On the existence and exponential stability for differential equations with multiple constant delays

On the existence and exponential stability for differential equations with multiple constant delays

and nonlinearity depending on fractional substantial integrals

Milan Medved’

and Michal Pospíšil

1Department of Mathematical Analysis and Numerical Mathematics,

Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovak Republic

2Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovak Republic Received 25 March 2019, appeared 27 June 2019

Communicated by Josef Diblík

Abstract. An existence result is proved for systems of differential equations with multi- ple constant delays, time-dependent coefficients and the right-hand side depending on fractional substantial integrals. Results on exponential stability for such equations are proved for linearly bounded nonlinearities and power type nonlinearities. An illustra- tive example is also given.

Keywords: multiple delays, fractional substantial integral, exponential stability, multi- delayed matrix exponential, logarithmic matrix norm.

2010 Mathematics Subject Classification: 34K20, 26A33.

1 Introduction

It is well known that the trivial solution of the linear fractional differential equation

CD^αx(t) =Ax(t), x(t)∈_R^N, α∈(0, 1), (1.1) where Ais a constant matrix and ^CD^αx(t)is the Caputo fractional derivative can be asymp- totically, but not exponentially stable. It is asymptotically stable if and only if |arg(λ)| > ^απ₂ for any eigenvalue of the matrix A(see e.g. [4,11,16]). However, for special types of fractional differential equations their solutions can be exponentially stable. In the paper [15], a suffi- cient condition for the exponential stability of the trivial solution of the nonlinear multi-delay fractional differential equation

CD^α h(t) (x˙(t)−Ax(t)−B₁x(t−τ₁)− · · · −B_mx(t−τ_m)) = f(x(t),x(t−τ₁), . . . ,x(t−τ_m))

BCorresponding author. Email: Michal.Pospisil@fmph.uniba.sk

was proved. In the paper [3], the equation

˙

x(t) =Ax(t) + f

t,x(t),^RLI^α1x(t), . . . ,^RLI^αmx(t), (1.2) where^RLI^α1x(t), . . . ,^RLI^αmx(t)are the Riemann–Liouville integrals, was studied. An existence result and a sufficient condition for the exponential stability of the trivial solution of this equation was proved. In the paper [2], an analogous problem was solved for an equation of the form (1.2) with Caputo–Fabrizio fractional integrals instead of the Riemann–Liouville integrals.

In this paper, we study systems of differential equations with multiple constant delays, time-dependent coefficients and the right-hand side depending on fractional substantial inte- grals, defined below. Originally, the formula for a solution of the initial-function problem

˙

x(t) =Ax(t) +B₁x(t−τ₁) +· · ·+Bnx(t−τn) + f(t), t ≥0, (1.3)

x(t) =ϕ(t), t∈[−τ, 0] (1.4)

whereτ =max_i=1,...,nτ_i, was stated in [14, Theorem 10] using so-called multi-delayed matrix exponential, which is an inductively built matrix polynomial of a degree depending on time.

This result was later simplified in [18] using the unilateral Laplace transform to obtain a closed-form formula (see Theorem2.1below). We remark that the delayed matrix exponential for the equation with one constant delay was introduced in the paper [7].

In the present paper, we make use of this formula to prove existence and exponential stability results for delayed differential equation (DDE) with multiple constant delays and nonlinearity depending on fractional substantial integrals of order β > 0 with a positive parameterγ(see e.g. [4,6]),

I^(β,γ)x(t) = ¹ Γ(β)

Z _t

0

(t−s)^β−1e^−γ(t−s)x(s)ds.

In particular, we consider the Cauchy problem

˙

x(t) =A(t)x(t) +B₁(t)x(t−τ₁) +· · ·+B_n(t)x(t−τ_n) +F t,x(t),x(t−τ₁), . . . ,x(t−τ_n),

I^(β01,γ01)x(t), . . . ,I^{(β0m0,γ0m0)}x(t),

I^(β11,γ11)x(t−τ₁)_{, . . . ,}I^{(βnmn,γnmn)}x(t−τ_n)_, t ≥_0.

x(t) = ϕ(t), t∈ [−τ, 0],

(1.5)

where A,B₁, . . . ,B_n are continuous matrix functions,

F(t,u0, . . . ,un,v00, . . . ,v0m₀,v₁₁, . . . ,vnmn)

is a continuous function of all its variables andϕ∈C([−τ, 0],R^N). This work is a continuation of [12,13], where an analogous problem was investigated without the presence of delays.

