Solution and stability of a reciprocal type functional equation in several variables

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1 Available online at J. Nonlinear Sci. Appl. 7 04, 8 7 Research Article Solution and stability of a reciprocal type functional equation in several variables K. Ravi a,, E. Thandapani b, B.V. Senthil Kuar c a PG & Research Departent of Matheatics, Sacred Heart College, Tirupattur , TailNadu, India b Raanujan Institute of Advance Study in Matheatics, University of Madras, Chepauk, Chennai Tail Nadu, India c Departent of Matheatics, C. Abdul Hakee College of Engg. and Tech., Melvishara , TailNadu, India Counicated by P. Gavruta Abstract In this paper, we obtain the general solution and investigate the generalized Hyers-Ula stability of a reciprocal type functional equation in several variables of the f r + i [ = j=,j i r + j ] r i [ r j=,j i j] r + r i where is a positive integer with 3. c 04 All rights reserved. Keywds: Rassias reciprocal functional equation, General reciprocal functional equations, Adjoint and difference functional equations 00 MSC: Priary 39B, 39B5, 39B7. Introduction About seventy years ago, Ula [3] raised the well known stability proble of functional equations. In the net year, Ula s proble was partially answered by Hyers [4] in Banach spaces. T. Aoki [] generalized Hyers thee f additive appings in the year 950. In the year 978, a generalized version of the thee of Hyers f approiately linear appings was given by Th.M. Rassias []. During , J.M. Rassias [5], [6], [7] treated the Ula-Gavruta-Rassias stability on linear and non-linear appings and generalized Cresponding auth Eail addresses: shckravi@yahoo.co.in K. Ravi, ethandapani@yahoo.co.in E. Thandapani, bvssree@yahoo.co.in B.V. Senthil Kuar Received 0--0

2 K. Ravi, E. Thandapani, B.V. Senthil Kuar, J. Nonlinear Sci. Appl. 7 04, Hyers result. In 994, a further generalization of the Th.M. Rassias thee was obtained by P. Gavruta [3], who replaced the bound θ p + y p by a general control function φ, y. In the year 00, K. Ravi and B.V. Senthil Kuar [8] investigated the generalized Hyers-Ula stability f the reciprocal functional equation r + y = rry. r + ry where r : X Y is a apping on the spaces of non-zero real nubers. The reciprocal function r = c is a solution of the functional equation.. Later, K. Ravi, J.M. Rassias and B.V. Senthil Kuar [0] introduced the reciprocal difference functional equation + y r r + y = rry r + ry and the reciprocal adjoint functional equation + y r + r + y = 3rry r + ry and investigated the generalized Hyers-Ula stability f the above two functional equations. and.3. Recently, K. Ravi, J.M. Rassias and B.V. Senthil Kuar [9] discussed the generalized Hyers-Ula stability f the generalized reciprocal functional equation r α i i = r i [ α i j=,j i r j..3 ].4 f arbitrary but fied real nubers α i 0 f i =,,...,, so that 0 < α = α + α + + α = α i and r : X Y with X and Y are the sets of non-zero real nubers. Very recently, K. Ravi, E. Thandapani and B.V. Senthil Kuar [] obtained the general solution and investigated the generalized Hyers-Ula stability f the reciprocal type functional equations rk k + k k y = rk k yrk y k rk k y + rk y k where k and k are any integers with k k and rk + k + k + k y = rk + k yrk y + k rk + k y + rk y + k where k and k are any integers with k k. In this paper, we obtain the solution and investigate the generalized Hyers-Ula stability f a reciprocal type functional equation in several variables of the f r + i [ = j=,j i r + j ] r [ r i j=,j i j] r + r i.7 where is a positive integer with 3. Throughout this paper, we assue that X is the set of non-zero real nubers. F convenience, we define the difference operat D r : X R such that r + i D r,,..., = [ j=,j i r + j ] r i [ r j=,j i j] r + r i.5.6

3 K. Ravi, E. Thandapani, B.V. Senthil Kuar, J. Nonlinear Sci. Appl. 7 04, f,,..., X. In the following results, we will set 0 = 0 f 3 and assue 0 r + j 0, r r j + r i 0 j=,j i j=,j i f all i X; i =,,..., ; 3 and i, f all i; i ; 3.. General solution of functional equation.7 Thee.. A apping r : X R satisfies the functional equation.7 f all,,..., X if and only if there eists a reciprocal apping r : X R satisfying the reciprocal functional equation. f all, y X. Proof. Let the apping r : X R satisfy the functional equation.7. Replacing by and i by y f i =, 3..., in.7, we arrive.. Conversely, let the apping r : X R satisfy the functional equation.. Replacing, y by, + 3 in., we obtain r = r r + 3 r + r + 3 r r r 3 = r r + r r 3 + r r 3 3 = r i [ 3 r 3 j=,j i j] r + 3 r i f all,, 3 X. Using induction on a positive integer, we have r = r [ r i ] j=,j i r j + r i f all,,..., X. Now, replacing i by f i =,,..., in., we get r = r, f all X. Replacing i by i+ f i =,,..., in., we obtain r = r i [ i=3 r j=3,j i j] r +. i=3 r i f all, 3,..., X. Now, replacing i by + i f i =, 3..., in., we get r + i [ i=3 r + j=3,j i r + j ] + i=3 r + j. = r = = r i=3 r [ r + = r r i j=3,j i r j]+ i=3 r i r i j=3,j i r j]+ i=3 r i i=3 r [ [ r i j=,j i j] r + r i f all,,..., X. On further siplification of the above equation.3 yields the equation.7. This copletes the proof of Thee...3

