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Abstract:
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This thesis has been written under the supervision of my mentor Prof. dr. Julka
Knezevi c-Miljanovi c at the University of Belgrade in the academic year 2014-2015.
The aim of this study is to investigate Hyers-Ulam stability of some types of
differential equations, and to study a generalized Hyers-Ulam stability and as
well as a special case of the Hyers-Ulam stability problem, which is called the
superstability. Therefore, when there is a differential equation, we answer the
three main questions:
1- Does this equation have Hyers -Ulam stability?
2- What are the conditions under which the differential equation has stability ?
3- What is a Hyers-Ulam constant of the differential equation?
The thesis is divided into three chapters. Chapter 1 is divided into 3 sections. In
this chapter, we introduce some sufficient conditions under which each solution
of the linear differential equation u′′(t) +
(
1 + (t)
)
u(t) = 0 is bounded. Apart
from this we prove the Hyers-Ulam stability of it and the nonlinear differential
equations of the form u′′(t) + F(t; u(t)) = 0, by using the Gronwall lemma and
we prove the Hyers-Ulam stability of the second-order linear differential equations
with boundary conditions. In addition to that we establish the superstability
of linear differential equations of second-order and higher order with continuous
coefficients and with constant coefficients, respectively. Chapter 2 is divided into
2 sections. In this chapter, by using the Laplace transform method, we prove
that the linear differential equation of the nth-order y(n)(t) +
nΣ1
k=0
ky(k)(t) = f(t)
has the generalized Hyers-Ulam stability. And we prove also the Hyers-Ulam-
Rassias stability of the second-order linear differential equations with initial and
boundary conditions, as well as linear differential equations of higher order in the
form of y(n)(x) + (x)y(x) = 0, with initial conditions. Furthermore, we establish
the generalized superstability of differential equations of nth-order with initial
conditions and investigate the generalized superstability of differential equations
of second-order in the form of y′′(x)+p(x)y′(x)+q(x)y(x) = 0. Chapter 3 is divided
into 2 sections. In this chapter, by applying the xed point alternative method,
we give a necessary and sufficient condition in order that the rst order linear
Alqi ary Abstract ii
system of differential equations z_(t) + A(t)z(t) + B(t) = 0 has the Hyers-Ulam-
Rassias stability and nd Hyers-Ulam stability constant under those conditions.
In addition to that, we apply this result to a second-order differential equation
y (t) + f(t)y_(t) + g(t)y(t) + h(t) = 0. Also, we apply it to differential equations
with constant coefficient in the same sense of proofs. And we give a sufficient
condition in order that the rst order nonlinear system of differential equations
has Hyers-Ulam stability and Hyers-Ulam-Rassias stability. In addition, we present
the relation between practical stability and Hyers-Ulam stability and also Hyers-
Ulam-Rassias stability. |