- Email: [email protected]
Hyers–Ulam stability of associated Laguerre differential equations in a subclass of analytic functions
J. Math. Anal. Appl. 437 (2016) 605–612
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Hyers–Ulam stability of associated Laguerre differential equations in a subclass of analytic functions M.R. Abdollahpour a,∗ , R. Aghayari a , M.Th. Rassias b,c a
Department of Mathematics, Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran b Institute of Mathematics, University of Zurich, Winterthurerstr. 190, CH-8057 Zurich, Switzerland c Institute for Advanced Study, Program in Interdisciplinary Studies, 1 Einstein Dr, Princeton, NJ 08540, USA
a r t i c l e
i n f o
Article history: Received 13 October 2015 Available online 14 January 2016 Submitted by M.J. Schlosser
a b s t r a c t In this paper by the use of the series method we study the Hyers–Ulam stability of the associated homogeneous Laguerre differential equation in a subclass of analytic functions. © 2016 Elsevier Inc. All rights reserved.
Keywords: Hyers–Ulam stability Associated Laguerre differential equation
1. Introduction The question about the Hyers–Ulam stability of group homomorphisms was posed in 1940 by S.M. Ulam [24]. In 1941, D.H. Hyers [7] provided a solution of Ulam’s problem in the special case of approximately additive mappings on Banach spaces. Since then Ulam’s problem has inspired a large number of researchers to investigate this subject for a broad class of functional equations (cf. [5,8,14,21–23] and the references therein). Let X be a normed space over the field K = R or K = C and let I ⊂ R be an open interval. Let a0 , a1 , . . . , an : I → K and g : I → X be given continuous functions. If for any y : I → X satisfying the inequality an (x)y (n) (x) + an−1 (x)y (n−1) (x) + . . . + a1 (x)y (x) + a0 (x)y(x) + g(x) ε for all x ∈ I and for some ε > 0, there exists a function y0 : I → X such that * Corresponding author. E-mail addresses: [email protected], [email protected] (M.R. Abdollahpour), [email protected] (R. Aghayari), [email protected], [email protected] (M.Th. Rassias). http://dx.doi.org/10.1016/j.jmaa.2016.01.024 0022-247X/© 2016 Elsevier Inc. All rights reserved.
606
M.R. Abdollahpour et al. / J. Math. Anal. Appl. 437 (2016) 605–612
an (x)y (n) (x) + an−1 (x)y (n−1) (x) + . . . + a1 (x)y (x) + a0 (x)y(x) + g(x) = 0
(1.1)
and y(x) − y0 (x) K(ε) for any x ∈ I where K(ε) is an expression of ε with limε→0 K(ε) = 0, then we say that the differential equation (1.1) has the Hyers–Ulam stability. Some of the first results of Hyers–Ulam stability of differential equations were obtained by Obloza [19,20] and Alsina and Ger [4]. Hyers–Ulam stability of differential equations of first order was investigated by S.-M. Jung [9–11]. Hyers– Ulam stability of differential equations of second and third order has been studied in the papers [1,3,2,6,18]. S.-M. Jung [15] solved the inhomogeneous differential equation of the form xy + (1 − x)y + ny =
∞
am x m ,
m=0
where n is a positive integer, and applied this result to obtain a partial solution to the Hyers–Ulam stability of the differential equation xy + (1 − x)y + ny = 0. In this paper, using the ideas of [12,13,16,17], we investigate the general solution of the inhomogeneous associated Laguerre differential equation
xy + (1 + ν − x)y + λy =
∞
am x m ,
(1.2)
m=0
where λ > 0 and ν is a positive non-integer number and we intend to show that the associated Laguerre equation xy + (1 + ν − x)y + λy = 0 has the Hyers–Ulam stability in a subclass of analytic functions. 2. Inhomogeneous associated Laguerre equation If λ > 0 and ν is a positive non-integer number then the associated Laguerre differential equation xy + (1 + ν − x)y + λy = 0
(2.1)
has a regular singular point at 0. Thus, it can be solved by using a series expansion and its general solution is given by yh (x) = Ay1 (x) + By2 (x),
(2.2)
where ∞
(−λ)n xn , n!(ν + 1) n n=1
(2.3)
∞ (−λ − ν)n n−ν + d0 x , n!(1 − ν)n n=1
(2.4)
y1 (x) = b0 + b0 and y2 (x) = d0 x
−ν
M.R. Abdollahpour et al. / J. Math. Anal. Appl. 437 (2016) 605–612
607
are independent solutions of (2.1), and A, B are constants. It is necessary to mention that we used the notation (t)n = t(t + 1)(t + 2) . . . (t + n − 1), in (2.3) and (2.4). Let us consider c0 = 0, c1 =
a0 ν+1
n ∈ N, t ∈ R
and
cm =
m−1 i=0
m−1 ai i! j−λ , m! (i + ν + 1) j=i+1 j + ν + 1
(2.5)
for all m 2, where we refer to (1.2) for the am ’s. Here we have made the convention m−1 j=m
(j − λ) =1. (j + ν + 1)
A simple computation shows that (m + 1)(m + ν + 1)cm+1 − (m − λ)cm = am ,
(2.6)
for each m ∈ N ∪ {0}. ∞ Proposition 2.1. Let ρ > 1. If the power series m=0 am xm converges for all x ∈ (−ρ, ρ) then the series ∞ m converges for all x ∈ (−1, 1) and there exists a positive constant C1 such that m=1 cm x ∞ cm xm <
C1 , 1 − |x|
m=1
x ∈ (−1, 1),
where cm ’s are given by (2.5). Proof. Since the series
∞ m=0
am is absolutely convergent, there exists M1 > 0 such that ∞
|am | < M1 .
