conditions of theorem 3.4 of [14]. Replacing x by ax in (3.16) we get ρ(
f (a2 x) 1 − f (ax)) ≤ φ(0, ax) a a
(3.17)
for all x ∈ E. Hence 1 f (a2 x) f (a2 x) − f (x)) = ρ( ( − af (x))) ρ( 2 a a a κ f (a2 x) ≤ [ρ( − f (ax)) + ρ(f (ax) − af (x))] 2a a κ 1 (3.18) ≤ 2 φ(0, ax) + φ(0, x) 2a a for all x ∈ E. By using (3.16), (3.18) and the principle of mathematical induction, we can easily see that n
ρ(
X κi−1 f (an x) φ(0, x) (1 − Ln )φ(0, x) i−1 ≤ (3.19) − f (x)) ≤ φ(0, a x) ≤ i−1 ai an 2 a(1 − L) a(1 − L) i=1
for all x ∈ E. Next, we assert that δρe(f ) = sup{e ρ(J n f − J m f ); m, n ∈ N} < ∞. It follow from (3.19) that ρ(
f (an x) f (am x) κ f (an x) f (am x) − ) ≤ [ρ( − f (x)) + ρ(f (x) − )] n m n a a 2 a am κ φ(0, x) ≤ a(1 − L)
for all x ∈ E. Hence
κ a(1 − L) for all m, n ∈ N. By the definition of δρe(f ), we have δρe(f ) < ∞. Lemme 3.3 of [14] schows that {J n f } is ρe− converges to T ∈ Mρe. Since ρ has Fatou property inequality (3.19) give ρe(JT − f ) < ∞. Replacing x by an x in (3.16) we obtain ρe(J n f − J m f ) ≤
ρ(
f (an+1 x) 1 − f (an x)) ≤ φ(0, an x) a a
for all x ∈ E. Thus f (an+1 x) f (an x) 1 Ln n ρ( − ) ≤ φ(0, a x) ≤ φ(0, x) ≤ φ(0, x) an+1 an an+1 a for all x ∈ E. Therefore ρe(JT − T ) < ∞. It follow from [[14], Theorem 3.4] that, the ρe−limit T ∈ Mρe of {J n f } is a fixed point of J. If we replace x by an x and y by an y in (3.1) and use (O3) we get Df (an x, an y) 1 1 ) ≤ n ρ(Df (an x, an y) ≤ n φ(an x, an y) ≤ Ln φ(x, y) n a a a for all x, y ∈ E. If n → ∞ and since 0 < L < 1, then DT (x, y) = 0 for all x, y ∈ E. ρ(
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It follow from (3.19) that ρe(T − f ) ≤
1 . a(1 − L)
If T 0 is another fixed point of J, then κ κ ρe(T − T 0 ) = ρe(T − f + f − T 0 ) ≤ [e ρ(T − f ) + ρe(T 0 − f )] ≤ < ∞. 2 a(1 − L) Since J is ρe−strict contraction, we get ρe(T − T 0 ) = ρe(JT − JT 0 ) ≤ Le ρ(T − T 0 ), which implies that ρe(T − T 0 ) = 0 or T = T 0 . This completes the proof of theorem. Corollary 3.4. [2] Assume that (X, k.k) is Banach space. Let f : E → X be an odd mapping satisfies the condition f (0) = 0 and an inequality of the form kDf (x, y)k ≤ φ(x, y), x⊥y
(3.20)
2
for all x, y ∈ E, where φ : E → [0, ∞) is a given function such that φ(ax, ay) ≤ aLφ(x, y) for all x, y ∈ E and a constant 0 < L < 1. Then there exist unique quadratic mapping T : E → X such that 1 φ(0, x) kf (x) − T (x)k ≤ a(1 − L) for all x ∈ E. Proof. It is well known the every normed space in a modular space with the modular ρ(x) = kxk. Theorem 3.5. Let f : E → Xρ be a mapping satisfies the condition f (0) = 0 and an inequality of the form ρ(Df (x, y)) ≤ φ(x, y), x⊥y
(3.21)
2
for all x, y ∈ E, where φ : E → [0, ∞) is a given function such that φ(ax, ay) ≤ aLφ(x, y)
(3.22)
for all x, y ∈ E and a constant 0 < L < 1. Then there exist unique additive mapping T : E → Xρ and unique quadratic mapping Q : E → Xρ such that ρ(f (x) − T (x) − Q(x)) ≤
κ2 (1 + a − 2L) (φ(0, x) + φ(0, −x)) 8a(a − L)(1 − L)
(3.23)
for all x ∈ E. (−x) Proof. Let f e be an even mapping such that f e (x) = f (x)+f . We have f e (0) = 0 2 and 1 1 ρ(Df e (x, y)) = ρ( [Df (x, y) + Df (−x, −y)]) ≤ ρ(Df (x, y) + Df (−x, −y)) 2 2 κ ≤ (φ(x, y) + φ(−x, −y)) 4 for all x, y ∈ E. Putting 0 < L0 = La < 1 we get
φ(ax, ay) ≤ a2 L0 φ(x, y)
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for all x, y ∈ E. So, by Theorem 3.1, there exist a unique orthogonally quadratic mapping Q : E → Xρ such that κ 1 (φ(0, x) + φ(0, −x)) a2 (1 − L0 ) 4 κ = (φ(0, x) + φ(0, −x)) 4a(a − L)
ρ(f e (x) − Q(x)) ≤
(3.24)
(−x) . for all x ∈ E. Similarly, let f o be an odd mapping such that f o (x) = f (x)−f 2 We have f o (0) = 0 and κ ρ(Df o (x, y)) ≤ (φ(x, y) + φ(−x, −y)) 4 for all x, y ∈ E. By Theorem 3.3, there exist a unique orthogonally additive mapping T : E → Xρ such that κ (φ(0, x) + φ(0, −x)) (3.25) ρ(f o (x) − T (x)) ≤ 4a(1 − L)
for all x ∈ E. It follow from (3.24) and (3.25) that ρ(f (x) − T (x) − Q(x)) = ρ(f o (x) − T (x) + f e − Q(x)) κ κ ≤ ρ(f o (x) − T (x)) + ρ(f e (x) − Q(x)) 2 2 κ2 κ2 (φ(0, x) + φ(0, −x)) + (φ(0, x) + φ(0, −x)) ≤ 8a(1 − L) 8a(a − L) κ2 κ2 = + (φ(0, x) + φ(0, −x)) 8a(1 − L) 8a(a − L) for all x ∈ E. This completes the proof of theorem.
