Fixed Points and the Stability of an AQCQ-Functional Equation in Non-Archimedean Normed Spaces

and Applied Analysis 3 is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof 7 for mappings f : X → Y , where X is a normed space and Y is a Banach space. Cholewa 8 noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik 9 proved the generalized Hyers-Ulam stability of the quadratic functional equation. In 10 , Jun and Kim considered the following cubic functional equation: f ( 2x y ) f ( 2x − y) 2f(x y) 2f(x − y) 12f x , 1.7 which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic mapping. In 11 , Lee et al. considered the following quartic functional equation: f ( 2x y ) f ( 2x − y) 4f(x y) 4f(x − y) 24f x − 6f(y), 1.8 which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem see 12–27 . Let X be a set. A function d : X × X → 0,∞ is called a generalized metric on X if d satisfies 1 d x, y 0 if and only if x y; 2 d x, y d y, x for all x, y ∈ X; 3 d x, z ≤ d x, y d y, z for all x, y, z ∈ X. We recall a fundamental result in fixed point theory. Theorem 1.3 see 28, 29 . Let X, d be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ X, either d ( Jx, J 1x ) ∞ 1.9 for all nonnegative integers n or there exists a positive integer n0 such that 1 d Jx, J 1x < ∞, for all n ≥ n0; 2 the sequence {Jnx} converges to a fixed point y∗ of J ; 3 y∗ is the unique fixed point of J in the set Y {y ∈ X | d J0x, y < ∞}; 4 d y, y∗ ≤ 1/ 1 − L d y, Jy for all y ∈ Y . In 1996, Isac and Rassias 30 were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors see 31–36 . 4 Abstract and Applied Analysis This paper is organized as follows: in Section 2, using the fixed point method, we prove the generalizedHyers-Ulam stability of the additive-quadratic-cubic-quartic functional equation f ( x 2y ) f ( x − 2y) 4f(x y) 4f(x − y) − 6f x f(2y) f(−2y) − 4f(y) − 4f(−y) 1.10 in non-Archimedean Banach spaces for an odd case. In Section 3, using the fixed point method, we prove the generalized Hyers-Ulam stability of the additive-quadratic-cubicquartic functional equation 1.10 in non-Archimedean Banach spaces for an even case. Throughout this paper, assume thatX is a non-Archimedean normed vector space and that Y is a non-Archimedean Banach space. 2. Generalized Hyers-Ulam Stability of the Functional Equation 1.10 : An Odd Case One can easily show that an odd mapping f : X → Y satisfies 1.10 if and only if the odd mapping f : X → Y is an additive-cubic mapping, that is, f ( x 2y ) f ( x − 2y) 4f(x y) 4f(x − y) − 6f x . 2.1 It was shown in Lemma 2.2 of 37 that g x : f 2x − 2f x and h x : f 2x − 8f x are cubic and additive, respectively, and that f x 1/6 g x − 1/6 h x . One can easily show that an even mapping f : X → Y satisfies 1.10 if and only if the even mapping f : X → Y is a quadratic-quartic mapping, that is, f ( x 2y ) f ( x − 2y) 4f(x y) 4f(x − y) − 6f x 2f(2y) − 8f(y). 2.2 It was shown in Lemma 2.1 of 38 that g x : f 2x − 4f x and h x : f 2x − 16f x are quartic and quadratic, respectively, and that f x 1/12 g x − 1/12 h x . For a given mapping f : X → Y , we define Df ( x, y ) : f ( x 2y ) f ( x − 2y) − 4f(x y) − 4f(x − y) 6f x − f(2y) − f(−2y) 4f(y) 4f(−y) 2.3 for all x, y ∈ X. We prove the generalized Hyers-Ulam stability of the functional equationDf x, y 0 in non-Archimedean Banach spaces: an odd case. Theorem 2.1. Let φ : X2 → 0,∞ be a function such that there exists an L < 1 with φ ( x, y ) ≤ L |8| ( 2x, 2y ) 2.4 Abstract and Applied Analysis 5 for all x, y ∈ X. Let f : X → Y be an odd mapping satisfyingand Applied Analysis 5 for all x, y ∈ X. Let f : X → Y be an odd mapping satisfying ∥ ∥Df ( x, y )∥ ∥ ≤ φ(x, y) 2.5 for all x, y ∈ X. Then there is a unique cubic mapping C : X → Y such that ∥ ∥f 2x − 2f x − C x ∥ ≤ L |8| − |8|L max {|4|φ x, x , φ 2x, x } 2.6 for all x ∈ X. Proof. Letting x y in 2.5 , we get ∥ ∥f ( 3y ) − 4f(2y) 5f(y)∥∥ ≤ φ(y, y) 2.7 for all y ∈ X. Replacing x by 2y in 2.5 , we get ∥ ∥f ( 4y ) − 4f(3y) 6f(2y) − 4f(y)∥∥ ≤ φ(2y, y) 2.8 for all y ∈ X. By 2.7 and 2.8 , ∥ ∥f ( 4y ) − 10f(2y) 16f(y)∥∥ ≤ max{∥∥4(f(3y) − 4f(2y) 5f(y))∥∥,∥∥f(4y) − 4f(3y) 6f(2y) − 4f(y)∥∥} ≤ max{|4| · ∥∥f(3y) − 4f(2y) 5f(y)∥∥,∥∥f(4y) − 4f(3y) 6f(2y) − 4f(y)∥∥} ≤ max{|4|φ(y, y), φ(2y, y)} 2.9 for all y ∈ X. Letting y : x/2 and g x : f 2x − 2f x for all x ∈ X, we get ∥ ∥ ∥g x − 8g x 2 )∥ ∥ ∥ ≤ max { |4|φ x 2 , x 2 ) , φ ( x, x 2 )} 2.10 for all x ∈ X. Consider the set S : { g : X −→ Y}, 2.11 and introduce the generalized metric on S d ( g, h ) inf { μ ∈ R : ∥ ∥g x − h x ∥ ≤ μ(max{|4|φ x, x , φ 2x, x , ∀x ∈ X})}, 2.12 6 Abstract and Applied Analysis where, as usual, infφ ∞. It is easy to show that S, d is complete. See the proof of Lemma 2.1 of 39 . Now we consider the linear mapping J : S → S such that Jg x : 8g x 2 ) 2.13 for all x ∈ X. Let g, h ∈ S be given such that d g, h ε. Then ∥ ∥g x − h x ∥ ≤ ε ·max{|4|φ x, x , φ 2x, x } 2.14 for all x ∈ X. Hence ∥ ∥Jg x − Jh x ∥ ∥ ∥ ∥8g x 2 ) − 8h x 2 )∥ ∥ ∥ ≤ |8|ε L |8| max {|4|φ x, x , φ 2x, x } 2.15 for all x ∈ X. So d g, h ε implies that d Jg, Jh ≤ Lε. This means that d ( Jg, Jh ) ≤ Ld(g, h) 2.16 for all g, h ∈ S. It follows from 2.10 that ∥ ∥ ∥g x − 8g x 2 )∥ ∥ ∥ ≤ L |8| ( max {|4|φ x, x , φ 2x, x }) 2.17 for all x ∈ X. So d g, Jg ≤ L/|8|. By Theorem 1.3, there exists a mapping C : X → Y satisfying the following. 1 C is a fixed point of J , that is, C x 2 ) 1 8 C x 2.18 for all x ∈ X. The mapping C is a unique fixed point of J in the set M { h ∈ S : d(g, h) < ∞}. 2.19 This implies that C is a unique mapping satisfying 2.18 such that there exists a μ ∈ 0,∞ satisfying ∥ ∥g x − C x ∥ ≤ μ ·max{|4|φ x, x , φ 2x, x } 2.20 for all x ∈ X; since g : X → Y is odd, C : X → Y is an odd mapping. Abstract and Applied Analysis 7 2 d Jg, C → 0 as n → ∞. This implies the equalityand Applied Analysis 7 2 d Jg, C → 0 as n → ∞. This implies the equality lim n→∞ 8g ( x 2n ) C x 2.21 for all x ∈ X. 3 d g,C ≤ 1/ 1 − L d g, Jg , which implies the inequality d ( g,C ) ≤ L |8| − |8|L 2.22 This implies that the inequality 2.6 holds. By 2.5 , ∥ ∥ ∥8Dg ( x 2n , y 2n )∥ ∥ ∥ ≤ |8| max { φ ( 2x 2n , 2y 2n ) , |2|φ ( x 2n , y 2n )} 2.23 for all x, y ∈ X and all n ∈ N. So ∥ ∥ ∥8Dg ( x 2n , y 2n )∥ ∥ ∥ ≤ |8| L n |8| max { φ ( 2x, 2y ) , |2|φ(x, y)} 2.24 for all x, y ∈ X and all n ∈ N. So ∥ ∥DC ( x, y )∥ ∥ 0 2.25 for all x, y ∈ X. Thus the mapping C : X → Y is cubic, as desired. Corollary 2.2. Let θ and p be positive real numbers with p < 3. Let f : X → Y be an odd mapping satisfying ∥ ∥Df ( x, y )∥ ∥ ≤ θ(‖x‖p ∥∥y∥∥p) 2.26 for all x, y ∈ X. Then there exists a unique cubic mapping C : X → Y such that ∥ ∥f 2x − 2f x − C x ∥ ≤ max{2 · |4|, |2| 1} θ |2| − |8| ‖x‖ p 2.27 for all x ∈ X. Proof. The proof follows from Theorem 2.1 by taking φ ( x, y ) : θ (‖x‖p ∥∥y∥∥p) 2.28 for all x, y ∈ X. Then we can choose L |8|/|2|p and we get the desired result. 