Stability Properties With Cone Perturbing Liapunov Function method

Theoretical Mathematics & Applications, vol. 6, no., 016, 139-151 ISSN: 179-9687(print), 179-9709(online) Scienpress Ltd, 016 Stability Properties With Cone Perturbing Liapunov Function method A.A. Soliman 1 and W.F. Seyam Abstract φ 0 L P equistability, integrally φ 0 equistability, eventually φ 0 equistability, eventually equistability of a system of differential equations are studied, perturbing Laipunov function. Our methods are cone valued perturbing Liapunov function method and comparison methods. Some results of these concepts are given. Mathematics Subject Classification:34D0 Keyword: φ 0 L P equistable, integrallyφ 0 equistable, eventually φ 0 equistable and eventually equistable,cone valued perturbing Liapunov function. 1 Introduction Stability concepts of differential equations has been interested important from many authors, Lakshmikantham and Leela [4] discussed some different concepts of stability of system of ordinary differential equations namely, eventually stability, integrally stability, totally stability, L P stability, partially stability, strongly stability, practically stability of the zero solution of systems of ordinary differential equations, Liapunov function method [6] that extend to perturbing Liapunov functional method in [3] play essential role to determine stability 1 Departmenf Math.faculty of sciences,benha university,13518,egypt. Abdelkarim.Mohamed@fsc.bu.edu.eg, a_a_soliman@hotmail.com E-mail: W_F_Seyam@yahoo.com Article Info: Received: November, 015. Revised: December 1, 015. Published online : April 5, 016.

140 Stability Properties With Cone Perturbing Liapunov Function method properties. Akpan et, al [1] discussed new concept namely, φ 0 equitable of the zero solution of systems of ordinary differential equations using cone -valued Liapunov function method. Soliman [7] extent perturbing Liapunov function to so-called cone-perturbing Liapunov function method that lies between perturbing Liapunov function and perturbing Liapunov function. In [], and [3] El-Shiekh et.al discussed and improved some concepts stability of [4] and discussed new concepts mix between φ 0 equitable and the previous kinds of stability [3-5],[8-11] In this paper, we discuss and improve the concepf L P equistability of the system of ordinary differential equations with cone perturbing Liapunov function method and comparison technique. Furthermore, we prove that some results of φ 0 L P equitability of the zero solution of the nonlinear system of function differential equations with cone -valued Liapunov function method. Also we discuss some results of φ 0 L P equitability of the zero solution of ordinary differential equations using a cone - perturbing Liapunov function method. Let R n be Euclidean n dimensional real space with any convenient norm, and scalar product (.,. )... Let for some ρ > 0 S ρ = {x R n, x < ρ}. Consider the nonlinear system of ordinary differential equations x = f(t, x), x(t 0 ) = x o, (1.1) where f C[J S ρ, R n ], J = [0, ) and C[J S ρ, R n ] denotes the space of continuous mappings J S ρ into R n. Consider the differential equation u = g(t, u) u(t 0 ) = u 0 (1.) where g C[J R n, R n ], E be an open (t, u) set in R n+1. The following definitions [1] will be needed in the sequel. Definition 1.1 A proper subset K of R n is called a cone if (i)λk K, λ 0. (ii)k + K K, (iii)k = K, (iv)k 0, (v)k ( K) = {0}. where K and K 0 denotes the closure and interior of K respectively, and K denote the boundary of K. Definition 1..The set K = {φ R n, (φ, x) 0, x K} is called the adjoint cone if it satisfies the properties of the definition 1.1. x K if (φ, x) = 0 for some φ K 0, K 0 = K/{0}. Definition 1.3. A function g: D K, D R n is called quasimonotone relative to the cone K if x, y D, y x K then there existsφ 0 K 0 such that (φ 0, y x) = 0 and (φ 0, g(y) g(x)) > 0. Definition 1.4. A function a(. ) is said to belong to the class K if a [R +, R + ], a(0) = 0 and a(r) is strictly monotone increasing in r.

