International Journal of Analysis and Applications ISSN 2291-8639 Volume 14, Number 2 (2017), 167-174 http://www.etamaths.com GENERALIZED STABILITIES OF EULER-LAGRANGE-JENSEN (a,b)-SEXTIC FUNCTIONAL EQUATIONS IN QUASI-β-NORMED SPACES JOHN MICHAEL RASSIAS1,∗, KRISHNAN RAVI2 AND BERI VENKATACHALAPATHY SENTHIL KUMAR3 Abstract. The aim of this paper is to investigate generalized Ulam-Hyers stabilities of the following Euler-Lagrange-Jensen-(a,b)-sextic functional equation f(ax + by) + f(bx + ay) + (a− b)6 [ f ( ax− by a− b ) + f ( bx−ay b−a )] = 64(ab)2 ( a2 + b2 ) [ f ( x + y 2 ) + f ( x−y 2 )] + 2 ( a2 − b2 ) ( a4 − b4 ) [f(x) + f(y)] where a 6= b, such that k ∈ R; k = a+b 6= 0,±1 and λ = 1+(a−b)6−2 ( a6 + b6 ) −62(ab)2 ( a2 + b2 ) 6= 0, in quasi-β-normed spaces by using fixed point method. In particular, we prove generalized stabil- ities involving the sum of powers of norms, product of powers of norms and the mixed product-sum of powers of norms of the above functional equation in quasi-β-normed spaces by using fixed point method. A counter-example for a singular case is also indicated. 1. Introduction The classical theory of stability of functional equations was instigated by the question of Ulam [42] in the year 1940. In the subsequent year 1941, Hyers [16] was the foremost mathematician to establish the pioneering result connected with the stability of functional equations. The result obtained by Hyers is called as Hyers-Ulam stability of functional equation. Later in the year 1950, Aoki [4] made a further simplification to the theorem of Hyers. In the year 1978, Th.M. Rassias [41] took a broad view in the Hyers result by taking the upper bound as a sum of powers of norms. The result obtained by Th.M. Rassias is recognized as Hyers-Ulam-Rassias stability of functional equation. John M. Rassias ( [29], [30], [31]) provided a further generalization of the result of Hyers by using weaker conditions controlled by a product of different powers of norms. The result proved by John M. Rassias is termed as Ulam-Gavruta-Rassias stability of functional equation. Further, in the year 1994, Gavruta [14] provided a generalization of Th.M. Rassias theorem by replacing a general control function as an upper bound. The stability result ascertained by Gavruta is celebrated as generalized Ulam-Hyers stability of functional equation. In the year 2008, Ravi et al. [39] investigated the stability of the following quadratic functional equation q(`x + y) + q(`x−y) = 2q(x + y) + 2q(x−y) + 2 ( `2 − 2 ) q(x) − 2q(y) for any arbitrary but fixed real constant ` with ` 6= 0; ` 6= ±1; ` 6= ± √ 2 using mixed product-sum of powers of norms. This stability result acquired by Ravi et al. is known as J.M. Rassias stability involving mixed product-sum of powers of norms. Several stability results have recently been obtained for various functional equations and functional inequalities, also for mappings with more general domains and ranges (see [6], [7], [8], [11], [13], [22], [23], [24], [40]). Many research monographs are also available on functional equations, one can see ( [1], [2], [3], [10], [17], [20], [21]). Received 6th March, 2017; accepted 27th April, 2017; published 3rd July, 2017. 