The present paper is concerned with the notions of Lipschitz and asymptotic stability for perturbed nonlinear differential system knowing the corresponding stability of nonlinear differential system. We investigate Lipschitz and asymtotic stability for perturbed nonlinear differential systems. The main tool used is integral inequalities of the Bihari-type, in special some consequences of an extension of Bihari's result to Pinto and Pachpatte, and all that sort of things. The notion of uniformly Lipschitz stability (ULS) was introduced by Dannan and Elaydi [8] . For linear systems, the notions of uniformly Lipschitz stability and that of uniformly stability are equivalent. However, for nonlinear systems, the two notions are quite distinct. In fact, uniformly Lipschitz stability lies somewhere between uniformly stability on one side and the notions of asmptotic stability in variation of Brauer [4] and uniformly stability in variation of Brauer and Strauss [3] on the other side. Gonzalez and Pinto [9] proved theorems which relate the asymptotic behavior and boundedness of the solutions of nonlinear differential systems. In this paper, we investigate Lipschitz and asymptotic stability for solutions of the nonlinear differential systems. To do this we need some integral inequalities. The method incorporating integral inequalities takes an important place among the methods developed for the qualitative analysis of solutions to linear and nonlinear system of differential equations. In the presence the method of integral inequalities is as efficient as the direct Lyapunov’s method. We consider the nonlinear nonautonomous differential system where and is the Euclidean n -space. We assume that the Jacobian matrix f _x = ∂ f / ∂ x exists and is continuous on and f ( t , 0) = 0. Also, consider the perturbed differential system of (2.1) where , g ( t , 0) = 0. For , let For an n × n matrix A , define the norm | A | of A by | A | = sup _|x|≤1 | Ax |. Let x ( t , t ₀ , x ₀ ) denote the unique solution of (2.1) with x ( t ₀ , t ₀ , x ₀ ) = x ₀ , existing on [ t ₀ ,∞). Then we can consider the associated variational systems around the zero solution of (2.1) and around x ( t ), respectively, and The fundamental matrix Φ( t , t ₀ , x ₀ ) of (2.4) is given by and Φ( t , t ₀ , 0) is the fundamental matrix of (2.3). Before giving further details, we give some of the main definitions that we need in the sequel [8] . Definition 2.1. The system (2.1) (the zero solution x = 0 of (2.1)) is called (S) stable if for any 𝜖 > 0 and t ₀ ≥ 0, there exists 𝛿 = 𝛿( t ₀ , 𝜖) > 0 such that if | x 0| < 𝛿, then | x ( t )| < 𝜖 for all t ≥ t ₀ ≥ 0, (US) uniformly stable if the 𝛿 in (S) is independent of the time t ₀ , (ULS) uniformly Lipschitz stable if there exist M > 0 and 𝛿 > 0 such that | x ( t )| ≤ M | x ₀ | whenever | x ₀ | ≤ 𝛿 and t ≥ t ₀ ≥ 0 (ULSV) uniformly Lipschitz stable in variation if there exist M > 0 and 𝛿 > 0 such that |Φ( t , t ₀ , x ₀ )| ≤ M for | x ₀ | ≤ 𝛿 and t ≥ t ₀ ≥ 0, (EAS) exponentially asymptotically stable if there exist constants K > 0 , c > 0, and 𝛿 > 0 such that |x(t)| ≤ K |x₀|e^-c(t-t₀), 0 ≤ t₀ ≤ t provided that | x ₀ | < 𝛿, (EASV) exponentially asymptotically stable in variation if there exist constants K > 0 and c > 0 such that |Φ(t,t₀,x₀)| ≤ K e^-c(t-t₀), 0 ≤ t₀ ≤ t provided that | x ₀ | < ∞. We give some related properties that we need in the sequel. We need Alekseev formula to compare between the solutions of (2.1) and the solutions of perturbed nonlinear system where and g ( t , 0) = 0. Let y ( t ) = y ( t , t ₀ , y ₀ ) denote the solution of (2.5) passing through the point ( t ₀ , y ₀ ) in × . The following is a generalization to nonlinear system of the variation of constants formula due to Alekseev [1] . Lemma 2.2. Let x and y be a solution of ( . 