A.A.Solimanand and W.F.Seyam 141 On φ 0 L P equistability Perturbing Liapunov function method was introduced in [] to discuss φ 0 equitability properties for ordinary differential equations. In this section, we will discuss φ 0 L P equistability of the zero solution of the non linear system of ordinary differential equations using cone valued perturbing Liapunov functions method. The following definitions will be needed in the sequel and related with []. Definition.1.The zero solution of the system (1.1) is said to be φ 0 equistable, if for ϵ > 0, t 0 J there exists a positive function δ(t 0, ϵ) > 0 that is continuous in t 0 such that for t t 0. (Φ 0, x 0 ) δ, implies (ϕ 0, x(t, t 0, x 0 )) < ε. where x(t, t 0, x 0 ) is the maximal solution of the system (1.1). In case of uniformly φ o -equistable, the δ is independenf. Definition.. The zero solution of the system (1.1) is said to be ϕ 0 L p equistable and P > 0, if it is ϕ 0 equistable and for each ϵ > 0, t 0 J there exists a positive function δ 0 = δ 0 (t 0, ϵ) > 0 continuous in t 0 such that the inequality (ϕ 0, x 0 ) δ 0, implies ( ϕ 0, x(s, t 0, x 0 ) P ds) < ϵ. In case of uniformly ϕ 0 L P equistable, the δ 0 is independenf t 0. Let for some ρ > 0ϕ 0 L p equistability of (1.1), integrally ϕ 0 equistability S ρ = {x R n, (ϕ 0, x) < ρ, ϕ 0 K 0 }. We define for V C[J S ρ, K], the function D + V(t, x)by D + V(t, x) = lim sup 1 (V(t + h, x + hf(t, x)) V(t, x)). h 0 h The following result will discuss the concepf ϕ 0 L p equistability of (1.1) using comparison principle method. Theorem.1. Suppose that there exist two functions g 1 C[J R, R] and g C[J R, R] with g 1 (t, 0) = g (t, 0) = 0 are monotone non decreasing functions, and there exist two Liapunov functions where V 1 (t, 0) = V η (t, 0) = 0, and S ρ = {x R n ; (ϕ 0, x) < η, ϕ 0 K 0 } and S C ρ denotes the complemenf S ρ, satisfying the following conditions: (H 1 )V 1 (t, x) is locally Lipschitzian in x and D + (ϕ 0, V 1 (t, x)) g 1 (t, V 1 (t, x)) for (t, x) J S ρ. (H )V η (t, x) is locally Lipschitzian in x and b(ϕ 0, x) (ϕ 0, V η (t, x)) a(ϕ 0, x) (.1)

14 Stability Properties With Cone Perturbing Liapunov Function method t (ϕ 0, x(s, t 0, x 0 ) P ds) t (ϕ0, Vη(t, x(t0, x0)) a1(ϕ0, x(s, t0, x0) P ds), (.) where a, a 1, b, b 1 K for (t, x t ) J S ρ S C ρ. (H 3 )D + (ϕ 0, V 1 (t, x)) + D + (ϕ 0, V η (t, x)) g (t, V 1 (t, x) + V η (t, x)) for (t, x) J S ρ S ρ C. (H 4 )If the zero solution of the equation u = g 1 (t, u), u(t 0 ) = u 0. (.3) is ϕ 0 equistable, and the zero solution of the equation ω = g (t, ω), ω(t 0 ) = ω 0 (.4) is uniformly ϕ 0 equistable. Then the zero solution of the system (1.1) is ϕ 0 L P equistable. Proof. Since the zero solution of (.4) is uniformly φ 0 equistable, given 0 < ε < ρ and b 1 (ε) > 0 there exists δ 0 = δ 0 (ε) > 0 such that t t 0 (ϕ 0, ω 0 ) δ 0, implies (ϕ 0, r (t, t 0, ω 0 )) < b 1 (ϵ). (.5) where r (t, t 0, ω 0 ) is the maximal solution of the system (.4). From the condition (H ), there exists δ = δ (ε) > 0 such that a(δ ) δ 0 (.6) From our assumption that the zero solution of the system (.3) is φ 0 equistable, given δ 0 and t 0 R +, there exists δ = δ (t 0, ε) > 0 such that (φ 0, u 0 ) δ, implies ( φ 0, r 1 (t, t 0, u 0 )) < δ 0, for t t (.7) 0 where r 1 (t, t 0, u 0 ) is the maximal solution of the system (.3). From the conditions (H 1 ), (.1), (H 3 ), (H 4 ) and applying Theorem () of [6], it follows the zero solution of the system (1.1) is φ 0 equistable. To show that there exists δ 0 = δ 0 (t 0, ε) > 0, such that (φ 0, x 0 ) δ 0, implies (φ 0, x(s, t 0, x 0 ) P ds) < ε. Suppose this is false, then there exists t 1 > t > t 0. such that for (φ 0, ψ) δ 0. t 1 (φ 0, x(s, t 0, x 0 ) P ds) = δ, (φ 0, x(s, t 0, x 0 ) P ds) = ε (.8) δ (φ 0, x(s, t 0, x 0 ) P ds) ε t for t [t 1, t ]. Let δ = η and setting m(t, x) = V 1 (t, x) + V η (t, x)for t [t 1, t ]. From the condition (H 3 ), we obtain D + (φ 0, m(t, x)) g (t, m(t, x)). We can choose m(t 1, x(t 1 )) = V 1 (t 1, x(t 1 )) + V η (t 1, x(t 1 )) = ω 0. t

A.A.Solimanand and W.F.Seyam 143 Applying Theorem (8.1.1) of [5], we get (φ 0, m(t, x)) (φ 0, r (t, t 1, m(t 1, x(t 1 ))) for t [t 1, t ]. (.9) Choosing u 0 = V 1 (t 0, x 0 ). From the condition (H 1 ) and applying the comparison Theorem, we get (φ 0, V 1 (t, x)) (φ 0, r 1 (t, t 0, u 0 ) ) Let t = t 1 and from (.7),we get (φ 0, V 1 (t 1, x(t 1 )) (φ 0, r 1 (t 1, t 0, u 0 )) < δ 0. From the condition(h ), (.6) and(.8), we obtain t 1 (φ 0, V η (t 1, x(t 1 )) a 1 ( φ 0, x(s, t 0, x 0 ) P ds) a t 1 (δ ) δ 0.. o So we get (φ 0, ω 0 ) = (φ 0, V 1 (t 1, x(t 1 )) + V η (t 1, x t1 x(t 1 )) δ 0. Then from (.5) and (.9), we get (φ 0, m(t, x t )) (φ 0, r (t, t 1, ω(t 1 )) < b 1 (ε). (.10) From the condition (H ), (.8) and (.10) at t = t t b 1 (ε) = b 1 (φ 0, x(s, t 0, x 0 ) P ds) (φ 0, V η (t, x(t )) < (φ 0, m(t, x(t )) b 1 (ε). This is a contradiction, therefore it must be (φ 0, x(s, t 0, x 0 ) P ds) < ε provided that (φ 0, x 0 ) δ 0. Then the zero solution of the system (1.1) is φ 0 L P equistable. 3 On Integrally φ 0 -equistable In this section, we discuss the concepf Integrally φ o - equistability of the zero solution of non linear system of ordinary diffrential equations using cone valued perturbing liapunow functions method and comparison principle method. Consider the non linear system of differential equation(1.1) and the perturbed system x = f(t, x) + R(t, x), x(t 0 ) = x 0 (3.1) where f, R C [J S ρ, R n ], J = [0, ] and C[J S ρ, R n ] denotes the space of continuous mapping J S ρ into R n. Consider the scalar differetail equation (.3), (.4) and the perturbing equations u = g 1 (t. u) + φ 1 (t), u(t 0 ) = u 0 (3.) ω = g (t. ω) + φ (t), ω(t 0 ) = ω 0 (3.3) where g 1, g C [J R, R], φ 1, φ C [J, R] respectively. The following definitions [4] will be needed in the sequal. Definition 3.1. The zero solution of the system (1.1) is said to be integrally

