A new fixed point approach to hyperstability of radical-type functional equations in quasi- (2,β) -Banach spaces
aa r X i v : . [ m a t h . F A ] J u l A NEW FIXED POINT APPROACH TO HYPERSTABILITY OFRADICAL-TYPE FUNCTIONAL EQUATIONS IN QUASI- (2 , β ) -BANACHSPACES IZ-IDDINE EL-FASSI ⋆ Abstract.
The main focus of this paper is to define the notion of quasi- (2 , β ) -Banachspace and show some properties in this new space, by help of it and under some naturalassumptions, we prove that the fixed point theorem [16, Theorem 2.1] is still valid in thesetting of quasi-(2 , β )-Banach spaces, this is also an extension of the fixed point resultof Brzdęk et al. [12, Theorem 1] in 2-Banach spaces to quasi-(2 , β )-Banach spaces. Inthe next part, we give a general solution of the radical-type functional equation (1.2). Inaddition, we study the hyperstability results for these functional equation by applyingthe aforementioned fixed point theorem, and at the end of this paper we will derive someconsequences. Introduction
Let E and F be linear spaces over a real or complex scalar field K . We recall that afunction g : E → F satisfies the general quadratic equation if g ( ax + by ) + g ( ax − by ) = cg ( x ) + dg ( y ) , x, y ∈ E, (1.1)where a, b, c, d ∈ K \ { } . are fixed numbers. We see that for c = d = a + b in (1.1) we getthe Euler-Lagrange functional equation investigated by J.M. Rassias [42, 41] (see also [38]),while the quadratic functional equation corresponds to a = b = 1 and c = d = 2 . In the sequel, N , R and K denote the sets of all positive integers, real numbers and thefield of real or complex numbers, respectively. Also, W V denotes the set of all functions froma set V = ∅ to a set W = ∅ . We put R := R \ { } , R + := [0 , ∞ ) and N := N ∪ { } .First of all, let us recall the history in the stability theory for functional equations. Thestory of the stability of functional equations dates back to 1925 when a stability resultappeared in the celebrated book by Póolya and Szeg [39]. In 1940, Ulam [43] posed thefamous Ulam stability problem which was partially solved by Hyers [27] in the frameworkof Banach spaces. Later, Hyers’ result was extended by Aoki [5] and next by Rassias [40].Since then numerous papers on this subject have been published and we refer to [2, 14,15, 19, 28, 32, 29, 35] for more details. On the other hand, fixed point theorems have beenalready applied in the theory of Hyers-Ulam stability by several authors (see for instance ⋆ Corresponding author.2010
Mathematics Subject Classification.
Primary 47H10, 46A16; Secondary 39B82, 65Q20.
Key words and phrases.
Fixed Point Theorems, Quasi-(2 , β )-Banach Space; Hyperstability; FunctionalEquations. [13, 18, 30, 31]) and it seems that Baker (see [6]) has used this tool for the first time in thisfield.According to our best knowledge, the first hyperstability result was published in [7], andconcerned ring homomorphisms. However, it seems that the term hyperstability was usedfor the Ąrst time in [33] (quite often it is confused with superstability , which admits alsobounded functions). There are many researchers investigating the hyperstability results forfunctional equations in many areas (see, e.g., [8, 9, 16, 21, 26]).The radical functional equation is one of the popular topics for investigating in the theoryof stability. Nowadays, a lot of papers concerning the stability and the hyperstability of theradical functional equation in various spaces was appeared (see in [3, 4, 17, 19, 20, 32] andreferences therein).In [32], Khodaei et al. solved the following functional equations f (cid:16)p ax + by (cid:17) = af ( x ) + bf ( y ) , x, y ∈ R ,f (cid:16)p ax + by (cid:17) + f (cid:16)p | ax − by | (cid:17) = a f ( x ) + b f ( y ) , x, y ∈ R , where a, b are fixed positive reals and proved generalized Ulam stability of these functionalequations in 2-normed spaces.In 2018, Dung et al. [16] extended the fixed point theorem of Brzdęk et al. [11, Theorem1] in metric spaces to b -metric spaces, in particular to quasi-Banach spaces, and studied thehyperstability for the general linear equation in the setting of quasi-Banach spaces.A function f : R → X is a solution of the radical-type functional equation if and only if f (cid:16) p ax + by (cid:17) + f (cid:16) p ax − by (cid:17) = cf ( x ) + df ( y ) , x, y ∈ R , (1.2)where a, b, c and d are nonzero real constants.We say that a function f : R → X fulfills the radical-type functional equation (1.2) on R if and only if f (cid:16) p ax + by (cid:17) + f (cid:16) p ax − by (cid:17) = cf ( x ) + df ( y ) , x, y ∈ R , (1.3)where √ ax = ± √ by and a, b, c, d ∈ R . The contents of the paper are as follows: ◦ In Sect. 2, we introduce a new space called quasi- (2 , β ) -Banach space and we inves-tigate also some results about this space. ◦ In Sect. 3, we prove that fixed point theorem [16, Theorem 2.1] remains valid in thesetting of quasi-(2 , β )-Banach space and we derive from it many particular cases. ◦ In Sect. 4, we achieve the general solution of the functional equation (1.2). ◦ In the last Sect. 5, we will apply our fixed point theorem to show the hyperstabilityresults for the radical-type functional equation (1.2), and we finish this paper withsome consequences.It is well-known that the theory of 2-normed spaces was initially introduced by G¨ahler[24, 25] in the mid 1960s, and has been developed extensively in different subjects by others,
IXED POINT THEOREM IN QUASI-(2 , β )-BANACH SPACES AND ITS APPLICATIONS 3 for example [22, 23, 37, 44]. Now, we recall by the definition of a (2 , β )-normed space andsome preliminary results.Let 0 < β ¬ X be a linear space over K with dim X k· , ·k β : X × X → R + is called a (2 , β ) -norm on X if and only if it satisfies:(N1) k x, y k β = 0 if and only if x and y are linearly dependent;(N2) k x, y k β = k y, x k β for all x, y ∈ X ;(N3) k λx, y k β = | λ | β k x, y k β for all x, y ∈ X and λ ∈ K ;(N4) k x, y + z k β ¬ k x, y k β + k x, z k β for all x, y, z ∈ X .The pair ( X, k· , ·k β ) is called a (2 , β )- normed space . • If x ∈ X and k x, y k β = 0 for all y ∈ X , then x = 0. Moreover; the functions x → k x, y k β are continuous functions of X into R + for each fixed y ∈ E . • If β = 1, then ( X, k· , ·k β ) becomes a linear 2-normed space.In 2006, Park [36] introduced the concept of quasi-2-normed spaces and quasi-(2 , p )-normed spaces and studied the properties of these spaces. Definition 1.1.
Let X be a linear space over K with dim X k· , ·k q : X × X → R + be a function such that(D1) k x, y k q = 0 if and only if { x, y } is linearly dependent;(D2) k x, y k q = k y, x k q for all x, y ∈ X ;(D3) k λx, y k q = | λ |k x, y k q for all x, y ∈ X and all λ ∈ K ;(D4) There is a constant κ k x, y + z k q ¬ κ ( k x, y k q + k x, z k q ) for all x, y, z ∈ X .Then k· , ·k q is called a quasi- -norm on X , the smallest possible κ is called the modulus ofconcavity and ( X, k· , ·k q , κ ) is called a quasi- -normed space .A quasi-2-norm k· , ·k q is called a quasi- p - -norm (0 < p ¬
1) if k x + y, z k pq ¬ k x, z k pq + k y, z k pq for all x, y, z ∈ X. The first difference between a quasi-2-norm and a 2-norm is that themodulus of concavity of a quasi-norm is greater than or equal to 1, while that of a 2-normis equal to 1. This causes the quasi-2-norm to be not continuous in general, while a norm isalways continuous. Moreover, by Aoki-Rolewicz Theorem [34], each quasi-norm is equivalentto some p-norm. In [36], Park has shown the following theorem.
