We also prove an existence and uniqueness theorem for this type of equations, which we use in the proof of the stability theorem

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXPONENTIAL STABILITY OF SOLUTIONS OF NONLINEAR FRACTIONALLY PERTURBED ORDINARY DIFFERENTIAL

EQUATIONS

EVA BRESTOVANSK ´A, MILAN MEDVE ˇD Communicated by Mokhtar Kirane

Abstract. The main aim of this paper is to prove a theorem on the expo- nential stability of the zero solution of a class of integro-differential equations, whose right-hand sides involve the Riemann-Liouville fractional integrals of dif- ferent orders and we assume that they are polynomially bounded. Equations of that type can be obtained e.g. from fractionally damped pendulum equa- tions, where the fractional damping terms depend on the Caputo fractional derivatives of solutions. The set of initial values of solutions that converge to the origin is also determined. We also prove an existence and uniqueness theorem for this type of equations, which we use in the proof of the stability theorem.

1. Introduction

Recently, fractional differential equations with fractional derivatives and frac- tional integrals of different types have attracted many scientists from various disci- plines due to their wide applications. The most known are the Riemann-Liouville and the Caputo derivatives and differential equations with these derivatives. The basic theory of fractional differential equations and many references can be found in the monographs [8, 23, 29]. These derivatives are defined as follows:

The Riemann-Liouville fractional derivative of a function u: [0,∞)→ Rof an orderα∈(0,1) is

RLD^αu(t) := 1 Γ(α)

d dt

Z t

0

(t−s)^α−1u(s)ds (1.1)

and the Caputo derivative is

CD^αu(t) := 1 Γ(1−α)

Z t

0

(t−s)^−αu⁰(s)ds, (1.2)

2010Mathematics Subject Classification. 34A08, 34A12, 34D20, 37C75.

Key words and phrases. Riemann-Liouville integral; Riemann-Liouville derivative;

Caputo derivative; fractional differential equation; exponential stability.

c

2017 Texas State University.

Submitted May 5, 2017. Published November 10, 2017.

1

where u⁰(t) = ^du(t)_dt ,Γ(z) = R∞

0 τ^z−1e^−τdτ is the Euler Gamma function and the integral

RLI^αu(t) = 1 Γ(α)

Z t

0

(t−s)^α−1u(s)ds (1.3)

is called the Riemann-Liouville fractional integral of order αof the functionu(t), provided that the right-hand sides of (1.1), (1.2) and (1.3), respectively, exist.

Ifu: [0,∞)→R^N,u(t) = u1(t), u2(t), . . . , uN(t)

, then, we define the Riemann- Liouville fractional derivative of the mappingu(t) of orderαas

RLD^αu(t) = ^RLD^αu1(t),^RLD^αu2(t), . . . ,^RLD^αuN(t)

(1.4) and the Riemann-Liouville fractional integral as

RLI^αu(t) = ^RLI^αu1(t),^RLI^αu2(t), . . . ,^RLI^αuN(t)

. (1.5)

An influence of viscous fluids on vibrating systems is often modeled by using the Riemann-Liouville or Caputo fractional derivative. These derivatives play the role of damping force, called the fractional damping. The well known Bargley-Torvik equation (see [2])

u⁰⁰(t) +A^CD³²u(t) =au(t) +φ(t), (1.6) modelling the motion of a rigid plate immersing in a viscous liquid, is one of the equations descibing the motion with the fractional damping termA^CD³²u(t).

It is well known that the system of linear fractional differential equations D^αx(t) =Ax(t), x(t)∈R^N, α∈(0,1), (1.7) whereD^αx(t) is the Riemann-Liouville or the Caputo derivative ofx(t) of the order α∈(0,1) andAis a constant matrix, do not have exponentially stable solutions, but asymptoticall stable only. The equilibriumx= 0 of this equation is asymptotically stable if and only if|arg(λ)|> ^απ₂ for all eigenvalues of the matrixA. In this case all components ofx(t) decay towards 0 liket^−α (see [11, 12, 20, 21, 9]).

We will show that fractional integro-differential equations of the form

˙

x(t) =Ax(t) +f t, x(t),^RLI^α1x(t), . . . ,^RLI^αmx(t)

, x(t)∈R^N (1.8) can have exponentially stable solutions, where the solution is defined as in the next definition.

Definition 1.1. A mappingx: [0, T)→R^N,where 0< T ≤ ∞, is a solution of the equation (1.8) satisfying the initial conditionx(0) =x0∈R^N if it is continuously differentiable on the interval (0, T), continuous on [0, T) and it satisfies the equality (1.8) for allt∈(0, T). IfT =∞, then this solution is called global.

