Research Article Approximation of Analytic Functions by Bessel s Functions of Fractional Order

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1 Abstract and Applied Analysis Volume 20, Article ID , 3 pages doi:0.55/20/ Research Article Approximation of Analytic Functions by Bessel s Functions of Fractional Order Soon-Mo Jung Mathematics Section, College of Science and Technology, Hongik University, Jochiwon , Republic of Korea Correspondence should be addressed to Soon-Mo Jung, smjung@hongik.ac.kr Received 3 March 20; Accepted 6 June 20 Academic Editor: Yong Zhou Copyright q 20 Soon-Mo Jung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We will solve the inhomogeneous Bessel s differential equation y xxy x ν 2 yx a mx m,whereν is a positive nonintegral number and apply this result for approximating analytic functions of a special type by the Bessel functions of fractional order.. Introduction The stability problem for functional equations starts from the famous talk of Ulam and the partial solution of Hyers to the Ulam problem see, 2. Thereafter, Rassias 3 attempted to solve the stability problem of the Cauchy additive functional equation in a more general setting. The stability concept introduced by Rassias s theorem significantly influenced a number of mathematicians to investigate the stability problems for various functional equations see 4 4 and the references therein. Assume that Y is a normed space and I is an open subset of R. If for any function f : I Y satisfying the differential inequality a n xy n x a n xy n x a xy x a 0 xyx hx ε,. for all x I and for some ε 0, there exists a solution f 0 : I Y of the differential equation a n xy n x a n xy n x a xy x a 0 xyx hx 0,.2

2 2 Abstract and Applied Analysis such that fx f 0 x Kε for any x I, where Kε depends on ε only, then we say that the above differential equation satisfies the Hyers-Ulam stability or the local Hyers-Ulam stability if the domain I is not the whole space R. We may apply these terminologies for other differential equations. For more detailed definition of the Hyers-Ulam stability, refer to 2, 3, 6, 8, 0 2, 4. Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations see 5, 6. Here, we will introduce a result of Alsina and Ger see 7: Ifadifferentiable function f : I R is a solution of the differential inequality y x yx ε, where I is an open subinterval of R, then there exists a constant c such that fx ce x 3ε for any x I. This result of Alsina and Ger has been generalized by Takahasi et al. They proved in 8 that the Hyers-Ulam stability holds for the Banach space-valued differential equation y x λyxsee also 9. Using the conventional power series method, the author has investigated the general solution of the inhomogeneous Legendre differential equation under some specific condition, and this result was applied to prove the Hyers-Ulam stability of the Legendre differential equation see 20. In a recent paper, he has also investigated an approximation property of analytic functions by the Legendre functions see 2. This study has been continued to various special functions including the Airy functions, the exponential functions, the Hermite functions, and the power functions see Recently, the author and Kim tried to prove the Hyers-Ulam stability of the Bessel differential equation y x xy x ν 2) yx 0..3 However, the obtained theorem unfortunately does not describe the Hyers-Ulam stability of the Bessel differential equation in a strict sense see 26. In Section 2 of this paper, by using the ideas from 2, we will determine the general solution of the inhomogeneous Bessel differential equation y x xy x ν 2) yx a m x m,.4 where the parameter ν is a positive nonintegral number. Section 3 will be devoted to the investigation of an approximation property of the Bessel functions. Throughout this paper, we denote by x the largest integer not exceeding x for any x R, and we define I ρ ρ, 0 0,ρ for any ρ>0. 2. Inhomogeneous Bessel s Differential Equation A function is called a Bessel function of fractional order if it is a solution of the Bessel differential equation.3, where ν is a positive nonintegral number. The Bessel differential equation plays a great role in physics and engineering. In particular, this equation is most useful for treating the boundary value problems exhibiting cylindrical symmetries.

3 Abstract and Applied Analysis 3 The convergence of the power series a mx m seems not to guarantee the existence of solutions to the inhomogeneous Bessel differential equation.4. Thus, we adopt an additional condition to ensure the existence of solutions to the equation. Theorem 2.. Let ν be a positive nonintegral number, and let ρ be a positive constant. Assume that the radius of convergence of power series a mx m is at least ρ and there exists a constant σ>0 satisfying the condition a m2 m2 σ 2 c m, 2. for all sufficiently large integers m, where c m m/2 m/2 a 2i m/2 ji a 2i m/2 ji ν 2 2j ) 2 ν 2 2j ) 2 for even m, for odd m, 2.2 for all m N 0.Letρ 0 min{ρ, σ}. Then every solution y : I ρ0 equation.4 can be expressed by C of the Bessel s differential yx y h x c m x m, 2.3 for all x I ρ0,wherey h x is a solution of the homogeneous Bessel equation.3. Proof. We assume that y : I ρ0 C is a function given in the form 2.3, and we define y p x yx y h x c mx m. Then, it follows from 2. and 2.2 that lim m c m2 c lim m m m 2 2 ν a m2 c m σ, since we can deduce the relation c m2 a m2 c m /m 2 2 ν 2 from 2.2 by some manipulations. That is, the power series for y p x converges for all x I ρ0. Hence, we see that the domain of yx is well defined.