144 Stability Properties With Cone Perturbing Liapunov Function method ϕ 0 -equistable if for every α 0 and t 0 J, there exists a positive function β = β(t 0, α) which in continuous in t 0, for each α and β K, such that for ϕ 0 K 0 every solution x(t, t 0, x 0 ) of pertubing differential equation (3.1), the inequality (ϕ 0, x(t, t 0, x 0 )) < β, t t 0 holds, provided that (ϕ 0, x 0 ) α, and every T> 0, (φ 0, t 0 sup x <β R(s, x) ds) α. Definition 3..The zero solution of (3.) is said to be integrally ϕ 0 -equistable if, for every α 1 0 and t 0 J, there exists a positive function β 1 = β 1 (t 0, α) which in continuous in t 0, for each α 1 and β 1 K, such that for ϕ 0 K 0 every solution u(t, t 0, u 0 ) of perturbing differential equation (.3), the inequality (ϕ 0, u(t, t 0, u 0 )) < β 1, t t 0 holds, provided that (ϕ 0, u 0 ) α 1, and for every T> 0, (φ 0, t 0 φ 1 (s)ds) α. In the case of uniformly integrally ϕ 0 -equistable, β 1 is independenf t 0. We define for a cone valued Liapunov function V(t, x) C[J S ρ, K] is Lipschitzian in x. The function D + V(t, x) 3.1 = lim sup 1 (V(t + h, x + h(f(t, x) + R(t, x))) V(t, x)). h 0 h The following result is related with thaf [5]. Theorem 3.1. Let the function g (t, ω) be nonincreasig in ω for each t R +, and the assumptions (H 1 ), (H ) (.1) and (H 3 ) be satisfied. If the zero solution of (.3) is integrally ϕ 0 -equistable, and the zero solution of (.4) is uniformly integrally ϕ 0 -equistable. Then the zero solution of (1.1) is integrally ϕ 0 -equistable. Proof. Since the zero solution of (.4) is integrally φ 0 equistable, given α 1 0 and t 0 0 there exists β 0 = β 0 (t 0, α 1 ) such that t t 0 such that for any φ 0 K 0 and for any solution u(t, t 0, u 0 ) of the perturbed system (3.) satisfies the inequality (φ 0, u(t, t 0, u 0 )) < β 0 (3.4) holds provided that (φ 0, u 0 ) α 1, and for every T> 0, (φ 0, φ 1 (s)ds) α 1. t 0 From our assumption that the zero solution of the system (.3) is uniformly integrally φ o - equistable given α 0, there exists β 1 = β 1 (α ) such that every solution ω(t, t 0, ω 0 ) of the perturbed equation (3.3) satisfies the inequality (φ 0, ω(t, t 0, ω 0 )) < β 1 (3.5) holds provided that (φ 0, ω 0 ) α and for every T> 0,

A.A.Solimanand and W.F.Seyam 145 (φ 0, φ (s)ds) α. t 0 Suppose that there exists α > 0 such that α = a(α) + β 0 (3.6) sinceb(u) as u then we can find β(t 0, α) such that b(β) > β 1 (α ) (3.7) To prove that the zero solution of (1.1) is integrally φ 0 equistable, it must be for every α 0 andt 0 J, there exists a positive function β = β(t 0, α) which in continuous in t 0, for each α and β K, such that for ϕ 0 K 0 every solution x(t, t 0, x 0 ) of pertubing differential equation (3.1), the inequality (ϕ 0, x(t, t 0, x 0 )) < β, t t 0 holds, provided that (ϕ 0, x 0 ) α and every T> 0, (φ 0, t 0 sup x <β R(s, x) ds) α. Suppose this is false, then there exists t > t 1 > t 0 such that (φ 0, x(t 1, t 0, x 0 )) = α, (φ 0, x(t, t 0, x 0 ) ) = β (3.8) α (φ 0, x(t, t 0, x 0 )) β for t [t 1, t ] Let