Equations of the form (1.8) can be obtained from the following linear multi- fractional pendulum equation

u⁰⁰(t) +λ1(t)^CD^β1u(t) +· · ·+λm(t)^CD^βmu(t) +λu⁰(t) +ω²u(t) = 0 (1.9) with m fractional and one ordinary damping terms, which can be written as a system of the form (1.8) with

A=

0 1

−ω² −λ

, x(t) = x1(t)

x2(t)

,

f(t, x(t),^RLI^α1x(t), . . . ,^RLI^αmx(t))

=

0

−λ1(t)^RLI^α1x2(t)− · · · −λm(t)^RLI^αmx2(t)−λ(t)x2(t)

,

wherex₁(t) =u(t),x₂(t) =u⁰(t), β_i ∈(0,1), α_i = 1−β_i,λ_i(t)i= 1,2, . . . , m are continuous functions on [0,∞) andλ >0,ω >0 are constants.

Let us recall an analysis of the fractional vibration equation

u⁰⁰(t) +b^CD^αu(t) +cu(t) = 0, α∈(0,1), (1.10) whereb >0,c >0 are constants, given in the papers [10] and [18]. It is proven in [18] that ifu(t) is a solution of the equation (1.10) satisfying the initial conditions u(0) =u0,u⁰(0) =u1, then its Laplace transform is

L[u(t)] =U(s) =s+bs^α−1+c

s²+bs^α+c u0+ 1

s²+bs^α+cu1, the characteristic equation

s²+bs^α+c= 0 (1.11)

has a couple of complex conjugate roots

s1,2=β±iσ=r^±iΘ, β <0, σ >0, r=p

β²+σ²>0, π

2 <Θ< π and the fundamental solutionφ1(t) with

L[φ1(t)] = Φ1(s) = s+bs^α−1+c s²+bs^α+c has the form

φ1(t) =Ce^βtcosσt+De^βtsinσt+ Z ∞

0

Kα(τ)e^{−τ t}dτ, whereC, D are functions of the variablesr,Θ. The function

f₁(t) =Ce^βtcosσt+De^βtsinσt

represents a decaying oscillation along the t-axis, where the amplitude decays ex- ponentially. The function L[φ1(t)] = Φ1(s) has the asymptotic representation Φ₁(s) ∼ ^b_cs^α−1 as s → 0 and hence the function f₂(t) = R∞

0 K_α(τ)e^{−τ t}dτ has the asymptotic representation f₂(t) ∼ ^b_cΓ(1−α)^t−α as t → ∞. The derivative u⁰(t) has a similar asymptotic representation u⁰(t) ∼ ^b_c^(−α)t_Γ(1−α)^−α−1 as t → ∞. We con- clude that the solutionu(t) of the equation (1.10) decays towards 0 ast→ ∞ like t^−α and u⁰(t) has similar asymptotic properies. This means that the equilibrium x= (x1, x2) = (u, u⁰) = (0,0) of the system of equations, corresponding to the the solutionu(t) of the equation (1.10), is asymptotically stable, but not exponentially.

We were motivated by the paper [25], where an existence and uniqueness result for the initial value problem

Au⁰⁰+

N

X

k=1

BkcD^αku(t) =f(t), u(0) =u0, u⁰(0) =c1 (1.12) with 0< αk <2, k= 1,2, . . . , N is proved. The Caputo fractional derivatives in the equation (1.12) play there the role of damping terms.

In the paper [5] the initial value problem

RLD^αx(t) =f(t, x(t)), t >0, lim

t→0t^1−αx(t) =b, α∈(0,1), b∈R, (1.13)

where f: R⁺×R → R, φ:R⁺ → R⁺ are continuous functions, is studied. It is assumed there that

|f(t, x)| ≤t^µφ(t)e^−σt|u|^m, for all (t, u)∈R⁺×R, (1.14) whereµ≥0,m >1,σ >0. Using the desingularization method, proposed in [14]

and the Bihari inequality, it is proven there that if kφkq =

Z ∞

0

φ(s)^qds < L:= Γ(α) 2^1+m−α|b|^m−1

2^m m−1

1/qh (σp)^λ1 Γ(λ1)(1 +^λ_λ¹

2) i1/p

, wherepq=p+q,λ1= 1 +p(µ−(1−α)m],λ2= 1 +p(α−1), then any solutionx(t) of the initial value problem (1.13) is global and there exists a constantc >0 such that|x(t)| ≤ _t1−α^c for allt∈(0,∞). This means that the trivial solutionx0(t)≡0 is asymptotically stable. It is obvious that similar results for more general type of power nonlinearities are extraordinary complicated. In the paper [15] the equation

˙

x(t) =Ax(t) +f t, x(t),^RLI^α1[g1x](t), . . . ,^RLI^αm[gmx](t)

, x(t)∈R^N, (1.15) wheref:R×R^N×R^N →R^N is a continuous mapping,gi:R×R^N →R^N,(t, x)7→

gi(t, x),i= 1,2, . . . , mare continuous mappings and

RLI^αi[gix](t) := 1 Γ(α_i)

