On the Stability Problem of Differential Equations in the Sense of Ulam

In this paper we consider the stability problem of a general class of differential equations in the sense of Hyers–Ulam and Hyers–Ulam–Rassias with the aid of a fixed point technique. We extend and improve the literature by dropping some assumptions of some well known and commonly cited results in this topic. Some illustrative examples are also given to visualize the improvement.

[1]  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.

[2]  S. Ulam A collection of mathematical problems , 1960 .

[3]  Choonkill Park,et al.  Hyers–Ulam Stability of General Jensen-Type Mappings in Banach Algebras , 2014 .

[4]  C. Borelli On Hyers— Ulam Stability of Hosszú’s Functional Equation , 1994 .

[5]  Tosio Aoki,et al.  On the Stability of the linear Transformation in Banach Spaces. , 1950 .

[6]  Dorian Popa Hyers-Ulam-Rassias stability of a linear recurrence , 2005 .

[7]  A fixed point approach to the stability of differential equations . , 2010 .

[8]  Sze-Bi Hsu,et al.  ORDINARY DIFFERENTIAL EQUATIONS WITH APPLICATIONS , 2005 .

[9]  Yonghong Shen The Ulam Stability of First Order Linear Dynamic Equations on Time Scales , 2017 .

[10]  Florin Bojor Note on the stability of first order linear differential equations , 2012 .

[11]  R. Bellman Stability theory of differential equations , 1953 .

[12]  G. Forti,et al.  Comments on the core of the direct method for proving Hyers–Ulam stability of functional equations , 2004 .

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

[14]  J. B. Díaz,et al.  A fixed point theorem of the alternative, for contractions on a generalized complete metric space , 1968 .

[15]  Dorian Popa,et al.  The Hyers–Ulam stability of nonlinear recurrences , 2007 .

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

[17]  Hiroyuki Takagi,et al.  The Hyers–Ulam stability constants of first order linear differential operators , 2004 .

[18]  Themistocles M. Rassias,et al.  Handbook of functional equations : stability theory , 2014 .

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

[20]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

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

[22]  J.-C. Yao,et al.  ULAM-HYERS STABILITY FOR OPERATORIAL EQUATIONS AND INCLUSIONS VIA NONSELF OPERATORS , 2011 .

[23]  D. Popa,et al.  Hyers–Ulam Stability of Euler’s Differential Equation , 2016 .

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

[25]  Viorel Radu,et al.  Fixed Point Methods for the Generalized Stability of Functional Equations in a Single Variable , 2007 .

[26]  J. Sousa,et al.  Ulam–Hyers–Rassias Stability for a Class of Fractional Integro-Differential Equations , 2018, Results in Mathematics.

[27]  C. Tunç,et al.  Hyers-Ulam-Rassias Stability for a First Order Functional Differential Equation , 2015 .

[28]  Vimal Singh,et al.  Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions , 1979, IEEE Transactions on Systems, Man, and Cybernetics.