8 Abstract and Applied Analysis Theorem 2.3. Let φ : X2 → 0,∞ be a function such that there exists an L < 1 with φ ( x, y ) ≤ |8|Lφ x 2 , y 2 ) 2.29 for all x, y ∈ X. Let f : X → Y be an odd mapping satisfying 2.5 . Then there is a unique cubic mapping C : X → Y such that ∥ ∥f 2x − 2f x − C x ∥ ≤ 1 |8| − |8|L max {|4|φ x, x , φ 2x, x } 2.30 for all x ∈ X. Proof. It follows from 2.10 that ∥ ∥ ∥ ∥ g x − 1 8 g 2x ∥ ∥ ∥ ∥ ≤ 1 |8| max {|4|φ x, x , φ 2x, x } 2.31 for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.1. Theorem 2.4. Let φ : X2 → 0,∞ be a function such that there exists an L < 1 with φ ( x, y ) ≤ L |2| ( 2x, 2y ) 2.32 for all x, y ∈ X. Let f : X → Y be an odd mapping satisfying 2.5 . Then there is a unique additive mapping A : X → Y such that ∥ ∥f 2x − 8f x −A x ∥ ≤ L |2| − |2|L max {|4|φ x, x , φ 2x, x } 2.33 for all x ∈ X. Proof. Letting y : x/2 and g x : f 2x − 8f x for all x ∈ X in 2.9 , we get ∥ ∥ ∥g x − 2g x 2 )∥ ∥ ∥ ≤ max { |4|φ x 2 , x 2 ) , φ ( x, x 2 )} 2.34 for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.1. Abstract and Applied Analysis 9 Corollary 2.5. Let θ and p be positive real numbers with p < 1. Let f : X → Y be an odd mapping satisfying 2.26 . Then there exists a unique additive mapping C : X → Y such thatand Applied Analysis 9 Corollary 2.5. Let θ and p be positive real numbers with p < 1. Let f : X → Y be an odd mapping satisfying 2.26 . Then there exists a unique additive mapping C : X → Y such that ∥ ∥f 2x − 8f x −A x ∥ ≤ max{2 · |4|, |2| 1} θ |2| − |2| ‖x‖ p 2.35 for all x ∈ X. Proof. The proof follows from Theorem 2.4 by taking φ ( x, y ) : θ (‖x‖p ∥∥y∥∥p) 2.36 for all x, y ∈ X. Then we can choose L |2|/|2|p and we get the desired result. Theorem 2.6. Let φ : X2 → 0,∞ be a function such that there exists an L < 1 with φ ( x, y ) ≤ |2|Lφ x 2 , y 2 ) 2.37 for all x, y ∈ X. Let f : X → Y be an odd mapping satisfying 2.5 . Then there is a unique additive mapping A : X → Y such that ∥ ∥f 2x − 8f x −A x ∥ ≤ 1 |2| − |2|L max {|4|φ x, x , φ 2x, x } 2.38 for all x ∈ X. Proof. It follows from 2.34 that ∥ ∥ ∥ ∥ g x − 1 2 g 2x ∥ ∥ ∥ ∥ ≤ 1 |2| max {|4|φ x, x , φ 2x, x } 2.39 for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.1. 3. Generalized Hyers-Ulam Stability of the Functional Equation 1.10 : An Even Case Now we prove the generalized Hyers-Ulam stability of the functional equation Df x, y 0 in non-Archimedean Banach spaces: an even case. Theorem 3.1. Let φ : X2 → 0,∞ be a function such that there exists an L < 1 with φ ( x, y ) ≤ L |16| ( 2x, 2y ) 3.1 10 Abstract and Applied Analysis for all x, y ∈ X. Let f : X → Y be an even mapping satisfying 2.5 and f 0 0. Then there is a unique quartic mapping Q : X → Y such that ∥ ∥f 2x − 4f x −Q x ∥ ≤ L |16| − |16|L max {|4|φ x, x , φ 2x, x } 3.2 for all x ∈ X. Proof. Letting x y in 2.5 , we get ∥ ∥f ( 3y ) − 6f(2y) 15f(y)∥∥ ≤ φ(y, y) 3.3 for all y ∈ X. Replacing x by 2y in 2.5 , we get ∥ ∥f ( 4y ) − 4f(3y) 4f(2y) 4f(y)∥∥ ≤ φ(2y, y) 3.4 for all y ∈ X. By 3.3 and 3.4 , ∥ ∥f ( 4y ) − 20f(2y) 64f(y)∥∥ ≤ max{∥∥4(f(3y) − 6f(2y) 15f(y))∥∥,∥∥f(4y) − 4f(3y) 4f(2y) 4f(y)∥∥} ≤ max{|4| · ∥∥f(3y) − 6f(2y) 15f(y)∥∥,∥∥f(4y) − 4f(3y) 4f(2y) 4f(y)∥∥} ≤ max{|4|φ(y, y), φ(2y, y)} 3.5 for all y ∈ X. Letting y : x/2 and g x : f 2x − 4f x for all x ∈ X, we get ∥ ∥ ∥g x − 16g x 2 )∥ ∥ ∥ ≤ max { |4|φ x 2 , x 2 ) , φ ( x, x 2 )} 3.6 for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.1. Corollary 3.2. Let θ and p be positive real numbers with p < 4. Let f : X → Y be an even mapping satisfying 2.26 and f 0 0. Then there exists a unique quartic mapping Q : X → Y such that ∥ ∥f 2x − 4f x −Q x ∥ ≤ max{2 · |4|, |2| 1} θ |2| − |16| ‖x‖ p 3.7