Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 20, pp. 1–12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXPONENTIAL STABILITY OF SOLUTIONS TO NONLINEAR TIME-VARYING DELAY SYSTEMS OF NEUTRAL TYPE EQUATIONS WITH PERIODIC COEFFICIENTS INESSA I. MATVEEVA Abstract. We consider a class of nonlinear time-varying delay systems of neutral type differential equations with periodic coefficients in the linear terms, d dt y(t) = A(t)y(t) +B(t)y(t− τ(t)) + C(t) d dt y(t− τ(t)) + F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) ) , where A(t), B(t), C(t) are T -periodic matrices, and ‖F (t, u, v, w)‖ ≤ q1‖u‖+ q2‖v‖+ q3‖w‖, q1, q2, q3 ≥ 0, t > 0. We obtain conditions for the exponential stability of the zero solution and estimates for the exponential decay of the solutions at infinity. 1. Introduction There is a large number of works devoted to the study of delay differential equations (see [1, 4, 14, 17, 18, 19, 20, 23, 24, 29] and the bibliography therein). The intense interest in such equations is due to their arise in many applied problems describing the processes whose speeds are defined not only by the present, but also by the previous states (for example, see [15, 25] and the bibliography therein). One of the important problems is that of the exponential stability of solutions to the equations of such kind. Unlike autonomous equations, the exponential stability for nonautonomous equations has been studied less. We consider nonlinear time-varying delay systems of the form d dt y(t) = A(t)y(t) +B(t)y(t− τ(t)) + C(t) d dt y(t− τ(t)) + F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) ) , t > 0, (1.1) where A(t), B(t), C(t) are (n×n) matrices with continuous T -periodic entries; i.e., A(t+ T ) ≡ A(t), B(t+ T ) ≡ B(t), C(t+ T ) ≡ C(t), 2010 Mathematics Subject Classification. 34K20. Key words and phrases. Time-varying delay equation; neutral equation; periodic coefficient; exponential stability. c©2020 Texas State University. Submitted April 30, 2019. Published February 14, 2020. 1 2 I. I. MATVEEVA EJDE-2020/20 τ(t) ∈ C1([0,∞)) is the time-varying delay, 0 < τ1 ≤ τ(t) ≤ τ2, τ3 ≤ d dt τ(t) ≤ τ4 < 1. (1.2) and F is a continuous vector-function mapping [0,∞) × Rn × Rn × Rn into Rn. We assume that F (t, u, v, w) satisfies the Lipschitz condition with respect to u on every compact set G ⊂ [0,∞)× Rn × Rn × Rn, ‖F (t, u, v, w)‖ ≤ q1‖u‖+ q2‖v‖+ q3‖w‖, t ≥ 0, u, v, w ∈ Rn, (1.3) for some constants q1, q2, q3 ≥ 0. Hereafter we use the following dot product and vector norm 〈x, z〉 = n∑ j=1 xj z̄j , ‖x‖ = √ 〈x, x〉. In this article we continue the study of exponential stability of solutions to delay differential equations presented in [7, 8, 26, 9, 10, 11, 12, 27]. We investigated linear and nonlinear time-delay systems with C(t) ≡ 0 in [7, 8, 26], with a constant matrix C(t) ≡ C in [9, 10, 