m=0
We have |cm | ≤
m−1 i=0
=
m−1 i=0
m−1 |j − λ| |ai | i! · m! i + ν + 1 j=i+1 j + ν + 1 m−1 |ai | |j − λ| , m(i + ν + 1) j=i+1 j(j + ν + 1)
for any m ∈ N. Suppose that q ≤ λ < q + 1 for some q ∈ N ∪ {0}. Let j ∈ {i + 1, . . . , m − 1}, then if j ≥ q + 1 and
|j−λ| j
< q + 1 if j < q + 1. Since |cm | <
m−1 i=0
1 j+ν+1
< 1, we have
|ai | (q + 1)q ≤ M1 (q + 1)q ≡ C1 , m(i + ν + 1)
|j−λ| j
<1
M.R. Abdollahpour et al. / J. Math. Anal. Appl. 437 (2016) 605–612
608
for all m ∈ N. Therefore n n ∞ cm xm ≤ |cm ||x|m < C1 |x|m = m=1
m=0
m=0
C1 , 1 − |x|
x ∈ (−1, 1)
for all n ∈ N, so ∞ cm xm < m=1
C1 , 1 − |x|
x ∈ (−1, 1).
2
∞ Proposition 2.2. Let ρ ≤ 1 and cm ’s be given by (2.5). If the power series m=0 am xm converges for all ∞ x ∈ (−ρ, ρ) then for any 0 < ρ0 < ρ, the series m=1 cm xm converges for all x ∈ [−ρ0 , ρ0 ] and there exists a positive constant C2 such that ∞ cm xm ≤ C2 ,
x ∈ [−ρ0 , ρ0 ].
m=1
Proof. Let x ∈ [−ρ0 , ρ0 ]. Then ∞
∞
|am xm | ≤
m=1
|am |ρm 0 = M2 .
m=1
Therefore, we have n n cm xm ≤ |cm |ρm 0 m=k
m=k
≤
=
n
ρm 0
m−1
m=k
i=0
n
m−1
ρm 0
i=0
m=k
≤
n m−1 m=k i=0
m−1 |j − λ| |ai | i! · m! i + ν + 1 j=i+1 j + ν + 1 m−1 |j − λ| i!|ai | · m!(m + ν) j=i+1 j + ν
ρi0 |ai |
m−1 |j − λ| 1 · , m(m + ν) j=i+1 j(j + ν)
for all n, k ∈ N with k ≤ n. Let q ∈ N ∪ {0} and q ≤ λ < q + 1, then m−1
|j − λ| ≤ (q + 1)q . j(j + ν) j=i+1 Hence n n cm xm ≤ M2 (q + 1)q m=k
Since
∞
1 m=1 m(m+ν)
is convergent,
m=k
∞
m=1 cm x
m
∞ ∞ m q cm x ≤ M2 (q + 1) m=1
1 , m(m + ν)
n, k ∈ N.
is convergent for all x ∈ [−ρ0 , ρ0 ]. Also
1 = C2 , m(m + ν) m=1
x ∈ [−ρ0 , ρ0 ].