δ = α, and setting m(t, x) = V 1 (t, x) + V η (t, x)for t [t 1, t ]. Since V 1 (t, x)andv η (t, x) are Lipschitizian in x for constants M and K respectively. Then D + (φ 0, V 1 (t, x)) 3.1 + D + (φ 0, V η (t, x)) 3.1 D + (φ 0, V 1 (t, x)) 1.1 + D + (φ 0, V η (t, x)) 1.1 + N(φ 0, R(t, x)). where N = M + K. From the condition (H 3 ), we obtain D + (φ 0, m(t, x)) g (t, m(t, x)) + N(φ 0, R(t, x)). We can choose m(t 1, x(t 1 )) = V 1 (t 1, x(t 1 )) + V η (t 1, x(t 1 )) = ω 0. Applying Theorem (8.1.1) of [5], we get (φ 0, m(t, x)) (φ 0, r (t, t 1, m(t 1, x(t 1 ))) for t [t 1, t ] (3.9) where r (t, t 1, m(t 1, x(t 1 )) is the maximal solution of the perturbed system (3.3), where φ (t) = NR(t, x). To prove that (φ 0, r (t, t 1, ω 0 )) < β 1 (α ), it must be shown that (φ 0, ω 0 ) α, (φ 0, φ (s)ds) α t 0 Choosing u 0 = V 1 (t 0, x 0 ), since V 1 (t, x) is a Lipschitizian in x for a constant M > 0,then φ 0 V 1 (t 0, x 0 ) M φ 0 x 0 (φ 0, u 0 ) = (φ 0, V 1 (t 0, x 0 )) M(φ 0, x 0 ) Mα = α 1 (3.10) Also we get D + (φ 0, V 1 (t, x)) 3.1 D + (φ 0, V 1 (t, x)) 1.1 + M(φ 0, R(t, x)). From the condition (H 1 ) we get (φ 0, V 1 (t, x)) (φ 0, r 1 (t, t 0, u 0 ) ) for t [t 1, t ]. where r 1 (t, t 0, u 0 ) is the maximal solution of (3.) and define φ 1 (t) = MR(t, x).

146 Stability Properties With Cone Perturbing Liapunov Function method Integrating it,we get ds = M R(s, x) ds M sup R(s, x) ds φ 1 (s) t 0 t 0 t 0 x <β which leads to (φ 0, φ 1 (s) ds) M (φ 0, sup x <β R(s, x) ds) M = α 1 (3.11) t 0 t 0 From (3.4),(3.10) and (3.11) at t = t 1,we get (φ 0, V 1 (t 1, x(t 1 )) (φ 0, r 1 (t 1, t 0, u 0 )) < β 0. From the condition (.1) and (3.7), we obtain (φ 0, V η (t 1, x(t 1 )) a 1 ( φ 0, x(t 1 )) a(α). From (3.6), we get (φ 0, ω 0 ) = (φ 0, V 1 (t 1, x(t 1 )) + V η (t 1, x t1 x(t 1 )) α (3.1) Since φ (t) = NR(t, x), then integrating both sides N sup R(t, x) ds φ (s)ds = N R(t, s) ds t 0 t 0 t 0 x <β which leads to (φ 0, φ (s) ds) N (φ 0, sup x <β R(s, x) ds) N = α (3.13) t 0 t 0 Then from (3.5), (3.1) and (3.13), we get (φ 0, m(t, x)) (φ 0, r (t, t 1, ω(t 1 )) < β 1 (α ). (3.14) From the condition (.1), (3.7) and (3.14) at t = t,we have b(β) = b(φ 0, x(t )) (φ 0, V η (t, x(t )) < (φ 0, m(t, x(t )) β 1 (α ) < b(β). That is a contradiction, therefore it must be (ϕ 0, x(t, t 0, x 0 )) < β, t t 0 Then the zero solution of the system (1.1) is integrally φ 0 equistable. 4 Eventually equistable In this section, we discuss the notion of eventually-equistable of the zero solution of non linear system (1.1) using perturbing liapunow functions method and comparison principle method. The following definition will be needed in the sequel and related with that [3]. Definition 4.1. The zero solution of the system (1.1) is said to eventually uniformly equistable if for ϵ > 0, there exists a positive function δ(ϵ) > 0 and τ = τ(ε) such that the inequality x 0 δ, implies x(t, t 0, x 0 ) < ε, t t 0 τ(ϵ) where x(t, t 0, x 0 ) is any solution of the system (1.1).

