ON THE HYERS-ULAM STABILITY OF THE BANACH SPACE-VALUED DIFFERENTIAL EQUATION y

Title & Authors
ON THE HYERS-ULAM STABILITY OF THE BANACH SPACE-VALUED DIFFERENTIAL EQUATION y
Takahasi, Sin-Ei; Miura, Takeshi; Miyajima, Shizuo;

Abstract
Let I be an open interval and X a complex Banach space. Let$\small{\varepsilon\geq0\;and\;\lambda}$ a non-zero complex number with Re $\small{\lambda\neq0}$. If $\small{\varphi}$ is a strongly differentiable map from I to X with $\small{\parallel\varphi^(t)-\lambda\varphi(t)\parallel\leq\varepsilon\;for\;all\;t\in\;I}$, then we show that the distance between $\small{\varphi}$ and the set of all solutions to the differential equation y
Keywords
exponential functions;Hyers-Ulam stability;
Language
English
Cited by
1.
HYERS-ULAM-RASSIAS STABILITY OF THE BANACH SPACE VALUED LINEAR DIFFERENTIAL EQUATIONS y′ = λy,;;;

대한수학회지, 2004. vol.41. 6, pp.995-1005
2.
ESSENTIAL NORMS AND STABILITY CONSTANTS OF WEIGHTED COMPOSITION OPERATORS ON C(X),;;;

대한수학회보, 2003. vol.40. 4, pp.583-591
3.
ON HYERS-ULAM STABILITY OF NONLINEAR DIFFERENTIAL EQUATIONS,;;;