Theorem 1.1. [36, Theorem 3]
Let ( X, k· , ·k q , κ ) be a quasi- -normed space. There is p ∈ (0 , and an equivalent quasi- -norm k|· , ·|k q on X satisfying k| x + y, z |k pq ¬ k| x, z |k pq + k| y, z |k pq for all x, y, z ∈ X, with k| x, z |k q := inf n(cid:16) n X i =1 k x i , z k pq (cid:17) /p : x = n X i =1 x i , x i ∈ X, z ∈ X, n ∈ N o and p = log κ . IZ. EL-FASSI Some properties of quasi- (2 , β ) -Banach spaces In this section, we generalize the concept of quasi-2-normed spaces and we prove someproperties of the quasi-(2 , β )-Banach spaces.
Definition 2.1.
Let 0 < β ¬ X be a linear space over K with dim X k· , ·k q,β : X × X → R + be a function such that(B1) k x, y k q,β = 0 if and only if { x, y } is linearly dependent;(B2) k x, y k q,β = k y, x k q,β for all x, y ∈ X ;(B3) k λx, y k q,β = | λ | β k x, y k q,β for all x, y ∈ X and all λ ∈ K ;(B4) There is a constant κ k x, y + z k q,β ¬ κ ( k x, y k q,β + k x, z k q,β ) for all x, y, z ∈ X .Then k· , ·k q,β is called a quasi- (2 , β ) -norm on X , the smallest possible κ is called the modulusof concavity and ( X, k· , ·k q,β , κ ) is called a quasi- (2 , β ) -normed space .A quasi-2-norm k· , ·k q,β is called a quasi- p - (2 , β ) -norm (0 < p ¬
1) if k x + y, z k pq,β ¬ k x, z k pq,β + k y, z k pq,β , x, y, z ∈ X, and we have (cid:12)(cid:12) k x, z k pq,β − k y, z k pq,β (cid:12)(cid:12) ¬ k x + y, z k pq,β , x, y, z ∈ X. Example 1.
Let X be a linear space with dimX
2, and let k· , ·k β be a (2 , β )-norm on X. Then k x, y k q,β = C k x, y k β , x, y ∈ X (with C > , is a quasi-(2 , β )-norm on X , and ( X, k· , ·k q,β ) is a quasi-(2 , β )-normed space Example 2. If X is a quasi-2-norm space with the quasi-2-norm k· , ·k q and the modulus ofconcavity κ
1, then it is a quasi-(2 , β )-normed space with the quasi-(2 , β )-norm k x, y k q,β = k x, y k βq for all x, y ∈ X and 0 < β ¬ Proof.
Indeed, for every x, y, z ∈ X and λ ∈ K , we have k x, y k q,β = 0 ⇔ k x, y k q = 0 ⇔ { x, y } is linearly dependent , k y, x k q,β = k y, x k βq = k x, y k βq = k x, y k q,β k λx, y k q,β = k λx, y k βq = | λ | β k x, y k q,β and k x + y, z k q,β = k x + y, z k βq ¬ κ β ( k x, z k q + k y, z k q ) β ¬ κ β ( k x, z k βq + k y, z k βq )= κ β ( k x, z k q,β + k y, z k q,β ) . As κ β , then ( X, k· , ·k q,β , κ β ) is a quasi-(2 , β )-normed space. (cid:3) Lemma 2.1.
Let ( X, k· , ·k q,β , κ ) be a quasi- (2 , β ) -normed space with < β ¬ . If x ∈ X and k x, y k q,β = 0 for all y ∈ X , then x = 0 . Proof.
Suppose that x = 0. Since dimX
2, choose y ∈ X such that { x, y } is linearlyindependent, and by Definition 2.1 (B1) we have k x, y k q,β = 0. This is a contradiction, thus x = 0 . (cid:3) IXED POINT THEOREM IN QUASI-(2 , β )-BANACH SPACES AND ITS APPLICATIONS 5
Definition 2.2.
Let X be a quasi-(2 , β )-normed space with 0 < β ¬ { x n } in X is called a Cauchy sequence if there are two points y, z ∈ X such that y and z are linearly independent,lim m,n →∞ k x m − x n , y k q,β = 0 = lim m,n →∞ k x m − x n , z k q,β . (2) A sequence { x n } in X is called a convergent sequence if there is an x ∈ X such thatlim n →∞ k x n − x, y k q,β = 0 for all y ∈ X. In this case we write we also write lim n →∞ x n = x. (3) A quasi-(2 , β )-normed space in which every Cauchy sequence is a convergent se-quence is called a quasi- (2 , β ) -Banach space . Remark 1.
The functions x → k x, y k q,β are not necessary continuous functions of X into R for each fixed y ∈ X. Theorem 2.1.
Let ( X, k· , ·k q,β , κ ) be a quasi- (2 , β ) -normed space, with < β ¬ . There is p ∈ (0 , and an equivalent quasi-2-norm k|· , ·|k q,β on X satisfying k| x + y, z |k pq,β ¬ k| x, z |k pq,β + k| y, z |k pq,β for all x, y, z ∈ X, with k| x, z |k q,β := inf n(cid:16) n X i =1 k x i , z k p/βq,β (cid:17) β/p : x = n X i =1 x i , x i ∈ X, z ∈ X, n ∈ N o and p = β log κ . Proof.
Let ( X, k· , ·k q,β , κ ) be a quasi-(2 , β )-normed space with 0 < β ¬ , and let x, z ∈ X such that k| x, z |k q := inf n(cid:16) n X i =1 k x i , z k p/βq,β (cid:17) /p : x = n X i =1 x i , x i ∈ X, n ∈ N o ,p = β log κ . We will demonstrate that k· , ·k q := k· , ·k β q,β is a quasi-2-norm on X with the modulus ofconcavity (2 κ ) β . For that, we assume that k· , ·k q,β is a quasi-(2 , β )-norm on X . Then for every x, y, z ∈ X and for every λ ∈ K , we have k x, y k q = 0 ⇔ k x, y k β q,β = 0 ⇔ k x, y k q,β = 0 ⇔ { x, y } is linearly dependent; k λx, y k q = k λx, y k β q,β = | λ |k x, y k q , k y, x k q = k x, y k q ;and k x + y, z k q = k x + y, z k /βq,β ¬ κ /β ( k x, z k q,β + k y, z k q,β ) /β . (2.1)Let g ( t ) = t r for all t > r
1. It is easy to check that g is a convex function, henceby the definition of convexity, we have g (cid:18) t + s (cid:19) ¬
12 ( g ( t ) + g ( s )) for all t, s > , IZ. EL-FASSI i.e., ( t + s ) r ¬ r − ( t r + s r ) . So, (2.1) becomes k x + y, z k q ¬ (2 κ ) /β (cid:0) k x, z k /βq,β + k y, z k /βq,β (cid:1) = (2 κ ) /β (cid:0) k x, z k q + k y, z k q (cid:1) . It follows that (cid:16) X, k· , ·k q , (2 κ ) /β (cid:17) is a quasi-2-normed space. From Theorem 1.1, we get aquasi-2-norm k|· , ·|k q on X satisfying k| x + y, z |k pq ¬ k| x, z |k pq + k| y, z |k pq , x, y, z ∈ X, and there exist µ , µ > µ k x, z k /βq,β = µ k x, z k q ¬ k| x, z |k q ¬ µ k x, z k q = µ k x, z k /βq,β ∀ x, y ∈ X. Since 0 < β ¬ k|· , ·|k q,β := k|· , ·|k βq is also a quasi-(2 , β )-normon X and k| x + y, z |k pq,β = k| x + y, z |k pβq ¬ ( k| x, z |k pq + k| y, z |k pq ) β ¬ k| x, z |k pβq + k| y, z |k pβq = k| x, z |k pq,β + k| y, z |k pq,β for all x, y, z ∈ X. Also we have µ β k x, z k q,β ¬ k| x, z |k βq ¬ µ β k x, z k q,β ∀ x, z ∈ X, i.e., C k x, z k q,β ¬ k| x, z |k q,β ¬ C k x, z k q,β ∀ x, z ∈ X, (2.2)with C i = µ βi for i ∈ { , } . This completes the proof. (cid:3) Remark 2.