Z t

0

(t−s)^αi−1gi(s, x(s))ds, 0< αi<1, i= 1,2, . . . , m (1.16) is studied. A sufficient condition for the exponential stability of the trivial solution x(t)≡0 of this equation is proven there. In the paper [16] a sufficient condition for the non-existence of blow-up solutions for a fractional functional-differential equations of the form

˙

x(t) =Ax(t) +h

t, x(t), xt,(I^α1[g1x])(t), . . . ,(I^αm[gmx])(t)

, t >0, x(t) = Φ(t), t∈[−r,0],

(1.17) where r > 0, Φ ∈ C_r := C([−r,0], X), X is a Banach space, x(t) ∈ X, x_t ∈ C, x_t(Θ) :=x(t+ Θ)t >0, Θ∈[−r,0],Ais the infinitesimal generator of a strongly continuous semigroup{S(t)}_t≥0,S(t) :=e^At,h:R+×X×C_r×X^m→X,X^m:=

X × · · · ×X (mtimes) is a continuous map, R⁺ = [0,∞), gi:R⁺ ×X → X, (t, x)7→gi(t, x),i= 1,2, . . . , mare continuous maps, is proved

In this paper, we study equation (1.15) with gi(t, x) ≡ x(t), i.e., we have the Riemann-Liouville fractional integrals of x(t) in the equation (1.8) instead of the nonlinear functions gi(t, x(t)). Moreover, the mapping f is more general than in [15]. The aim is to give some conditions under which the trivial solution of this equation is exponentially stable.

2. Existence and uniqueness result

In this section, we prove a local existence and uniqueness result concerning the initial value problem

˙

x(t) =Ax(t) +f

t, x(t),^RLI^α1x(t), . . . ,^RLI^αmx(t)

, t >0, x(t)∈R^N, x(t0) =x0.

(2.1) Many papers are devoted to the fractional initial value problem

RLD^αx(t) =f(t, x(t)), x(t₀) =x₀, (2.2)

where the right-hand side is independent of fractional derivatives and fractional integrals, respectivelly. Some basic existence results for the problem (2.2) can be found in the monograph [8], where they are proved by using the classical Picard method of successive approximations. Many local and global existence results for various classes of fractional differential equations are proved by using fixed point theorems (see, e.g., the monograph [29], [6] and the paper [30]). Some existence results for the second-order abstract differential equations on Banach spaces, in- volving several fractional derivatives on their right-hand sides are proved in the papers [7, 27, 28]. In its proof, we use the method of Picard successive approx- imations. There is a problem to apply the Banach fixed point theorem without the assumption of the global boundedness of the mapping f, because there are fractional integrals of the unknown functionx(t) in its arguments.

Theorem 2.1. Let G⊂R×R^N be a region,Hm⊂R^m is a region with 0∈Hm

andf ∈C(G×Hm,R^N) be a continuous locally Lipschitz mapping. Then for any (t0, x0)∈ G, t0 ≥ 0, there exists a δ >0 such that the initial value problem (2.1) has a unique solutionx(t)on the intervalI_δ = [t₀, t₀+δ).

Proof. Let

G₀=

(t, x, u₁, . . . , u_m)∈G×H_m:t₀≤t≤t₀+a, t₀≥0,

kx−x₀k ≤b,ku_ik ≤ kx₀k+b, i= 1,2, . . . , m , (2.3) for somea >0,b >0. Let

M₁= max

kx−x0k≤bkAxk, M₂= max

(t,x,u1,...,um)∈G0

kf(t, x, u₁, . . . , u_m)k and the mappingf satisfies the condition

kf(t, x, u1, u2, . . . , um)−f(t, y, v1, v2, . . . , vm)k ≤L0kx−yk+

m

X

i=1

Likui−vik (2.4) for all (t, x, u1, u2, . . . , um),(t, y, v1, v2, . . . , vm)∈G0. Let

0< δ= min

a, b

M1+M2

, c, 1

kAk+L0+Pm i=1Li

, where c = min1≤i≤m

Γ(αi)αi

_αi¹

. Let Cδ := C(Iδ,R^N) be the Banach space of continuous mappings fromIδ intoR^N endowed with the metricd(h, g) :=kh−gk:=

max_t∈Iδkh(t)−g(t)k. Let us define the successive approximations {xn}^∞_n=0, xn ∈ Cδ:=C(Iδ,R^N),Iδ = [t0, t0+δ], by

x₀(t)≡x₀, xn+1(t)

=x0+ Z t

t₀

Axn(s)ds+ Z t

t₀

f

s, xn(s), 1 Γ(α1)

× Z s

0

(s−τ)^α1−1xn(τ)dτ, . . . , 1 Γ(α_m)

Z s

0

(s−τ)^αm−1xn(τ)dτ ds, n= 0,1,2, . . . t∈I_δ.