2
M.R. Abdollahpour et al. / J. Math. Anal. Appl. 437 (2016) 605–612
609
∞ Corollary 2.3. Suppose that the power series m=0 am xm converges for all x ∈ (−ρ, ρ) with some positive ρ. ∞ m Let ρ1 = min{1, ρ}. Then the power series with cm ’s given in (2.5) is convergent for all m=1 cm x x ∈ (−ρ1 , ρ1 ). Also, for any positive ρ0 < ρ1 , it holds ∞ cm xm ≤ C,
x ∈ [−ρ0 , ρ0 ]
m=1
for some positive constant C which depends on ρ0 . In the next proposition we show that Laguerre equation (1.2).
∞
m=1 cm x
m
is a particular solution of the inhomogeneous associated
Proposition 2.4. Let λ > 0 and ν be a positive non-integer number and ρ > 0 be the radius of convergence ∞ of the power series m=0 am xm and ρ1 = min{1, ρ}. Then y : (−ρ1 , ρ1 ) → C,
y(x) = b0 + b0
∞
∞ (−λ)n xn + cm xm , n!(ν + 1) n n=1 m=1
(2.7)
is a solution of differential equation (1.2), where cm ’s are given by (2.5). ∞ Proof. By Corollary 2.3, the power series m=1 cm xm is convergent for each x ∈ (−ρ1 , ρ1 ). Substituting ∞ m for y in (1.2), by (2.6) we have m=1 cm x
xy + (1 + v − x)y + λy =
∞
am x m ,
x ∈ (−ρ1 , ρ1 ).
m=0
Therefore, y(x) = yh (x) +
∞
cm xm ,
m=1
where yh (x) is given by (2.2). Since y(x) is defined on (−ρ1 , ρ1 ) so y(x) is in the form (2.7).
2
3. Stability of associated Laguerre differential equation In this section, we show that the associated Laguerre differential equation (2.1) has the Hyers–Ulam stability in a subclass of analytic functions. Theorem 3.1. Let ρ, λ > 0 and ν be a positive non-integer number. Let y : (−ρ, ρ) → C be an analytic function which can be represented by a power-series expansion centered at x = 0. Suppose there exists a constant ε > 0 such that |xy + (1 + ν − x)y + λy| ≤ ε,
x ∈ (−ρ, ρ).
Also, suppose that
xy + (1 + ν − x)y + λy =
∞ m=0
and
am x m ,
x ∈ (−ρ, ρ),
(3.1)
610
M.R. Abdollahpour et al. / J. Math. Anal. Appl. 437 (2016) 605–612 ∞ m=0
∞ |am xm | ≤ K am xm ,
x ∈ (−ρ, ρ)
(3.2)
m=0
for some constant K. Let ρ1 = min{1, ρ}. Then there exists a solution of (2.1), z : (−ρ1 , ρ1 ) → C such that |y(x) − z(x)| ≤ Cε for all x ∈ [−ρ0 , ρ0 ], where ρ0 < ρ1 is any positive number and C is some constant which depends on ρ0 . Proof. Since y(x) satisfies the following differential equation xy + (1 + v − x)y + λy =
∞
am x m ,
m=0
by Proposition 2.4, we obtain y(x) = b0 + b0
∞ (−λ)n xn + cm xm , n!(ν + 1) n n=1 m=1 ∞
x ∈ (−ρ1 , ρ1 ),
where cm ’s are given by (2.5). Also by (3.1) and (3.2), we get ∞
|am xm | ≤ Kε,
x ∈ (−ρ, ρ).
m=0
Let us define the function z(x) = b0 + b0
∞
(−λ)n xn n!(ν + 1)n n=1
for all x ∈ (−ρ1 , ρ1 ). Then an argument similar to the proof of Proposition 2.2 shows that ∞ |y(x) − z(x)| = cm xm ≤ Cε,
x ∈ (−ρ0 , ρ0 )
m=1
where ρ0 < 1 is any positive number and C is some constant. 2 We show that there exist functions which satisfy all assumptions of Theorem 3.1. Example 3.2. Let ε > 0, n ∈ N and 0
1 −1 n(n + 1)(8n + 7) + n2 ε. 6
Let y = (−1, 1) → C be given by 3 y(x) = − x + t x2k . 2 n
k=1
Then ∞ 3 xy + ( − x)y + y = am x m , 2 m=0
M.R. Abdollahpour et al. / J. Math. Anal. Appl. 437 (2016) 605–612
611
where
am
⎧ 3 ⎪ m = 1, 3, . . . , 2n − 1 ⎨ (m + 1)(m + 2 )t, = −(m − 1)t, m = 2, 4, . . . , 2n ⎪ ⎩ 0, otherwise.
