Stability of an Additive-Cubic-Quartic Functional Equation in Multi-Banach Spaces

and Applied Analysis 3 for some natural number n0. Moreover, if the second alternative holds, then i the sequence {Jnx} is convergent to a fixed point y∗ of J ; ii y∗ is the unique fixed point of J in the set Y : {y ∈ X | d J0x, y < ∞} and d y, y∗ ≤ 1/ 1 − L d y, Jy , for all , x, y ∈ Y . Following 30, 31 , we recall some basic facts concerning multi-normed spaces and some preliminary results. Let E, ‖ · ‖ be a complex normed space, and let k ∈ N. We denote by Ek the linear space E⊕ · · · ⊕E consisting of k-tuples x1, . . . , xk , where x1, . . . , xk ∈ E. The linear operations Ek are defined coordinatewise. The zero element of either E or Ek is denoted by 0. We denote by Nk the set {1, 2, . . . , k} and by Sk the group of permutations on k symbols. Definition 2.2 cf. 30, 31 . A multi-norm on {Ek : k ∈ N} is a sequence ‖ ·‖k ‖ ·‖k : k ∈ N such that ‖ · ‖k is a norm on Ek for each k ∈ N, ‖x‖1 ‖x‖ for each x ∈ E, and the following axioms are satisfied for each k ∈ N with k ≥ 2: N1 ‖ xσ 1 , . . . , xσ k ‖k ‖ x1, . . . , xk ‖k, for σ ∈ Sk, x1, . . . , xk ∈ E; N2 ‖ α1x1, . . . , αkxk ‖k ≤ maxi∈Nk |αi| ‖ x1, . . . , xk ‖k, for α1, . . . , αk ∈ C, x1, . . . , xk ∈ E; N3 ‖ x1, . . . , xk−1, 0 ‖k ‖ x1, . . . , xk−1 ‖k−1, for x1, . . . , xk−1 ∈ E; N4 ‖ x1, . . . , xk−1, xk−1 ‖k ‖ x1, . . . , xk−1 ‖k−1, for x1, . . . , xk−1 ∈ E. In this case, we say that Ek, ‖ · ‖k : k ∈ N is a multi-normed space. Suppose that Ek, ‖ · ‖k : k ∈ N is a multi-normed space, and take k ∈ N. We need the following two properties of multi-norms. They can be found in 30 a ‖ x, . . . , x ‖k ‖x‖, for x ∈ E, b maxi∈Nk‖xi‖ ≤ ‖ x1, . . . , xk ‖k ≤ ∑k i 1‖xi‖ ≤ kmaxi∈Nk‖xi‖, for x1, . . . , xk ∈ E. It follows from b that if E, ‖ · ‖ is a Banach space, then Ek, ‖ · ‖k is a Banach space for each k ∈ N; in this case, Ek, ‖ · ‖k : k ∈ N is a multi-Banach space. Lemma 2.3 cf. 30, 31 . Suppose that k ∈ N and x1, . . . , xk ∈ Ek. For each j ∈ {1, . . . , k}, let x n n 1,2,... be a sequence in E such that limn→∞ x n xj . Then limn→∞ ( x1 n − y1, . . . , x n − yk ) ( x1 − y1, . . . , xk − yk ) 2.4 holds for all y1, . . . , yk ∈ Ek. Definition 2.4 cf. 30, 31 . Let Ek, ‖ · ‖k : k ∈ N be a multi-normed space. A sequence xn in E is a multi-null sequence if for each ε > 0, there exists n0 ∈ N such that sup k∈N ‖ xn, . . . , xn k−1 ‖k ≤ ε n ≥ n0 . 2.5 Let x ∈ E, we say that the sequence xn is multi-convergent to x in E and write limn→∞ xn x if xn − x is a multi-null sequence. 4 Abstract and Applied Analysis 3. Main Results Throughout this section, let ε ≥ 0, E be a linear space, and let Fn, ‖ · ‖n : n ∈ N be a multi-Banach space. For convenience, we use the following abbreviation for a given mapping f : E → F: Df ( x, y ) 11 [ f ( x 2y ) f ( x − 2y − 44fx y fx − y − 12f3y 48f2y − 60fy 66f x . 