We note that in [14] and [17] the matrices A,B₁, . . . ,B_n were supposed to be pairwise permutable, i.e., AB_i = B_iA, B_iB_j = B_jB_i for each i,j = _{1, . . . ,}n. But our existence result, Theorem3.1, holds without any permutability assumption. For the stability results, Theorems 4.1and5.1, we only assume that the matrix functions A(t),B₁(t), . . . ,B_n(t)are permutable at some pointst₀,t₁, . . . ,t_n, respectively.

In the whole paper, we shall denote k · k the norm of a vector and the corresponding induced matrix norm. Further, N and N0 denote the set of all positive and nonnegative integers, respectively. We also assume the property of an empty sum, ∑i∈_∅z(i) = 0 for any functionz.

To make our stability results more applicable, we use the logarithmic matrix norm in assumptions. Analogous results can be obtained using the largest real value of all the eigen- values of A(t₀), max_{λA∈σ(A(t0))}Reλ_A, or a weighted logarithmic matrix norm [8]. However, then one has to work with the estimation

ke^Atk ≤c₁e^c2t (1.6)

with some positive constants c₁, c₂, where c₁ is not immediately known. So, the area of exponential stability can not be predetermined. By the logarithmic norm, (1.6) holds with c₁ =1.

The paper is organized as follows. In the following section, we collect some known results and definitions. Section3is devoted to the existence result of a unique solution of the initial- function problem (1.5). Sections4and5contain results on the exponential stability of a trivial solution of a class of nonlinear DDEs with the linearly bounded nonlinearity and nonlinearity bounded by some powers of its arguments, respectively. In final Section 6, we present an example illustrating the theoretical results.

2 Preliminary results

Let us recall a result from [18, Theorem 3.3] (see also [17, Theorem 2.15] for the case with variable delays) on the representation of a solution of a DDE with multiple delays.

Theorem 2.1. Let n ∈_N, 0< τ₁, . . . ,τn ∈ _R,τ := max{τ₁,τ₂, . . . ,τn}, A,B₁, . . . ,Bnbe pairwise permutable constant N×N matrices,ϕ∈C([−τ, 0],R^N), and f : [0,∞)→_R^N be a given function.

Then the solution of the Cauchy problem(1.3),(1.4)has the form x(t) =

(

ϕ(t), −τ≤t <0,

B(t)ϕ(₀) +_∑ⁿ_j=1B_jRτ_j

0 B(t−s)ϕ(s−τ_j)ds+Rt

0 B(t−s)f(s)ds, 0≤t where

B(t) =e^At

∑

∑ⁿm=1kmτm≤t k1,...,kn≥0

(t−_∑ⁿ_m=1kmτm)^∑nm=1km k₁! . . .k_n!

∏

Be_m^km for any t ∈_{R, and}Be_m = B_me^−Aτm for each m=1, . . . ,n.

Combining an estimation of the multi-delayed matrix exponential, [14, Lemma 13], with the representations of solutions of (1.3), (1.4) from [14] and Theorem 2.1, we obtain the fol- lowing statement.

Lemma 2.2. Let n ∈ _N, 0 < τ₁, . . . ,τ_n ∈ _{R, B1}, . . . , B_n be pairwise permutable constant N×N matrices. If α₁, . . . ,α_n∈_Rare such thatkB_ik ≤α_ie^αiτi for each i=_{1, . . . ,}n, then

∑ⁿm=1

∑

(t−_∑ⁿ_m=1k_mτ_m)^∑nm=1km k₁! . . .k_n!

∏

B^k_m^m

≤e^{(α1+···+αn)t}

for any t ∈_R.

We will investigate the exponential stability with respect to a ball in the sense of the next definition.