Also for each x ∈ (−1, 1), ∞ m=0
∞ |am xm | ≤ 3 am xm m=0
and ∞ ∞ 1 3 m |xy + ( − x)y + y| = | am x | ≤ |am | ≤ n(n + 1)(8n + 7) + n2 t < ε. 2 6 m=0 m=0
Example 3.3. Let ε > 0, n ∈ N and −1 4 31 0 < t < (n + 1)( n2 + n + 3) ε. 3 6 Let y = (−1, 1) → C be given by y(x) = 1 +
∞ n (− 12 )m m x + t x2k+1 . 5 m!( ) m 2 m=1 k=0
Then ∞ 5 1 xy + ( − x)y + y = am x m , 2 2 m=0
where
am
⎧ 5 ⎪ m = 0, 2, . . . , 2n ⎨ (m + 1)(m + 2 )t, 1 = −(m − 2 )t, m = 1, 3, . . . , 2n + 1 ⎪ ⎩ 0, otherwise.
Then for each x ∈ (−1, 1), ∞ m=0
∞ m |am x | ≤ 2 am x m
m=0
and ∞ ∞ 5 1 4 31 |xy + ( − x)y + y| = | am x m | ≤ |am | ≤ (n + 1)( n2 + n + 3) t < ε. 2 2 3 6 m=0 m=0
Acknowledgments The third author (M.Th. Rassias) expresses his gratitude to Professor P.-O. Dehaye for granting him financial support through his SNF grant: SNF PP00P2_138906, to conduct postdoctoral research at the University of Zurich during the academic year 2015–2016.
612
M.R. Abdollahpour et al. / J. Math. Anal. Appl. 437 (2016) 605–612
References [1] M.R. Abdollahpour, A. Najati, Stability of linear differential equations of third order, Appl. Math. Lett. 24 (2011) 1827–1830. [2] M.R. Abdollahpour, A. Najati, C. Park, T.M. Rassias, D.Y. Shin, Approximate perfect differential equations of second order, Adv. Difference Equ. (2012) 2012:225. [3] M.R. Abdollahpour, C. Park, Hyers–Ulam stability of a class of differential equations of second order, J. Comput. Anal. Appl. 18 (5) (2015) 899–903. [4] C. Alsina, R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl. 2 (1998) 373–380. [5] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002. [6] M. Eshaghi Gordji, Y. Cho, M.B. Ghaemi, B. Alizadeh, Stability of the exact second order partial differential equations, J. Inequal. Appl. 2011 (2011) 306275. [7] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941) 222–224. [8] D.H. Hyers, G. Isac, Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Boston, 1998. [9] S.-M. Jung, Hyers–Ulam stability of linear differential equations of first order, Appl. Math. Lett. 17 (2004) 1135–1140. [10] S.-M. Jung, Hyers–Ulam stability of linear differential equations of first order, III, J. Math. Anal. Appl. 311 (2005) 139–146. [11] S.-M. Jung, Hyers–Ulam stability of linear differential equations of first order, II, Appl. Math. Lett. 19 (2006) 854–858. [12] S.-M. Jung, Legendre’s differential equation and its Hyers–Ulam stability, Abstr. Appl. Anal. 2007 (2007) 56419, http:// dx.doi.org/10.1155/2007/56419. [13] S.-M. Jung, Approximation of analytic functions by Legendre functions, Nonlinear Anal. 71 (12) (2009) 103–108. [14] S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Optimization and Its Applications, vol. 48, 2011. [15] S.-M. Jung, Approximation of analytic functions by Laguerre functions, Appl. Math. Comput. 218 (2011) 832–835. [16] S.-M. Jung, B. Kim, Bessel’s differential equation and its Hyers–Ulam stability, J. Inequal. Appl. (2007) 21640. [17] S.-M. Jung, B. Kim, Chebyshev’s differential equation and its Hyers–Ulam stability, Differ. Equ. Appl. 1 (2009) 199–207. [18] Y. Li, Y. Shen, Hyers–Ulam stability of linear differential equations of second order, Appl. Math. Lett. 23 (2010) 306–309. [19] M. Obłoza, Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat. 13 (1993) 259–270. [20] M. Obłoza, Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.-Dydakt. Prace Mat. 14 (1997) 141–146. [21] D. Popa, I. Rasa, On the Hyers–Ulam stability of the linear differential equation, J. Math. Anal. Appl. 381 (2011) 530–537. [22] D. Popa, I. Rasa, Hyers–Ulam stability of the linear differential operator with nonconstant coefficients, Appl. Math. Comput. 291 (2012) 1562–1568. [23] Th.M. Rassias, On the stability of linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300. [24] S.M. Ulam, Problems in Modern Mathematics, Chapter VI, science ed., Wiley, New York, 1940.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Hyers–Ulam stability of associated Laguerre differential equations in a subclass of analytic functions M.R. Abdollahpour a,∗ , R. Aghayari a , M.Th. Rassias b,c a
Department of Mathematics, Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran b Institute of Mathematics, University of Zurich, Winterthurerstr. 190, CH-8057 Zurich, Switzerland c Institute for Advanced Study, Program in Interdisciplinary Studies, 1 Einstein Dr, Princeton, NJ 08540, USA
a r t i c l e
i n f o
Article history: Received 13 October 2015 Available online 14 January 2016 Submitted by M.J. Schlosser
a b s t r a c t In this paper by the use of the series method we study the Hyers–Ulam stability of the associated homogeneous Laguerre differential equation in a subclass of analytic functions. © 2016 Elsevier Inc. All rights reserved.