11, 12]. with a T -periodic matrix C(t) in [27]. Conditions for exponential stability of the zero solution were established and estimates of exponen- tial decay of solutions at infinity have been obtained. However, all the mentioned articles consider systems of the form (1.1) with the constant delay τ(t) ≡ τ . Linear time-varying delay systems of the form (1.1) with F (t, u, v, w) ≡ 0 were considered in [28]. This article deals with the initial value problem d dt y(t) = A(t)y(t) +B(t)y(t− τ(t)) + C(t) d dt y(t− τ(t)) + F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) ) , t > 0, y(t) = ϕ(t), t ∈ [−τ2, 0], y(+0) = ϕ(0), (1.4) where ϕ(t) ∈ C1[−τ2, 0] is a given real-valued vector-function. The solution to (1.4) is defined as a continuous function on [−τ2,∞), continuously differentiable on [−τ2,∞) except for points kτ1, k = 0, 1, 2, . . . , and satisfying (1.1) everywhere on [0,∞) except for points kτ1, k = 0, 1, 2, . . . . Our aim is to establish conditions for the exponential stability of the zero solution to (1.1) and to obtain estimates for the decay rate of solutions to (1.4) at infinity, without using any spectral information (like roots of quasipolynomials in the case of constant coefficients). To establish conditions of stability, researchers often use various Lyapunov– Krasovskii functionals (of Lyapunov type functionals) (see the bibliography in [3, 16]). However, not every Lyapunov–Krasovskii functional allows us to obtain estimates characterizing the decay rate of solutions at infinity. In recent years, the research in this direction has been actively developing. Many works are devoted to time-delay systems with constant coefficients, including systems of neutral type (see the bibliography in [21]). In the nonautonomous case, the most studied systems are the systems of the form (1.1), where C(t) is a constant matrix (see bibliogra- phy in [17]). There are very few studies of the systems, where the matrix C(t) is not constant [2, 5, 13, 17, 22, 30]; moreover, the authors of these works require that ‖C(t)‖ < 1. A Lyapunov-Krasovskii functional was proposed in [27], which allowed us to obtain the conditions for the exponential stability and the estimates for the solutions to the linear systems of the form (1.1) (F (t, u, v, w) ≡ 0) with the EJDE-2020/20 EXPONENTIAL STABILITY OF SOLUTIONS 3 constant delay τ(t) ≡ τ , without any additional restrictions on the norm ‖C(t)‖. A generalization of this functional was used in [28] for studying the exponential stability of the linear time-varying delay systems of neutral type; i.e. the systems of the form (1.1) with F (t, u, v, w) ≡ 0. In this article we use the functional proposed in [28]. We introduce the necessary notation and formulate the main results in Section 2. Their proofs are given in Section 3. 