Research Article Approximation of Analytic Functions by Solutions of Cauchy-Euler Equation

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1 Funtion Spaes Volume 2016, Artile ID , 5 pages Researh Artile Approimation of Analyti Funtions by Solutions of Cauhy-Euler Equation Soon-Mo Jung Mathematis Setion, College of Siene and Tehnology, Hongik University, Sejong 30016, Republi of Korea Correspondene should be addressed to Soon-Mo Jung; smjung@hongik.a.kr Reeived 2 November 2015; Aepted 15 May 2016 Aademi Editor: Gestur Ólafsson Copyright 2016 Soon-Mo Jung. This is an open aess artile distributed under the Creative Commons Attribution Liense, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited. We investigate the approimation properties of a speial lass of twie ontinuously differentiable funtions by solutions of the Cauhy-Euler equation. 1. Introdution Throughout this paper, let n be a positive integer, let I be a nondegenerate interval of R,and let K denote either R or C. Wewill onsider the (linear) differential equation of nth order F (y (n),y (n 1),...,y,y,) =0 (1) defined on I, wherey:i K is an n times ontinuously differentiable funtion. For arbitrary >0, assume that an n times ontinuously differentiable funtion y : I K satisfies the differential inequality F (y(n),y (n 1),...,y,y,) (2) for all I. If for eah funtion y : I K satisfying inequality (2) there eists a solution y 0 : I K of the differential equation (1) y () y 0 () K() (3) for any I,whereK() depends on only and satisfies 0 K() = 0, then we say that the differential equation (1) satisfies (or has) the Hyers-Ulam stability (or the loal Hyers- Ulam stability if the domain I is not the whole spae R). If the above statement also holds when we replae and K() with some appropriate φ() and Φ(), respetively, then we say that the differential equation (1) satisfies the generalized Hyers-Ulam stability (or the Hyers-Ulam-Rassias stability). We may apply these terminologies for other differential equations. For more detailed definition of the Hyers-Ulam stability and reent papers on this subjet, refer to [1 4]. Obłoza seems to be the first author who investigated the Hyers-Ulam stability of linear differential equations (see [5, 6]). Let g, r : (a, b) R be ontinuous funtions with b g() d <, wherea and b are real onstants. Assume a that >0isan arbitrary real number. Obłoza proved that if a differentiable funtion y:(a,b) Rsatisfies the inequality y ()+g()y() r() for all (a, b) and if a funtion y 0 : (a, b) R satisfies y 0 () + g()y 0() = r() for all (a, b) and y(τ) = y 0 (τ) for some τ (a, b), then there eists a onstant δ>0 y() y 0 () δ for all (a, b). Thereafter, Alsina and Ger [7] proved that if a differentiable funtion f : (a,b) R satisfies the differential inequality y () y(), then there eists a solution f 0 : (a, b) R of the differential equation y () = y() f() f 0 () 3 for any (a, b).thisresultofalsinaand GerwasgeneralizedbyTakahasietal.Theyprovedin[8]that the Hyers-Ulam stability holds for the Banah spae valued differential equation y () = λy() (see also [9 13]). For a reent result on the Hyers-Ulam stability for seond-order linear differential equations, we refer to [14, 15]. In this paper, we onsider the (inhomogeneous) Cauhy- Euler equation 2 y () +αy () +βy() =r(), (4)