It follows from Theorem 2.1 that(i) (cid:12)(cid:12) k| x, z |k pq,β − k| y, z |k pq,β (cid:12)(cid:12) ¬ k| x − y, z |k pq,β , x, y, z ∈ X. (ii) The functions x → k| x, y |k q,β are continuous functions of X into R for each fixed y ∈ X. (iii) If β = 1, we get the Theorem 1.1.3. A new fixed point theorem
In this section, we prove that the fixed point theorem [16, Theorem 2.1] remains valid inthe setting of quasi-(2 , β )-Banach space. Let us introduce the following four hypotheses: (A1) W is a nonempty set, ( X, k· , ·k q,β , κ ) is a quasi-(2 , β )-Banach space. (A2) f i : W → W and L i : W × X → R + are given maps for i = 1 , . . . , j . (A3) T : X W → X W is an operator satisfying the inequality k ( T ξ )( x ) − ( T µ )( x ) , y k q,β ¬ j X i =1 L i ( x, y ) k ξ ( f i ( x )) − µ ( f i ( x )) , y k q,β for all ξ, µ ∈ X W and ( x, y ) ∈ W × X . IXED POINT THEOREM IN QUASI-(2 , β )-BANACH SPACES AND ITS APPLICATIONS 7 (A4)
Λ : R W × X + → R W × X + is a linear operator defined by(Λ δ )( x, y ) := j X i =1 L i ( x, y ) δ ( f i ( x ) , y ) , δ ∈ R W × X + , ( x, y ) ∈ W × X. Theorem 3.1.
Let hypotheses (A1) - (A4) be valid, and let ε : W × X → R + , ϕ : W → X satisfy the conditions k ( T ϕ )( x ) − ϕ ( x ) , y k β ¬ ε ( x, y ) , ( x, y ) ∈ W × X, (3.1) ε ∗ ( x, y ) := ∞ X n =0 (Λ n ε ) θ ( x, y ) < ∞ , ( x, y ) ∈ W × X, (3.2) where θ = β log κ . Then the limit ψ ( x ) = lim n →∞ T n ϕ ( x ) , x ∈ W (3.3) exists and the function ψ : W → X defined by (3.3) is a fixed point of T satisfying k ϕ ( x ) − ψ ( x ) , y k θq,β ¬ Kε ∗ ( x, y ) , ( x, y ) ∈ W × X. (3.4) for some constant K > . Moreover, if ε ∗ ( x, y ) ¬ M ∞ X n =0 (Λ n ε )( x, y ) ! θ < ∞ (3.5) for some positive real number M , then ψ is the unique fixed point of T satisfying (3.4) .Proof. It is easy to show by induction that, for any n ∈ N (cid:13)(cid:13) ( T n +1 ϕ )( x ) − ( T n ϕ )( x ) , y (cid:13)(cid:13) q,β ¬ (Λ n ε )( x, y ) , ( x, y ) ∈ W × X. (3.6)Indeed, by (3.1), we have (cid:13)(cid:13) ϕ ( x ) − ( T ϕ )( x ) , y (cid:13)(cid:13) q,β ¬ ε ( x, y ) = (Λ ε )( x, y ) , ( x, y ) ∈ W × X. Then the case n = 0 is true. Now, fix an n ∈ N and suppose that (3.6) is true. Then, by(A3)-(A4), for any ( x, y ) ∈ W × X , we get (cid:13)(cid:13) ( T n +1 ϕ )( x ) − ( T n +2 ϕ )( x ) , x , y (cid:13)(cid:13) q,β = (cid:13)(cid:13) T ( T m +1 ϕ )( x ) − T ( T m ϕ )( x ) , y (cid:13)(cid:13) q,β ¬ X ¬ i ¬ j L i ( x, y ) (cid:13)(cid:13) T n ϕ ( f i ( x )) − T n +1 ϕ ( f i ( x )) , y (cid:13)(cid:13) q,β ¬ X ¬ i ¬ j L i ( x, y )(Λ n ε )( f i ( x ) , y ) = (Λ n +1 ε )( x, y ) . So, (3.6) holds for all n ∈ N . IZ. EL-FASSI
Next, from (3.6), (2.2) and Theorem 2.1, for every k ∈ N , n ∈ N and ( x, y ) ∈ W × X ,we have (cid:13)(cid:13)(cid:12)(cid:12) ( T n ϕ )( x ) − ( T n + k ϕ )( x ) ,y (cid:12)(cid:12)(cid:13)(cid:13) θq,β = (cid:13)(cid:13)(cid:12)(cid:12) k − X i =0 (cid:0) ( T n + i ϕ )( x ) − ( T n + i +1 ϕ )( x ) (cid:1) , y (cid:12)(cid:12)(cid:13)(cid:13) θq,β ¬ k − X i =0 (cid:13)(cid:13)(cid:12)(cid:12) ( T n + i ϕ )( x ) − ( T n + i +1 ϕ )( x ) , y (cid:12)(cid:12)(cid:13)(cid:13) θq,β ¬ C θ k − X i =0 (cid:13)(cid:13) ( T n + i ϕ )( x ) − ( T n + i +1 ϕ )( x ) , y (cid:13)(cid:13) θq,β ¬ C θ k − X i =0 (Λ n + i ε ) θ ( x, y ) = C θ n + k − X i = n (Λ i ε ) θ ( x, y ) ¬ C θ ε ∗ ( x, y ) for some C > . (3.7)By the convergence of the series P n (Λ n ε ) θ ( x, y ), it follows from (3.7) that, for every x ∈ W , (cid:8) ( T n ϕ )( x ) (cid:9) n ∈ N is a Cauchy sequence in ( X, k|· , ·|k q,β ) . By Theorem 2.1, (cid:8) ( T n ϕ )( x ) (cid:9) n ∈ N is also a Cauchy sequence in ( X, k· , ·k q,β , κ ). As ( X, k· , ·k q,β , κ ) is a quasi-(2 , β )-Banach space,then the limit ψ ( x ) := lim m →∞ ( T m ϕ )( x ) exists for any x ∈ W , so (3.3) holds.Since k|· , ·|k q,β is continuous and taking n = 0 and k → ∞ in (3.7), we get k| ϕ ( x ) − ψ ( x ) , y |k θq,β ¬ C θ ε ∗ ( x, y ) (3.8)for all ( x, y ) ∈ W × X and for some C > . From (2.2), we find k ϕ ( x ) − ψ ( x ) , y k θq,β ¬ C − θ k| ϕ ( x ) − ψ ( x ) , y |k θq,β ¬ ( C /C ) θ ε ∗ ( x, y )for all ( x, y ) ∈ W × X and for some C , C > , so (3.4) holds with K := ( C /C ) θ .By applying (A3), (3.3) and (2.2), we get (cid:13)(cid:13)(cid:0) T n +1 ϕ )( x ) − ( T ψ )( x ) ,y (cid:13)(cid:13) q,β ¬ j X i =1 L i ( x, y ) (cid:13)(cid:13)(cid:0) T n ϕ )( f i ( x )) − ψ ( f i ( x )) , y (cid:13)(cid:13) q,β ¬ C − j X i =1 L i ( x, y ) (cid:13)(cid:13)(cid:12)(cid:12)(cid:0) T n ϕ )( f i ( x )) − ψ ( f i ( x )) , y (cid:12)(cid:12)(cid:13)(cid:13) q,β (3.9)for all ( x, y ) ∈ W × X , n ∈ N and for some C > . Letting