Keywords: Hyers–Ulam stability Associated Laguerre differential equation
1. Introduction The question about the Hyers–Ulam stability of group homomorphisms was posed in 1940 by S.M. Ulam [24]. In 1941, D.H. Hyers [7] provided a solution of Ulam’s problem in the special case of approximately additive mappings on Banach spaces. Since then Ulam’s problem has inspired a large number of researchers to investigate this subject for a broad class of functional equations (cf. [5,8,14,21–23] and the references therein). Let X be a normed space over the field K = R or K = C and let I ⊂ R be an open interval. Let a0 , a1 , . . . , an : I → K and g : I → X be given continuous functions. If for any y : I → X satisfying the inequality an (x)y (n) (x) + an−1 (x)y (n−1) (x) + . . . + a1 (x)y (x) + a0 (x)y(x) + g(x) ε for all x ∈ I and for some ε > 0, there exists a function y0 : I → X such that * Corresponding author. E-mail addresses: [email protected], [email protected] (M.R. Abdollahpour), [email protected] (R. Aghayari), [email protected], [email protected] (M.Th. Rassias). http://dx.doi.org/10.1016/j.jmaa.2016.01.024 0022-247X/© 2016 Elsevier Inc. All rights reserved.
606
M.R. Abdollahpour et al. / J. Math. Anal. Appl. 437 (2016) 605–612
an (x)y (n) (x) + an−1 (x)y (n−1) (x) + . . . + a1 (x)y (x) + a0 (x)y(x) + g(x) = 0
(1.1)
and y(x) − y0 (x) K(ε) for any x ∈ I where K(ε) is an expression of ε with limε→0 K(ε) = 0, then we say that the differential equation (1.1) has the Hyers–Ulam stability. Some of the first results of Hyers–Ulam stability of differential equations were obtained by Obloza [19,20] and Alsina and Ger [4]. Hyers–Ulam stability of differential equations of first order was investigated by S.-M. Jung [9–11]. Hyers– Ulam stability of differential equations of second and third order has been studied in the papers [1,3,2,6,18]. S.-M. Jung [15] solved the inhomogeneous differential equation of the form xy + (1 − x)y + ny =
∞
am x m ,
m=0
where n is a positive integer, and applied this result to obtain a partial solution to the Hyers–Ulam stability of the differential equation xy + (1 − x)y + ny = 0. In this paper, using the ideas of [12,13,16,17], we investigate the general solution of the inhomogeneous associated Laguerre differential equation
xy + (1 + ν − x)y + λy =
∞
am x m ,
(1.2)
m=0
where λ > 0 and ν is a positive non-integer number and we intend to show that the associated Laguerre equation xy + (1 + ν − x)y + λy = 0 has the Hyers–Ulam stability in a subclass of analytic functions. 2. Inhomogeneous associated Laguerre equation If λ > 0 and ν is a positive non-integer number then the associated Laguerre differential equation xy + (1 + ν − x)y + λy = 0
(2.1)
has a regular singular point at 0. Thus, it can be solved by using a series expansion and its general solution is given by yh (x) = Ay1 (x) + By2 (x),
(2.2)
where ∞
(−λ)n xn , n!(ν + 1) n n=1
(2.3)
∞ (−λ − ν)n n−ν + d0 x , n!(1 − ν)n n=1
(2.4)
y1 (x) = b0 + b0 and y2 (x) = d0 x
−ν
M.R. Abdollahpour et al. / J. Math. Anal. Appl. 437 (2016) 605–612
607
are independent solutions of (2.1), and A, B are constants. It is necessary to mention that we used the notation (t)n = t(t + 1)(t + 2) . . . (t + n − 1), in (2.3) and (2.4). Let us consider c0 = 0, c1 =
a0 ν+1
n ∈ N, t ∈ R
and
cm =
m−1 i=0
m−1 ai i! j−λ , m! (i + ν + 1) j=i+1 j + ν + 1
(2.5)
for all m 2, where we refer to (1.2) for the am ’s. Here we have made the convention m−1 j=m
(j − λ) =1. (j + ν + 1)
A simple computation shows that (m + 1)(m + ν + 1)cm+1 − (m − λ)cm = am ,
(2.6)
for each m ∈ N ∪ {0}. ∞ Proposition 2.1. Let ρ > 1. If the power series m=0 am xm converges for all x ∈ (−ρ, ρ) then the series ∞ m converges for all x ∈ (−1, 1) and there exists a positive constant C1 such that m=1 cm x ∞ cm xm <
C1 , 1 − |x|
m=1
x ∈ (−1, 1),
where cm ’s are given by (2.5). Proof. Since the series
∞ m=0
am is absolutely convergent, there exists M1 > 0 such that ∞
|am | < M1 .