1 ) and ( . 5 ), respectively. If then for all t such that Lemma 2.3 ( [7] ) . Let u , λ ₁ , λ ₂ , w ( u ) be nondecreasing in u and w ( u ) ≤ w( ) for some v > 0. If , for some c > 0, then where u > 0, u ₀ > 0 W ^-1 ( u ) is the inverse of W ( u ) and Lemma 2.4 ( [10] ) . Let u, p, q,w, and r ∈ C ( ) and suppose that, for some c ≥ 0, we have Then Lemma 2.5 ( [15] ) . Let u ( t ), f ( t ), and g ( t ) be real-valued nonnegative continuous functions defined on , for which the inequality holds, where u ₀ is a nonnegative constant. Then, Lemma 2.6 ( [12] ) . Let u , λ ₁ , λ ₂ , λ ₃ ∈ C ( ), w ∈ C ((0,∞)) and w ( u ) be nondecreasing in u, u ≤ w ( u ). Suppose that for some c > 0, Then where W, W ^-1 are the same functions as in Lemma 2.3 and Lemma 2.7 ( [13] ) . Let u, p, q,w, r ∈ C ( ), w ∈ C ((0,∞)) and w ( u ) be nondecreasing in u. Suppose that for some c ≥ 0, Then where and Lemma 2.8 ( [14] ) . Let the following condition hold for functions u ( t ), v ( t ) ∈ C [[ t ₀ ,∞) ) and k ( t, u ) ∈ C [[ t ₀ ,∞) × , ) : t ≥ t ₀ and k ( s , u ) is strictly increasing in u for each fixed s ≥ 0. If u ( t ₀ ) < v ( t ₀ ), then u ( t ) < v ( t ), t ≥ t ₀ ≥ 0. Lemma 2.9 ( [5] ) . Let u, λ ₁ , λ ₂ , λ ₃ ∈ C ( ), w ∈ C ((0,∞)) and w ( u ) be nondecreasing in u. Suppose that for some c > 0, Then where u > 0, u ₀ > 0, W ^-1 ( u ) is the inverse of W ( u ) and In this section, we investigate Lipschitz and asymptotic stability for solutions of the nonlinear perturbed differential systems. Theorem 3.1. Assume that x = 0 of ( . 1) is ULS. Let the following condition hold for ( . ): where W ( t , u ) ∈ C ( × , ) is monotone nondecreasing in u with W ( t , 0) = 0. Suppose that u ( t ) is any solution of the scalar differential equation existing on such that m ( t ₀ ) < u ( t ₀ ). If u = 0 of (3.1) is ULS, then y = 0 of ( . ) is also ULS whenever M | y ₀ | < u ₀ . Proof. Let x ( t ) = x ( t , t ₀ , y ₀ ) and y ( t ) = y( t , t ₀ , y ₀ ) be solutions of (2.1) and (2.2), respectively. Using the variation of constants formula, we have where Φ( t , t ₀ , y ₀ ) is the fundemental matrix of (2.4). Since x = 0 of (2.1) is ULS, it is ULSV by Corollary 3.6 [5] . Thus there exist M > 0 and 𝛿 > 0 such that |Φ( t , t ₀ , y ₀ )| ≤ M for t ≥ t ₀ ≥ 0. Therefore, by the assmption, we have Hence | y ( t )| < u ( t ) by Lemma 2.8. Since u = 0 of (3.1) is ULS, it easily follows that y = 0 of (2.2) is ULS. Corollary 3.2. Assume that x = 0 of ( . 1 ) is ULS. Consider the scalar differential equation where u ₀ ≥ 1, K ≥ 1 and a , k ∈ C ( ) satisfy the conditions ( a ) where ( b ) Then y = 0 of ( . ) is ULS. Proof. Let u ( t ) = u ( t , t ₀ , x ₀ ) be any solution of (3.2). Then, by Lemma 2.5 , we have Hence u = 0 of (3.2) is ULS. This implies that the solution y = 0 of (2.2) is ULS by Theorem 3.1. Remark 3.3. In Corollary 3.2, it is needed that b ₁ = ∞. The condition W (∞) = ∞ is too strong and it represents situations which are not stable. For example, if w ( u ) = u ^𝛼 , then only 𝛼 ≤ 1 satisfies W (∞) = ∞ and 𝛼 < 1 is not stable. See [18] . Corollary 3.4. Assume that x = 0 of ( . 