Definition 2.3. The zero solution of equation (1.3) is exponentially stable with respect to the ball Ω(r) := {h ∈ _R^N | khk ≤ r} if there are positive constants c₁, c₂ such that any solution xof (1.3) satisfying initial condition (1.4) with ϕ(t)∈ _Ω(r) for allt ∈ [−τ, 0]fulfills kx(t)k ≤c₁e^−c2t for allt ≥0.

Exponential stability of a trivial solution of other delay equations is understood analo- gously.

The logarithmic norm of a square matrix Ais defined by µ(A) = lim

ε→0⁺

kI+_εAk −1

ε .

The properties we need are concluded in the following lemma (see e.g. [5]).

Lemma 2.4. The logarithmic norm of a matrix A satisfies:

1. −kAk ≤ −µ(−A)≤Reσ(A)≤ µ(A)≤ kAk, 2. ke^Atk ≤e^µ(A)tfor all t≥0.

We shall also need the following integral inequality, which was proved in [10] for integer powers. The authors did not realize/mention that their proof works even in the more general setting with real exponents.

Lemma 2.5. Let 2 ≤ n ∈ _{N, c} ≥ 0, f_i(t) for i = 1, . . . ,n be nonnegative continuous functions defined on [a,b] and 1 = q₁ < q₂ ≤ q₃ ≤ · · · ≤ q_n be real numbers. If a positive differentiable real-valued function z(t)satisfies

z(t)≤c+

Z _t

a

∑

f_i(s)z^qi(s)ds, t ∈[a,b] and

1−(q_n−1)

Z _b

a

∑

c^qi−1f_i(s)exp

(q_n−1)

Z _s

a f₁(σ)dσ

ds>0, then

z(t)≤ ^cexp

Rt

a f₁(s)ds

1−(q_n−1)Rt

a ∑ⁿi=2c^qi−1f_i(s)exp (q_n−1)Rs

a f₁(σ)dσ

ds_qn¹₋₁. Proof. The proof is exactly the same as the proof of [10, Theorem 2.6].

3 Existence result

Here we prove an existence and uniqueness result for a solution of the initial-function problem (1.5).

Theorem 3.1. Let I = [_0,A]⊂_Rfor some A>0, G⊂_R^N be a region, H⊂_R^m0× · · · ×_R^mn be a region containing0∈ _R^m0× · · · ×_R^mn,F ∈C(I×Gⁿ⁺¹×H,R^N)is a continuous locally Lipschitz function. Then for any ϕ ∈ C([−τ, 0],G) there exists δ > 0 such that the initial function problem (1.5)has a unique solution x(t)on the interval I_δ = [−_τ,δ].

Proof. Letb_i,b_ij >0, i=0, . . . ,n, j=1, . . . ,m_i be such that

G_bi :={x ∈_R^N | kx−ϕ(−τ_i)k ≤b_i} ⊂G, i=0, . . . ,n forτ₀=0, and

V :={(v₀₁, . . . ,vnmn)∈_R^m0 × · · · ×_R^mn | kv_ijk ≤b_ij, i=0, . . . ,n, j=1, . . . ,m_i} ⊂H.

Let 0< a< Abe such that max

σ∈[0,min{a,τi}]kϕ(σ−τ_i)−ϕ(−τ_i)k ≤b_i, i=_{1, . . . ,n. (3.1)} From now on, we shall assume without any loss of generality that a≤_mini=1...,nτ_i. Note that (3.1) then implies

max

σ∈[0,a]

kϕ(σ−τ_i)k ≤b_i+kϕ(−τ_i)k, i=1, . . . ,n. (3.2) So, we haveG₀:= [0,a]×G_b0× · · · ×G_bn×V⊂ I×Gⁿ⁺¹×H. Let us denote

M0 := max

t∈[0,a],x∈G_b0kA(t)xk, M_A:= max

t∈[0,a]kA(t)k, M_i := max

t∈[0,a],x∈G_bi

kB_i(t)xk, i=1, . . . ,n,

MF := max

(t,u₀,...,un,v₀₁,...,v_nmn)∈G₀F(t,u₀, . . . ,un,v₀₁, . . . ,vnmn). Let L_i,L_ij >0,i=0, . . . ,n, j=1, . . . ,m_i be such that

kF(t,u₀, . . . ,un,v₀₁, . . . ,vnmn)− F(t, ˜u₀, . . . , ˜un, ˜v₀₁, . . . , ˜vnmn)k