3.1 Before proceeding to the proof of the main results in this section, we shall need the following two lemmas. Lemma 3.1 cf. 34 . If an even mapping f : X → Y satisfies 1.1 , then f is quartic. Lemma 3.2 cf. 34 . If an odd mapping f : X → Y satisfies 1.1 , then f is cubic-additive. Theorem 3.3. Suppose that an even mapping f : E → F satisfies f 0 0 and sup k∈N ∥Df ( x1, y1 ) , . . . , Df ( xk, yk ))∥∥ k ≤ ε 3.2 for all x1, . . . , xk, y1, . . . , yk ∈ E. Then there exists a unique quartic mapping Q : E → F satisfying 1.1 and sup k∈N ∥f x1 −Q x1 , . . . , f xk −Q xk )∥∥ k ≤ 13 330 ε 3.3 for all x1, . . . , xk ∈ E. Proof. Letting x1 · · · xk 0 in 3.2 , we get sup k∈N ∥−12f3y1 ) 70f ( 2y1 ) − 148fy1 ) , . . . ,−12f3yk ) 70f ( 2yk ) − 148fyk ))∥∥ k ≤ ε 3.4 for all y1, . . . , yk ∈ E. Replacing x1, . . . , xk with y1, . . . , yk in 3.2 , we obtain sup k∈N ∥−f3y1 ) 4f ( 2y1 ) 17f ( y1 ) , . . . ,−f3yk ) 4f ( 2yk ) 17f ( yk ))∥∥ k ≤ ε 3.5 for all y1, . . . , yk ∈ E. It follows from 3.4 and 3.5 that sup k∈N ∥f 2x1 − 16f x1 , . . . , f 2xk − 16f xk )∥∥ k ≤ 13 22 ε 3.6 for all x1, . . . , xk ∈ E. Abstract and Applied Analysis 5 Let E : {gg : E → F, g 0 0}, and introduce the generalized metric d defined on E byand Applied Analysis 5 Let E : {gg : E → F, g 0 0}, and introduce the generalized metric d defined on E by d ( g, h ) inf { c ∈ 0, ∞ | sup k∈N ∥g x1 − h x1 , . . . , g xk − h xk )∥∥ k ≤ c, ∀x1, . . . , xk ∈ E } . 3.7 Then, it is easy to show that d is a complete generalized metric on E see the proof in 36 or 5 . We now define a function J1 : E → E by J1g x 1 16 g 2x , ∀x ∈ E. 3.8 We assert that J1 is a strictly contractivemapping. Given g, h ∈ E, let c ∈ 0,∞ be an arbitrary constant with d g, h ≤ c. From the definition of d, it follows that sup k∈N ∥g x1 − h x1 , . . . , g xk − h xk )∥∥ k ≤ c 3.9 for all x1, . . . , xk ∈ E. Therefore, sup k∈N ∥J1g x1 − J1h x1 , . . . , J1g xk − J1h xk )∥∥ k sup k∈N ∥∥∥∥ ( 1 16 g 2x1 − 1 16 2x1 , . . . , 1 16 g 2xk − 1 16 2xk )∥∥∥∥ k ≤ 1 16 c 3.10 for all x1, . . . , xk ∈ E. Hence, it holds that d J1g , J1h ≤ 1/16 c, that is, d J1g, J1h ≤ 1/16 d g, h for all g, h ∈ E. By using 3.6 , we have d J1f, f ≤ 13/352 ε. According to Lemma 2.1, we deduce the existence of a fixed point of J1, that is, the existence of a mapping Q : X → Y such that Q 2x 16Q x for all x ∈ E. Moreover, we have d J 1 f, Q → 0, which implies that Q x lim n→∞ ( J 1 f ) x lim n→∞ f 2x 16n , ∀x ∈ E. 3.11 Also, d f, Q ≤ 1/ 1 − L d J1f, f implies the inequality d ( f, Q ) ≤ 1 1 − 1/16 d ( J1f, f ) ≤ 13 330 ε. 3.12 6 Abstract and Applied Analysis Set x1 · · · xk 2x, y1 · · · yk 2y in 3.2 , and divide both sides by 16. Then, using property a , we obtain ∥DQ ( x, y )∥∥ lim n→∞ 1 16n ∥Df ( 2x, 2y )∥∥ ≤ lim n→∞ ε 16n 0 3.13 for all x, y ∈ E. Hence, by Lemma 3.1, Q is quartic. The uniqueness of Q follows from the fact that Q is the unique fixed point of J1 with the property that there exists l ∈ 0, ∞ such that sup k∈N ∥f x1 −Q x1 , . . . , f xk −Q xk )∥∥ k ≤ l 3.14 for all x1, . . . , xk ∈ E. This completes the proof of the theorem. Theorem 3.4. Suppose that an odd mapping f : E → F satisfies sup k∈N ∥Df ( x1, y1 ) , . . . , Df ( xk, yk ))∥∥ k ≤ ε 3.15 for all x1, . . . , xk, y1, . . . , yk ∈ E. Then there exists a unique additive mapping A : E → F and a unique cubic mapping C : E → F satisfying 1.1 and sup k∈N ∥f 2x1 − 8f x1 −A x1 , . . . , f 2xk − 8f xk −A xk )∥∥ k ≤ 17 33 ε, sup k∈N ∥f 2x1 − 2f x1 − C x1 , . . . , f 2xk − 2f xk − C xk )∥∥ k ≤ 17 231 ε 3.16 for all x1, . . . , xk ∈ E. Proof. Put x1 · · · xk 0 in 3.15 . Then, by the oddness of f , we have sup k∈N ∥12f ( 3y1 ) − 48f2y1 ) 60f ( y1 ) , . . . 