m=0
We have |cm | ≤
m−1 i=0
=
m−1 i=0
m−1 |j − λ| |ai | i! · m! i + ν + 1 j=i+1 j + ν + 1 m−1 |ai | |j − λ| , m(i + ν + 1) j=i+1 j(j + ν + 1)
for any m ∈ N. Suppose that q ≤ λ < q + 1 for some q ∈ N ∪ {0}. Let j ∈ {i + 1, . . . , m − 1}, then if j ≥ q + 1 and
|j−λ| j
< q + 1 if j < q + 1. Since |cm | <
m−1 i=0
1 j+ν+1
< 1, we have
|ai | (q + 1)q ≤ M1 (q + 1)q ≡ C1 , m(i + ν + 1)
|j−λ| j
<1
M.R. Abdollahpour et al. / J. Math. Anal. Appl. 437 (2016) 605–612
608
for all m ∈ N. Therefore n n ∞ cm xm ≤ |cm ||x|m < C1 |x|m = m=1
m=0
m=0
C1 , 1 − |x|
x ∈ (−1, 1)
for all n ∈ N, so ∞ cm xm < m=1
C1 , 1 − |x|
x ∈ (−1, 1).
2
∞ Proposition 2.2. Let ρ ≤ 1 and cm ’s be given by (2.5). If the power series m=0 am xm converges for all ∞ x ∈ (−ρ, ρ) then for any 0 < ρ0 < ρ, the series m=1 cm xm converges for all x ∈ [−ρ0 , ρ0 ] and there exists a positive constant C2 such that ∞ cm xm ≤ C2 ,
x ∈ [−ρ0 , ρ0 ].
m=1
Proof. Let x ∈ [−ρ0 , ρ0 ]. Then ∞
∞
|am xm | ≤
m=1
|am |ρm 0 = M2 .
m=1
Therefore, we have n n cm xm ≤ |cm |ρm 0 m=k
m=k
≤
=
n
ρm 0
m−1
m=k
i=0
n
m−1
ρm 0
i=0
m=k
≤
n m−1 m=k i=0
m−1 |j − λ| |ai | i! · m! i + ν + 1 j=i+1 j + ν + 1 m−1 |j − λ| i!|ai | · m!(m + ν) j=i+1 j + ν
ρi0 |ai |
m−1 |j − λ| 1 · , m(m + ν) j=i+1 j(j + ν)
for all n, k ∈ N with k ≤ n. Let q ∈ N ∪ {0} and q ≤ λ < q + 1, then m−1
|j − λ| ≤ (q + 1)q . j(j + ν) j=i+1 Hence n n cm xm ≤ M2 (q + 1)q m=k
Since
∞
1 m=1 m(m+ν)
is convergent,
m=k
∞
m=1 cm x
m
∞ ∞ m q cm x ≤ M2 (q + 1) m=1
1 , m(m + ν)
n, k ∈ N.
is convergent for all x ∈ [−ρ0 , ρ0 ]. Also
1 = C2 , m(m + ν) m=1
x ∈ [−ρ0 , ρ0 ].
2
M.R. Abdollahpour et al. / J. Math. Anal. Appl. 437 (2016) 605–612
609
∞ Corollary 2.3. Suppose that the power series m=0 am xm converges for all x ∈ (−ρ, ρ) with some positive ρ. ∞ m Let ρ1 = min{1, ρ}. Then the power series with cm ’s given in (2.5) is convergent for all m=1 cm x x ∈ (−ρ1 , ρ1 ). Also, for any positive ρ0 < ρ1 , it holds ∞ cm xm ≤ C,
x ∈ [−ρ0 , ρ0 ]
m=1
for some positive constant C which depends on ρ0 . In the next proposition we show that Laguerre equation (1.2).