(2.5)

First, let us prove thatkxn(t)−x0k ≤bfor alln≥1,t∈Iδ. From the definition of the numberc it follows that

1 Γ(αi)

Z t

0

(t−s)^αi−1ds≤ 1 Γ(αi)

δ^αi αi

≤ 1 Γ(αi)

c^αi δ^αi ≤ 1

Γ(αi)

Γ(α_i)α_i αi

= 1 (2.6) fori= 1,2, . . . , m; and so, we have

k 1 Γ(αi)

Z t

t₀

(t−τ)^αi−1x0dτk ≤ 1 Γ(αi)

δ^αi αi

[kx0k+b]≤ kx0k+b, (2.7) fori= 1,2, . . . , m,t∈Iδ, and i= 1,2, . . . m. Hence, the first approximationx1(t) is well defined and

kx₁(t)−x₀k ≤M₁δ+M₂δ= (M₁+M₂)δ≤(M₁+M₂) b

M₁+M₂ =b, fort∈Iδ. This yields the inequality

kx1(t)k ≤ kx0k+b for allt∈I_δ. and thus

t, x1(t), 1 Γ(α1)

Z t

0

(t−τ)^α1−1x1(τ)dτ, . . . , 1 Γ(αm)

Z t

0

(t−τ)^αm−1x1(τ)dτ

∈G0

for all t ∈Iδ. Now, we find by using the Lipschitz condition (2.4) and inequality (2.6) that

kx2(t)−x1(t)k ≤δ(kAk+L0)kx1(t)−x0(t)k +

m

X

i=1

L_i Γ(αi)

Z t

t₀

Z s

0

(s−τ)^αi−1kx₁(τ)−x₀(τ)kdτ ds

≤δ(kAk+L0)kx1−x0k +

m

X

i=1

L_i Γ(αi)

Z t

t₀

Z s

0

(s−τ)^αi−1dτ ds

kx1−x₀k

≤δkkx1−x0k,

(2.8)

wherek=kAk+L0+Pm

i=1Li and so, we get kx2−x1k ≤kδkx1−x0k.

Now assume that the estimate

kxn(t)−x_n−1(t)k ≤(kδ)ⁿ⁻¹

holds forn >2. Then, using this inequality, the Lipschitz condition (2.4) and the inequality (2.6), one can get

kxn+1(t)−x_n(t)k ≤(kδ)ⁿkx1−x₀k and so, we have

kxn+1−xnk ≤(kδ)ⁿkx1−x0k.

Since

x_n(t) =x₀(t) +

n

X

i=1

[x_i(t)−x_i−1(t)] with x₀(t)≡x₀, (2.9)

we obtain kx₀(t) +

n

X

i=1

[x_i(t)−x_i−1(t)]k ≤ kx₀k+

n

X

i=1

kx_i(t)−x_i−1(t)k

≤ kx0k+

n

X

i=1

(kδ)ⁱ

kx1−x0k,

(2.10)

for all t ∈ I_δ. From the definition of δ it follows that kδ < 1, and so the series kx0k+P∞

i=1(kδ)ⁱis convergent. This yields the uniform convergence of the sequence {x_n(t)}^∞_i=0 on the intervalI_δ to a continuous mappingx∈C_δ. This implies

n→∞lim 1 Γ(αi)

Z s

0

(s−τ)^αi−1x_n(τ)dτ

= 1

Γ(αi) Z s

0

(s−τ)^αi−1x(τ)dτ fori= 1,2, . . . , m, s∈I_δ

(2.11)

and therefore

n→∞lim f

s, xn(s), 1 Γ(α1)

Z s

0

(s−τ)^α1−1xn(τ)dτ, . . . , 1

Γ(αm) Z s

0

(s−τ)^αm−1xn(τ)dτ ds

=f

s, x(s), 1 Γ(α₁)

Z s

0

(s−τ)^α1−1x(τ)dτ, . . . , 1

Γ(α_m) Z s

0

(s−τ)^αm−1x(τ)dτ

ds for alls∈Iδ.

(2.12)

Therefore from (2.5) it follows that x(t) is a solution of the initial value problem (2.1), defined on the interval Iδ. Now let us prove its uniqueness. Assume that there are two different solutions x, y ∈ Cδ of the initial value problem (2.1). Let w(t) :=kx(t)−y(t)k,t∈Iδ andW = max_t∈Iδw(t). Then, by using the Lipschitz condition (2.4) and the inequality (2.6) we obtain

w(t)≤(kAk+L₀) Z t

t₀

w(s)ds+

m

X

i=0

Li

Γ(αi) Z t

t₀

Z s

0

(s−τ)^αi−1kx(τ)−y(τ)kdτ ds

+δ

kAk+L0+

m

X

i=0

Li

Γ(α_i) Z t

t0

(t−τ)^αi−1dτ W

≤δ

kAk+L0+

m

X

i=0

Li

W

= (δk)W for allt∈Iδ

(2.13) and this yields the inequalityW ≤(kδ)W < W. This is a contradiction and hence,

we havex(t) =y(t) for allt∈Iδ.