2. Main results At first we formulate the result on the exponential stability of the zero solution to (1.1) with F (t, u, v, w) ≡ 0: d dt y(t) = A(t)y(t) +B(t)y(t− τ(t)) + C(t) d dt y(t− τ(t)), t > 0. (2.1) Theorem 2.1 ([28]). Suppose that there are matrices H(t) ∈ C1[0, T ], K(s), and L(s) in C1[0, τ2]: H(t) = H∗(t), t ∈ [0, T ], H(0) = H(T ) > 0, (2.2) K(s) = K∗(s) > 0, d ds K(s) < 0, s ∈ [0, τ2], (2.3) L(s) = L∗(s) > 0, d ds L(s) < 0, s ∈ [0, τ2], (2.4) such that the matrix Q(t) = Q11(t) Q12(t) Q13(t) Q∗12(t) Q22(t) Q23(t) Q∗13(t) Q∗23(t) Q33(t)  (2.5) with entries Q11(t) = − d dt H(t)−H(t)A(t)−A∗(t)H(t)−K(0)−A∗(t)L(0)A(t), Q12(t) = −H(t)B(t)−A∗(t)L(0)B(t), Q13(t) = −H(t)C(t)−A∗(t)L(0)C(t), Q22(t) = (1− τ4)K(τ2)−B∗(t)L(0)B(t), Q23(t) = −B∗(t)L(0)C(t), Q33(t) = (1− τ3)−1L(τ2)− C∗(t)L(0)C(t), (2.6) is positive definite for t ∈ [0, T ]. Then the zero solution to (2.1) is exponentially stable. Assuming that the conditions of Theorem 2.1 are satisfied, we establish condi- tions for the exponential stability of the zero solution to the nonlinear system (1.1). To state our results, we introduce some notation. If the matrix H(t) satisfies the conditions of Theorem 2.1 then d dt H(t) +H(t)A(t) +A∗(t)H(t) < −K(0)−A∗(t)L(0)A(t); i.e., H(t) is a solution to a special boundary value problem for the Lyapunov dif- ferential equation d dt H +HA(t) +A∗(t)H = −G(t), t ∈ [0, T ], 4 I. I. MATVEEVA EJDE-2020/20 H(0) = H(T ) > 0, where G(t) is a positive definite Hermitian matrix with continuous entries. In this case, it follows from the results of [6] that H(t) > 0 on the whole segment [0, T ]. Let us extend this matrix T -periodically on the whole semi-axis {t ≥ 0}, keeping the same notation. Using this matrix H(t) together with the matrices K(s), L(s) satisfying the conditions of Theorem 2.1, we introduce the functions β1(t) = 2‖H(t)‖+ (2‖A(t)‖+ q1)‖L(0)‖, β2(t) = (2‖B(t)‖+ q2)‖L(0)‖, β3(t) = (2‖C(t)‖+ q3)‖L(0)‖, (2.7) α1(t) = q1β1(t) + q1β2(t) + q2β1(t) 2 + q1β3(t) + q3β1(t) 2 , α2(t) = q2β2(t) + q2β1(t) + q1β2(t) 2 + q2β3(t) + q3β2(t) 2 , α3(t) = q3β3(t) + q3β1(t) + q1β3(t) 2 + q3β2(t) + q2β3(t) 2 , (2.8) and the matrix Qα(t) = Q(t)− α1(t)I 0 0 0 α2(t)I 0 0 0 α3(t)I  , (2.9) where I is the unit matrix. Theorem 2.2. Let the conditions of Theorem 2.1 be satisfied. Suppose that q1, q2, q3 are such that the matrix Qα(t) is positive definite for t ∈ [0, T ]. Then the zero solution to (1.1) is exponentially stable. Below we present the estimate for the exponential decay rate of the solution to the initial value problem (1.4) as t→∞. We use the following notation V (0, ϕ) = 〈H(0)ϕ(0), ϕ(0)〉+ ∫ 0 −τ(0) 〈K(−s)ϕ(s), ϕ(s)〉ds + ∫ 0 −τ(0) 〈 L(−s) d ds ϕ(s), d ds ϕ(s) 〉 ds, (2.10) P (t) = Q11(t)− α1(t)I − [ Q12(t)−Q13(t)(Q33(t)− α3(t)I)−1Q∗23(t) ] × [ Q22(t)− α2(t)I −Q23(t)(Q33(t)− α3(t)I)−1Q∗23(t) ]−1 × [ Q12(t)−Q13(t)(Q33(t)− α3(t)I)−1Q∗23(t) ]∗ −Q13(t)(Q33(t)− α3(t)I)−1Q∗13(t), (2.11) where the matrices Qij(t) are defined by (2.6). It is not difficult to show that P (t) is positive definite if Qα(t) in (2.9) is positive definite (see Lemma 3.1). We denote by pmin(t) > 0 the minimal eigenvalue of the matrix P (t), and by hmin(t) > 0 the minimal eigenvalue of the matrix H(t). Theorem 2.3. Suppose that the conditions of Theorem 2.2 are satisfied. Let k, l > 0 be maximal numbers such that d ds K(s) + kK(s) ≤ 0, d ds L(s) + lL(s) ≤ 0, s ∈ [0, τ2]. (2.12) EJDE-2020/20 EXPONENTIAL STABILITY OF SOLUTIONS 5 Then the following estimate holds for the solution to (1.4), ‖y(t)‖ ≤ √ V (0, ϕ) hmin(t) exp ( − 1 2 ∫ t 0 γ(ξ)dξ ) , t > 0, (2.13) where γ(t) = min {pmin(t) ‖H(t)‖ , k, l } > 0. The existence of k, l > 0 in Theorem 2.3 is provided by using conditions (2.3) and (2.4). 3. Proof of the main results Obviously, the assertion of Theorem 2.2 follows immediately from estimate (2.13). Therefore, it suffices to prove Theorem 2.3. Proof of Theorem 2.3. We follow the scheme from [8]. Let y(t) be the solution to the initial value problem (1.4). Using the matrices H(t), K(s), and L(s) defined in Section 2, we consider the following Lyapunov-Krasovskii functional on the solution V (t, y) = 〈H(t)y(t), y(t)〉+ ∫ t t−τ(t) 〈K(t− s)y(s), y(s)〉ds + ∫ t t−τ(t) 〈 L(t− s) d ds y(s), d ds y(s) 〉 ds. (3.1) Differentiating we obtain d dt V (t, y) = 〈 d dt H(t)y(t), y(t) 〉 + 〈 H(t) d dt y(t), y(t) 〉 + 〈 H(t)y(t), d dt y(t) 〉 + 〈K(0)y(t), y(t)〉 − ( 1− d dt τ(t) ) 〈K(τ(t))y(t− τ(t)), y(t− τ(t))〉 + ∫ t t−τ(t) 〈 d dt K(t− s)y(s), y(s) 〉 ds+ 〈 L(0) d dt y(t), d dt y(t) 〉 − ( 1− d dt τ(t) )−1〈 L(τ(t)) d dt y(t− τ(t)), d dt y(t− τ(t)) 〉 + ∫ t t−τ(t) 〈 d dt L(t− s) d ds y(s), d ds y(s) 〉 ds. We introduce the notation z(t) = A(t)y(t) +B(t)y(t− τ(t)) + C(t) d dt y(t− τ(t)). Taking into account that y(t) satisfies (1.1), we have d dt V (t, y) = 〈 d dt H(t)y(t), y(t) 〉 + 〈 H(t)z(t), y(t) 〉 + 〈 H(t)F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) ) , y(t) 〉 + 〈 H(t)y(t), z(t) 〉 + 〈 H(t)y(t), F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) )〉 + 〈K(0)y(t), y(t)〉 − ( 1− d dt τ(t) ) 〈K(τ(t))y(t− τ(t)), y(t− τ(t))〉 6 I. I. MATVEEVA EJDE-2020/20 + ∫ t t−τ(t) 〈 d dt K(t− s)y(s), y(s) 〉 ds+ 〈 L(0)z(t), z(t) 〉 + 〈 L(0)F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) ) , z(t) 〉 + 〈 L(0)z(t), F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) )〉 + 〈 L(0)F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) ) , F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) )〉 − ( 1− d dt τ(t) )−1〈 L(τ(t)) d dt y(t− τ(t)), d dt y(t− τ(t)) 〉 + ∫ t t−τ(t) 〈 d dt L(t− s) d ds y(s), d ds y(s) 〉 ds. By (1.2), (2.3), (2.4), we obtain( 1− d dt τ(t) ) 〈K(τ(t))y(t− τ(t)), y(t− τ(t))〉 ≥ (1− τ4)〈K(τ2)y(t− τ(t)), y(t− τ(t))〉,( 1− d dt τ(t) )−1〈 L(τ(t)) d dt y(t− τ(t)), d dt y(t− τ(t)) 