∞
m=1 cm x
m
is a particular solution of the inhomogeneous associated
Proposition 2.4. Let λ > 0 and ν be a positive non-integer number and ρ > 0 be the radius of convergence ∞ of the power series m=0 am xm and ρ1 = min{1, ρ}. Then y : (−ρ1 , ρ1 ) → C,
y(x) = b0 + b0
∞
∞ (−λ)n xn + cm xm , n!(ν + 1) n n=1 m=1
(2.7)
is a solution of differential equation (1.2), where cm ’s are given by (2.5). ∞ Proof. By Corollary 2.3, the power series m=1 cm xm is convergent for each x ∈ (−ρ1 , ρ1 ). Substituting ∞ m for y in (1.2), by (2.6) we have m=1 cm x
xy + (1 + v − x)y + λy =
∞
am x m ,
x ∈ (−ρ1 , ρ1 ).
m=0
Therefore, y(x) = yh (x) +
∞
cm xm ,
m=1
where yh (x) is given by (2.2). Since y(x) is defined on (−ρ1 , ρ1 ) so y(x) is in the form (2.7).
2
3. Stability of associated Laguerre differential equation In this section, we show that the associated Laguerre differential equation (2.1) has the Hyers–Ulam stability in a subclass of analytic functions. Theorem 3.1. Let ρ, λ > 0 and ν be a positive non-integer number. Let y : (−ρ, ρ) → C be an analytic function which can be represented by a power-series expansion centered at x = 0. Suppose there exists a constant ε > 0 such that |xy + (1 + ν − x)y + λy| ≤ ε,
x ∈ (−ρ, ρ).
Also, suppose that
xy + (1 + ν − x)y + λy =
∞ m=0
and
am x m ,
x ∈ (−ρ, ρ),
(3.1)
610
M.R. Abdollahpour et al. / J. Math. Anal. Appl. 437 (2016) 605–612 ∞ m=0
∞ |am xm | ≤ K am xm ,
x ∈ (−ρ, ρ)
(3.2)
m=0
for some constant K. Let ρ1 = min{1, ρ}. Then there exists a solution of (2.1), z : (−ρ1 , ρ1 ) → C such that |y(x) − z(x)| ≤ Cε for all x ∈ [−ρ0 , ρ0 ], where ρ0 < ρ1 is any positive number and C is some constant which depends on ρ0 . Proof. Since y(x) satisfies the following differential equation xy + (1 + v − x)y + λy =
∞
am x m ,
m=0
by Proposition 2.4, we obtain y(x) = b0 + b0
∞ (−λ)n xn + cm xm , n!(ν + 1) n n=1 m=1 ∞
x ∈ (−ρ1 , ρ1 ),
where cm ’s are given by (2.5). Also by (3.1) and (3.2), we get ∞
|am xm | ≤ Kε,
x ∈ (−ρ, ρ).
m=0
Let us define the function z(x) = b0 + b0
∞
(−λ)n xn n!(ν + 1)n n=1
for all x ∈ (−ρ1 , ρ1 ). Then an argument similar to the proof of Proposition 2.2 shows that ∞ |y(x) − z(x)| = cm xm ≤ Cε,
x ∈ (−ρ0 , ρ0 )
m=1
where ρ0 < 1 is any positive number and C is some constant. 2 We show that there exist functions which satisfy all assumptions of Theorem 3.1. Example 3.2. Let ε > 0, n ∈ N and 0
1 −1 n(n + 1)(8n + 7) + n2 ε. 6
Let y = (−1, 1) → C be given by 3 y(x) = − x + t x2k . 2 n
k=1
Then ∞ 3 xy + ( − x)y + y = am x m , 2 m=0
M.R. Abdollahpour et al. / J. Math. Anal. Appl. 437 (2016) 605–612
611
where
am
⎧ 3 ⎪ m = 1, 3, . . . , 2n − 1 ⎨ (m + 1)(m + 2 )t, = −(m − 1)t, m = 2, 4, . . . , 2n ⎪ ⎩ 0, otherwise.