4 K. Ravi, E. Thandapani, B.V. Senthil Kuar, J. Nonlinear Sci. Appl. 7 04, Generalized Hyers-Ula stability of equation.7 Thee 3.. Let ϕ : X R be a function satisfying i+ ϕ i+, i+,..., i+ < + 3. f all,,..., X. If a function f : X R satisfies the functional inequality D f,,..., ϕ,,..., 3. f all,,..., X, then there eists a unique reciprocal apping r : X R which satisfies.7 and the inequality r f i+ ϕ i+, i+,..., i+ 3.3 f all X. Proof. Replacing i by f i =,,..., in 3. and ultiplying by, we get f f ϕ,,..., 3.4 f all X. Now, replacing by in 3.4, dividing by and suing the resulting inequality with 3.4, we obtain f f i+ ϕ i+, i+,..., i+ f all X. Proceeding further and using induction arguents on a positive integer n, we arrive f n n f n i+ ϕ i+, i+,..., i+ f all X. F any positive integer i and X, we have i+ f i+ i f i = i f i f i+ i+ ϕ i+, i+,..., i+. Hence f any integers l, k with l > k > 0, we obtain by using the triangular inequality l f l k f k = l f l l f l + l f l + k+ f k+ l ϕ l, l,..., l + + k+ ϕ k+, k+,..., k+ l i ϕ i, i,..., i i=k+ l i+ ϕ i+, i+,..., i+ i=k

5 K. Ravi, E. Thandapani, B.V. Senthil Kuar, J. Nonlinear Sci. Appl. 7 04, 8 7 f all X. Taking the liit as k + in 3.6 and considering 3., it follows that the sequence { f n } is a Cauchy sequence f each X. Since R is coplete, we can define r : X R by n r = li n f n. To show that r satisfies.7, replacing n,,..., by n, n,..., n in 3. and dividing by n, we obtain n D f n, n,..., n n ϕ n, n,..., n 3.7 f all,,..., X and f all positive integer n. Using 3. and 3.5 in 3.7, we see that r satisfies.7, f all,,..., X. Taking liit n in 3.5, we arrive 3.3. Now, it reains to show that r is uniquely defined. Let r : X R be another reciprocal apping which satisfies.7 and the inequality 3.3. Clearly, r n = n r, r n = n r and using 3.3, we arrive r r = n r n r n 4 n+i+ ϕ n+i+, n+i+,..., n+i+ 4 i=n i+ ϕ i+, i+,..., i+ 3.8 f all X. Allowing n in 3.8, we find that r is unique. This copletes the proof of Thee 3.. Thee 3.. Let ϕ : X R be a function satisfying i ϕ i, i,..., i < f all,,..., X. If a function f : X R satisfies the functional inequality D f,,..., ϕ,,..., 3.0 f all,,..., X, then there eists a unique reciprocal apping r : X R which satisfies.7 and the inequality r f i ϕ i, i,..., i 3. f all X. Proof. The proof is obtained by replacing i by f i =,,..., in 3.0 and proceeding further by siilar arguents as in Thee 3.. Collary 3.3. F any fied c 0 and p > p <, if f : X R satisfies D f,,..., c i p f all,,..., X, then there eists a unique reciprocal apping r : X R such that { c p f p > r f p+ c p f p < p+ f all X.

6 K. Ravi, E. Thandapani, B.V. Senthil Kuar, J. Nonlinear Sci. Appl. 7 04, Proof. If we choose ϕ,,..., = c i p, f all,,..., X, then by Thee 3., we arrive r f c p+ p, f all X and p > and using Thee 3., we arrive r f c p+ p, f all X and p <. Collary 3.4. Let f : X R be a apping and there eists p such that p > p <. If there eists c 0 such that D f,,..., c i p f all,,..., X, then there eists a unique reciprocal apping r : X R satisfying the functional equation.7 and { c r f p+ p f p > c p f p < p+ f all X. Proof. Considering ϕ,,..., = c i p, f all,,..., X, then by Thee 3., we arrive r f c p+ p, f all X and p > and using Thee 3., we arrive r f c p+ p, f all X and p <. Collary 3.5. Let c 3 > 0 and α > α < be real nubers, and f : X R be a apping satisfying the functional inequality { } D f,,..., c 3 i α + i α f all,,..., X. Then there eists a unique reciprocal apping r : X R satisfying the functional equation.7 and { +c3 α f α > r f α+ +c 3 α f α < α+ f all X. Proof. Choosing ϕ,,..., = c 3 { i α + i α }, f all,,..., X, then by Thee 3., we arrive and using Thee 3., we arrive r f + c 3 α+ α, f all X and α > r f + c 3 α+ α, f all X and α <.