〉 ≥ (1− τ3)−1 〈 L(τ2) d dt y(t− τ(t)), d dt y(t− τ(t)) 〉 . Consequently, d dt V (t, y) ≤ − 〈 Q(t)  y(t) y(t− τ(t)) d dty(t− τ(t))  ,  y(t) y(t− τ(t)) d dty(t− τ(t)) 〉 + 〈 H(t)F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) ) , y(t) 〉 + 〈 H(t)y(t), F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) )〉 + 〈 L(0)F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) ) , z(t) 〉 + 〈 L(0)z(t), F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) )〉 + 〈 L(0)F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) ) , F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) )〉 + ∫ t t−τ(t) 〈 d dt K(t− s)y(s), y(s) 〉 ds + ∫ t t−τ(t) 〈 d dt L(t− s) d ds y(s), d ds y(s) 〉 ds, where the matrix Q(t) is defined in (2.5). EJDE-2020/20 EXPONENTIAL STABILITY OF SOLUTIONS 7 Consider the group of the summands containing F ( t, y(t), y(t − τ(t)), ddty(t − τ(t)) ) and denote them by W (t). Then, d dt V (t, y) ≤ − 〈 Q(t)  y(t) y(t− τ(t)) d dty(t− τ(t))  ,  y(t) y(t− τ(t)) d dty(t− τ(t)) 〉+W (t) + ∫ t t−τ(t) 〈 d dt K(t− s)y(s), y(s) 〉 ds + ∫ t t−τ(t) 〈 d dt L(t− s) d ds y(s), d ds y(s) 〉 ds. (3.2) Obviously, W (t) ≤ ( 2‖H(t)‖ ‖y(t)‖+ 2‖L(0)‖ ∥∥A(t)y(t) +B(t)y(t− τ(t)) + C(t) d dt y(t− τ(t)) ∥∥ + ‖L(0)‖ ∥∥∥F(t, y(t), y(t− τ(t)), d dt y(t− τ(t)) )∥∥∥) × ∥∥∥F(t, y(t), y(t− τ(t)), d dt y(t− τ(t)) )∥∥∥. Using (1.3), we have W (t) ≤ ( β1(t)‖y(t)‖+ β2(t)‖y(t− τ(t))‖+ β3(t)‖ d dt y(t− τ(t))‖ ) × ( q1‖y(t)‖+ q2‖y(t− τ(t))‖+ q3‖ d dt y(t− τ(t))‖ ) , where βj(t), j = 1, 2, 3, are defined in (2.7). Obviously, W (t) ≤ α1(t)‖y(t)‖2 + α2(t)‖y(t− τ(t))‖2 + α3(t)‖ d dt y(t− τ(t))‖2, (3.3) where the functions αj(t), j = 1, 2, 3, are defined in (2.8). By (3.3), from (3.2) we obtain d dt V (t, y) ≤ − 〈 Qα(t)  y(t) y(t− τ(t)) d dty(t− τ(t))   y(t) y(t− τ(t)) d dty(t− τ(t)) 〉 + ∫ t t−τ(t) 〈 d dt K(t− s)y(s), y(s) 〉 ds + ∫ t t−τ(t) 〈 d dt L(t− s) d ds y(s), d ds y(s) 〉 ds, (3.4) where the matrix Qα(t) is given in (2.9). For further transformations, we use an auxiliary lemma from matrix theory. Lemma 3.1. Let R(t) be a positive definite Hermitian matrix with continuous entries R(t) = R11(t) R12(t) R13(t) R∗12(t) R22(t) R23(t) R∗13(t) R∗23(t) R33(t)  , t ∈ [0, T ], 8 I. I. MATVEEVA EJDE-2020/20 Then the following representation holds R(t) = I R̃1(t)R̃−12 (t) R13(t)R−133 (t) 0 I R23(t)R−133 (t) 0 0 I  × R11(t)− R̃1(t)R̃−12 (t)R̃∗1(t)−R13(t)R−133 (t)R∗13(t) 0 0 0 R̃2(t) 0 0 0 R33(t)  ×  I 0 0 R̃−12 (t)R̃∗1(t) I 0 R−133 (t)R∗13(t) R−133 (t)R∗23(t) I  , where R̃1(t) = R12(t)−R13(t)R−133 (t)R∗23(t), R̃2(t) = R22(t)−R23(t)R−133 (t)R∗23(t); moreover, the matrices R11(t)− R̃1(t)R̃−12 (t)R̃∗1(t)−R13(t)R−133 (t)R∗13(t), R̃2(t), R33(t) are positive definite. By the above lemma, for the matrix Qα(t) in (2.9), we have〈 Qα(t)  y(t) y(t− τ(t)) d dty(t− τ(t))  ,  y(t) y(t− τ(t)) d dty(t− τ(t)) 〉 ≥ 〈P (t)y(t), y(t)〉, where P (t) is the positive