Corollary 3.6. [2] Assume that (X, k.k) is Banach space. Let f : E → X be a mapping satisfies the condition f (0) = 0 and an inequality of the form kDf (x, y)k ≤ φ(x, y), x⊥y for all x, y ∈ E, where φ : E 2 → [0, ∞) is a given function such that φ(ax, ay) ≤ aLφ(x, y) for all x, y ∈ E and a constant 0 < L < 1. Then there exist unique additive mapping T : E → X and unique quadratic mapping Q : E → X such that kf (x) − T (x) − Q(x)k ≤
(1 + a − 2L) (φ(0, x) + φ(0, −x)) 2a(a − L)(1 − L)
for all x ∈ E. Proof. It is well known the every normed space in a modular space with the modular ρ(x) = kxk and κ = 2. A convex function ϕ defined on the interval [0, ∞), non-decreasing and continuous for α ≥ 0 and such that ϕ(0) = 0, ϕ(α) > 0 for α > 0 , ϕ(α) → ∞ as α → ∞, is called an Orlicz function. The Orlicz function ϕ satisfies the ∆2 -condition if there exist κ > 0 such that ϕ(2α) ≤ κϕ(α) for all α > 0. Let (Ω, Σ, µ) be a measure space. Let us consider the space L0µ consisting of all measurable real-valued (or
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complex-valued) function on Ω. Define for every f ∈ L0µ the Orlicz modular ρϕ (f ) by the formula Z ϕ(|f |)dµ ρϕ (f ) = Ω
The associated modular function space with respect to this modular is called an ϕ Orlicz space, and will be denoted by Lϕ µ (Ω) or briefly L . In other words Lϕ = f ∈ L0µ : ρϕ (λf ) → 0 as λ → 0 or equivalently as Lϕ = f ∈ L0µ : ρϕ (λf ) < ∞ for some λ > 0 . It is known that the Orlicz space Lϕ is ρϕ -complete. Moreover, (Lϕ , k.kρϕ ) is a Banach space, where the Luxemburg norm k.kρϕ is defined as follows Z |f | kf kρϕ = inf λ > 0 : ϕ( )dµ ≤ 1 . λ Ω Moreover, if ` is the space of sequences x = (xi )∞ i=1 with real or complex terms xi , ∞ ϕ ϕ = (ϕi )∞ i=1 , ϕi are Orlicz functions and πϕ (x) = Σi=1 ϕi (|xi |), we shall write ` ϕ ϕ in place of L . The space ` is called the generalized Orlicz sequence space. The motivation for the study of modular spaces (and Orlicz spaces) and many examples are detailed in [18, 26, 27, 28]. Now, we give a following examples. Example 3.7. Let ϕ is an Orlicz function and satisfy the ∆2 -condition with 0 < κ ≤ 2. Let f : E → Lϕ be mappings satisfying Z ϕ(|Df (x, y)|)dµ ≤ φ(x, y), x⊥y Ω
for all x, y ∈ E and f(0)=0, where φ : E 2 → [0, ∞) is a given function such that φ(ax, ay) ≤ aLφ(x, y) for all x, y ∈ E and a constant 0 < L < 1. Then there exist unique additive mapping T : E → Lϕ and unique quadratic mapping Q : E → Lϕ such that Z κ2 (1 + a − 2L) (φ(0, x) + φ(0, −x)) ϕ(|f (x) − T (x) − Q(x)|)dµ ≤ 2a(a − L)(1 − L) Ω for all x ∈ E. Example 3.8. Let ϕ b = (ϕi ) be sequnece of Orlicz functions and satisfy the ∆2 condition with 0 < κ ≤ 2 and let (`ϕb, πϕb) be generalized Orlicz sequence space associated to ϕ b = (ϕi ). Let f : E → `ϕb be mappings satisfying πϕb(Df (x, y)) ≤ φ(x, y), x⊥y for all x, y ∈ E and f(0)=0, where φ : E 2 → [0, ∞) is a given function such that φ(ax, ay) ≤ aLφ(x, y) for all x, y ∈ E and a constant 0 < L < 1. Then there exist unique additive mapping T : E → `ϕb and unique quadratic mapping Q : E → `ϕb such that πϕb(f (x) − T (x) − Q(x)) ≤
κ2 (1 + a − 2L) (φ(0, x) + φ(0, −x)) 2a(a − L)(1 − L)
for all x ∈ E.
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[email protected]
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