3. Stability Theorem

In this section, we prove a result on the exponential stability of the trivial solution x(t)≡0 of the equation (1.8). In its proof, we apply a desingularization method, proposed in the paper [14], where it is applied in the study of nonlinear integral inequalities with weakly singular kernels.

We assume that the following conditions are satisfied:

(A1)

ke^Atxk ≤Ke^−atkxk for allt≥0, x∈R^N, whereK >0,a >0 are constants;

(A2) The mapping f: R×R^N ×R^mN →R^N is continuous and it satisfies the condition

kf(t, x, u1, u2, . . . , um)k ≤µ1(t)e^−γ1tkxk+

m

X

i=1

λi1(t)e^−γi1tkuik

+

l

X

j=2

µ_j(t)kxk^kj +

l

X

j=2 m

X

i=1

λ_ij(t)ku_ik^kj

(3.1)

for all (t, x, u1, u2, . . . , um) ∈ R×R^N ×R^mN, where µj(t), λij(t), i = 1,2, . . . , m, j = 1,2, . . . , l are nonnegative continuous functions on [0,∞), γ1 > 0, γi1 > a+ 1, i = 1,2, . . . , m, 1 = k1 < k2 < · · · < kl, kzk = max{|z1|,|z2|, . . . ,|zN|};

(A3) There exist numberspi >1,i= 1,2, . . . , msuch that pi(αi−1) + 1>0, i= 1,2, . . . , m and

ω_j :=

Z ∞

0

µ_j(s)^qds <∞, j = 1,2,3, . . . , l, (3.2) whereq=q1q2. . . qm,qi= _p^pi

i−1,i= 1,2, . . . , m;

(A4)

ηi1:=

Z ∞

0

e^{−[γi1−(a+1)]s}λi1(s)ds <∞, η_ij :=

Z ∞

0

e^(a+kj)ss^kj

−1

qi λ_ij(s)ds <∞, i= 1,2, . . . , m, j= 2,3, . . . , l,

(3.3)

whereq1, q2, . . . , qm are defined as in (A3).

(A5) The mappingf(t, x, u1, u2, . . . , um) is locally lipschitz with respect to the variablesx, u1, . . . , um.

In the proof of the main result, we use the following corollary of the Pinto’s inequality (see [22, Theorem 1], [1, Theorem 10.2] and [26, Example 5]). We present it in the form of the next lemma also with its proof, because we did not find this formulation in literature.

Lemma 3.1. Let c >0 be a constant,Ψ_j(t),j= 1,2, . . . , lbe continuous, nonneg- ative functions on[a,∞)andu(t)be a continuous nonnegative function satisfying the integral inequality

u(t)≤c+

l

X

j=1

Z t

a

Ψ_j(s)u(s)^kjds, t∈[a,∞),

wherea∈R,1 =k1≤k2<· · · ≤kl. Let the following conditions be satisfied:

(kj−1)(cDj)^kj−1 Z ∞

a

Ψjs)ds <1, j= 2,3, . . . , l, Z ∞

0

Ψ1(s)ds <∞, (3.4)

where

D1=e^{R0∞Ψ1(s)ds}, Dj =

1−(kj−1)(Dj−1c)^kj−1 Z ∞

a

Ψj(s)ds−_kj¹₋₁

, j = 2,3, . . . , l.

(3.5) Then

u(t)≤cD₁D₂. . . D_l, for all t∈[a,∞). (3.6) Proof. By [22, Theorem 1] (see also [26, Example 5]),

u(t)≤W_l⁻¹h

W_l(c_l−1(t)) + Z t

a

Ψ_l(s)dsi

, (3.7)

where

c0(t)≡c, ci(t) =W_i⁻¹h

Wi(c_i−1(t)) + Z t

a

Ψi(s)dsi , W_i(z) =

Z z

u_i

dy

y^ki, y≥u_i>0, i= 1,2, . . . , l.

(3.8)

One can calculate that W₁(y) = 1

1−k1

y^1−k1−c^1−k1

, W₁⁻¹(u) =

c^1−k1−(k₁−1)u−_k¹

1−1

and

c1(t) =W₁⁻¹h

W1(c) + Z t

a

Ψ1(s)dsi

≤cD1. (3.9)

From the assumption (3.4) it follows that 0< D₁<∞. Using the inequality (3.9), we obtain

c₂(t)≤W₂⁻¹

W₂(c₁(t)) + Z t

a

Ψ₂(s)ds

≤W₂⁻¹

W₂(cD₁) + Z t

a

Ψ₂(s)ds

≤h

(D1c)^1−k2−(k2−1) Z t

a

Ψ2(s)dsi−_k¹

2−1

≤cD1D2,

(3.10)

where

D2=h

1−(k2−1)(cD1)^k2−1 Z ∞

a

Ψ2(s)dsi−_k¹

2−1

. Now, assume that

c_l−1(t)≤cD₁D₂· · ·D_l−1.