≤

∑

L_iku_i−u˜_ik+

∑

mi

j

∑

L_ijkv_ij−v˜_ijk

for all (t,u₀, . . . ,u_n,v₀₁, . . . ,v_nmn),(t, ˜u₀, . . . , ˜u_n, ˜v₀₁, . . . , ˜v_nmn)∈ G₀. Finally, let 0<δ <min

a,c, b0

M₀+· · ·+M_n+MF,κ⁻¹

with

c≤ min

i=0,...,n j=1,...,mi

b_ijΓ(1+β_ij) b_i+kϕ(−τ_i)k

_β¹

ij , κ= M_A+L₀+

m0

j

∑

L_0jc^β0j Γ(1+β_0j)^.

Consider the Banach space C_δ := C(I_δ,R^N)endowed with the maximum norm, i.e., kxk = max_t∈I_δkx(t)kforx∈ C_δ, and define the successive approximations{x_k}^∞_k=0 ⊂C_δ by

x0(_t) = (

ϕ(t), t ∈[−τ, 0), ϕ(0), t ∈[0,δ],

x_k+1(t) =



































ϕ(t)_, t∈ [−_{τ, 0})_,

ϕ(0) +Rt

0 A(s)x_k(s)ds+_∑ⁿ_i=1Rt

0B_i(s)x_k(s−τ_i)ds +Rt

0Fs,x_k(s),x_k(s−τ₁), . . . ,x_k(s−τ_n), . . . ,

Γ(β101)

Rs

0(s−σ)^β01−1e^{−γ01(s−σ)}x_k(σ)dσ, . . . ,

Γ(β10m0)

Rs

0(s−σ)^β0m0−1e^{−γ0m0(s−σ)}x_k(σ)dσ,

Γ(β1₁₁)

Rs

0(s−σ)^β11−1e^{−γ11(s−σ)}x_k(σ−τ₁)dσ, . . . ,

Γ(β1_nmn)

Rs

0(s−σ)^βnmn−1e^{−γnmn(s−σ)}x_k(σ−τ_n)dσ

, t∈ [0,δ]

fork =0, 1, . . .

First, we show thatx₁(t)is well defined. For anys ∈[0,t]⊂[0,δ]we haves∈[0,a], kx₀(s)−ϕ(₀)k ≤ _max

σ∈[0,δ]

kx₀(σ)−ϕ(₀)k=kϕ(₀)−ϕ(₀)k=₀≤ b₀, i.e.,x₀(s)∈ G_b0, and

kx0(s−τ_i)−ϕ(−τ_i)k ≤ max

σ∈[0,δ]kx0(σ−τ_i)−ϕ(−τ_i)k

≤ max

σ∈[0,a]kϕ(σ−τ_i)−ϕ(−τ_i)k ≤b_i (3.3) for eachi=1, . . . ,nby (3.1), i.e.,x0(s−τ_i)∈G_bi. Next, using the estimation

1 Γ(β_ij)

Z _s

0

(s−σ)^βij−1e^{−γij(s−σ)}dσ= ¹ Γ(β_ij)

Z _s

0 σ^βij−1e^−γijσdσ

≤ ¹ Γ(β_ij)

Z _s

0 σ^βij−1dσ= ^s

βij

β_ijΓ(β_ij) = ^s

βij

Γ(1+β_ij)

≤ ^δ

β_ij

Γ(1+β_ij) ≤ ^c

β_ij

Γ(1+β_ij) for alls∈ [0,t]⊂[0,δ]and eachi=0, . . . ,n, j=1, . . . ,m_i, we derive

1 Γ(β_0j)

Z _s

0

(s−σ)^β0j−1e^{−γ0j(s−σ)}x₀(σ)dσ

≤ max

σ∈[0,δ]kx0(σ)k ^c

β_0j

Γ(1+β_0j) = kϕ(0)kc^β0j

Γ(1+β_0j) ≤ kϕ(0)kb_0j

b₀+kϕ(0)k ≤b_0j for each j=1, . . . ,m₀, and

1 Γ(β_ij)