A.A.Solimanand and W.F.Seyam 147 Theorem 4.1. Suppose that there exist two functions g 1, g C[J R +, R] with g 1 (t, 0) = g (t, 0) = 0, and two Liapunov functions V 1 (t, x) C[J S ρ, R n ] and V η (t, x) C[J S ρ S η C, R n ] where V 1 (t, 0) = V η (t, 0) = 0, and S η = {x R n ; x < η} and S η C denotes the complemenf S η, satisfying the following conditions (h 1 )V 1 (t, x) is locally Lipschitzian in x and D + V 1 (t, x) g 1 (t, V 1 (t, x)) for (t, x t ) J S ρ. (h )V η (t, x) is locally Lipschitzian in x, and b( x ) V η (t, x) a( x ) for 0 < r < x < ρ and t θ(r), where θ(r) is a continuous monotone decreasing in r, for 0 < r < ρ where a, b K. For (t, x) J S ρ S η C. (h 3 )D + V 1 (t, x) + D + V η (t, x) g (t, V 1 (t, x) + V η (t, x)) for (t, x)) J S ρ S η C. (h 4 ) If the zero solution of (.3) is uniformly equistable, and the zero solution of (.4) is eventually uniformly equistable, then the zero solution of the system (1.1) is uniformly eventually equistable. Proof. Since the zero solution of (.4) is eventually uniformly equistable, given b(ϵ) > 0 there exists τ 1 = τ 1 (ϵ) > 0 and δ 0 = δ 0 (ϵ) > 0 such that ω 0 δ 0, implies ω(t, t 0, ω 0 ) < b(ϵ), t t 0 τ 1 (ε) (4.1) whereω(t, t 0, ω 0 ) is any solution of the system (.4). Since a(u) as u for a K, it is possible to choose δ 1 = δ 1 (ϵ) > 0 such that a(δ 1 ) δ 0 (4.) From our assumption that the zero solution of the system (.3) is uniformly equistable. Given δ 0, there exists δ = δ (ϵ) > 0 such that u 0 δ, implies u(t, t 0, u 0 ) < δ 0 (4.3) whereu(t, t 0, u 0 ) is any solution of the system (.3). Choosing u 0 = V 1 (t 0, x 0 ), since V 1 (t, x) is a Lipschitizian function for a contant M. Then there exists δ = δ (ε) > 0 such that x 0 δ, implies V 1 (t 0, x 0 ) M x 0 Mδ δ max [τ 1 (ε), τ (ε)]. To prove theorem, it must be shown that set δ = min(δ 1, δ ) and suppose x 0 δ, define τ (ε) = θ(δ(ε)) and let τ = τ(ε) x 0 δ implies x(t, t 0, x 0 ) < ε, t t 0 τ(ε) Suppose that is false, then there exists t > t 1 > t 0 such that x(t 1 ) = δ 1, x(t ) = ε (4.4) δ 1 x(t) ε for t [t 1, t ]. Let δ 1 = η and setting m(t, x) = V 1 (t, x) + V η (t, x) for t [t 1, t ].