Using the same arguments as above one can prove inequality (3.6).

Theorem 3.2. Let conditions(A1)–(A5)be satisfied andkx0k< ρ, whereρ=∞, if l= 1 and if l >1, then

ρ= sup

z∈R:|C(z)Dj−1(z)|<h 1 (kj−1)Gj

i_kj¹−1

, i= 2,3, . . . , l , (3.11) where

C(z) =d^qK^qz^q, d=m(l+ 1) + 2, D₁(z)≡G₁, D₂(z) =

1−(k₂−1)[C(z)D₁(z)]^k2−1G₂−_k¹

2−1, D_j(z) =

1−(k_j−1)

C(z)D_j−1(z)^kj−1

G_j−_kj¹₋₁

, j= 3, . . . , l,

(3.12)

G₁=d^q−1K^qn 1 γ1p

^q/p ω₁+

m

X

i=1

Q_i Γ(αi)

^q η_i1^q 1

aqipˆi

_piˆ¹ 1 q

o,

G_j=d^q−1K^qnh 1 (k_j−1)ap

iq/p

ω_j+

m

X

i=1

Qi

Γ(α_i) kjq

η_ij^q 1 aq_ipˆ_ik_j

_piˆ¹ 1 k_jq

o ,

(3.13)

j = 2,3, . . . , l, pi > 1, qi = _p^pi

i−1, i = 1,2, . . . , m, q = q1q2. . . qm, p = _q−1^q , ˆ

q_i=q₁. . . q_i−1q_i+1. . . q_m,pˆ_i= _qˆ^qˆi

i−1,

Qi=Γ(p_i(α_i−1) + 1) p^p_i^i(αi−1)+1

^1/pi

. (3.14)

Then for the solutionx(t)of the equation (1.8), satisfying the conditionx(0) =x₀, the following inequality holds:

kx(t)k ≤Ψ(kx₀k)e^−at for allt∈[0,∞), (3.15) where the function Ψ(w) is defined as

Ψ(w) =h

C(w)D1(w)D2(w)D2(w). . . Dl(w)i^1/q

, |w|< ρ, for whichlim_w→0Ψ(w) = 0 and ifl >1, then lim_w→ρ−Ψ(w) =∞.

Proof. Letx(t) be the maximal solution of the equation (1.8), defined on the interval [0, d), satisfying the condition x(0) =x0∈R^N. From Theorem 2.1 it follows that this solution exists. Then

x(t) =e^Atx0+ Z t

0

e^A(t−s)f s, x(s),^RLI^α1x(s), . . . ,^RLI^αmx(s)

ds, t∈[0, d) and the conditions (A1), (A2) yield

kx(t)k ≤Ke^−atkx0k+Ke^−at Z t

0

e^{−(γ1−a)s}µ₁(s)kx(s)kds +Ke^−at

m

X

i=1

Z t

0

e^−(γi1a)sλi1(s)k^RLI^αix(s)kds

+Ke^−at Z t

0

e^asX^l

j=2

µ_j(s)kx(s)k^kj +

l

X

j=2 m

X

i=1

λ_ij(s)k^RLI^αix(s)k^kj ds.

(3.16) Ifu(t) =e^atkx(t)k, thenkx(t)k=e^−atu(t),kx(t)k^kj =e^−akjtu(t)^kj,

k^RLI^αix(s)k= 1 Γ(αi)

Z t

0

(t−s)^αi−1x(s)ds

≤ 1 Γ(αi)

Z t

0

(t−s)^αi−1kx(s)kds

≤ 1 Γ(αi)

Z t

0

(t−s)^αi−1e^−asu(s)ds,

(3.17)

k^RLI^αix(s)k^kj = 1 Γ(αi)^kj

Z t

0

(t−s)^αi−1x(s)ds

k_j

≤ 1

Γ(αi)^kj Z t

0

(t−s)^αi−1kx(s)kds^kj

≤ 1

Γ(αi)^kj Z t

0

(t−s)^αi−1e^−asu(s)ds^kj

.