n → ∞ in (3.9), we findlim n →∞ (cid:13)(cid:13)(cid:0) T n +1 ϕ )( x ) − ( T ψ )( x ) , y (cid:13)(cid:13) q,β = 0for all ( x, y ) ∈ W × X , that is, lim n →∞ (cid:0) T n +1 ϕ )( x ) = ( T ψ )( x ) for all x ∈ W . This prove T ψ = ψ. So ψ is a fixed point of T that satisfies (3.4).It remains to prove the uniqueness of ψ . Let γ be also a fixed point of T satisfying (3.4).For every m ∈ N , we show that k ψ ( x ) − γ ( x ) , y k q,β = (cid:13)(cid:13)(cid:0) T m ψ )( x ) − ( T m γ )( x ) , y (cid:13)(cid:13) q,β ¬ (2 K ) /θ M ∞ X i = m (Λ i ε )( x, y ) (3.10) IXED POINT THEOREM IN QUASI-(2 , β )-BANACH SPACES AND ITS APPLICATIONS 9 for all ( x, y ) ∈ W × X and for some K, M > . Indeed, for m = 0 and from (3.8), we have k| ψ ( x ) − γ ( x ) , y |k θq,β ¬ k| ψ ( x ) − ϕ ( x ) , y |k θq,β + k| ϕ ( x ) − γ ( x ) , y |k θq,β ¬ C θ ε ∗ ( x, y )for all ( x, y ) ∈ W × X and for some C > . From (2.2) and (3.5) we have k ψ ( x ) − γ ( x ) , y k θq,β ¬ C − θ k| ψ ( x ) − γ ( x ) , y |k θq,β ¬ ( C /C ) θ ε ∗ ( x, y ) ¬ K M ∞ X i =0 (Λ i ε )( x, y ) ! θ , ( x, y ) ∈ W × X. Thus, (3.10) holds for m = 0 . Now, assume that (3.10) is valid for some m ∈ N . By (A3) and (3.10), we obtain (cid:13)(cid:13)(cid:0) T m +1 ψ )( x ) − ( T m +1 γ )( x ) , y (cid:13)(cid:13) q,β ¬ j X k =1 L k ( x, y ) (cid:13)(cid:13)(cid:0) T m ψ )( f k ( x )) − ( T m γ )( f k ( x )) , y (cid:13)(cid:13) q,β ¬ j X k =1 L k ( x, y )(2 K ) /θ M ∞ X i = m (Λ i ε )( x, y )= (2 K ) /θ M ∞ X i = m +1 (Λ i ε )( x, y ) , x, y ∈ X. So, (3.10) holds for all m ∈ N . It follows from (2.2) and (3.10) that (cid:13)(cid:13)(cid:12)(cid:12)(cid:0) T m ψ )( x ) − ( T m γ )( x ) , y (cid:12)(cid:12)(cid:13)(cid:13) θq,β ¬ C θ (cid:13)(cid:13)(cid:0) T m ψ )( x ) − ( T m γ )( x ) , y (cid:13)(cid:13) θq,β ¬ KC θ M ∞ X i = m (Λ i ε )( x, y ) ! θ (3.11)for all ( x, y ) ∈ W × X and all m ∈ N . Letting m → ∞ in (3.11) and from (3.5), we get (cid:13)(cid:13)(cid:12)(cid:12) ψ ( x ) − γ ( x ) , y (cid:12)(cid:12)(cid:13)(cid:13) q,β = 0 for all y ∈ X. That is ψ ≡ γ . This completes the proof. (cid:3) Corollary 3.1.
Assume that hypotheses (A1) - (A4) are satisfied. Suppose that there existtwo functions ε : W × X → R + and ϕ : W → X such that (3.1) holds and (Λ ε )( x, y ) = qε ( x, y ) , ( x, y ) ∈ W × X, q ∈ [0 , . (3.12) Then the limit (3.3) exists and the function ψ : W → X so defined is the unique fixed pointof T with k ϕ ( x ) − ψ ( x ) , y k ¬ Kε θ ( x, y )1 − q θ , ( x, y ) ∈ W × X, q ∈ [0 , for some K > , where θ = β log κ . Proof.
It follows from (3.12) that(Λ n ε )( x, y ) = q n ε ( x, y ) , ( x, y ) ∈ W × X, n ∈ N . Therefore, for every ( x, y ) ∈ W × X,ε ∗ ( x, y ) = ∞ X n =0 (Λ n ε ) θ ( x, y ) = ∞ X n =0 q nθ ( ε ( x, y )) θ = ε θ ( x, y )1 − q θ < ∞ where θ = β log κ . So, condition (3.2) holds. Moreover ∞ X n =0 (Λ n ε )( x, y ) ! θ = ∞ X n =0 q n ε ( x, y ) ! θ = ε θ ( x, y )(1 − q ) θ < ∞ for all ( x, y ) ∈ W × X. As ε θ ( x, y )1 − q θ ¬ ε θ ( x, y )(1 − q ) θ , ( x, y ) ∈ W × X, then condition (3.5) holds and our assertion follows from Theorem 3.1 and its proof. (cid:3) Remark 3. (i) If ( X, k· , ·k , κ ) is a quasi-2-Banach in Theorem 3.1, then θ = log κ β = 1) and Theorem 3.1 remains true.(ii) If ( X, k· , ·k β ) is a (2 , β )-Banach in Theorem 3.1, then κ = 1 and Theorem 3.1 remainstrue with k|· , ·|k β = k· , ·k β .(iii) If ( X, k· , ·k ) is a 2-Banach in Theorem 3.1, then κ = 1 = β we obtain [12, Theorem1] with k|· , ·|k = k· , ·k . Moreover, we have k ϕ ( x ) − ψ ( x ) , y k ¬ ε ∗ ( x, y ) . General Solution of Eq. (1.2)In this section, we give the general solution of the radical-type functional equation (1.2)by using some results that are reported in [10].
Proposition 4.1.
Let a, b, c, d ∈ R \ { } be fixed numbers and V a real vector space. Afunction f : R → V satisfies (1.2) if and only if there exists a solution g : R → V of theequation g ( ax + by ) + g ( ax − by ) = cg ( x ) + dg ( y ) , x, y ∈ R , (4.1) such that f ( x ) = g (cid:0) x (cid:1) for all x ∈ R . Proof.
It is clear that if f : R → V has form f ( x ) = g (cid:0) x (cid:1) for x ∈ R , with g satisfies (4.1),then it is a solution of (1.2).On the other hand, if f : R → V is a solution to (1.2), and setting h ( x ) = f (cid:0) √ x (cid:1) for all x ∈ R , then from (1.2), we get h (cid:0) ax + by (cid:1) + h (cid:0) ax − by (cid:1) = f (cid:16) p ax + by (cid:17) + f (cid:16) p ax − by (cid:17) = cf ( x ) + df ( y ) = ch (cid:0) x (cid:1) + dh (cid:0) y (cid:1) , x, y ∈ R , where a, b, c, d ∈ R \ { } . So, h satisfies (4.1). Thus it suffices to take g ≡ h . (cid:3) Lemma 4.1.