148 Stability Properties With Cone Perturbing Liapunov Function method From the condition(h 3 ), we obtain D + m(t, x) G (t, m(t, x)). we can choose m(t 1, x(t 1 )) = V 1 (t 1, x(t 1 )) + V η (t 1, x(t 1 )) = ω 0. Applying Theorem (8.1.1) of [5], we get m(t, x) r (t, t 1, m(t 1, x(t 1 ))) (4.5) wherer (t, t 1, m(t 1, x(t 1 ))) is the maximal solution of (.4). Choosing u 0 = V 1 (t 0, x 0 ). From the condition (h 1 ) and applying the comparison Theorem, we get V 1 (t, x) r 1 (t, t 0, u 0 )for t [t 0, t 1 ]. (4.6) Let t = t 1 and from (4.3), we get V 1 (t 1, x(t 1 )) r 1 (t 1, t 0, u 0 ) < δ 0. From the condition (h ), (4.) and (4.4) V η (t 1, x(t 1 )) a( x(t 1 ) ) a(δ 1 ) δ 0. So we get ω 0 = V 1 (t 1, x(t 1 )) + V η (t 1, x(t 1 )) δ 0. Then from (4.1) and (4.5), we get m(t, x) r (t, t 1, ω(t 1 )) < b(ϵ) (4.7) From (h ),(4.4) and (4.7) at t = t b(ϵ) = b( x(t ) V η (t, x(t )) < m(t, x(t )) b(ϵ). This is a contradiction, therefore it must be x(t, t 0, x 0 ) < ε, t t 0 τ(ε) Provided that x 0 δ.then the zero solution of the system (1.1) is uniformly eventually equistable. 5 Eventually φ 0 equistable In this section, we discuss the notion of eventuallyφ 0 -equistable of the zero solution of non linear system (1.1) using cone valued perturbing liapunow functionsmethod and comparison principle method. The following definition is somewhat new and related with that [3]. Definition 5.1. The zero solution of the system (1.1) is said to eventually uniformly ϕ 0 -equistable if for ϵ > 0,there exists a positive function δ(ϵ) > 0 and τ = τ(ε) such that the inequality (ϕ 0, x 0 ) δ, implies (ϕ 0, x(t, t 0, x 0 )) < ε, t t 0 τ(ϵ) where x(t, t 0, x 0 ) is the maximal solution of the system (1.1). Theorem 5.1.Let the assumptions (H 1 ), (H ) (.1) and (H 3 ) be satisfied for 0 < r < (φ 0, x) < ρ and t θ(r), where θ(r) is a continuous monotone

A.A.Solimanand and W.F.Seyam 149 decreasing in r for 0 < r < ρ wherea, b K. If the zero solution of (.3) is uniformlyϕ 0 -equistable and the zero solution of (.4) is uniformly eventually ϕ 0 -equistable. Then the zero solution of (1.1) is uniformly eventually ϕ 0 -equistable. Proof. Since the zero solution of (.4) is eventually uniformly φ 0 equistable, given b(ϵ) > 0 there exists τ 1 = τ 1 (ϵ) > 0 and δ 0 = δ 0 (ϵ) > 0 such that (φ 0, ω 0 ) δ 0, implies (φ 0, r (t, t 0, ω 0 )) < b(ϵ), t t 0 τ 1 (ε) (5.1) where r (t, t 0, ω 0 ) is the maximal solution of the system (.4). Since a(u) as u for a K, it is possible to choose δ 1 = δ 1 (ϵ) > 0 such that a(δ 1 ) δ 0 (5.) From our assumption that the zero solution of the system (.3) is uniformly φ 0 equistable.given δ 0, there exists δ = δ (ϵ) > 0 such that (φ 0, u 0 ) δ, implies (φ 0, r 1 (t, t 0, u 0 )) < δ 0 (5.3) where r 1 (t, t 0, u 0 ) is the maximal solution of the system (.3). Choosing u 0 = V 1 (t 0, x 0 ), since V 1 (t, x) is a Lipschitizian function for a constant M. Then there exists δ = δ (ε) > 0 such that (φ 0, x 0 ) δ implies (φ 0, V 1 (t 0, x 0 )) M (φ 0, x 0 ) Mδ δ Set δ = min(δ 1, δ ) and suppose(ϕ 0, x 0 ) δ, then define τ (ϵ) = θ(δ(ϵ)) and let τ(ϵ) = max [τ 1 (ϵ), τ (ϵ)]. To prove the zeo solution of (1.1) is uniformly eventually ϕ 0 -equistable, it must be shown that (φ 0, x 0 ) δ, implies(φ 0, x(t, t 0, x 0 )) < ε, t t 0 τ(ε) Suppose that is false, then there exists t > t 1 > t 0 such that (φ 0, x(t 1 )) = δ 1, (φ 0, x(t ) = ε (5.4) δ 1 (φ 0, x(t, t 0, x 0 )) ε for t [t 1, t ]. Let δ 1 = η, and setting m(t, x) = V 1 (t, x) + V η (t, x)for t [t 1, t ]. From the condition (H 3 ), we obtain D + (φ 0, m(t, x)) g (t, m(t, x)). Choose m(t 1, x(t 1 )) = V 1 (t 1, x(t 1 )) + V η (t 1, x(t 1 )) = ω 0. Applying Theorem (8.1.1) of [5], we get A (φ 0, m(t, x)) (φ 0, r (t, t 1, m(t 1, x(t 1 ))) (5.5) Choosingu 0 = V 1 (t 0, x 0 ), from the condition (H 1 ) and applying the comparison Theorem 1.4.1 of [3], we get (φ 0, V 1 (t, x)) (φ 0, r 1 (t, t 0, u 0 ) ) for t [t 0, t 1 ]. (5.6) Let t = t 1 and from (5.3), we get (φ 0, V 1 (t 1, x(t 1 ))) (φ 0, r 1 (t 1, t 0, u 0 )) < δ 0.