2010 Mathematics Subject Classification. 39B82, 39B72. Key words and phrases. Quasi-β-normed spaces; Sextic mapping; (β,p)-Banach spaces; Generalized Ulam-Hyers stabilities. c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 167 168 RASSIAS, RAVI AND KUMAR In 1996, Isac and Th.M. Rassias [18] were the first to provide applications of the stability theory of functional equations for the proof of new fixed point theorems with applications. The stability problems of several various functional equations have been extensively investigated by a number of authors using fixed point methods (see [5], [26], [43], [45]). John M. Rassias [32] introduced Euler-Lagrange type quadratic functional equation of the form f(ax + by) + f(bx−ay) = ( a2 + b2 ) (f(x) + f(y)) (1.1) motivated from the following pertinent algebraic equation |ax + by|2 + |bx−ay|2 = ( a2 + b2 )( |x|2 + |y|2 ) . (1.2) The solution of the functional equation (1.1) is called an Euler-Lagrange quadratic type mapping. In addition, John M. Rassias ( [32], [33], [34], [35], [36]) generalized the standard quadratic equation to the following quadratic equation m1m2 |a1x1 + a2x2| 2 + |m2a2x1 −m1a1x2| 2 = ( m1 |a1| 2 + m2 |a2| 2 )( m2 |x1| 2 + m1 |x2| 2 ) . He introduced and investigated the general pertinent Euler-Lagrange quadratic mappings. These Euler-Lagrange mappings are named Euler-Lagrange-Rassias mappings, and the corresponding Euler- Lagrange equations are called Euler-Lagrange-Rassias equations (see [15], [25], [27], [28]). These no- tions provide a cornerstone in analysis, because of their particular interest in probability theory and stochastic analysis in marrying these fields of research to functional equations via the pioneering in- troduction of the Euler-Lagrange-Rassias quadratic weighted means and fundamental mean equations ( [15], [34], [35]). John M. Rassias [38] introduced the cubic functional equation, as follows: f(x + 2y) − 3f(x + y) + 3f(x) −f(x−y) = 6f(y). (1.3) This inspiring cubic functional equation was the transition from the following famous Euler-Lagrange- Rassias quadratic functional equation: f(x + y) − 2f(x) + f(x−y) = 2f(y) to the cubic functional equations. John M. Rassias [37] introduced also the following quartic functional equation: f(x + 2y) + f(x− 2y) + 6f(x) = 4f(x + y) + 4f(x−y) + 24f(y). (1.4) It is easy to see that f(x) = x4 is a solution of equation (1.4). For this reason, the equation (1.4) is called a quartic functional equation. The general solution of (1.4) is determined without assuming any regularity conditions on the unknown function (refer [9]). Since the solution of equation (1.4) is even, we can rewrite (1.4) as f(2x + y) + f(2x−y) = 4f(x + y) + 4f(x−y) + 24f(x) − 6f(y). (1.5) In 2010, Xu et al. [44] achieved the