(3.18)

Now, we apply the desingularization method as follows:

Z t

0

(t−s)^αi−1e^se^−se^−asu(s)ds

≤Z t 0

(t−s)^pi(αi−1)e^pisds^1/piZ t 0

e^−qise^−aqisu(s)^qids^1/qi

(3.19)

withqi= _p^pi

i−1,pi(αi−1) + 1>0. By the following inequality, proved in [14] (see also [17]),

Z t

0

(t−s)^pi(αi−1)e^pisds1/pi

≤Q_ie^t, where the numberQi is given by (3.14),we obtain the inequality

Z t

0

(t−s)^αi−1kx(s)kds^kj

≤Q^k_i^je^kjtZ t 0

e^−qise^−aqisu(s)^qids^kj/q_i

. (3.20) Then the H¨older inequality yields

Z t

0

e^−qise^−aqisu(s)^qids^kj

≤t^kj−1 Z t

0

e^−qikjse^−aqikjsu(s)^kjqids, j >2 and hence, we have the inequality

k^RLI^αix(s)k^kj ≤ Qi

Γ(α_i) kj

e^kjss

kj−1 qi

Z s

0

e^−qikjτe^−aqikjτu(τ)^kjqidτ1/qi

. Thus, we obtain the inequality

u(t)≤Kkx0k+K Z t

0

e^−γ1sµ1(s)u(s)ds +K

m

X

i=1

Qi

Γ(αi) Z t

0

e^{−[γi1−(a+1))s]}λi1(s)Z s 0

e^−qiτe^−aqiτu(τ)^qidτ^1/qi

ds

+K

l

X

j=2

Z t

0

e^{−(kj−1)a)s}µ_j(s)u(s)^kj ds+K

l

X

j=2 m

X

i=1

Q_i Γ(αi)

^kj

× Z t

0

λij(s)e^(a+kj)ss

kj−1 qi

Z s

0

e^−qikjτe^−aqikjτu(τ)^kjqidτ^1/qi

ds.

Ifq=q₁q₂. . . q_mandd=m(l+ 1) + 2, then the inequality (z₁+z₂+· · ·+z_d)^q≤ d^q−1(z₁^q+z₂^q+· · ·+z_m^q), valid for any z₁, z₂, . . . , z_d≥0, yields

u(t)^q

≤d^q−1K^qkx0k^q+d^q−1K^qZ t 0

e^−γ1sµ1(s)u(s)dsq

+d^q−1K^q

m

X

i=1

Q_i Γ(αi)

qhZ t

0

e^{−[γi1−(a+1)]s}λ_i1(s)Z s 0

e^−qiτe^−aqiτu(τ^qidτ dsiq

+d^q−1K^q

l

X

j=2

hZ t

0

e^{−(kj−1)as}µj(s)u(s)^kjdsiq

+d^q−1K^q

l

X

j=2 m

X

i=1

Qi

Γ(α_i) kjq

×hZ t 0

λ_ij(s)e^(a+kj)ss^kj

−1 qi

Z s

0

e^−aqikjτe^−kjqiτu(τ)^kjqidτ^1/qi

dsi^q

≤d^q−1K^qkx0k^q+d^q−1K^q 1 γ₁p

q/pZ t

0

µ1(s)^qu(s)^qds +d^q−1K^q

m

X

i=1

Q_i Γ(αi)

q

η_i1Z t 0

e^−qiτe^−aqiτu(τ)^qidτqˆi

+d^q−1K^q

l

X

j=2 m

X

i=1

Qi

Γ(αi) ^kjq

η^q_ijZ t 0

e^−qikjτe^−aqikjτu(τ)^qikjdτ^qˆi

,

where ˆqi=q1q2. . . q_i−1qi+1. . . qm.

Using H¨older’s inequality with ˆqi,pˆi=_qˆ^qˆi

i−1 and withp, q, we obtain Z t

0

e^−γ1sµ1(s)u(s)ds^q

≤Z t 0

e^−γ1psds^q/pZ t 0

µ1(s)^qu(s)^qds

≤ 1 γ1p

^q/pZ t

0

µ₁(s)^qu(s)^qds,

(3.21)

Z t

0

e^{−[γi1−(a+1)]s}λi1(s)u(s)ds^q

≤ 1

[γi1−(a+ 1)]p

^q/pZ t

0

µi1(s)^qu(s)^qds,

Z t

0

e^asµ_j(s)e^−kjasu(s)^kjdsq

≤

e^{−(kj−1)aps}dsq/pZ t 0

µ_j(s)^qu(s)^kjqds

≤h 1 (kj−1)ap

i^q/pZ t

0

µj(s)^qu(s)^kjqds,

hZ t

0

λij(s)e^(a+kj)ss

kj−1 qi

Z s

0

e^−aqikjτe^−kjqiτu(τ)^kjqidτ1/qi

dsiq

≤Z t 0

λij(s)e^(a+kj)ss

kj−1

qi ds^qZ t 0

e^−aqikjτe^−kjqiτu(τ)^kjqidτ^qˆi

,

(3.22)

Z t

0

e^−aqikτe^−kjqiτu(τ)^kqidτ^qˆi

≤Z t 0

e^{−aqipˆikτ}dτ_piˆ¹Z t 0

e^−kjqτu(τ)^kjqdτ

≤ 1 aqipˆikj

_piˆ¹ Z t

0

e^−kjqτu(τ)^kjqdτ.