2 2 Funtion Spaes where α and β are real-valued onstants and r:r R is a ontinuous funtion, and we investigate the approimation properties of twie ontinuously differentiable funtions by solutions of the Cauhy-Euler equation 2 y () +αy () +βy() =0, (5) whih is assoiated with (4). 2. Preinaries Reently, Choi and Jung [16, Corollary 4.2] proved the Hyers- Ulam stability of the Cauhy-Euler equation (4) for the ase of (α 1) 2 4β>0. Theorem 1. Assume that the real-valued onstants α, β are given with (α 1) 2 4β > 0and is an arbitrarily given positive onstant. Let be a positive real-valued onstant and let m 1, m 2 be given as m 1 = (α 1) (α 1) 2 4β, 2 m 2 = (α 1) + (α 1) 2 4β. 2 If r:(0, ) R is a differentiable funtion and y:(0, ) R is a twie ontinuously differentiable funtion the inequality 2 y () +αy () +βy() r() (7) holds for any (0, ), then there eists a solution y : (0, ) R of the inhomogeneous Cauhy-Euler equation (4) (6) y () y () + ( 1 ( 1 ( ) (for m 1 m 2 m 2 m 1 m 2 )m2 m 1 )m1 m1 =0 =m 2 ), m 2 2 m 2 1 (( )m 2 m 2 (( )m 1 m 1 (for m 1 =0), (for m 2 =0) (8) For the ase of (α 1) 2 4β = 0,theHyers-Ulamstability of the inhomogeneous Cauhy-Euler equation (4) was proved in [16, Corollary 4.4]. Theorem 2. Assume that the real-valued onstants α and β are given with α =1, β = (α 1) 2 /4 and is an arbitrarily given positive onstant. Let be a positive real-valued onstant and let λ = (α 1)/2.Ifr:(0, ) Ris a differentiable funtion and y:(0, ) Risatwie ontinuously differentiable funtion the inequality 2 y () +αy () + (α 1)2 y () r() (9) 4 holds for all (0, ), then there eists a solution y : (0, ) R of the inhomogeneous Cauhy-Euler equation (4) with β = (α 1) 2 /4 y () y () λ 2 + λ ( )λ ( 1 λ ) (10) Finally, the Hyers-Ulam stability of the Cauhy-Euler equation (4) was also proven in [16, Theorem 4.5] for the ase of (α 1) 2 4β<0. Theorem 3. Assume that the real-valued onstants α and β are given with (α 1) 2 4β < 0and is an arbitrarily given positive onstant. Let >0be a given real-valued onstant and let λ= α 1 2, μ= 1 2 4β (α 1) 2. (11) If a differentiable funtion r:(0, ) R and a twie ontinuously differentiable funtion y:(0, ) R satisfy inequality (7) for all (0, ), then there eists a solution y :(0, ) R of the inhomogeneous Cauhy-Euler equation (4) suh that y () y () μ λ ζ λ+1 sin (μ ln ζ ) dζ (12) Remark 4. Cîmpean and Popa [14] proved the Hyers-Ulam stability of the linear differential equations of nth order with onstant oeffiients. Indeed, they proved a general theorem for the Hyers-Ulam stability whih inludes Theorems 1, 2, and 3 as its orollaries with the inequality

3 Funtion Spaes 3 y () y () β (for either α 2 4β>0, β 4 α 2 (for α 2 4β<0, α=0). =0or α 2 4β=0, α =0), (13) However, Theorems 1, 2, and 3 have the advantage of more eat loal approimation over the result of Cîmpean and PopaasweseeinTheorems5,6,and7. 3. Approimation Properties We denote by B(α; β) the set of all twie ontinuously differentiable funtions y:(0, ) R for whih there eists a onstant >0 2 y () +αy () +βy() (14) for all (0, ),whereα and β are real-valued onstants. If we define (y 1 +y 2 ) () fl y 1 () +y 2 (), (γy 1 ) () fl γy 1 () (15) for all y 1,y 2 B(α; β) and γ R, thenb(α; β) is a vetor spae over R. This fat implies that the set B(α; β) is large enough to be a vetor spae. In the following theorems, we investigate approimation properties of funtions of B(α; β) by solutions of the Cauhy- Euler equation (5). Theorem 5. Let >0be a given real number and let α, β R be given with (α 1) 2 4β > 0.Ify B(α; β), then there eists asolutiony :(0, ) R of the Cauhy-Euler equation (5) as. y () y () =o( ) (16) Proof. We define m 1 and m 2 by the formulas given in Theorem 1; that is, m 1 and m 2 are the distint roots of the indiial equation m 2 +(α 1)m+β = 0.Sine(α 1) 2 4β > 0,wehave m 1 <m 2.Siney B(α; β), there eists a onstant >0suh that inequality (14) holds Aording to Theorem 1 with r() 0, there eists a solution y :(0, ) R of the Cauhy-Euler equation (5) y () y () + ( 1 ( 1 ( ) (for m 1 m 2 m 2 m 1 m 2 )m2 m 1 )m1 m1 =0 =m 2 ), m 2 2 m 2 1 (( )m 2 m 2 (( )m 1 m 1 (for m 1 =0), (for m 2 =0) (17) for any (0, ). We will only estimate the following it for the ase of m 1 =0by applying L Hospital s rule: y () y () (/m 2 2 )((/)m 2 (/m 2 ) ln (/) = ( m 2 ()m2 1 1 m 2 ) =0, whih implies the validity of this theorem. (18) We now onsider the ase of (α 1) 2 4β = 0and use Theorem 2 to prove the following theorem. Theorem 6. Let > 0 and α = 1 be real numbers and let λ= (α 1)/2.Ify B(α; λ 2 ), then there eists a solution y :(0, ) RoftheCauhy-Eulerequation(5)withβ=λ 2 as. y () y () =o( ) (19) Proof. Sine y B(α; λ 2 ), there eists a onstant >0suh that inequality (14) holds Aording to Theorem 2 with r() 0, there eists a solution y :(0, ) R of the Cauhy-Euler equation (5) with β=λ 2 y () y () λ 2 + λ ( )λ ( 1 λ ) (20)