Let V be a real vector space and f : R → V be a function satisfying (1.2) with a, b, c, d ∈ R \ { } and c + d = 2 . Then (i) f satisfies the functional equation f (cid:16) p x + y (cid:17) + f (cid:16) p x − y (cid:17) = 2 f ( x ) + 2 f ( y ) , x, y ∈ R . (4.2)(ii) f is a sextic mapping and if f is continuous, then f ( x ) = x f (1) for all x ∈ R + . IXED POINT THEOREM IN QUASI-(2 , β )-BANACH SPACES AND ITS APPLICATIONS 11
Proof. (i) Suppose that f satisfies (1.2), then by letting x = y = 0 in (1.2), we have f (0) = 0because c + d = 2 . Replacing y by − y in (1.2), we get f (cid:16) p ax − by (cid:17) + f (cid:16) p ax + by (cid:17) = cf ( x ) + df ( − y ) , x, y ∈ R . (4.3)If we compare (1.2) with (4.3), we obtain that the function f is even.Taking first y = 0 and next x = 0 in (1.2), we get f (cid:0) √ ax (cid:1) = c f ( x ) , f (cid:0) √ by (cid:1) = d f ( y ) . So, f (cid:0) √ abx (cid:1) = cd f ( x ) , x ∈ R . (4.4)Replacing ( x, y ) by (cid:0) √ bx, √ ay (cid:1) in (1.2), we obtain f (cid:16) p abx + aby (cid:17) + f (cid:16) p abx − aby (cid:17) = cf (cid:0) √ bx (cid:1) + df (cid:0) √ ay (cid:1) x ∈ R . (4.5)It follows from (4.4) and (4.5) that f is a solution of (4.2).(ii) As f satisfies (4.2) and by [17, Remark 1], we get the desired results. (cid:3) Theorem 4.2.
Let a, b, c, d ∈ R \ { } be fixed numbers and V a real vector space. A function f : R → V satisfies (1.2) if and only if: (i) In the case c + d = 2 , there exists a quadratic mapping Q : R → V such that f ( x ) = Q (cid:0) x (cid:1) for all x ∈ R and Q ( ax ) = c Q ( x ) , Q ( bx ) = d Q ( x ) , x ∈ R . (4.6)(ii) In the case c + d = 2 , there exist w ∈ V and a quadratic mapping Q : R → V , suchthat (4.6) holds and f ( x ) = Q (cid:0) x (cid:1) + w for all x ∈ R . Proof. (i) Assume that f satisfies (1.2) and c + d = 2. Then by Lemma 4.1, we obtain that f satisfies (4.2). Applying [17, Theorem 2.1], there exists a quadratic mapping Q : R → V such that f ( x ) = Q (cid:0) x (cid:1) for all x ∈ R . Also, we have f (0) = 0 and f (cid:0) √ ax (cid:1) = c f ( x ) , f (cid:0) √ bx (cid:1) = d f ( x ) , x ∈ R . So, Q ( ax ) = c Q ( x ) and Q ( bx ) = d Q ( x ) , x ∈ R . (ii) Suppose that f satisfies (1.2) and c + d = 2. Putting f ( x ) := f ( x ) − f (0) , x ∈ R . Then f (0) = 0 and f satisfies (1.2). Similarly to the proof of Lemma 4.1, we get f (cid:0) √ ax (cid:1) = c f ( x ) , f (cid:0) √ bx (cid:1) = d f ( x ) , x ∈ R , and f satisfies (4.2), so by [17, Theorem 2.1], there exists a quadratic mapping Q : R → V such that f ( x ) = Q (cid:0) x (cid:1) and Q satisfies (4.6). Hence f ( x ) = Q (cid:0) x (cid:1) + w, with w := f (0) . The converse is easy to check. This completes the proof. (cid:3)
Remark 4.
It is well known (see, e.g., [1]) that a mapping Q : R → V is quadratic if andonly there exists B : R → V that is symmetric (i.e., B ( x, y ) = B ( y, x ) for all x, y ∈ R ) andbiadditive (i.e., B ( x + y, z ) = B ( x, z ) + B ( y, z ) for all x, y, z ∈ R ) such that Q ( x ) = B ( x, x )for all x ∈ R .We derive from Proposition 4.1, Theorem 4.2 and Remark 4 the following corollaries. Corollary 4.1.
Let a, b, c, d ∈ R \{ } be fixed numbers and V a real vector space. A function f : R → V satisfies (1.2) if and only if: (i) In the case c + d = 2 , there is a symmetric biadditive mapping B : R → V such that f ( x ) = B (cid:0) x , x (cid:1) for all x ∈ R , and B ( ax, ay ) = c B ( x, y ) , B ( bx, by ) = d B ( x, y ) , x, y ∈ R . (4.7)(ii) In the case c + d = 2 , there are w ∈ V and a symmetric biadditive mapping B : R → V such that (4.7) holds and f ( x ) = B (cid:0) x , x (cid:1) + w for all x ∈ R . Corollary 4.2.
Let a, b, c, d ∈ R \{ } be fixed numbers and V a real vector space. A function g : R → V satisfies (4.1) if and only if: (i) In the case c + d = 2 , there is a symmetric biadditive mapping B : R → V such that g ( x ) = B (cid:0) x, x (cid:1) and B ( ax, ay ) = c B ( x, y ) , B ( bx, by ) = d B ( x, y ) , x, y ∈ R . (4.8)(ii) In the case c + d = 2 , there are w ∈ V and a symmetric biadditive mapping B : R → V such that (4.8) holds and g ( x ) = B (cid:0) x, x (cid:1) + w for all x ∈ R . Hyperstability criterion of Eq. (1.3) and some consequences
By using Theorem 3.1, we study the hyperstability results of Eq. (1.3) in quasi-(2 , β )-Banach space. In the following, we consider ( X, k· , ·k q,β , κ ) is a quasi-(2 , β )-Banach space, a, b, c, d ∈ R \ { } are fixed numbers, θ = β log κ N m := { m ∈ N : m m } with m ∈ N . A function f : R → X fulfilling Eq. (1.3) γ -approximately if (cid:13)(cid:13)(cid:13) f (cid:16) p ax + by (cid:17) + f (cid:16) p ax − by (cid:17) − cf ( x ) − df ( y ) , z (cid:13)(cid:13)(cid:13) q,β ¬ γ ( x, y, z ) (5.1)for all ( x, y, z ) ∈ R × X , where √ ax = ± √ by and γ : R × X → R + is a given function. Theorem 5.1.