definite Hermitian matrix given in (2.11). Then 〈P (t)y(t), y(t)〉 ≥ pmin(t)‖y(t)‖2, where pmin(t) > 0 is the minimal eigenvalue of the matrix P (t). Consequently, from (3.4) we obtain d dt V (t, y) ≤ −〈pmin(t)y(t), y(t)〉+ ∫ t t−τ(t) 〈 d dt K(t− s)y(s), y(s) 〉 ds + ∫ t t−τ(t) 〈 d dt L(t− s) d ds y(s), d ds y(s) 〉 ds. Clearly, hmin(t)‖y(t)‖2 ≤ 〈 H(t)y(t), y(t) 〉 ≤ ‖H(t)‖ ‖y(t)‖2, (3.5) where hmin(t) > 0 is the minimal eigenvalue of the matrix H(t). Hence, d dt V (t, y) ≤ −pmin(t) ‖H(t)‖ 〈 H(t)y(t), y(t) 〉 + ∫ t t−τ(t) 〈 d dt K(t− s)y(s), y(s) 〉 ds + ∫ t t−τ(t) 〈 d dt L(t− s) d ds y(s), d ds y(s) 〉 ds. Using the condition (2.12), we arrive at d dt V (t, y) ≤ −pmin(t) ‖H(t)‖ 〈 H(t)y(t), y(t) 〉 − k ∫ t t−τ(t) 〈 K(t− s)y(s), y(s) 〉 ds − l ∫ t t−τ(t) 〈 L(t− s) d ds y(s), d ds y(s) 〉 ds. EJDE-2020/20 EXPONENTIAL STABILITY OF SOLUTIONS 9 By the definition of the functional in (3.1), we have d dt V (t, y) ≤ −γ(t)V (t, y), where γ(t) = min {pmin(t) ‖H(t)‖ , k, l } . From this differential inequality, we obtain the estimate V (t, y) ≤ V (0, ϕ) exp ( − ∫ t 0 γ(ξ)dξ ) , where V (0, ϕ) is defined by (2.10). Using (3.5) and taking into account the definition of the functional (3.1), we infer ‖y(t)‖2 ≤ 1 hmin(t) 〈 H(t)y(t), y(t) 〉 ≤ V (t, y) hmin(t) ≤ V (0, ϕ) hmin(t) exp ( − ∫ t 0 γ(ξ)dξ ) . Hence, we have the required inequality (2.13). This completes the proof. � Repeating the reasoning of the proof of Theorem 2.3 for F (t, u, v, w) ≡ 0, we arrive immediately to the statement of Theorem 2.1. Theorem 3.2. Suppose that there are matrices H(t) ∈ C1[0, T ], K(s), L(s) ∈ C1[0, τ2] satisfying (2.2)–(2.4) and such that P (t) > 0, Q22(t)− α2(t)I −Q23(t)(Q33(t)− α3(t)I)−1Q∗23(t) > 0, and Q33(t)− α3(t)I > 0 for t ∈ [0, T ]. Then the zero solution to (1.1) is exponen- tially stable. Proof. By Lemma 3.1, the matrix Qα(t) in (2.9) is positive definite if and only if the matrices P (t), Q22(t)−α2(t)I−Q23(t)(Q33(t)−α3(t)I)−1Q∗23(t), and Q33(t)−α3(t)I are positive definite. � Remark 3.3. The results obtained above give us the assertions on the robust stability for (2.1). Indeed, consider uncertain systems of the form d dt y(t) = A(t)y(t) +B(t)y(t− τ(t)) + C(t) d dt y(t− τ(t)) + ∆A(t)y(t) + ∆B(t)y(t− τ(t)) + ∆C(t) d dt y(t− τ(t)), (3.6) where ∆A(t), ∆B(t), and ∆C(t) are unknown (n× n) matrices such that ‖∆A(t)‖ ≤ q1, ‖∆B(t)‖ ≤ q2, ‖∆C(t)‖ ≤ q3 . Obviously, in this case the vector-function F (t, u, v, w) = ∆A(t)u+ ∆B(t)v + ∆C(t)w satisfies (1.3). Then Theorem 2.2 gives us the conditions of the robust exponential stability for (2.1). From Theorems 2.3 we have the estimates of the exponential decay of solutions to (3.6). 