7 K. Ravi, E. Thandapani, B.V. Senthil Kuar, J. Nonlinear Sci. Appl. 7 04, Counter-eaples The following eaple illustrates the fact that the functional equation.7 is not stable f p = in Collary 3.3. We present the following counter-eaple odified by the well-known counter-eaple of Z. Gajda []. Eaple 4.. Let ϕ : X R be a apping defined by { µ ϕ = f, µ otherwise where µ > 0 is a constant, and define a apping f : X R by f = Then the apping f satisfies the inequality n=0 ϕ n n, f all X. D f,,..., 6µ i f all,,..., X. Therefe there do not eist a reciprocal apping r : X R and a constant β > 0 such that f r β 4. f all X. Proof. f ϕ n n=0 n µ n=0 = µ = µ. Hence f is bounded by µ. If n i, then the left hand side of 4. is less than 6µ. Now, suppose that 0 < i <. Then there eists a positive integer k such that k+ i < k. 4.3 Hence i < k iplies k i < i > f i =,,..., k i k > > f i =,,..., i > > f i =,,..., k and consequently k, k i, k + i > f i =, 3...,. Therefe, f each value of n = 0,,,..., k, we obtain n, n i, n + i > f i =, 3,..., 4.

8 K. Ravi, E. Thandapani, B.V. Senthil Kuar, J. Nonlinear Sci. Appl. 7 04, and D ϕ n, n,..., n = 0 f n = 0,,,..., k. Using 4.3 and the definition of f, we obtain D f,,..., = f + i [ j=,j i f + j ] f i [ f j=,j i j] f + f i µ n=k n 3 µ k 6µ k+ 6µ i µ n=k n + n=k µ n n=k f all,,..., X. Therefe the inequality 4. holds true. We clai that the reciprocal functional equation.7 is not stable f p = in Collary 3.3. Assue that there eists a reciprocal apping r : X R satisfying 4.. Therefe, we have µ n f β However, we can choose a positive integer with µ > β +. If, then n, f all n = 0,,,..., and therefe f = n=0 ϕ n n µ n n = µ > β + n=0 which contradicts 4.4. Therefe, the reciprocal type functional equation.7 is not stable f p = in Collary 3.3. The following eaple illustrates the fact that the functional equation.7 is not stable f α = in Collary 3.5. Eaple 4.. Let φ : X R be a apping defined by { δ φ = f, δ otherwise where δ > 0 is a constant, and define a apping f : X R by φ n f = n, f all X. Then the apping f satisfies the inequality n=0 D f,,..., 6δ { } i + i f all,,..., X. Therefe there do not eist a reciprocal apping r : X R and a constant β > 0 such that f r β 4.6 f all X. 4.5

9 K. Ravi, E. Thandapani, B.V. Senthil Kuar, J. Nonlinear Sci. Appl. 7 04, Proof. It is easy to show that f is bounded by δ, by siilar arguents as in Eaple 4.. If { } i + i, then the left hand side of 4.5 is less than 6δ. Now, suppose that { } 0 < i + i <. Then there eists a positive integer k such that { k+ } i + i < k. 4.7 Hence { } i + i < iplies k { } k i + k i < k < f i =,,..., i i > f i =,,..., k i k > > f i =,,..., i > > f i =,,..., k and consequently k, k i, k + i > f i =, 3,...,. Therefe, f each value of n = 0,,,..., k, we obtain n, n i, n + i > f i =, 3,..., and D φ n, n,..., n = 0 f n = 0,,,..., k. Using 4.7, the definition of f and siilar arguents as in Eaple 4., we obtain the inequality 4.5. The reaining part of the proof is obtained by siilar arguents as in Eaple 4.. Hence, the reciprocal type functional equation.7 is not stable f α = in Collary 3.5. References [] T. Aoki, On the stability of the linear transfation in Banach spaces, J. Math. Soc. Japan, 950, [] Z. Gajda, On stability of additive appings, Int. J. Math. Math. Sci. 4 99, [3] P. Gavruta, A generalization of the Hyers-Ula-Rassias stability of approiately additive appings, J. Math. Anal. Appl , [4] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 7 94, 4. [5] J.M. Rassias, On approiation of approiately linear appings by linear appings, J. Funct. Anal. 6 98, [6] J.M. Rassias, On approiation of approiately linear appings by linear appings, Bull. Sci. Math , [7] J.M. Rassias, Solution of a proble of Ula, J. Appro.They , [8] K. Ravi and B.V. Senthil Kuar, Ula-Gavruta-Rassias stability of Rassias Reciprocal functional equation, Global Journal of App. Math. and Math. Sci. 3 Jan-Dec 00,

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