Abstract:
The concept of stability in differential equations is of immense importance particularly for determination of properties of solutions of nonlinear equations which
cannot be readily solved to obtain closed analytic solutions. Stability in the sense
of Hyers-Ulam(H-U) and Hyers-Ulam-Rassias(H-U-R) has been considered for linear Ordinary Differential Equations(ODE) due to their solutions which are easily
determined. However, cases of nonlinear ODE of second and third orders have
received little attention. This research was therefore designed to establish the stability of nonlinear second and third order in the sense of H-U and H-U-R.
Variants of the perturbed second order ODE of the form
u00(t) + f(t; u(t); u0(t)) = P(t; u(t); u0(t)) and third order ODE of the form
u000(t) + f(t; u(t); u0(t); u00(t)) = P(t; u(t); u0(t)) were reduced to their equivalent integral equations, where t is an independent real variable; f; P; and u are continuous
functions of their argument. Extension of Gronwall-Bellman-Bihari(G-B-B) type
inequalities having the same number of integrals as the equivalent integral equations were developed. These integral inequalities were used to prove the existence
of H-U and H-U-R stability. They were also used to estimate the H-U and H-U-R
constants for each of the variants of the equations.
The newly developed G-B-B type inequalities of nonlinear integrals obtained were:
u(t) ≤ u0 + L Rtt0 f(s)u(s)ds + Rtt0 f(s)r(s)(Rts0 ρ(τ)$(u(τ))dτ)ds
and u(t) ≤ u0 + T Rtt0 r(s)β(s)ds + L Rtt0 h(s)$u(s))ds
where f; r; ρ; $; β and h are continuous functions and T; L and u0 are positive
constants. The nonlinear second order ODE were found to possess H-U stability
and H-U constants
K21 = L(1 + 1 2λ2 + jq(ξ; u(ξ); u0(ξ); u00(ξ))j + jp(u(ρ); u0(ρ))jΩ−1(Ω(1) + neM)
vand K22 = (αL(ξ) + λ2
2α(ξ))Ω−1(Ω(1) + λ2
α(ξ)n$(F−1(F(1) + λ2
α(ξ)m))F−1(F(1) + m λ2
α(ξ));
respectively, where q; p; Ω; Ω−1; F; F−1; $; h are functions of their argument and
λ; ξ; ρ; M; n and m are constants. The newly developed G-B-B type inequality for
two nonlinear integrals u(t) ≤ ρ(t) + T Rtt0 r(s)β(s)ds + L Rtt0 h(s)$(u(s))ds: where
ρ(t) a monotonic, nonnegative, continuous function, with this nonlinear second order ODE were found to possess H-U-R stability and the H-U-R constants
C’
21 = Ω(Ω(1) + m(η + η2)$(F−1(F(1) + l))F−1(F(1) + l) and
C’
22 = Γ−1(Γ(1) + mηnγ(F−1(F(1) + l)))F−1(F(1) + l) where
Γ; Γ−1 and γ are functions of their argument and η; l are constants. The new G-B-B
type inequality for three nonlinear integrals was:
u(t) ≤ D + T Rtt0 r(s)β(u(s))ds + B Rtt0 h(s)$(u(s))ds + L Rtt0 g(s)γ(u(s))ds; where
B and D are positive constants. The nonlinear third order ODE were found
to possess H-U stability with H-U constants given by K31 = L+δL F−1(F(1) +
d(λ)λδ φ(SX))SX; where S = Ω−1(Ω(1) + nλδ γ(X))X and X = F−1(F(1) + mλδ2)
and K32 = L+jr(u(κ))jL
2φju0(ξ)j Γ−1(Γ(1) + C4qr(Ω−1(Ω(1) + C3mg(H))Ω−1(Ω(1) + C2n)))
Ω−1(Ω(1) + C3mg)H; where H = F−1(F(1) + C2n); C2 = 2ju0(η)jλ
2φju0(ξ)j; C3 = λ2
2φju0(ξ)j
and C4 = λn+1
2φju0(ξ)j; δ; constants and g; r; φ are functions of their argument. The
newly developed G-B-B type inequality with the three nonlinear integrals :
u(t) ≤ ρ(t)+T Rtt0 r(s)β(u(s))ds+B Rtt0 h(s)$(u(s))ds+L Rtt0 g(s)γ(u(s))ds; where
ρ(t) a monotonic, nonnegative, nondecreasing and continuous function. The nonlinear third order ODE were also found to possess H-U-R stability with H-U-R
constants
C’
31 =
1δ
Υ−1(Υ(1) + ηδnρ1γ(Ω−1(Ω(1) + 1δρ3α(Y ))y)Ω−1(Ω(1) + 1δρ3α(Y ))Y; where
Y = F−1(F(1)+ηδρ1) and C’32 = 1δΩ−1(Ω(1)+d1λn
η
!(F−1(F(1)+d2h(λ)
η
)))F−1(F(1)+
d2h(λ)
η
) with ρ1; ρ2 and ρ3 are constants.
A generalisation of the existing results on Hyers-Ulam and Hyers-Ulam-Rassias
stability to nonlinear ordinary differential equations was achieved. This can also
be used to achieve the stability of the other differential equations.