150 Stability Properties With Cone Perturbing Liapunov Function method From the condition (H ), (5.) and (5.4) (φ 0, V η (t 1, x(t 1 ))) a( φ 0, x(t 1 )) a(δ 1 ) δ 0 So we get (φ 0, ω 0 ) = (φ 0, V 1 (t 1, x(t 1 ))) + (φ 0, V η (t 1, x(t 1 ))) δ 0. Then from (5.1) and (5.5), we get (φ 0, m(t, x)) (φ 0, r (t, t 1, ω(t 1 ))) < b(ϵ). (5.7) From (H ),(5.4) and (5.7) at t = t b(ϵ) = b(φ 0, x(t )) (φ 0, V η (t, x(t )) < (φ 0, m(t, x(t )) b(ϵ). This is a contradiction, therefore it must be (φ 0, x(t, t 0, x 0 )) < ε, t t 0 τ(ε) Provided that (φ 0, x 0 ) δ. Then the zero solution of the system (1.1) is uniformly eventually φ 0 equistable. References [1] E.P.Akpan, O.Akinyele,"On ϕ 0 stability of comparison differential systems" J. Math. Anal. Appl. 164(199)307-34. [] M.M.A.El Sheikh, A. A. Soliman, M. H. AbdAlla, "On stability of nonlinearsystems of ordinary differential equations", Applied Mathematics and Computation,113(000)175-198. [3] V.Lakshmikantham, S. Leela, "On Differential and Integral Inequalities", Vol.I, Academic Press, New York,(1969). [4] A. A. Soliman, M. H. AbdAlla, "On integral stability criteria of nonlinear differential systems", Applied Mathematics and Computation, 48(00), 58-67. [5] A.A.Soliman,"On φ 0 -boundedness and stability properties with perturbing Liapunov functional", IMA Journal of Applied Mathematics (00),67,551-557. [6] M.M.A.El Sheikh, A.A.Soliman and M.H.AbdAlla "Onstability of nonlinear differential systems via cone valued Liapunov function method" Applied Math.and Computations(001), 65-81.

A.A.Solimanand and W.F.Seyam 151 [7] A.A.Soliman "On practical stability of perturbed differential systems" Appl.Math. and Comput. (005),vol.163,N.3,1055-1060. [8] A.A.Soliman "Eventual -Stability of Nonlinear Systems of Differential 0 Equations", Pan American Math.J.(1999) Vol.3.N.5 11-10. [9] A.A.Soliman "On total - Stability of Nonlinear Systems of Differential 0 Equations" Appl.Math. and Comput. (00),Vol.130,9-38. [10] A.A.Soliman On eventual Stability of Impulsive Systems of Differential Equations International Journal of Math. and Math. Sciences(001),Vol.7,No.8,485-494. [11] A.A.Soliman "On total Stability for Perturbed Systems of Differential Equations" Applied Math. Letters (003) vol.16, 1157-116.