12f ( 3yk ) − 48f2yk ) 60f ( yk ))∥∥ k ≤ ε 3.17 for all y1, . . . , yk ∈ E. Replacing x1, . . . , xk with 2y1, . . . , 2yk in 3.15 , we obtain sup k∈N ∥11f ( 4y1 ) − 56f3y1 ) 114f ( 2y1 ) − 104fy1 ) , . . . , 11f ( 4yk ) − 56f3yk ) 114f ( 2yk ) − 104fyk ))∥∥ k ≤ ε 3.18 for all y1, . . . , yk ∈ E. By 3.17 and 3.18 , we have sup k∈N ∥f 4x1 − 10f 2x1 16f x1 , . . . , f 4xk − 10f 2xk 16f xk )∥∥ k ≤ 17 33 ε 3.19 Abstract and Applied Analysis 7 for all x1, . . . , xk ∈ E. Putting α x : f 2x − 8f x for all x ∈ E, we getand Applied Analysis 7 for all x1, . . . , xk ∈ E. Putting α x : f 2x − 8f x for all x ∈ E, we get sup k∈N ‖ α 2x1 − 2α x1 , . . . , α 2xk − 2α xk ‖k ≤ 17 33 ε 3.20 for all x1, . . . , xk ∈ E. Let the same definitions for E and d as in the proof of Theorem 3.3 such that E, d becomes a complete generalized metric space. We now define a function J2 : E → E by J2g x 1 2 g 2x , ∀x ∈ E. 3.21 Applying a similar technique as in the proof of Theorem 3.3, we obtain d J2g, J2h ≤ 1/2 c, that is, d J2g, J2h ≤ 1/2 d g, h for all g, h ∈ E. By 3.20 , we have d J2α, α ≤ 17/66 ε. According to Lemma 2.1, we deduce the existence of a fixed point of J2, that is, the existence of a mapping A : X → Y such that A 2x 2A x for all x ∈ E. Moreover, we have d J 2 α,A → 0, which implies that A x lim n→∞ ( J 2 α ) x lim n→∞ α 2x 2n , ∀x ∈ E. 3.22 Also, d α,A ≤ 1/ 1 − L d J2α, α implies the inequality d α, A ≤ 1 1 − 1/2 d J2α, α ≤ 17 33 ε. 3.23 Hence, it follows that ∥DA ( x, y )∥∥ lim n→∞ 1 2n ∥Dα ( 2x, 2y )∥∥ lim n→∞ 1 2n ∥∥Df ( 2 1x, 2 1y ) − 8Df2nx, 2ny ∥∥ ≤ lim n→∞ ε 2n 0 3.24 for all x, y ∈ E. This means that A satisfies 1.1 . Then, by Lemma 3.2, x → A 2x − 8A x is additive. Thus, by A 2x 2A x , we conclude that A is additive. Putting β x : f 2x − 2f x for all x ∈ E, we get sup k∈N ∥β 2x1 − 8β x1 , . . . , α 2xk − 8β xk )∥∥ k ≤ 17 33 ε 3.25 for all x1, . . . , xk ∈ E. We now define a function J3 : E → E by J3g x 1 8 g 2x , ∀x ∈ E. 3.26 8 Abstract and Applied Analysis Applying a similar technique as in the proof of Theorem 3.3, we obtain d J3g, J3h ≤ 1/8 c, that is, d J3g, J3h ≤ 1/8 d g, h for all g, h ∈ E. By 3.25 , we have d J3β, β ≤ 17/264 ε. According to Lemma 2.1, we deduce the existence of a fixed point of J3, that is, the existence of a mapping C : X → Y such that C 2x 8C x for all x ∈ E. Moreover, we have d J 3 β, C → 0, which implies that C x lim n→∞ ( J 3 β ) x lim n→∞ β 2x 8n , ∀x ∈ E. 3.27 Also, d β, C ≤ 1/ 1 − L d J3β, β implies the inequality d ( β, C ) ≤ 1 1 − 1/8 d ( J3β, β ) ≤ 17 231 ε. 3.28 Then we have ∥DC ( x, y )∥∥ lim n→∞ 1 8n ∥Dβ ( 