대한수학회보, 2015. vol.52. 2, pp.685-697
1.
The Ulam Stability of First Order Linear Dynamic Equations on Time Scales, Results in Mathematics, 2017
2.
On Some Recent Developments in Ulam's Type Stability, Abstract and Applied Analysis, 2012, 2012, 1
3.
Laplace transform and Hyers–Ulam stability of linear differential equations, Journal of Mathematical Analysis and Applications, 2013, 403, 1, 244
4.
Invariance of Hyers-Ulam stability of linear differential equations and its applications, Advances in Difference Equations, 2015, 2015, 1
5.
Hyers-Ulam-Rassias Stability for the Heat Equation, Applied Mathematics, 2013, 04, 07, 1001
6.
Hyers-Ulam Stability of the Delay Equationy'(t)=λy(t-τ), Abstract and Applied Analysis, 2010, 2010, 1
7.
Hyers–Ulam stability of the time independent Schrödinger equations, Applied Mathematics Letters, 2017, 74, 147
8.
An approximation property of exponential functions, Acta Mathematica Hungarica, 2009, 124, 1-2, 155
9.
Power series method and approximate linear differential equations of second order, Advances in Difference Equations, 2013, 2013, 1, 76
10.
Optimal Hyers-Ulam’s constant for the linear differential equations, Journal of Inequalities and Applications, 2016, 2016, 1
11.
Implicit function theorem and its application to a Ulam’s problem for exact differential equations, Acta Mathematica Sinica, English Series, 2010, 26, 11, 2085
12.
On the Hyers–Ulam stability of the linear differential equation, Journal of Mathematical Analysis and Applications, 2011, 381, 2, 530
13.
A characterization of Hyers–Ulam stability of first order linear differential operators, Journal of Mathematical Analysis and Applications, 2003, 286, 1, 136
14.
On Approximate Euler Differential Equations, Abstract and Applied Analysis, 2009, 2009, 1
15.
Approximation of Analytic Functions by Bessel's Functions of Fractional Order, Abstract and Applied Analysis, 2011, 2011, 1
16.
On the stability of Laplace’s equation, Applied Mathematics Letters, 2013, 26, 5, 549
17.
Hyers-Ulam Stability of Nonhomogeneous Linear Differential Equations of Second Order, International Journal of Mathematics and Mathematical Sciences, 2009, 2009, 1
18.
Hyers–Ulam stability of first-order homogeneous linear differential equations with a real-valued coefficient, Applied Mathematics Letters, 2017, 63, 102
19.
E α -Ulam type stability of fractional order ordinary differential equations, Journal of Applied Mathematics and Computing, 2014, 45, 1-2, 449
20.
Hyers-Ulam stability of linear differential equations of first order, Applied Mathematics Letters, 2004, 17, 10, 1135
21.
Hyers–Ulam stability of linear differential equations of second order, Applied Mathematics Letters, 2010, 23, 3, 306
22.
The Hyers–Ulam stability constants of first order linear differential operators, Journal of Mathematical Analysis and Applications, 2004, 296, 2, 403
23.
Hyers–Ulam stability of linear differential equations of first order, Applied Mathematics Letters, 2008, 21, 10, 1024
24.
An approximation property of simple harmonic functions, Journal of Inequalities and Applications, 2013, 2013, 1, 3
25.
Approximation solution of nonlinear Stratonovich Volterra integral equations by applying modification of hat functions, Journal of Computational and Applied Mathematics, 2016, 302, 272
26.
Hyers-Ulam Stability of a Generalized Second-Order Nonlinear Differential Equation, Applied Mathematics, 2012, 03, 12, 1857
27.
New stability results for fractional integral equation, Computers & Mathematics with Applications, 2012, 64, 10, 3468
28.
On the Ulam stability of first order linear fuzzy differential equations under generalized differentiability, Fuzzy Sets and Systems, 2015, 280, 27
29.
Hyers-Ulam Stability of Nonlinear Integral Equation, Fixed Point Theory and Applications, 2010, 2010, 1, 927640
30.
Hyers–Ulam Stability of Euler’s Differential Equation, Results in Mathematics, 2016, 69, 3-4, 317
31.
Laplace Transform Method for the Ulam Stability of Linear Fractional Differential Equations with Constant Coefficients, Mediterranean Journal of Mathematics, 2017, 14, 1
32.
Ulam’s type stability of impulsive ordinary differential equations, Journal of Mathematical Analysis and Applications, 2012, 395, 1, 258
33.
The linear differential equations with complex constant coefficients and Schrödinger equations, Applied Mathematics Letters, 2017, 66, 23
34.
Hyers–Ulam stability of a system of first order linear differential equations with constant coefficients, Journal of Mathematical Analysis and Applications, 2006, 320, 2, 549
35.
On the Stability of Nonautonomous Linear Impulsive Differential Equations, Journal of Function Spaces and Applications, 2013, 2013, 1
36.
Hyers-Ulam Stability of the First-Order Matrix Differential Equations, Journal of Function Spaces, 2015, 2015, 1
37.
Approximation by First-Order Linear Differential Equations with an Initial Condition, Journal of Function Spaces, 2016, 2016, 1
38.
Approximation of analytic functions by bessel functions of fractional order, Ukrainian Mathematical Journal, 2012, 63, 12, 1933
39.
Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces, Advances in Difference Equations, 2016, 2016, 1
40.
Hyers–Ulam stability of Euler’s equation, Applied Mathematics Letters, 2011, 24, 9, 1539
41.
Bessel's Differential Equation and Its Hyers-Ulam Stability, Journal of Inequalities and Applications, 2007, 2007, 1, 021640
42.
Hyers-Ulam-Rassias stability of first order linear partial fuzzy differential equations under generalized differentiability, Advances in Difference Equations, 2015, 2015, 1
43.
Simple Harmonic Oscillator Equation and Its Hyers-Ulam Stability, Journal of Function Spaces and Applications, 2012, 2012, 1
44.
Stability of the second order partial differential equations, Journal of Inequalities and Applications, 2011, 2011, 1, 81
45.
Nonlinear impulsive problems for fractional differential equations and Ulam stability, Computers & Mathematics with Applications, 2012, 64, 10, 3389
46.
On the Stability of Heat Equation, Abstract and Applied Analysis, 2013, 2013, 1
47.
Approximation of Analytic Functions by Chebyshev Functions, Abstract and Applied Analysis, 2011, 2011, 1
48.
Fixed-point results and the Hyers–Ulam stability of linear equations of higher orders, Pacific Journal of Mathematics, 2015, 273, 2, 483
49.
Hyers–Ulam stability of linear differential equations of first order, III, Journal of Mathematical Analysis and Applications, 2005, 311, 1, 139
50.
Hyers–Ulam stability of the linear differential operator with nonconstant coefficients, Applied Mathematics and Computation, 2012, 219, 4, 1562
51.
Hyers–Ulam stability of a first order partial differential equation, Journal of Mathematical Analysis and Applications, 2012, 385, 1, 86
52.
Generalized Hyers-Ulam Stability of the Second-Order Linear Differential Equations, Journal of Applied Mathematics, 2011, 2011, 1
53.
On the Ulam stability of a class of Banach space valued linear differential equations of second order, Advances in Difference Equations, 2014, 2014, 1, 294
54.
On the Hyers-Ulam Stability of First-Order Impulsive Delay Differential Equations, Journal of Function Spaces, 2016, 2016, 1
55.
On the stability of the linear differential equation of higher order with constant coefficients, Applied Mathematics and Computation, 2010, 217, 8, 4141
56.
Connections between Hyers-Ulam stability and uniform exponential stability of 2-periodic linear nonautonomous systems, Advances in Difference Equations, 2017, 2017, 1
57.
A fixed point approach to the Ulam stability of fuzzy differential equations under generalized differentiability, Journal of Intelligent & Fuzzy Systems, 2016, 30, 6, 3253
58.
Hyers–Ulam stability of linear differential equations of first order, II, Applied Mathematics Letters, 2006, 19, 9, 854
59.
Approximation of Analytic Functions by Solutions of Cauchy-Euler Equation, Journal of Function Spaces, 2016, 2016, 1
60.
Ulam’s problem for approximate homomorphisms in connection with Bernoulli’s differential equation, Applied Mathematics and Computation, 2007, 187, 1, 223
61.
ON HYERS-ULAM STABILITY OF NONLINEAR DIFFERENTIAL EQUATIONS, Bulletin of the Korean Mathematical Society, 2015, 52, 2, 685
62.
On the $$\beta$$ β -Ulam–Hyers–Rassias stability of nonautonomous impulsive evolution equations, Journal of Applied Mathematics and Computing, 2015, 48, 1-2, 461
63.
A Uniform Method to Ulam–Hyers Stability for Some Linear Fractional Equations, Mediterranean Journal of Mathematics, 2016, 13, 2, 625
64.
Fuzzy Laplace transform method for the Ulam stability of linear fuzzy differential equations of first order with constant coefficients, Journal of Intelligent & Fuzzy Systems, 2017, 32, 1, 671
65.
On the stability of the heat equation with an initial condition, Journal of Inequalities and Applications, 2013, 2013, 1, 475
66.
A Fixed Point Approach to the Stability of Linear Differential Equations, Bulletin of the Malaysian Mathematical Sciences Society, 2015, 38, 2, 855
67.
The Stability of Some Differential Equations, Mathematical Problems in Engineering, 2011, 2011, 1
68.
On the Hyers-Ulam Stability of Differential Equations of Second Order, Abstract and Applied Analysis, 2014, 2014, 1
69.
Legendre's Differential Equation and Its Hyers-Ulam Stability, Abstract and Applied Analysis, 2007, 2007, 1
70.
Hyers-Ulam stability of nonlinear differential equations with fractional integrable impulses, Mathematical Methods in the Applied Sciences, 2017, 40, 15, 5502
71.
New concepts and results in stability of fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 2012, 17, 6, 2530
References
1.
C. Alsina and R. Ger, On some inequalities and stability results related to theexponential function, J. Inequal. Appl. 2 (1998), 373-380.

2.
T. Miura, S. -E. Takahasi, and H. Choda, On the Hyers-Ulam stability of realcontinuous function valued differentiable map, Tokyo J. Math., 24 (2001), 467-476.

3.
T. Miura, On the Hyers-Ulam stability of a differentiable map, Sci. Math. Japon. 55 (2002), 17-24.

4.
W. Rudin, Real and Complex Analysis (Third Edition), McGraw-Hill.

5.
K. Yosida, Functional Analysis (Sixth Edition), Springer Verlag, 1980.