10 I. I. MATVEEVA EJDE-2020/20 4. Examples Consider the following time-delay equation of the form (1.1), d dt y(t) = (0.1 cos t−2)y(t)−0.1y(t−τ(t))+q cos ( d dt y(t−τ(t)) ) d dt y(t−τ(t)), (4.1) where τ(t) = 0.5 sin t + 1. Obviously, τ(t) satisfies (1.2) with τ1 = 0.5, τ2 = 1.5, τ3 = −0.5, and τ4 = 0.5. The function F (t, u, v, w) = q cos(w)w satisfies (1.3) with q1 = q2 = 0, q3 = q. At first we consider the linear case (F (t, u, v, w) ≡ 0); i.e. q = 0. We choose the functions H(t), K(s), and L(s) as follows H(t) = 0.5− 0.1 sin t, K(s) = 0.27e−1.65s, L(s) = 0.001e−1.65s. Obviously, these functions satisfy (2.2)–(2.4), and (2.12) with k = l = 1.65. In this case the matrix Q(t) has entires Q11(t) = 1.73− 0.4 sin t+ 0.02 sin t cos t− 0.001(2− 0.1 cos t)2, Q12(t) = 0.0498− 0.01 sin t+ 0.00001 cos t, Q13(t) = 0, Q22(t) = 0.135e−2.475 − 0.00001, Q23(t) = 0, Q33(t) = 0.002 3 e−2.475. It is not difficult to verify that Q(t) is positive definite for t ∈ [0, 2π]. Then, by Theorem 2.1, the zero solution to (4.1) with q = 0 is exponentially stable. By Theorem 2.3, to estimate the decay rate of solutions to (4.1), we need to calculate (for q = 0) the functions P (t) and γ(t) = min { P (t)/‖H(t)‖, 1.65 } . In our case P (t) = Q11(t)−Q2 12(t)(Q22(t))−1, ‖H(t)‖ = |0.5− 0.1 sin t|. It is not difficult to show that P (t) ≥ 0.99047 and that ‖H(t)‖ ≤ 0.6. Therefore, P (t)/‖H(t)‖ ≥ 1.65078 and γ(t) = 1.65. By (2.13), we have the estimate ‖y(t)‖ ≤ ce−0.825t, c > 0, for the solutions to (4.1) with q = 0. We now consider (4.1) with q = 0.1. We choose the functions H(t), K(s), and L(s) as follows H(t) = 0.5− 0.1 sin t, K(s) = 0.06e−0.27s, L(s) = 0.28e−0.27s. Obviously, these functions satisfy (2.2)–(2.4), and (2.12) with k = l = 0.27. In this case the entries of the matrix Q(t) has entries Q11(t) = 1.94− 0.4 sin t+ 0.02 sin t cos t− 0.28(2− 0.1 cos t)2, Q12(t) = −0.006− 0.01 sin t+ 0.0028 cos t, Q13(t) = 0, Q22(t) = 0.03e−0.405 − 0.0028, Q23(t) = 0, Q33(t) = 0.56 3 e−0.405, and β1(t) = |1− 0.2 sin t|+ |1.12− 0.056 cos t|, β2(t) = 0.056, β3(t) = 0.028, α1(t) = |0.05− 0.01 sin t|+ |0.056− 0.0028 cos t|, α2(t) = 0.0028, α3(t) = |0.05− 0.01 sin t|+ |0.056− 0.0028 cos t|+ 0.0056. It is not difficult to verify that Qα(t) defined by (2.9) is positive definite for t ∈ [0, 2π]. Then, by Theorem 2.2, the zero solution to (4.1) with q = 0.1 EJDE-2020/20 EXPONENTIAL STABILITY OF SOLUTIONS 11 is exponentially stable. By Theorem 2.3, to estimate the decay rate of solu- tions to (4.1), we need to calculate (for q = 0.1) the functions P (t) and γ(t) = min { P (t)/‖H(t)‖, 0.27 } . In our case P (t) = Q11(t)− α1(t)−Q2 12(t)(Q22(t)− α2(t))−1. It is not difficult to show that P (t) ≥ 0.16319. 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Koptyug avenue, Novosibirsk 630090, Russia. Department of Mechanics and Mathematics, Novosibirsk State University, 2, Pirogov street, Novosibirsk 630090, Russia Email address: matveeva@math.nsc.ru 1. Introduction 2. Main results 3. Proof of the main results 4. Examples Acknowledgments References