4 4 Funtion Spaes Therefore, we estimate the it by applying L Hospital s rule: y () y () (/λ 2 )+(/λ)(/) λ (ln (/) 1/λ) (21) λ 1 = ln =0, λ whih implies the validity of this theorem. Finally, we investigate the approimation property of eah funtion of B(α; β) by a solution of the differential equation (5) when (α 1) 2 4β<0. Theorem 7. Let >0be a given real number and let α, β R be given with (α 1) 2 4β < 0.Ify B(α; β), then there eists asolutiony :(0, ) R ofthecauhy-eulerequation(5) as. Proof. Let us define y () y () =o( ) (22) λ= α 1 2, μ= 1 2 4β (α 1) 2. (23) Sine y B(α; β), there eists a onstant > 0 suh that inequality (14) holds Aording to Theorem 3 with r() 0, there eists a solution y :(0, ) R of the Cauhy-Euler equation (5) λ ζ λ+1 sin (μ ln ζ ) dζ y () y () μ If we substitute η=/ζ,thenwehave λ ζ λ+1 sin (μ ln ζ ) dζ (24) (25) = 1 η λ 1 λ sin (μ ln η ) dη. Hene, we further apply L Hospital s rule to obtain y () y () 1 μ λ ζ λ+1 sin (μ ln ζ ) dζ = μ λ 1 η λ 1 sin (μ ln η ) dη = μ λ λ 1 sin (μ ) =0, whih implies the validity of this theorem. (26) Competing Interests The author delares that there are no ompeting interests. Aknowledgments This researh was supported by Basi Siene Researh Program through the National Researh Foundation of Korea (NRF) funded by the Ministry of Eduation (no. 2013R1A1A ). This work was supported by 2014 Hongik University Researh Fund. Referenes [1] J.-H. Bae and W.-G. Park, On the Ulam stability of the Cauhy- Jensen equation and the additive-quadrati equation, Nonlinear Siene and Its Appliations,vol.8,no.5,pp , [2] D.H.Hyers,G.Isa,andTh.M.Rassias,Stability of Funtional Equations in Several Variables, Birkhäuser, Boston, Mass, USA, [3] S.-M. Jung, Hyers-Ulam-Rassias Stability of Funtional Equations in Nonlinear Analysis,vol.48ofSpringer Optimization and Its Appliations, Springer, New York, NY, USA, [4]S.O.Kim,A.Bodaghi,andC.Park, Stabilityoffuntional inequalities assoiated with the Cauhy-Jensen additive funtional equalities in non-arhimedean Banah spaes, Journal of Nonlinear Siene and Its Appliations, vol.8,no.5,pp , [5] M. Ob loza, Hyers stability of the linear differential equation, Roznik Naukowo-Dydaktyzny. Prae Matematyzne,no.13,pp , [6] M. Obłoza, Connetions between Hyers and Lyapunov stability of the ordinary differential equations, Roznik Nauk.- Dydakt. Prae Mat Journals,vol.14,pp ,1997. [7] C. Alsina and R. Ger, On some inequalities and stability results related to the eponential funtion, Inequalities and Appliations, vol. 2, no. 4, pp , [8] S.-E. Takahasi, T. Miura, and S. Miyajima, On the Hyers-Ulam stability of the Banah spae-valued differential equation y = λy, Bulletin of the Korean Mathematial Soiety, vol. 39, pp , [9] S.-M. Jung, Hyers-ulam stability of linear differential equations of first order, Applied Mathematis Letters, vol. 17, no. 10, pp , [10] S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order, II, Applied Mathematis Letters, vol.19,no. 9,pp ,2006. [11] D. Popa and I. Raşa, On the Hyers-Ulam stability of the linear differential equation, Mathematial Analysis and Appliations,vol.381,no.2,pp ,2011. [12] D. Popa and I. Raşa, Hyers-Ulam stability of the linear differential operator with non-onstant oeffiients, Applied Mathematis and Computation, vol.219,no.4,pp , [13] G. Wang, M. Zhou, and L. Sun, Hyers-Ulam stability of linear differential equations of first order, Applied Mathematis Letters, vol. 21, no. 10, pp , [14] D. S. Cîmpean and D. Popa, On the stability of the linear differential equation of higher order with onstant oeffiients,

5 Funtion Spaes 5 Applied Mathematis and Computation,vol.217,no.8,pp , [15] Y. Li and Y. Shen, Hyers-Ulam stability of linear differential equations of seond order, Applied Mathematis Letters,vol.23, no. 3, pp , [16] G. Choi and S.-M. Jung, Invariane of Hyers-Ulam stability of linear differential equations and its appliations, Advanes in Differene Equations,vol.2015,artile277,14pages,2015.

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