Z _s

0

(s−σ)^βij−1e^{−γij(s−σ)}x₀(σ−τ_i)dσ

≤ max

σ∈[0,δ]kx₀(σ−τ_i)k ^c

β_ij

Γ(1+β_ij) = max

σ∈[0,δ]kϕ(σ−τ_i)k ^c

β_ij

Γ(1+β_ij)

≤ (b_i+kϕ(−τ_i)k)c^βij Γ(₁+β_ij) ≤b_ij

(3.4)

for eachi=1, . . . ,n,j=1, . . . ,m_iwhere we applied (3.2). Note that estimations (3.3), (3.4) are valid forx_kinstead ofx₀without any respect tok, since it holdsx_k(σ−τ_i) = ϕ(σ−τ_i)for any σ∈[0,δ]as 0<δ ≤a≤min_i=1...,nτ_i. Therefore, the inclusion

s,x_k(s)_,x_k(s−τ₁)_{, . . . ,}x_k(s−τ_n)_,I^(β01,γ01)x_k(s)_{, . . . ,}I^{(β0m0,γ0m0)}x_k(s)_,

I^(β11,γ11)x_k(s−τ₁), . . . ,I^{(βnmn,γnmn)}x_k(s−τ_n)∈G₀, ∀s∈[0,δ] (3.5)_k holds fork =0, i.e., (3.5)₀holds. That means that the argument ofF in the definition ofx₁(t) is inG0. So,x₁(t)is well defined.

Now, assume (3.5)_k−1for somek∈N. We will show that (3.5)k follows, i.e.,x_k+1(t)is well defined on I_δ. By the above arguments, to show (3.5)_k it is enough to prove x_k(s) ∈ G_b0 and kI^(β0j,γ0j)x_k(s)k ≤b_0j for alls ∈[0,δ]andj=1, . . . ,m0. Firstly,

kx_k(s)−ϕ(0)k ≤ max

σ∈[0,δ]kx_k(σ)−ϕ(0)k ≤δ(M0+· · ·+Mn+MF)≤b0.

Secondly, using the latter estimation,

1 Γ(β_0j)

Z _s

0

(s−σ)^β0j−1e^{−γ0j(s−σ)}x_k(σ)dσ

≤ max

σ∈[0,δ]

kx_k(σ)k ^c

β0j

Γ(1+β_0j)

≤

max

σ∈[0,δ]kx_k(σ)−ϕ(0)k+kϕ(0)k

c^β0j

Γ(1+β_0j) ≤ (b0+kϕ(0)k)c^β0j Γ(1+β_0j) ≤b_0j. So, we have inductively proved that allx_k(t),k∈_Nare well-defined functions fromC_δ.

In the next step, we show that x_k(t) converges uniformly on I_δ to a solution of (1.5) as k → ∞. Using the identity x_k(s−τ_i)−x_k−1(s−τ_i) = 0 for all s ∈ [0,δ] and k ∈ _{N, we can} estimate

kx_k+1−x_kk= max

t∈[0,δ]

kx_k+1(t)−x_k(t)k

≤ max

t∈[0,δ]

"

M_A Z _t

0

kx_k(s)−x_k−1(s)kds+

Z _t

0

L₀kx_k(s)−x_k−1(s)k

+

m0

j

∑

L_0j Γ(β_0j)

Z _s

0

(s−σ)^β0j−1e^{−γ0j(s−σ)}kx_k(σ)−x_k−1(σ)kdσ

ds

#

≤δkx_k−x_k−1k M_A+L₀+

m0

j

∑

L_0jc^β0j Γ(1+β_0j)

!

=δκkx_k−x_k−1k for each k∈N. Therefore,

kx_k+1−x_kk ≤(δκ)^kkx₁−x₀k, k∈ _N0. Consequently,

∑

kx_i(t)−x_i−1(t)k ≤ kx₁−x0k

k−1 i

∑

(δκ)ⁱ, ∀t∈ [0,δ],k ∈_N.

Hence, ∑^∞i=0(δκ)ⁱ < _∞implies the uniform convergence of the series∑^∞i=1(x_i(t)−x_i−1(t))on I_δ. So, using x_k = x₀+_∑^k_i=1(x_i−x_i−1)for each k ∈ N, we see that the sequence{x_k(t)}^∞_k=0 converges uniformly on I_δ to the continuous functionx= x₀+_∑^∞_i=1(x_i−x_i−1)∈C_δ, which is a unique solution of (1.5).