Let h i : R × X → R + be a given function for i ∈ { , , , } , and let M := n n ∈ N | P n := max { A n , B n , C n } < o = ∅ , (5.2) where A n = κ | c | β s ( u n ) s ( u n ) + κ | d | β s ( v n ) s ( v n ) + κ s ( w n ) s ( w n ) B n = κ | c | β s ( u n ) + κ | d | β s ( v n ) + κ s ( w n ) C n = κ | c | β s ( u n ) + κ | d | β s ( v n ) + κ s ( w n ) u n = n √ a , v n = r − n b , w n = p n − IXED POINT THEOREM IN QUASI-(2 , β )-BANACH SPACES AND ITS APPLICATIONS 13 lim n →∞ max { s ( u n ) s ( v n ) , s ( u n ) , s ( v n ) } = 0 and s i ( ρ ) := inf (cid:8) t ∈ R + : h i (cid:0) ρx , z (cid:1) ¬ th i (cid:0) x , z (cid:1) for all ( x, z ) ∈ R × X (cid:9) for ρ ∈ R and i ∈ { , , , } , such that one of the conditions holds: (i) lim | ρ |→ + ∞ s ( ρ ) s ( ρ ) = 0 if P n = A n , (ii) lim | ρ |→ + ∞ s ( ρ ) = 0 if P n = B n , (iii) lim | ρ |→ + ∞ s ( ρ ) = 0 if P n = C n .If f : R → X satisfies (5.1) with γ ( x, y, z ) = h (cid:0) x , z (cid:1) h (cid:0) y , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) y , z (cid:1) , then (1.3) holds.Proof. It is clear that for ρ ∈ R and i = { , , , } , we have h i (cid:0) ρx , z (cid:1) ¬ s i ( ρ ) h i (cid:0) x , z (cid:1) , ( x, z ) ∈ R × X. (5.3)Replacing ( x, y ) by ( u m x, v m x ) in (5.1), with u m = m √ a and v m = q − m b , we get k cf ( u m x ) + df ( v m x ) − f ( w m x ) − f ( x ) , z k q,β ¬ h (cid:0) u m x , z (cid:1) h (cid:0) v m x , z (cid:1) + h (cid:0) u m x , z (cid:1) + h (cid:0) v m x , z (cid:1) (5.4)for all x ∈ R and all z ∈ X with w m = √ m − m ∈ N . For each m ∈ N , we willdefine an operator T m : X R → X R by( T m ξ )( x ) := cξ ( u m x ) + dξ ( v m x ) − ξ ( w m x ) , ξ ∈ X R , x ∈ R . Putting ε m ( x, z ) := h (cid:0) u m x , z (cid:1) h (cid:0) v m x , z (cid:1) + h (cid:0) u m x , z (cid:1) + h (cid:0) v m x , z (cid:1) for all x ∈ R and z ∈ X with m ∈ N . Then by (5.3) we have ε m ( x, z ) ¬ σ m (cid:2) h (cid:0) x , z (cid:1) h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1)(cid:3) , ( x, z ) ∈ R × X, (5.5)with σ m := max { s ( u m ) s ( v m ) , s ( u m ) , s ( v m ) } . Then the inequality (5.4) takes the form k ( T m f )( x ) − f ( x ) , z k q,β ¬ ε m ( x, z ) , ( x, z ) ∈ R × X, m ∈ N . For each m ∈ N , we find that the operator Λ m : R R × X + → R R + × X defined by(Λ m δ )( x, z ) = κ | c | β δ ( u m x, z ) + κ | d | β δ ( v m x, z ) + κ δ ( w m x, z ) , δ ∈ R R × X + , x ∈ R has the shape given in ( A4 ) with j = 3, f ( x ) ≡ u m x, f ( x ) ≡ v m x, f ( x ) ≡ w m x, L ( x, z ) ≡ κ | c | β ,L ( x, z ) ≡ κ | d | β , L ( x, z ) ≡ κ ; ( x, z ) ∈ R × X, m ∈ N . Moreover, for every ξ, µ ∈ X R , m ∈ N and ( x, z ) ∈ R × X , we obtain (cid:13)(cid:13) ( T m ξ )( x ) − ( T m µ )( x ) , z (cid:13)(cid:13) q,β = k cξ ( u m x ) + dξ ( v m x ) − ξ ( w m x ) − cµ ( u m x ) − dµ ( v m x ) + µ ( w m x ) , z k q,β ¬ κ | c | β k ξ ( f ( x )) − µ ( f ( x )) , z k q,β + κ | d | β k ξ ( f ( x )) − µ ( f ( x )) , z k q,β + κ k ξ ( f ( x )) − µ ( f ( x )) , z k q,β = X i =1 L i ( x, z ) (cid:13)(cid:13) ξ ( f i ( x )) − µ ( f i ( x )) , z (cid:13)(cid:13) q,β . So, (A3) is valid for T m with m ∈ N . It is not hard to show thatΛ m ε m ( x, z ) ¬ P m σ m (cid:2) h (cid:0) x , z (cid:1) h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1)(cid:3) (5.6)for all ( x, z ) ∈ R × X. By induction, we will show that for each n ∈ N and ( x, z ) ∈ R × X, Λ nm ε m ( x, z ) ¬ P nm σ m (cid:2) h (cid:0) x , z (cid:1) h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1)(cid:3) (5.7)where m ∈ M . From (5.5), we obtain that the inequality (5.7) holds for n = 0 . Next, wewill assume that (5.7) holds for n = r , where r ∈ N . Then,Λ r +1 m ε m ( x, z ) = Λ m (Λ rm ε m ( x, z ))= κ | c | β Λ rm ε m ( u m x, z ) + κ | d | β Λ rm ε m ( v m x, z ) + κ Λ rm ε m ( w m x, z ) ¬ P rm σ m κ | c | β (cid:2) h (cid:0) u m x , z (cid:1) h (cid:0) u m x , z (cid:1) + h (cid:0) u m x , z (cid:1) + h (cid:0) u m x , z (cid:1)(cid:3) + P rm σ m κ | d | β (cid:2) h (cid:0) v m x , z (cid:1) h (cid:0) v m x , z (cid:1) + h (cid:0) v m x , z (cid:1) + h (cid:0) v m x , z (cid:1)(cid:3) + P rm σ m κ (cid:2) h (cid:0) w m x , z (cid:1) h (cid:0) w m x , z (cid:1) + h (cid:0) w m x , z (cid:1) + h (cid:0) w m x , z (cid:1)(cid:3) ¬ P rm σ m (cid:2) A m h (cid:0) x , z (cid:1) h (cid:0) x , z (cid:1) + B m h (cid:0) x , z (cid:1) + C m h (cid:0) x , z (cid:1)(cid:3) ¬ P rm σ m max { A m , B m , C m } (cid:2) h (cid:0) x , z (cid:1) h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1)(cid:3) = P r +1 m σ m (cid:2) h (cid:0) x , z (cid:1) h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1)(cid:3) for all ( x, z ) ∈ R × X and m ∈ M . This shows that (5.7) holds for all n ∈ N .By the definition of M , we find that for each ( x, z ) ∈ R × X and m ∈ M , ε ∗ m ( x, z ) = ∞ X n =0 (Λ nm ε m ) θ ( x, z ) ¬ ∞ X n =0 P nθm σ θm (cid:2) h (cid:0) x , z (cid:1) h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1)(cid:3) θ = σ θm (cid:2) h (cid:0) x , z (cid:1) h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1)(cid:3) θ − P θm . Thus, according to Theorem 3.1, there exists a fixed point Q m : R → X of the operator T m satisfying (cid:13)(cid:13) f ( x ) − Q m ( x ) , z (cid:13)(cid:13) θq,β ¬ Kε ∗ m ( x, z ) ¬ Kσ θm (cid:2) h (cid:0) x , z (cid:1) h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1)(cid:3) θ − P θm (5.8) IXED POINT THEOREM IN QUASI-(2 , β )-BANACH SPACES AND ITS APPLICATIONS 15 for all ( x, z ) ∈ R × X, m ∈ M and for some constant K >
