Hyers-Ulam stability of linear differential equations of first order, III

Abstract Let X be a complex Banach space and let I = ( a , b ) be an open interval. In this paper, we will prove the generalized Hyers–Ulam stability of the differential equation t y ′ ( t ) + α y ( t ) + β t r x 0 = 0 for the class of continuously differentiable functions f : I → X , where α, β and r are complex constants and x 0 is an element of X. By applying this result, we also prove the Hyers–Ulam stability of the Euler differential equation of second order.

[1]  S.-M. Jung,et al.  Hyers-Ulam stability of linear differential equations of first order , 2004, Appl. Math. Lett..

[2]  Claudi Alsina,et al.  On Some Inequalities and Stability Results Related to the Exponential Function , 1998 .

[3]  Ilijas Farah,et al.  Approximate Homomorphisms , 1998, Comb..

[4]  Takeshi Miura,et al.  ON THE HYERS-ULAM STABILITY OF THE BANACH SPACE-VALUED DIFFERENTIAL EQUATION y'=λy , 2002 .

[5]  George Isac,et al.  Stability of Functional Equations in Several Variables , 1998 .

[6]  Soon-Mo Jung,et al.  HYERS-ULAM-RASSIAS STABILITY OF THE BANACH SPACE VALUED LINEAR DIFFERENTIAL EQUATIONS y′ = λy , 2004 .

[7]  Zbigniew Gajda,et al.  On stability of additive mappings , 1991 .

[8]  T. Rassias.  On the stability of the linear mapping in Banach spaces , 1978 .

[9]  P. Gǎvruţa,et al.  A Generalization of the Hyers-Ulam-Rassias Stability of Approximately Additive Mappings , 1994 .

[10]  Takeshi Miura,et al.  On the Hyers-Ulam Stability of Real Continuous Function Valued Differentiable Map , 2001 .

[11]  Roman Ger,et al.  The stability of the exponential equation , 1996 .

[12]  S. Ulam.  Problems in modern mathematics , 1964 .

[13]  D. H. Hyers.  On the Stability of the Linear Functional Equation. , 1941, Proceedings of the National Academy of Sciences of the United States of America.

[14]  Themistocles M. Rassias,et al.  On the Stability of Functional Equations and a Problem of Ulam , 2000 .

[15]  Takeshi Miura,et al.  A characterization of Hyers–Ulam stability of first order linear differential operators , 2003 .

[16]  Takeshi Miura,et al.  Hyers–Ulam stability of linear differential operator with constant coefficients , 2003 .

[17]  Takeshi Miura,et al.  ON THE HYERS-ULAM STABILITY OF A DIFFERENTIABLE MAP , 2002 .