4 Exponential stability for linearly bounded right-hand side

In this section, we prove a sufficient condition for the exponential stability of a trivial solu- tion of the DDE with variable coefficients, multiple delays and nonlinearity depending on fractional substantial integrals,

˙

x(t) =A(t)x(t) +B₁(t)x(t−τ₁) +· · ·+Bn(t)x(t−τn) +F(t,x(t),x(t−τ₁), . . . ,x(t−τn))

+ f

t,I^(β01,γ01)x(t)_{, . . . ,}I^{(β0m0,γ0m0)}x(t)_,

I^(β11,γ11)x(t−τ₁), . . . ,I^{(βnmn,γnmn)}x(t−τ_n), t≥0.

(4.1)

For better clarity, we conclude the main assumptions here:

(H1) there are positive numbersr_i andq_i,Θi,t_i ≥0,i=0, . . . ,nsuch that kA(t)−A(t₀)k ≤q₀e^{−r0|t−t0|}|t−t₀|^Θ0,

kB_i(t)−B_i(t_i)k ≤q_ie^{−ri|t−ti|}|t−t_i|^Θi, i=1, . . . ,n for allt≥0;

(H2) there are constantsα₁, . . . ,α_nsuch that

kB_i(t_i)e^−A(t0)τik ≤α_ie^αiτi for eachi=1, . . . ,n;

(H3) it holds γ_ij > ρ = −µ(A(t₀))−α > 0 for each i = 0, . . . ,n, j = 1, . . . ,m₀, where α= α₁+· · ·+αn andµ(A(t₀))is the logarithmic norm of the constant matrix A(t₀); (H4) for a constant 0< r ≤ _∞there are positive constantsϑ_i andδ_i ≥ 0 fori =0, . . . ,nsuch

that

kF(t,u₀, . . . ,u_n)k ≤

∑

δ_ie^−ϑitku_ik for allt≥0 andu_i ∈ _Ω(r),i=0, . . . ,n;

(H5) there are m_i ∈ _N positive constants µ_ij and η_ij ≥ 0 for i = _{0, . . . ,}n, j = _{1, . . . ,}m_i such that

kf(t,v₀₁, . . . ,v0m₀,v₁₁, . . . ,vnmn)k ≤

∑

m_i

∑

η_ije^−µijtkv_ijk for allt≥0 andv_ij ∈ _Ω(r),i=0, . . . ,n,j=1, . . . ,m_i.

Without conditions (H4), (H5), equation (4.1) could not have an exponentially stable trivial solution (see e.g. [3,9]).

Theorem 4.1. Let n ∈ _N,0 < τ₁, . . . ,τn ∈ _R,τ :=max{τ₁,τ2, . . . ,τn}, A,B₁, . . . ,Bn be N×N- matrix valued functions, and suppose that the assumptions(H1)–(H5)are satisfied. If A(t₀), B₁(t₁), . . . , B_n(t_n)are pairwise permutable, then the trivial solution of equation(4.1) is exponentially stable with respect to the ballΩ(λ)with

λ= ^rmin{1,γ} e^K

1+_∑ⁿ_j=1kB_j(t_j)k^eρτ_ρ^{j−1 (4.2)} whereγ=min_i=0,...,n

j=1,...,m_i

γ_ij^βij,

K= ^2q0Γ(_Θ0+1) r^Θ₀⁰⁺¹ + ^δ0

ϑ₀ +

m0

j

∑

η_0j µ_0j(γ_0j−ρ)^β0j +

∑

e^ρτi 2q_iΓ(_Θi+1) r^Θ_i ⁱ⁺¹ + ^δi

ϑ_i +

mi

∑

η_ij µ_ij(γ_ij−ρ)^βij

! .

(4.3)

Proof. For simplicity in notation, we shall write F(t) and f(t) omitting most of their argu- ments. Let x be a solution of equation (4.1) on the interval [0,T), 0< T < _∞with the initial function ϕ∈ C([−τ, 0],R^N)satisfying

kϕk= max

t∈[−τ,0]

kϕ(t)k ≤λ.