0. That is, Q m ( x ) = cQ m ( u m x ) + dQ m ( v m x ) − Q m ( w m x ) , x ∈ R , m ∈ M , and (5.8) holds for all ( x, z ) ∈ R × X . Moreover; Q m ( x ) := lim n →∞ T nm f ( x ) . (5.9)Now, we show that for every n ∈ N and ( x, y, z ) ∈ R × X with √ ax = ± √ by , (cid:13)(cid:13)(cid:13) T nm f (cid:16) p ax + by (cid:17) + T nm f (cid:16) p ax − by (cid:17) − c T nm f ( x ) − d T nm f ( y ) , z (cid:13)(cid:13)(cid:13) q,β (5.10) ¬ P nm (cid:2) h (cid:0) x , z (cid:1) h (cid:0) y , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) y , z (cid:1)(cid:3) where m ∈ M and γ ( x, y, z ) := (cid:2) h (cid:0) x , z (cid:1) h (cid:0) y , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) y , z (cid:1)(cid:3) .Indeed, if n = 0, then (5.10) is simply (5.1). So, fix n ∈ N and assume that (5.10) holdsfor n and every ( x, y, z ) ∈ R × X with √ ax = ± √ by . Then, by the definition of T m and(5.3), we get that (cid:13)(cid:13)(cid:13) T n +1 m f (cid:16) p ax + by (cid:17) + T n +1 m f (cid:16) p ax − by (cid:17) − c T n +1 m f ( x ) − d T n +1 m f ( y ) , z (cid:13)(cid:13)(cid:13) q,β = (cid:13)(cid:13)(cid:13) c T nm f (cid:16) u m p ax + by (cid:17) + d T nm f (cid:16) v m p ax + by (cid:17) − T nm f (cid:16) w m p ax + by (cid:17) + c T nm f (cid:16) u m p ax − by (cid:17) + d T nm f (cid:16) v m p ax − by (cid:17) − T nm f (cid:16) w m p ax − by (cid:17) − c (cid:2) c T nm f ( u m x ) + d T nm f ( v m x ) − T nm f ( w m x ) (cid:3) − d (cid:2) c T nm f ( u m y ) + d T nm f ( v m y ) − T nm f ( w m y ) (cid:3) , z (cid:13)(cid:13)(cid:13) q,β ¬ κ | c | β (cid:13)(cid:13)(cid:13) T nm f (cid:16) u m p ax + by (cid:17) + T nm f (cid:16) u m p ax − by (cid:17) − c T nm f ( u m x ) − d T nm f ( u m y ) , z (cid:13)(cid:13)(cid:13) q,β + κ | d | β (cid:13)(cid:13)(cid:13) T nm f (cid:16) v m p ax + by (cid:17) + T nm f (cid:16) v m p ax − by (cid:17) − c T nm f ( v m x ) − d T nm f ( v m y ) , z (cid:13)(cid:13)(cid:13) q,β + κ (cid:13)(cid:13)(cid:13) T nm f (cid:16) w m p ax + by (cid:17) + T nm f (cid:16) w m p ax − by (cid:17) − c T nm f ( w m x ) − d T nm f ( v m y ) , z (cid:13)(cid:13)(cid:13) q,β ¬ P nm κ | c | β (cid:8) h (cid:0) u m x , z (cid:1) h (cid:0) u m y , z (cid:1) + h (cid:0) u m x , z (cid:1) + h (cid:0) u m y , z (cid:1)(cid:9) + P nm κ | d | β (cid:8) h (cid:0) v m x , z (cid:1) h (cid:0) v m y , z (cid:1) + h (cid:0) v m x , z (cid:1) + h (cid:0) v m y , z (cid:1)(cid:9) + P nm κ (cid:8) h (cid:0) w m x , z (cid:1) h (cid:0) w m y , z (cid:1) + h (cid:0) w m x , z (cid:1) + h (cid:0) w m y , z (cid:1)(cid:9) ¬ P nm max { A m , B m , C m } (cid:2) h (cid:0) x , z (cid:1) h (cid:0) y , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) y , z (cid:1)(cid:9) = P n +1 m (cid:8) h (cid:0) x , z (cid:1) h (cid:0) y , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) y , z (cid:1)(cid:9) for all ( x, y, z ) ∈ R × X with √ ax = ± √ by . Thus, by induction, we have shown that (5.10)holds for all n ∈ N and for all ( x, y, z ) ∈ R × X with √ ax = ± √ by . From (5.10) and (2.2) we find that (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12) T nm f (cid:16) p ax + by (cid:17) + T nm f (cid:16) p ax − by (cid:17) − c T nm f ( x ) − d T nm f ( y ) , z (cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13) θq,β ¬ C θ (cid:13)(cid:13)(cid:13) T nm f (cid:16) p ax + by (cid:17) + T nm f (cid:16) p ax − by (cid:17) − c T nm f ( x ) − d T nm f ( y ) , z (cid:13)(cid:13)(cid:13) θq,β ¬ C θ P nθm (cid:2) h (cid:0) x , z (cid:1) h (cid:0) y , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) y , z (cid:1)(cid:3) θ (5.11)for some constant C >
0. We conclude from the continuity of k|· , ·|k θ , (5.9) and (5.11) that,for every ( x, y, z ) ∈ R × X with √ ax = ± √ by, (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12) Q m (cid:16) p ax + by (cid:17) + Q m (cid:16) p ax − by (cid:17) − cQ m ( x ) − dQ m ( y ) , z (cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13) θq,β (5.12)= lim n →∞ (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12) T nm f (cid:16) p ax + by (cid:17) + T nm f (cid:16) p ax − by (cid:17) − c T nm f ( x ) − d T nm f ( y ) , z (cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13) θq,β = 0 , this means Q m (cid:16) p ax + by (cid:17) + Q m (cid:16) p ax − by (cid:17) = cQ m ( x ) + dQ m ( y ) (5.13)for all x, y ∈ R , with √ ax = ± √ by. We want now to prove that the mapping Q m : R → X is unique. So, let G m : R → X be a solution of (5.13) and (cid:13)(cid:13) f ( x ) − G m ( x ) , z (cid:13)(cid:13) θq,β ¬ Kσ θm (cid:2) h (cid:0) x , z (cid:1) h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1)(cid:3) θ − P θm (5.14)for all ( x, z ) ∈ R × X. Thus, replacing ( x, y ) by ( u m x, v m y ) in (5.13), we get T m G m ( x ) = G m ( x ) for all x ∈ R and m ∈ M .Moreover, we have k| Q m ( x ) − G m ( x ) , z |k q,β ¬ C k Q m ( x ) − G m ( x ) , z k q,β ¬ κC (cid:0) k Q m ( x ) − f ( x ) , z k q,β + k G m ( x ) − f ( x ) , z k q,β (cid:1) ¬ L (cid:2) h (cid:0) x , z (cid:1) h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1)(cid:3) , (5.15)where L := κC K /θ σ m (1 − P θm ) /θ and m ∈ M . It is easy to prove by induction on n that k| Q m ( x ) − G m ( x ) , z |k q,β ¬ LP nm (cid:2) h (cid:0) x , z (cid:1) h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) x , z (cid:1)(cid:3) (5.16)for all x ∈ R and m ∈ M . Letting n → ∞ in (5.16), we get Q m ≡ G m . So, the fixed pointsatisfying (5.8) of T m is unique.Letting m → ∞ in (5.8), we obtain lim m →∞ k f ( x ) − Q m ( x ) , z k q,β = 0, i.e.,lim m →∞ Q m ( x ) = f ( x ) . (5.17) IXED POINT THEOREM IN QUASI-(2 , β )-BANACH SPACES AND ITS APPLICATIONS 17
Also, letting m → ∞ in (5.12), using (5.17) and the continuity of k|· , ·|k , we have (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12) f (cid:16) p ax + by (cid:17) + f (cid:16) p ax − by (cid:17) − cf ( x ) − df ( y ) , z (cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13) q,β = lim m →∞ (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12) Q m (cid:16) p ax + by (cid:17) + Q m (cid:16) p ax − by (cid:17) − cQ m ( x ) − dQ m ( y ) , z (cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13) q,β = 0for all ( x, y, z ) ∈ R × X , with √ ax = ± √ by , that is, f (cid:16) p ax + by (cid:17) + f (cid:16) p ax − by (cid:17) = cf ( x ) + df ( y )for all x, y ∈ R , with √ ax = ± √ by. This proves that f is a solution of the radical-typefunctional equation (1.3). The proof of the theorem is complete. (cid:3) Remark 5.
Theorem 5.1 also provide hyperstability outcomes in each of the following cases:(i) γ ( x, y, z ) = h (cid:0) x , z (cid:1) h (cid:0) y , z (cid:1) + h (cid:0) x , z (cid:1) (ii) γ ( x, y, z ) = h (cid:0) x , z (cid:1) h (cid:0) y , z (cid:1) + h (cid:0) y , z (cid:1) (iii) γ ( x, y, z ) = h (cid:0) x , z (cid:1) h (cid:0) y , z (cid:1) (iv) γ ( x, y, z ) = h (cid:0) x , z (cid:1) + h (cid:0) y , z (cid:1) (v) γ ( x, y, z ) = h (cid:0) x , z (cid:1) + h (cid:0) y , z (cid:1) (vi) γ ( x, y, z ) = h (cid:0) x , z (cid:1) (vii) γ ( x, y, z ) = h (cid:0) y , z (cid:1) for all ( x, y, z ) ∈ R × X. Applying Theorem 5.1 and the same technique, we get the following corollary.
Corollary 5.1.