general solution and proved the stability of the quintic functional equation f(x + 3y) − 5f(x + 2y) + 10f(x + y) − 10f(x) + 5f(x−y) −f(x− 2y) = 120f(y) (1.6) and the sextic functional equation f(x + 3y) − 6f(x + 2y) + 15f(x + y) − 20f(x) + 15f(x−y) − 6f(x + 2y) + f(x− 3y) = 720f(y) (1.7) in quasi-β-normed spaces using fixed point method. EULER-LAGRANGE-JENSEN (a,b)-SEXTIC FUNCTIONAL EQUATIONS 169 In this paper, the first author of this paper introduces a new Euler-Lagrange-Jensen (a,b; k = a+b)- sextic functional equation f(ax + by) + f(bx + ay) + (a− b)6 [ f ( ax− by a− b ) + f ( bx−ay b−a )] = 64(ab)2 ( a2 + b2 )[ f ( x + y 2 ) + f ( x−y 2 )] + 2 ( a2 − b2 )( a4 − b4 ) [f(x) + f(y)] (1.8) where a 6= b, such that k ∈ R; k = a+b 6= 0,±1 and λ = 1+(a−b)6−2 ( a6 + b6 ) −62(ab)2 ( a2 + b2 ) 6= 0. Then we investigate the generalized Ulam-Hyers stability of the equation (1.8) in quasi-β-normed spaces using fixed point method. We extend the stability results involving sum of powers of norms, product of powers of norms and mixed product-sum of powers of norms of the above functional equation. We also provide a counter-example to show that the functional equation (1.8) is not stable for singular case. It is easy to see that the function f(x) = kx6 is a solution of the equation (1.8). Hence we say that it is a sextic functional equation. 2. Preliminaries In this section, we recall some fundamental notions in association with quasi-β-normed spaces and m-additive symmetric mappings. Let β be a fixed real number with 0 < β ≤ 1 and let K denote either R or C. Definition 2.1. Let X be a linear space over K. A quasi-β-norm ‖·‖ is a real-valued function on X satisfying the following conditions: (i) ‖a‖≥ 0 for all a ∈X and ‖a‖ = 0 if and only if a = 0. (ii) ‖ηa‖ = |η|β · ‖a‖ for all η ∈ K and all a ∈X. (iii) There is a constant K ≥ 1 such that ‖a + b‖≤ K (‖a‖ + ‖b‖) for all a,b ∈X . The pair (X ,‖·‖) is called quasi-β-normed space if ‖·‖ is a quasi-β-norm on X. The smallest possible K is called the modulus of concavity of ‖·‖. Definition 2.2. A complete quasi-β-normed space is called a quasi-β-Banach space. Definition 2.3. A quasi-β-norm ‖·‖ is called a (β,p)-norm (0 < p < 1) if ‖x + y‖p ≤‖x‖p + ‖y‖p for all x,y ∈X. In this case, a quasi-β-Banach space is called a (β,p)-Banach space. 3. Generalized Ulam-Hyers stability of equation (1.8) Throughout this section, we assume that X is a linear space and Y is a (β,p)-Banach space with (β,p)-norm ‖·‖Y. Let K be the modulus of concavity of ‖·‖Y. For notational convenience, we define the difference operator for a given mapping f : X →Y as Dsf(x,y) = f(ax + by) + f(bx + ay) + (a− b)6 [ f ( ax− by a− b ) + f ( bx−ay b−a )] = 64(ab)2 ( a2 + b2 )[ f ( x + y 2 ) + f ( x−y 2 )] + 2 ( a2 − b2 )( a4 − b4 ) [f(x) + f(y)] for all x,y ∈X . Lemma 3.1. (see [44]). Let j ∈ {−1, 1} be fixed, m,b ∈ N with b ≥ 2 and Φ : X → [0,∞) be a function such that there exists an L < 1 with Φ ( bjx ) ≤ bjmβLΦ(x) for all x ∈X. Let g : X →Y