Let us rewrite equation (4.1) as follows:

˙

x(t) =A(t₀)x(t) +B₁(t₁)x(t−τ₁) +· · ·+B_n(t_n)x(t−τ_n) + (A(t)−A(t₀))x(t)

+ (B₁(t)−B₁(t₁))x(t−τ₁) +· · ·+ (Bn(t)−Bn(tn))x(t−τn) +F(t) + f(t), t≥0.

By Theorem2.1, xhas the form x(t) =B(t)ϕ(0) +

∑

B_j(t_j)

Z _τj

0

B(t−s)ϕ(s−τ_j)ds +

Z _t

0

B(t−s) (A(s)−A(t₀))x(s) + (B₁(s)−B₁(t₁))x(s−τ₁) +· · ·+ (_Bn(_s)−Bn(_tn))_x(_s−τ_n)_ds+

Z _t

0

B(t−s)(_F(_s) + _f(_s))_ds fort ∈[0,T], where

B(t) =e^A(t0)t

∑

∑ⁿm=1kmτm≤t k1,...,kn≥0

(t−_∑ⁿ_m=1k_mτ_m)^∑nm=1km k₁! . . .k_n!

∏

Be_m^km

andBem = Bm(tm)e^−A(t0)τm for each m=1, . . . ,n.

For now, let us assume thatr =∞. The caser <_∞is postponed to the end of the proof.

Using the assumptions and Lemmas2.2,2.4, we obtain

kB(t)k ≤ ke^A(t0)tke^αt ≤e^{(µ(A(t0))+α)t} =e^−ρt for any t≥_{0. Hence}

e^ρtkx(t)k ≤ kϕ(0)k+

∑

kB_j(t_j)k

Z _τj

0 e^ρskϕ(s−τ_j)kds +

Z _t

0 e^ρs kA(s)−A(t0)kkx(s)k+kB₁(s)−B₁(t₁)kkx(s−τ₁)k +· · ·+kB_n(s)−B_n(t_n)kkx(s−τ_n)kds+

Z _t

0 e^ρs(kF(s)k+kf(s)k)ds.

Note that

kI^(β,γ)h(t)k ≤ ¹ Γ(β)

Z _t

0

(t−s)^β−1e^−γ(t−s)kh(s)kds. (4.4) Therefore, denotingu(t)_:=_e^ρtkx(t)k_,

C:= kϕk 1+

∑

kB_j(t_j)k^e

ρτj−1 ρ

!

(4.5)

and using assumptions(H1),(H4),(H5), we obtain u(t)≤C+

Z _t

0

q₀e^{−r0|s−t0|}|s−t₀|^Θ0u(s) +

∑

q_ie^{−ri|s−ti|}|s−t_i|^Θie^ρτiu(s−τ_i)

ds +

Z _t

0 δ₀e^−ϑ0su(s) +

∑

δ_ie^−ϑise^ρτiu(s−τ_i)

! ds +

Z _t

0 e^ρs

m0

∑

η_0j Γ(β_0j)^e

−µ_0jsZ _s

0

(s−σ)^β0j−1e^{−γ0j(s−σ)}e^−ρσu(σ)dσ

+

∑

mi

∑

η_ij Γ(β_ij)^e

−µ_ijsZ _s

0

(s−σ)^βij−1e^{−γij(s−σ)}e^{−ρ(σ−τi)}u(σ−τ_i)dσ

! ds.

Let us denoteΨ(t)the right-hand side of the latter inequality. Clearly, it is a nondecreasing function satisfyingΨ(0) =C. To estimate the delayed terms, we use the inequality

u(s−τ_i)≤ max

σ∈[0,s]u(σ−τ_i)≤ max

σ∈[−τ,s]u(σ)

=max

max

σ∈[−τ,0]u(σ), max

σ∈[0,s]u(σ)

≤max

max

σ∈[−τ,0]e^ρσkϕ(σ)k, max

σ∈[0,s]Ψ(σ)