4 4 Abstract and Applied Analysis We now prove that the function y p x satisfies the inhomogeneous equation.4. Indeed, it follows from 2.2 that y px xy px ν 2) y p x mm c m x m mc m x m c m x m2 ν 2 c m x m m2 c x ν 2 c 0 ν 2 c x a 0 a x a m x m, m2 m [ c m 2 m2 m 2 ν 2) c m ] x m 2.5 since we obtain c 0 ν 2 a 0, c ν a, c 2 m 2 m 2 ν 2) c m a m, for m 2, 2.6 which proves that y p x is a particular solution of the inhomogeneous equation.4. On the other hand, since every solution to.4 can be expressed as a sum of a solution y h x of the homogeneous equation and a particular solution y p x of the inhomogeneous equation, every solution of.4 is certainly of the form Approximate Bessel s Differential Equation In this section, assume that ν is a positive nonintegral number and ρ is a positive constant. For a given K 0, we denote by C K the set of all functions y : I ρ C with the properties a and b: a yx is expressible by a power series b mx m whose radius of convergence is at least ρ; b a mx m K a mx m for any x I ρ, where a m b m 2 m 2 ν 2 b m for all m N 0 and set b 2 b 0. For a positive nonintegral number ν, define k M e x max ji ν 2 2j ) :0 i k μ, k M o x max ji ν 2 2j ) :0 i k μ, 3. Mx max{m e x,m o x, },

5 Abstract and Applied Analysis 5 where μ ν 2 /2 and L ν m ν < We remark that Mx as x 0. We will now solve the approximate Bessel differential equations in a special class of analytic functions, C K. Theorem 3.. Let ν be a positive nonintegral number, and let p be a nonnegative integer with p< ν<p. Assume that a function y C K satisfies the differential inequality y x xy x ν 2) yx ε, 3.3 for all x I ρ and for some ε 0. If the sequence {b m } satisfies the condition b m2 Ob m as m 3.4 with a Landau constant C 0, then there exists a solution y h x of the Bessel differential equation.3 such that yx yh x KLν Mxε, 3.5 for any x I ρ0,whereρ 0 min{ρ, / C } and C is a positive number larger than C. IfC and ρ are sufficiently small and large; respectively, then x x 2 x 2 Mx max ν 2 p 2, x x /2 ν 2 p ) 2 x /2 3.6 for all sufficiently large x. Proof. Since y belongs to C K, it follows from a and b that y x xy x ν 2) yx [ b m 2 m 2 ν 2) b m ] x m a m x m, 3.7 for all x I ρ. By considering 3.3 and 3.7, weget a m x m ε, 3.8

6 6 Abstract and Applied Analysis for any x I ρ. This inequality, together with b, yields a m x m K a m x m Kε, 3.9 for each x I ρ. Now, it follows from b that n a 2i ji ν 2 2j ) 2 n b 2i 2 ji ν 2 2j ) 2 n b 2i ji ν 2 2j ) 2 n b 2i i ji b 2 n ν 2 2j ) 2 n j0 ν 2 2j ) 2 b 2n n b 2i ji ν 2 2j ) b 2n, since b 2 0. Similarly, we obtain n a 2i ji ν 2 2j ) 2 b 2n, 3. for all n N 0,thatis, b m m/2 m/2 a 2i m/2 ji a 2i m/2 ji ν 2 2j ) 2 ν 2 2j ) 2 for even m, for odd m, 3.2 for all m N 0. On the other hand, by b and 3.4, we have a m2 [ b m m 2 2 ν 2] b m2 b m m 2 2 b m2 m 2 / C ) 2 b m, 3.3 for all sufficiently large integers m, where C is a positive number larger than C. Hence, in view of 3.2 and 3.3, the condition 2. is satisfied with σ / C. Moreover, we