be a mapping satisfying ‖g(bx) − bmg(x)‖Y ≤ Φ(x) (3.1) 170 RASSIAS, RAVI AND KUMAR for all x ∈X, then there exists a uniquely determined mapping G : X →Y such that G(bx) = bmG(x) and ‖g(x) −G(x)‖Y ≤ 1 bmβ |1 −Lj| Φ(x) (3.2) for all x ∈X. Theorem 3.1. Let i ∈ {−1, 1} be fixed. Let φ : X ×X → [0,∞) be a function such that there exists an L < 1 with φ ( kix,kiy ) ≤ k6iβLφ(x,y) for all x,y ∈X. Let f : X →Y be a mapping satisfying ‖Dsf(x,y)‖Y ≤ φ(x,y) (3.3) for all x,y ∈X. Then there exists a unique sextic mapping S : X →Y such that ‖f(x) −S(x)‖Y ≤ 1 k6β |1 −Li| Ψ(x) (3.4) for all x ∈X, where Ψ(x) = K 2β [ φ(x,x) + 32β(ab)2β ( a2 + b2 )β λβ φ(0, 0) ] . Proof. Plugging (x,y) into (0, 0) in (3.3), we obtain ‖f(0)‖Y ≤ 1 2βλβ φ(0, 0). (3.5) Switching (x,y) to (x,x) in (3.3), one finds∥∥f(kx) −k6f(x) − 32(ab)2 (a2 + b2)f(0)∥∥Y ≤ 12β φ(x,x) (3.6) for all x ∈X . Using (3.5) and (3.6), we arrive∥∥f(kx) −k6f(x)∥∥Y ≤ Ψ(x) (3.7) for all x ∈ X . By Lemma 3.1, there exists a unique mapping S : X → Y such that S(kx) = k6S(x) and ‖f(x) −S(x)‖Y ≤ 1 k6β |1 −Li| Ψ(x) for all x ∈X . It remains to show that S is a sextic map. By (3.3), we have∥∥∥∥ 1k6inDsf (kinx,kiny) ∥∥∥∥ Y ≤ k−6inβφ ( kinx,kiny ) ≤ k−6inβ ( k6iβL )n φ(x,y) = Lnφ(x,y) for all x,y ∈ X and n ∈ N. So ‖DsS(x,y)‖Y = 0 for all x,y ∈ X . Thus the mapping S : X → Y is sextic, which completes the proof of theorem. � Corollary 3.1. Let X be a quasi-α-normed space with quasi-α-norm ‖·‖X, and let Y be a (β,p)- Banach space with (β,p)-norm ‖·‖Y. Let k1,p be positive numbers with p 6= 6β α and f : X → Y be a mapping satisfying ‖Dsf(x,y)‖Y ≤ k1 (‖x‖ p X + ‖y‖ p X) for all x,y ∈X. Then there exists a unique sextic mapping S : X →Y such that ‖f(x) −S(x)‖Y ≤   k1K 2β(k6β−kpα) ‖x‖ p X , p ∈ ( 0, 6β α ) kpαk1K k6β2β(kpα−k6β) ‖x‖ p X , p ∈ ( 6β α ,∞ ) for all x ∈X. Proof. The proof is obtained by taking φ(x,y) = k1 (‖x‖ p X + ‖y‖ p X), for all x,y ∈ X and L = kpα k6β in Theorem 3.1. � EULER-LAGRANGE-JENSEN (a,b)-SEXTIC FUNCTIONAL EQUATIONS 171 Corollary 3.2. Let X be a quasi-α-normed space with quasi-α-norm ‖·‖X, and let Y be a (β,p)-Banach space with (β,p)-norm ‖·‖Y. Let k2,p,q be positive numbers with ρ = p + q 6= 6β α and f : X →Y be a mapping satisfying ‖Dsf(x,y)‖Y ≤ k2 ‖x‖ p X ‖y‖ q X for all x,y ∈ X. Then there exists a unique sextic mapping S : X →Y such that ‖f(x) −S(x)‖Y ≤   k2K 2β(k6β−kρα) ‖x‖ ρ X , ρ ∈ ( 0, 6β α ) kραk2K k6β2β(kρα−k6β) ‖x‖ ρ X , ρ ∈ ( 6β α ,∞ ) for all x ∈X. Proof. Letting φ(x,y) = k2 ‖x‖ p X ‖y‖ q X , for all x,y ∈ X and L = kρα k6β in Theorem 3.1, we obtain the required results. � Corollary 3.3. Let X be a quasi-α-normed space with quasi-α-norm ‖·‖X, and let Y be a (β,p)- Banach space with (β,p)-norm ‖·‖Y. Let k3,r be positive numbers r 6= 3β α and f : X → Y be a mapping satisfying ‖Dsf(x,y)‖Y ≤ k3 [ ‖x‖rX ‖y‖ r X + ( ‖x‖2rX + ‖y‖ 2r X )] for all x,y ∈X. Then there exists a unique sextic mapping S : X →Y such that ‖f(x) −S(x)‖Y ≤   3k3K 2β(k6β−k2rα) ‖x‖ 2r X , r ∈ ( 0, 3β α ) 3k2rαk3K