1 ) is ULS. Consider the scalar differential equation where u ₀ ≥ 1, K ≥ 1, u , w ∈ C ( ), w ( u ) be nondecreasing in u and w ( u )≤ w ( ) for some v > 0, and a , k ∈ C ( ) satisfy the conditions ( a ) where ( b ) and a, k ∈ L ₁ ( ). Then y = 0 of ( . ) is ULS. Proof. Let u ( t ) = u ( t , t ₀ , x ₀ ) be any solution of (3.3). Then, by Lemma 2.3, we have Hence u = 0 of (3.3) is ULS. By Theorem 3.1, the solution y = 0 of (2.2) is ULS. Corollary 3.5. Assume that x = 0 of ( . 1 ) is ULS. Consider the scalar differential equation where w ∈ C ((0,∞), w ( u ) is nondecreasing on u and u ≤ w ( u ), u ₀ ≥ 1, K ≥ 1 and a, b, k ∈ C ( ) satisfy the conditions ( a ) where ( b ) L ₁ ( ). Then y = 0 of ( . ) is ULS. Proof. Let u ( t ) = u ( t , t ₀ , x ₀ ) be any solution of (3.4). Then, Lemma 2.6, we have Hence u = 0 of (3.4) is ULS, and so by Theorem 3.1, the solution y = 0 of (2.2) is ULS. □ Theorem 3.6. For the perturbed ( . ), we asssume that where a, b, k ∈ C ( ), a, b, k ∈ L ₁ ( ), w ∈ C ((0,∞), and w ( u ) is nondecreasing in u,u ≤ w ( u ), and w ( u ) ≤ w ( ) for some v > 0, where M ( t ₀ ) < ∞ and b ₁ = ∞. Then the zero solution of ( . ) is ULS whenever the zero solution of ( . 1 ) is ULSV. Proof. Let x ( t ) = x ( t , t ₀ , y ₀ ) and y ( t ) = y ( t , t ₀ , y ₀ ) be solutions of (2.1) and (2.2), respectively. Since x = 0 of (2.1) is ULSV, it is ULS by Theorem 3.3 [8] . Applying Lemma 2.2, we have Set u ( t ) = | y ( t )|| y ₀ | ^-1 . Now an application of Lemma 2.6 yields Hence we have | y ( t )| ≤ M ( t ₀ )| y ₀ | for some M ( t ₀ ) > 0 whenever | y ₀ | < 𝛿. This completes the proof. □ Theorem 3.7. For the perturbed ( . ), we asssume that where a, b, k ∈ C ( ), a, b, k ∈ L ₁ ( ), w ∈ C ((0,∞), and w ( u ) is nondecreasing in u,u ≤ w ( u ), and w ( u ) ≤ w ( ) for some v > 0, where M ( t ₀ ) < ∞ and b ₁ = ∞. Then the zero solution of ( . ) is ULS whenever the zero solution of ( . 1 ) is ULSV. Proof. Let x ( t ) = x ( t , t ₀ , y ₀ ) and y ( t ) = y ( t , t ₀ , y ₀ ) be solutions of (2.1) and (2.2), respectively. Using the nonlinear variation of constants formula and the ULSV condition of x = 0 of (2.1), we have Set u ( t ) = | y ( t )|| y ₀ | ^-1 . Now an application of Lemma 2.7 yields Thus we have | y ( t )| ≤ M ( t ₀ )| y ₀ | for some M ( t ₀ ) > 0 whenever | y ₀ | < 𝛿, and so the proof is complete. □ Theorem 3.8. Let the solution x = 0 of ( . 1 ) be EAS. Suppose that the perturbing term g ( t , y ) satisfies where 𝛼 > 0, a, b, k ∈ C ( ), a, b, k ∈ L ₁ ( ), w ( u ) is nondecreasing in u, and w ( u ) ≤ w ( ) for some v > 0. If where c = | y ₀ | Me ^𝛼t₀ , then all solutions of ( . ) approch zero as t → ∞ Proof. Let x ( t ) = x ( t , t ₀ , y ₀ ) and y ( t ) = y ( t , t ₀ , y ₀ ) be solutions of (2.1) and (2.2), respectively. Since the solution x = 0 of (2.1) is EAS, we have |Φ( t , t ₀ , x ₀ )| ≤ Me ^{-𝛼(t-t₀)} for some M > 0 and c > 0(Theorem 2 [2] ). Using Lemma 2.2, we have since e ^𝛼t is increasing. Set u ( t ) = | y ( t )| e ^𝛼t . An application of Lemma 2.4 obtains The above estimation yields the desired result. □ Theorem 3.9. Let the solution x = 0 of ( . 1 ) be EAS. Suppose that the perturbing term g ( t , y ) satisfies where 𝛼 > 0, a, b, k , w ∈ C ( ), a, b, k ∈ L ₁ ( ) and w ( u ) is nondecreasing in u. If where c = M | y ₀ | e ^𝛼t₀ , then all solutions of ( . ) approch zero as t → ∞ Proof. Let x ( t ) = x ( t , t ₀ , y ₀ ) and y ( t ) = y ( t , t ₀ , y ₀ ) be solutions of (2.1) and (2.2), respectively. Using Lemma 2.2 and the assmptions, we have Set u ( t ) = | y ( t )| e ^𝛼t . Since w ( u ) is nondecreasing, an application of Lemma 2.9 obtains where c = M | y ₀ | e ^𝛼t₀ . From the above estimation, we obtains the desired result. □