7 Abstract and Applied Analysis 7 know that the radius of convergence of power series a mx m is at least ρ because the convergence radius of the power series expression for yx is at least ρ see a and 3.7. According to 3.7 and Theorem 2., there exists a solution y h x of the homogeneous Bessel differential equation.3 satisfying 2.3 for all x I ρ0. Thus, it follows from 2.2 and 3.2 that yx y h x c 2n n c 2n n n0 n0 n0 n n n a 2i n0 ji ν 2 2j ) n a 2 2i n0 ji ν 2 2j ) 2 n a n0 ν 2 2n 2 2i i ji ν 2 2j ) 2 n a ν 2 2n 2 2i i ν 2 2j ) 2, ji 3.4 for any x I ρ0. Moreover, we have yx yh x μ n0 ν 2 2n 2 a 2i i n ji ν 2 2j ) nμ2ν 2 2n 2 nμ2ν 2 2n 2 μ a 2i i n ji ν 2 2j ) a 2i i n ji ν 2 2j ) iμ 3.5 ν 2 2n 2 a 2i i n ji μ n0 nμ2 ν 2 2j ) μ ν 2 2n 2 a 2i i n ji ν 2 2j ) nμ2 ν 2 2n 2 a 2i i n ji ν 2 2j ), iμ for all x I ρ0, where μ ν 2 /2.

8 8 Abstract and Applied Analysis We know that / ν 2 2j 2 < and / ν 2 2j 2 < forj μand / ν 2 2j 2 or / ν 2 2j 2 is not perhaps less than for j μ. Then, we have n ji ν 2 2j ) μ ji ν 2 2j ) n jμν 2 2j ) M 2 e x Mx, 3.6 for all n μ 2andi 0,,...,μ. Similarly, we get n ji ν 2 2j ) Mx, 3.7 for all n μ 2andi 0,,...,μ. Thus, it follows from 3.9 and 3.5 that yx yh x μ Mx n n0 ν 2 2n 2 a 2i i μ a2n n n0 ν 2 2n 2 Mx μ ν 2 2n 2 a 2i i nμ2 nμ2 μ n0 ν 2 2n 2 a 2i i iμ Mx n ν 2 2n 2 a 2i i μ a2n n n0 ν 2 2n 2 Mx μ ν 2 2n 2 a 2i i nμ2 nμ2 ν 2 2n 2 a 2i i iμ Mx n0 ν 2 2n 2 a 2i i n0 ν 2 2n 2 a 2i i MxKε ν 2 m 2 KL ν Mxε, 3.8 for x I ρ0.

9 Abstract and Applied Analysis 9 Finally, assume that C is sufficiently small and ρ is sufficiently large. Then ρ 0 is sufficiently large. For any j 0andx I ρ0, we have ν 2 2j ) x max 2 ν 2 p 2, ν 2 p ). 3.9 If x is so large that μ [ ] ν 2 2 [ ] x x 2 2, 3.20 then it follows from the definition of Mx that x x 2 Mx max ν2 p 2, x /2 x x 2 ν 2 p ) 2 x /2, 3.2 for all sufficiently large x. If ν is large enough, then we can prove the local Hyers-Ulam stability of the Bessel differential equation.3 as we see in the following corollary. Corollary 3.2. Let ν be a positive nonintegral number and let p be a nonnegative integer with p< ν<p. Assume that a function y C K satisfies the differential inequality 3.3 for all x I ρ and for some ε 0. Suppose the sequence{b m } satisfies the condition 3.4 with a Landau constant C 0 and define ρ 0 min{ρ, / C } for a positive number C >C.If ν2 p 2, ν 2 p ), 3.22 for all x I ρ0, then there exists a solution y h x of the Bessel differential equation.3 such that yx yh x KLν ε, 3.23 for any x I ρ0. Proof. For any j 0andx I ρ0, we have x max 2 ν 2 2j ), ν 2 2j ) max ν2 p 2, ν 2 p ). 3.24

10 0 Abstract and Applied Analysis Thus, we get Mx max ν2 p 2, ν 2 p ),, 3.25 and the assertion is true due to Theorem Examples We will show that there exist functions yx which satisfy all the conditions given in Theorem 3. and Corollary 3.2. Let us define a function y : I 0 R by yx J 00.5 x c b m x m, 4. where J 00.5 x is the Bessel function of the first kind of order 00.5, n is a positive integer, and c is a constant satisfying c ε , 4.2 for some ε>0. It is obvious that the convergence radius of the power series b mx m is infinity. So we can set ρ 0. In fact, the infinite series b m m converges. So we have m b m lim sup m b m, m m2 b m2 lim sup m b m. m 4.3 Thus, it holds true that lim sup b m2 lim sup b m2 lim sup b m 2 m b m m b m m b m 4.4 which implies that the sequence {b m } satisfies the condition 3.4 with a Landau constant C /2. If we take C /00, then ρ 0 min{ρ, / C } 0. Since J 00.5 x is a particular solution of the Bessel differential equation.3 with ν 00.5, it follows from 4. that y x xy x ) yx c cx 4, 4.5 for any x I 0.