≤max{C,Ψ(s)}=_Ψ(s) for anys∈ [0,t]and eachi=1, . . . ,n. So we obtain

e^ρs Z _s

0

(s−σ)^β0j−1e^{−γ0j(s−σ)}e^−ρσu(σ)dσ

≤ _Ψ(_s)

Z _s

0

σ^β0j−1e^{−(γ0j−ρ)σ}dσ≤_Ψ(_s)

Z _∞

0

σ^β0j−1e^{−(γ0j−ρ)σ}dσ

= ^Ψ(s)_Γ(β_0j) (γ_0j−ρ)^β0j

(4.6)

for alls∈ [_0,t]_{and each} j=_{1, . . . ,}m₀. Analogously, e^ρs

Z _s

0

(s−σ)^βij−1e^{−γij(s−σ)}e^{−ρ(σ−τi)}u(σ−τ_i)dσ ≤ ^Ψ(s)e^ρτiΓ(β_ij)

(γ_ij−ρ)^{βij (4.7)} for alls∈ [_0,t]_{and each}i=_{1, . . . ,}n, j=_{1, . . . ,}m_i. Therefore, we arrive at

Ψ(t)≤C+

Z _t

0 b(s)_Ψ(s)ds, t∈[0,T] (4.8) where

b(s) =q₀e^{−r0|s−t0|}|s−t₀|^Θ0+δ₀e^−ϑ0s+

m0

j

∑

η_0je^−µ0js (γ_0j−ρ)^β0j +

∑

e^ρτi q_ie^{−ri|s−ti|}|s−t_i|^Θi +δ_ie^−ϑis+

mi

j

∑

η_ije^−µijs (γ_ij−ρ)^βij

! .

(4.9)

Note that

Z _t

0 e^{−ri|s−ti|}|s−t_i|^Θids≤

Z _∞

0 e^{−ri|s−ti|}|s−t_i|^Θids

=

Z ₀

−ti

e^−ri|s||s|^Θids+

Z _∞

0 e^−riss^Θids

≤2 Z _∞

0 e^−riss^Θids= ^2Γ(_Θi+1) r_i^Θi+1 for each i=0, . . . ,n. So, it holds

Z _t

0 b(s)ds≤

Z _∞

0 b(s)ds≤K.

Applying the Gronwall’s inequality to (4.8) then gives Ψ(t)≤Cexp

_{Z t}

0 b(s)ds

≤ Ce^K <_∞ for any t≥0. That means

kx(t)k=e^−ρtu(t)≤e^−ρtΨ(t)≤Ce^Ke^−ρt ∀t∈[0,T). (4.10) Since the right-hand side is independent ofT, the estimation holds for anyt ≥0.

The condition (4.2) on λ enables to apply estimations of kF(t)k and kf(t)k during the proof. If condition (4.2) holds, from (4.10), one can see that kx(t)k ≤ r for all t ∈ [0,T). Clearly, it is true also fort∈ [−τ, 0]. Next, from (4.4) and (4.10), we get

kI^(β,γ)x(t)k ≤ ^Ce

K

Γ(β)

Z _t

0

(t−s)^β−1e^−γ(t−s)ds≤ ^Ce

K

γ^β . (4.11)

The same holds with x(t−τ_i)for anyi=1, . . . ,ninstead ofx(t). So again, we can apply the estimation ofkf(t)kdue to (4.2).

Finally, ifr <∞, the statement follows from the previous case using the Urysohn’s lemma [1, Lemma 10.2].

We would like to emphasize that in the above theorem, the commutativity of matrix func- tions A,B₁, . . . , B_nat generaltis not required.

5 Exponential stability for power nonlinearities on right-hand side

Here we investigate the case of more general functions F and f on the right-hand side of equation (4.1). In particular, we consider the modified assumptions:

(H4’) for a constant 0 < r ≤ _∞there are ϑ_i > 0, δ_i, ˜δ_i, ˜ϑ_i ≥0 and ω_i >1 for i= 0, . . . ,nsuch that

kF(t,u₀, . . . ,u_n)k ≤

∑

δ_ie^−ϑitku_ik+δ^˜_ie^−ϑ˜itku_ik^ωi for allt≥_{0 and}u_i ∈ _Ω(r)_,i=_{0, . . . ,}n;