Under the hypotheses of Theorem 5.1, we consider two functions f : R → X and F : R → X such that (cid:13)(cid:13)(cid:13) f (cid:16) p ax + by (cid:17) + f (cid:16) p ax − by (cid:17) − cf ( x ) − df ( y ) − F ( x, y ) , z (cid:13)(cid:13)(cid:13) q,β ¬ γ ( x, y, z ) for all ( x, y, z ) ∈ R × X , with √ ax = ± √ by and γ ( x, y, z ) = h (cid:0) x , z (cid:1) h (cid:0) y , z (cid:1) + h (cid:0) x , z (cid:1) + h (cid:0) y , z (cid:1) . Assume that the functional equation ˜ f (cid:16) p ax + by (cid:17) + ˜ f (cid:16) p ax − by (cid:17) = c ˜ f ( x ) + d ˜ f ( y ) + F ( x, y ) , (5.18) x, y ∈ R , √ ax = ± √ by admits a solution f : R → X . Then f is a solution of (5.18) .Proof. Let g ( x ) := f ( x ) − f ( x ) for x ∈ R . Then (cid:13)(cid:13)(cid:13) g (cid:16) p ax + by (cid:17) + g (cid:16) p ax − by (cid:17) − cg ( x ) − dg ( y ) , z (cid:13)(cid:13)(cid:13) q,β ¬ κ (cid:13)(cid:13)(cid:13) f (cid:16) p ax + by (cid:17) + f (cid:16) p ax − by (cid:17) − cf ( x ) − df ( y ) − F ( x, y ) , z (cid:13)(cid:13)(cid:13) q,β + κ (cid:13)(cid:13)(cid:13) f (cid:16) p ax + by (cid:17) + f (cid:16) p ax + by (cid:17) − cf ( x ) − df ( y ) − F ( x, y ) , z (cid:13)(cid:13)(cid:13) q,β = κ (cid:13)(cid:13)(cid:13) f (cid:16) p ax + by (cid:17) + f (cid:16) p ax − by (cid:17) − cf ( x ) − df ( y ) − F ( x, y ) , z (cid:13)(cid:13)(cid:13) q,β ¬ κγ ( x, y, z ) , ( x, y, z ) ∈ R × X, √ ax = ± √ by. It follows from Theorem 5.1 that g satisfies the equation (1.3). Therefore, f (cid:16) p ax + by (cid:17) + f (cid:16) p ax − by (cid:17) − cf ( x ) − df ( y ) − F ( x, y )= g (cid:16) p ax + by (cid:17) + g (cid:16) p ax − by (cid:17) − cg ( x ) − dg ( y )+ f (cid:16) p ax + by (cid:17) + f (cid:16) p ax − by (cid:17) − cf ( x ) − df ( y ) − F ( x, y ) = 0for all x, y ∈ R with √ ax = − √ by . (cid:3) To end this section, we derive some particular cases from Theorem 5.1 and Corollary 5.1.Let ( X, k· , ·k q,β , κ ) be a quasi-(2 , β )-Banach space, ( Y, k· , ·k q,α , κ ′ ) a quasi-(2 , α )-normedspace. Also, let λ i : R → Y \ { } be additive functions continuous at a point for i = 1 , , , , and g : X → Y \ { } a surjective mapping. According to Theorem 5.1 and Corollary 5.1, wegive some consequences with h i ( x, z ) := c i k λ i ( x ) , g ( z ) k p i q,α , ( x, z ) ∈ R × X, where c i , p i ∈ R for i ∈ { , , , } . Corollary 5.2.
Let a, b, c, d ∈ R be fixed numbers, D , D , D be three constants andlet p i ∈ R such that p + p , p , p < for i ∈ { , , , } . If f : R → X satisfies the inequality (cid:13)(cid:13)(cid:13) f (cid:16) p ax + by (cid:17) + f (cid:16) p ax − by (cid:17) − cf ( x ) − df ( y ) , z (cid:13)(cid:13)(cid:13) q,β ¬ D (cid:13)(cid:13) λ ( x ) , g ( z ) (cid:13)(cid:13) p q,α (cid:13)(cid:13) λ ( y ) , g ( z ) (cid:13)(cid:13) p q,α + D (cid:13)(cid:13) λ ( x ) , g ( z ) (cid:13)(cid:13) p q,α + D (cid:13)(cid:13) λ ( y ) , g ( z ) (cid:13)(cid:13) p q,α for all ( x, y, z ) ∈ R × X with √ ax = ± √ by , then (1.3) holds.Proof. Define h i : R × X → R + by h i ( x , z ) = c i (cid:13)(cid:13) λ i ( x ) , g ( z ) (cid:13)(cid:13) p i q,α for some c i ∈ R + and p , p , p + p < i ∈ { , , , } and ( D , D , D ) = ( c c , c , c ).For each ρ ∈ R , we have s i ( ρ ) = inf n t ∈ R + : h i (cid:0) ρx , z (cid:1) ¬ th i (cid:0) x , z (cid:1) , ∀ ( x, z ) ∈ R × X o = inf n t ∈ R + : c i (cid:13)(cid:13) λ i ( ρx ) , g ( z ) (cid:13)(cid:13) p i q,α ¬ tc i (cid:13)(cid:13) λ i ( x ) g ( z ) (cid:13)(cid:13) p i q,α , ∀ ( x, z ) ∈ R × X o = | ρ | αp i for i ∈ { , , , } . So,lim n →∞ max { s ( u n ) s ( v n ) , s ( u n ) , s ( v n ) } = lim n →∞ max (cid:26) s (cid:16) n a (cid:17) s (cid:16) − n b (cid:17) , s (cid:16) n a (cid:17) , s (cid:16) − n b (cid:17)(cid:27) = lim n →∞ max (cid:26)(cid:12)(cid:12)(cid:12) n a (cid:12)(cid:12)(cid:12) αp (cid:12)(cid:12)(cid:12) − n b (cid:12)(cid:12)(cid:12) αp , (cid:12)(cid:12)(cid:12) n a (cid:12)(cid:12)(cid:12) αp , (cid:12)(cid:12)(cid:12) − n b (cid:12)(cid:12)(cid:12) αp (cid:27) = 0where u n = n √ a , v n = q − n b and n ∈ N . Similarly, we havelim n →∞ P n = lim n →∞ max { A n , B n , C n } = 0 IXED POINT THEOREM IN QUASI-(2 , β )-BANACH SPACES AND ITS APPLICATIONS 19 where A n , B n and C n are defined as in Theorem 5.1. Clearly, there is n ∈ N such that P n < , n n . Thus, all the conditions in Theorem 5.1 are fulfilled. (cid:3)
Corollary 5.3.
Let a, b, c, d ∈ R be fixed numbers, D , D , D be three constants andlet p i ∈ R such that p + p , p , p < for i ∈ { , , , } . Let f : R → X and F : R → X be two functions such that (cid:13)(cid:13)(cid:13) f (cid:16) p ax + by (cid:17) + f (cid:16) p ax − by (cid:17) − cf ( x ) − df ( y ) − F ( x, y ) , z (cid:13)(cid:13)(cid:13) q,β ¬ D (cid:13)(cid:13) λ ( x ) , g ( z ) (cid:13)(cid:13) p q,α (cid:13)(cid:13) λ ( y ) , g ( z ) (cid:13)(cid:13) p q,α + D (cid:13)(cid:13) λ ( x ) , g ( z ) (cid:13)(cid:13) p q,α + D (cid:13)(cid:13) λ ( y ) , g ( z ) (cid:13)(cid:13) p q,α for all ( x, y, z ) ∈ R × X with √ ax = ± √ by . Assume that the functional equation ˜ f (cid:16) p ax + by (cid:17) + ˜ f (cid:16) p ax − by (cid:17) = c ˜ f ( x ) + d ˜ f ( y ) + F ( x, y ) , (5.19) x, y ∈ R , √ ax = ± √ by admits a solution f : R → X . Then f is a solution of (5.19) . Note 1.
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