11 Abstract and Applied Analysis If we set c for m 2, 4 a m c for m 4, 0 otherwise, 4.6 then we have y x xy x ) yx a m x m, x2 y x xy x 4040 ) ) yx a m x m c x 4 < c ε, for all x I 0. Moreover, we have a mx m a mx m 40385/4 cx2 cx /4c cx 4 > , 4.8 and hence, we get a m x m < 857 a 7997 m x m, 4.9 for all x I 0.Thatis,{a m } satisfies the property b with K 857/7997. It holds true that ν2 p 2 <, ν 2 p ) <, 4.0

12 2 Abstract and Applied Analysis for all x I 0, and since L 00.5 m ) ζ2 8 π2 3 8, it follows from Corollary 3.2 that there exists a solution y h x of the Bessel differential equation.3 such that yx yh x L 00.5ε < 857 ) π ε, 4.2 for any x I 0. Acknowledgment This research was supported by the Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education, Science, and Technology no References S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York, NY, USA, D. H. Hyers, On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp , T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, vol. 72, no. 2, pp , T. Aoki, On the stability of the linear transformation in Banach spaces, the Mathematical Society of Japan, vol. 2, pp , L. P. Castro and A. Ramos, Hyers-Ulam-Rassias stability for a class of nonlinear Volterra integral equations, Banach Mathematical Analysis, vol. 3, no., pp , S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, Singapore, A. Dishliev and S. Hristova, Stability on a cone in terms of two measures for differential equations with maxima, Annals of Functional Analysis, vol., no., pp , G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Mathematicae, vol. 50, no. -2, pp , 995.

13 Abstract and Applied Analysis 3 9 P. Gǎvrutǎ, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, Mathematical Analysis and Applications, vol. 84, no. 3, pp , D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables,Birkhäuser, Boston, Mass, USA, 998. D. H. Hyers and T. M. Rassias, Approximate homomorphisms, Aequationes Mathematicae, vol. 44, no. 2-3, pp , S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, NY, USA, M. S. Moslehian and T. M. Rassias, Stability of functional equations in non-archimedean spaces, Applicable Analysis and Discrete Mathematics, vol., no. 2, pp , T. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Applicandae Mathematicae, vol. 62, no., pp , M. Obłoza, Hyers stability of the linear differential equation, Rocznik Naukowo-Dydaktyczny. Prace Matematyczne, no. 3, pp , M. Obłoza, Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Naukowo-Dydaktyczny. Prace Matematyczne, no. 4, pp. 4 46, C. Alsina and R. Ger, On some inequalities and stability results related to the exponential function, Inequalities and Applications, vol. 2, no. 4, pp , S.-E. Takahasi, T. Miura, and S. Miyajima, On the Hyers-Ulam stability of the Banach space-valued differential equation y λy, Bulletin of the Korean Mathematical Society, vol. 39, no. 2, pp , T. Miura, S.-M. Jung, and S.-E. Takahasi, Hyers-Ulam-Rassias stability of the Banach space valued linear differential equations y λy, the Korean Mathematical Society, vol. 4, no. 6, pp , S.-M. Jung, Legendre s differential equation and its Hyers-Ulam stability, Abstract and Applied Analysis, vol. 2007, Article ID 5649, 4 pages, S.-M. Jung, Approximation of analytic functions by Legendre functions, Nonlinear Analysis, vol. 7, no. 2, pp , S.-M. Jung, Approximation of analytic functions by Airy functions, Integral Transforms and Special Functions, vol. 9, no. -2, pp , S.-M. Jung, An approximation property of exponential functions, Acta Mathematica Hungarica, vol. 24, no. -2, pp , S.-M. Jung, Approximation of analytic functions by Hermite functions, Bulletin des Sciences Mathématiques, vol. 33, no. 7, pp , S.-M. Jung and S. Min, On approximate Euler differential equations, Abstract and Applied Analysis, vol. 2009, Article ID , 8 pages, B. Kim and S.-M. Jung, Bessel s differential equation and its Hyers-Ulam stability, Inequalities and Applications, vol. 2007, Article ID 2640, 8 pages, 2007.

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