2x, 2y )∥∥ lim n→∞ 1 8n ∥∥Df ( 2 1x, 2 1y ) − 2Df2nx, 2ny ∥∥ ≤ lim n→∞ ε 8n 0 3.29 for all x, y ∈ E. Hence, the mapping C satisfies 1.1 . Therefore, by Lemma 3.2, x → C 2x − 2C x is cubic. Thus, C 2x 8C x implies that the mapping C is cubic. The uniqueness of A and C can be proved in the same reasoning as in the proof of Theorem 3.3. This completes the proof of the theorem. Theorem 3.5. Suppose that an odd mapping f : E → F satisfies sup k∈N ∥Df ( x1, y1 ) , . . . , Df ( xk, yk ))∥∥ k ≤ ε 3.30 for all x1, . . . , xk, y1, . . . , yk ∈ E. Then there exists a unique additive mapping A : E → F and a unique cubic mapping C : E → F such that sup k∈N ∥f x1 −A x1 − C x1 , . . . , f xk −A xk − C xk )∥∥ k ≤ 68 693 ε 3.31 for all x1, . . . , xk ∈ E. Proof. By Theorem 3.4, there exist an additive mapping A0 : E → F and a cubic mapping C0 : E → F such that sup k∈N ∥f 2x1 − 8f x1 −A0 x1 , . . . , f 2xk − 8f xk −A0 xk )∥∥ k ≤ 17 33 ε, sup k∈N ∥f 2x1 − 2f x1 − C0 x1 , . . . , f 2xk − 2f xk − C0 xk )∥∥ k ≤ 17 231 ε 3.32 Abstract and Applied Analysis 9 for all x1, . . . , xk ∈ E. Combining the above two equations in 3.32 yields thatand Applied Analysis 9 for all x1, . . . , xk ∈ E. Combining the above two equations in 3.32 yields that sup k∈N ∥6f x1 A0 x1 − C0 x1 , . . . , f 6xk A0 xk − C0 xk )∥∥ k ≤ 136 231 ε 3.33 for all x1, . . . , xk ∈ E. So we obtain 3.31 by letting A −A0/6 and C C0/6. To prove the uniqueness of A and C, let A1, C1 : E → F be other additive and cubic mappings satisfying 3.31 . Let A′ A − A1 and C′ C − C1. Then, using property a , we obtain ∥A′ x C′ x ∥∥ ≤ ∥f x −A x − C x ∥∥ ∥f x −A1 x − C1 x ∥∥ ≤ 136 693 ε 3.34 for all x ∈ E, then 3.34 implies that lim n→∞ 1 8n ∥A′ 2x C′ 2x ∥∥ 0 3.35 for all x ∈ E. Therefore, C′ x 0 for all x ∈ E. By 3.34 , we have A′ x 0 for all x ∈ E. This completes the proof of the theorem. Theorem 3.6. Suppose that a mapping f : E → F satisfies f 0 0 and sup k∈N ∥Df ( x1, y1 ) , . . . , Df ( xk, yk ))∥∥ k ≤ ε 3.36 for all x1, . . . , xk, y1, . . . , yk ∈ E. Then there exists a unique additive mapping A : E → F, a unique cubic mapping C : E → F, and a unique quartic mapping Q : E → F such that sup k∈N ∥∥ f x1 −A x1 − C ( x1 −Q x1 , . . . , f xk −A xk − C xk −Q xk )∥∥ k ≤ 953 6930 ε 3.37 for all x1, . . . , xk ∈ E. Proof. Let fo x 1/2 f x − f −x for all x ∈ E, then fo 0 0, fo −x −fo x and sup k∈N ∥Dfo ( x1, y1 ) , . . . , Dfo ( xk, yk ))∥∥ k ≤ ε 3.38 for all x1, . . . , xk, y1, . . . , yk ∈ E. From Theorem 3.5, it follows that there exists a unique additive mapping A : E → F and a unique cubic mapping C : E → F satisfying 3.31 . Let fe x 1/2 f x f −x for all x ∈ E, then fe 0 0, fe −x fe x and sup k∈N ∥Dfe ( x1, y1 ) , . . . , Dfe ( xk, yk ))∥∥ k ≤ ε 3.39 10 Abstract and Applied Analysis for all x1, . . . , xk, y1, . . . , yk ∈ E. By Theorem 3.3, there exists a unique quartic mapping Q : E → F satisfying 3.3 . Now it is obvious that 3.37 holds for all x1, . . . , xk ∈ E. This completes the proof of the theorem. 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