Also for each x ∈ (−1, 1), ∞ m=0
∞ |am xm | ≤ 3 am xm m=0
and ∞ ∞ 1 3 m |xy + ( − x)y + y| = | am x | ≤ |am | ≤ n(n + 1)(8n + 7) + n2 t < ε. 2 6 m=0 m=0
Example 3.3. Let ε > 0, n ∈ N and −1 4 31 0 < t < (n + 1)( n2 + n + 3) ε. 3 6 Let y = (−1, 1) → C be given by y(x) = 1 +
∞ n (− 12 )m m x + t x2k+1 . 5 m!( ) m 2 m=1 k=0
Then ∞ 5 1 xy + ( − x)y + y = am x m , 2 2 m=0
where
am
⎧ 5 ⎪ m = 0, 2, . . . , 2n ⎨ (m + 1)(m + 2 )t, 1 = −(m − 2 )t, m = 1, 3, . . . , 2n + 1 ⎪ ⎩ 0, otherwise.
Then for each x ∈ (−1, 1), ∞ m=0
∞ m |am x | ≤ 2 am x m
m=0
and ∞ ∞ 5 1 4 31 |xy + ( − x)y + y| = | am x m | ≤ |am | ≤ (n + 1)( n2 + n + 3) t < ε. 2 2 3 6 m=0 m=0
Acknowledgments The third author (M.Th. Rassias) expresses his gratitude to Professor P.-O. Dehaye for granting him financial support through his SNF grant: SNF PP00P2_138906, to conduct postdoctoral research at the University of Zurich during the academic year 2015–2016.
612
M.R. Abdollahpour et al. / J. Math. Anal. Appl. 437 (2016) 605–612
References [1] M.R. Abdollahpour, A. Najati, Stability of linear differential equations of third order, Appl. Math. Lett. 24 (2011) 1827–1830. [2] M.R. Abdollahpour, A. Najati, C. Park, T.M. Rassias, D.Y. Shin, Approximate perfect differential equations of second order, Adv. Difference Equ. (2012) 2012:225. [3] M.R. Abdollahpour, C. Park, Hyers–Ulam stability of a class of differential equations of second order, J. Comput. Anal. Appl. 18 (5) (2015) 899–903. [4] C. Alsina, R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl. 2 (1998) 373–380. [5] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002. [6] M. Eshaghi Gordji, Y. Cho, M.B. Ghaemi, B. Alizadeh, Stability of the exact second order partial differential equations, J. Inequal. Appl. 2011 (2011) 306275. [7] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941) 222–224. [8] D.H. Hyers, G. Isac, Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Boston, 1998. [9] S.-M. Jung, Hyers–Ulam stability of linear differential equations of first order, Appl. Math. Lett. 17 (2004) 1135–1140. [10] S.-M. Jung, Hyers–Ulam stability of linear differential equations of first order, III, J. Math. Anal. Appl. 311 (2005) 139–146. [11] S.-M. Jung, Hyers–Ulam stability of linear differential equations of first order, II, Appl. Math. Lett. 19 (2006) 854–858. [12] S.-M. Jung, Legendre’s differential equation and its Hyers–Ulam stability, Abstr. Appl. Anal. 2007 (2007) 56419, http:// dx.doi.org/10.1155/2007/56419. [13] S.-M. Jung, Approximation of analytic functions by Legendre functions, Nonlinear Anal. 71 (12) (2009) 103–108. [14] S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Optimization and Its Applications, vol. 48, 2011. [15] S.-M. Jung, Approximation of analytic functions by Laguerre functions, Appl. Math. Comput. 218 (2011) 832–835. [16] S.-M. Jung, B. Kim, Bessel’s differential equation and its Hyers–Ulam stability, J. Inequal. Appl. (2007) 21640. [17] S.-M. Jung, B. Kim, Chebyshev’s differential equation and its Hyers–Ulam stability, Differ. Equ. Appl. 1 (2009) 199–207. [18] Y. Li, Y. Shen, Hyers–Ulam stability of linear differential equations of second order, Appl. Math. Lett. 23 (2010) 306–309. [19] M. Obłoza, Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat. 13 (1993) 259–270. [20] M. Obłoza, Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.-Dydakt. Prace Mat. 14 (1997) 141–146. [21] D. Popa, I. Rasa, On the Hyers–Ulam stability of the linear differential equation, J. Math. Anal. Appl. 381 (2011) 530–537. [22] D. Popa, I. Rasa, Hyers–Ulam stability of the linear differential operator with nonconstant coefficients, Appl. Math. Comput. 291 (2012) 1562–1568. [23] Th.M. Rassias, On the stability of linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300. [24] S.M. Ulam, Problems in Modern Mathematics, Chapter VI, science ed., Wiley, New York, 1940.