k6β2β(k2rα−k6β) ‖x‖ 2r X , r ∈ ( 3β α ,∞ ) for all x ∈X. Proof. By taking ϕ(x,y) = k3 [ ‖x‖rX ‖y‖ r X + ( ‖x‖2rX + ‖y‖ 2r X )] , for all x,y ∈ X and L = k 2rα k6β in Theorem 3.1, we arrive at the desired results. � 4. Counter-example In this section, using the idea of the well-known counter-example provided by Z. Gajda [12], we illustrate a counter-example that the functional equation (1.8) is not stable for p = 6β α in Corollary 3.1. We consider the function ϕ(x) = { x6, for |x| < 1 1, for |x| ≥ 1. (4.1) where ϕ : R → R. Let f : R → R be defined by f(x) = ∞∑ n=0 2−6nϕ(2nx) (4.2) for all x ∈ R. The function f serves as a counter-example for the fact that the functional equation (1.8) is not stable for p = 6β α in Corollary 3.1 in the following theorem. Theorem 4.1. If the function f defined in (4.2) satisfies the functional inequality |Dsf(x,y)| ≤ 643δ 63 ( |x|6 + |y|6 ) (4.3) where δ = 2 [ 1 + (a− b)6 − 2 ( a6 + b6 ) − 62(ab)2 ( a2 + b2 )] > 0, for all x,y ∈ R, then there do not exist a sextic mapping S : R → R and a constant � > 0 such that |f(x) −S(x)| ≤ � |x|6 , for all x ∈ R. Proof. First, we are going to show that f satisfies (4.3). |f(x)| = ∣∣∣∣∣ ∞∑ n=0 2−6nϕ(2nx) ∣∣∣∣∣ ≤ ∞∑ n=0 1 26n = 64 63 . 172 RASSIAS, RAVI AND KUMAR Therefore, we see that f is bounded by 64 63 on R. If |x|6 + |y|6 = 0 or |x|6 + |y|6 ≥ 1 64 , then |Dsf(x,y)| ≤ 64δ 63 ≤ 642δ 63 ( |x|6 + |y|6 ) . Now, suppose that 0 < |x|6 + |y|6 < 1 64 . Then there exists a non-negative integer k such that 1 64k+1 ≤ |x|6 + |y|6 < 1 64k . (4.4) Hence 64k |x|6 < 1, 64k |y|6 < 1 and 2n(ax + by), 2n(bx + ay), 2n ( ax−by a−b ) , 2n ( bx−ay b−a ) , 2n ( x+y 2 ) , 2n ( x−y 2 ) , 2nx, 2ny ∈ (−1, 1) for all n = 0, 1, 2, . . . ,k − 1. Hence for n = 0, 1, 2, . . . ,k − 1, ϕ (2n(ax + by)) + ϕ (2n(bx + ay)) + (a− b)6 [ ϕ ( 2n ( ax− by a− b )) + ϕ ( 2n ( bx−ay b−a ))] − 64(ab)2 ( a2 + b2 )[ ϕ ( 2n ( x + y 2 )) + ϕ ( 2n ( x−y 2 ))] − 2 ( a2 − b2 )( a4 − b4 ) [ϕ (2nx) + ϕ (2ny)] = 0. (4.5) From the definition of f and the inequality (4.4), we obtain that |Dsf(x,y)| = ∣∣∣ ∞∑ n=0 2−6nϕ (2n(ax + by)) + ∞∑ n=0 2−6nϕ (2n(bx + ay)) + (a− b)6 [ ∞∑ n=0 2−6nϕ ( 2n ( ax− by a− b )) + ∞∑ n=0 2−6nϕ ( 2n ( bx−ay b−a ))] − 64(ab)2 ( a2 + b2 )[ ∞∑ n=0 2−6nϕ ( 2n ( x + y 2 )) + ∞∑ n=0 2−6nϕ ( 2n ( x−y 2 ))] − 2 ( a2 − b2 )( a4 − b4 )[ ∞∑ n=0 2−6nϕ (2nx) + ∞∑ n=0 2−6nϕ (2ny) ]∣∣∣ ≤ ∞∑ n=0 2−6n ∣∣∣ϕ (2n(ax + by)) + ϕ (2n(bx + ay)) + (a− b)6 [ ϕ ( 2n ( ax− by a− b )) + ϕ ( 2n ( bx−ay b−a ))] − 64(ab)2 ( a2 + b2 )[ ϕ ( 2n ( x + y 2 )) + ϕ ( 2n ( x−y 2 ))] − 2 ( a2 − b2 )( a4 − b4 ) [ϕ (2nx) + ϕ (2ny)] ∣∣∣ ≤ ∞∑ n=0 2−6nδ = 26(1−k)δ 63 ≤ 643δ 63 ( |x|6 + |y|6 ) . (4.6) Therefore, f satisfies (4.3) for all x,y ∈ R. 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Abdul Hakeem College of Engineering and Technology, Melvisharam - 632 509, Tamil Nadu, INDIA ∗Corresponding author: jrassias@primedu.uoa.gr 1. Introduction 2. Preliminaries 3. Generalized Ulam-Hyers stability of equation (1.8) 4. Counter-example References