(3.23)

The above inequalities yield u(t)^q

≤d^q−1K^qkx₀k^q+d^q−1K^q 1 γ₁p

q p

Z t

0

µ₁(s)^qu(s)^qds +d^q−1K^q

m

X

i=1

Q_i Γ(αi)

^q η^q_i1 1

aqipˆi

_piˆ¹ Z t

0

e^−qτu(s)^qds

+d^q−1K^q

l

X

j=2

h 1 (k_j−1)ap

iq/pZ t

0

µj(s)^qu(s)^kjqds

+d^q−1K^q

l

X

j=2 m

X

i=1

Q_i Γ(αi)

^kjq

η^q_ij 1 aqipˆikj

_piˆ¹ Z t

0

e^−kjqτu(τ)^kjqdτ.

(3.24)

Thereforev(t) =u(t)^q satisfies the integral inequality v(t)≤d^q−1K^qkx0k^q+

l

X

j=1

Z t

0

Fj(s)v(s)^kjds, (3.25) where

F₁(t) =d^q−1K^qn 1 γ1p

^q/p

µ₁(t)^q+

m

X

i=1

Q_i Γ(αi)

^q η^q_i1 1

aqipˆi

_piˆ¹ e^−qto

,

F_j(t) =d^q−1K^qnh 1 (k_j−1)ap

iq/p

µ_j(s)^q+

m

X

i=1

Qi

Γ(α_i) kjq

η^q_ij 1 aq_ipˆ_ik_j

_piˆ¹

e^−kjqτo . Obviously,

Z ∞

0

F1(s)ds=d^q−1K^qn 1 γ1p

^q/p ω1+

m

X

i=1

Qi

Γ(αi) ^q

η^q_i1 1 aqipˆi

_piˆ¹ 1 q

o , Z ∞

0

Fj(s)ds=d^q−1K^qnh 1 (kj−1)ap

i^q/p ωj+

m

X

i=1

Q_i Γ(αi)

^kjq

η_ij^q 1 aqipˆikj

_piˆ¹ 1 kjq

o , i.e.,Gj=R∞

0 Fj(s)ds <∞,j= 1,2, . . . , l.

From Lemma 3.1 it follows that ifkx0k < ρ, where ρ >0 is defined by (3.11), then

v(t)≤Ψ₀kx0k) =C(kx0k)D1(kx0k)D2(kx0k)D2(kx0k). . . D_l(kx0k), (3.26) fort∈[0, d), where

C(kx0k) =d^qK^qkx0k^q, D₁(kx0k) = Z ∞

0

F₁(s)ds, D₂(kx₀k) =h

1−(k₁−1)[C(kx₀kD₁(kx₀k)]^k1−1 Z ∞

0

F₂(s)dsi−_k¹

1−1

, D_j(kx₀k) =h

1−(k_j−1)

C(kx₀k)D_j−1(kx₀k)^kj−1Z ∞ 0

F_j(s)dsi−_kj¹₋₁

,

(3.27)

forj= 3, . . . , l.

Obviously, the right hand side of the inequality (3.26) is finite if kx0k< ρ and it is going to∞forkx0k →ρ, if l >1. Ifl= 1, then it is defined for allx0∈R^N. Hence, we have the estimate

v(t) =u(t)^q= kx(t)ke^atq

≤Ψ0(kx0k), t∈[0, d), i.e.,

kx(t)k ≤Ψ(kx₀k)e^−at, t∈[0, d). (3.28) From this inequality it follows that lim_t→dx(t) = d₋ < ∞ and by Theorem 2.1 there is an >0 such that the inial value problem (2.1) has a unique solutionw(t), defined on the interval [0, d+). This is a contradiction with the maximality of the solutionx(t). Hence, the inequality (3.28) holds for all t∈[0,∞) and the proof is

complete.

4. Illustrative example

Let us apply Theorem 3.2 to the fractionally perturbed pendulum equation v⁰⁰(t) + 2v⁰(t) + 4v(t) + 3e^−4tCD^1/3v(t) + 5e^−4th

CD^1/2v(t)i²

= 0. (4.1) We can write this equation in the form of system (1.8), wherex(t) = (x1(t), x2(t)) = (v(t), v⁰(t)),m= 2, α1= 1−¹₃= ²₃,α2= 1−¹₂ =¹₂,

A=

0 1

−4 −2

, f(t, x, u₁, u₂) =

0

−3e^−4tu12−5e^−4tu²₂₂,

, whereu1= (u11, u12),u2= (u21, u22),

f(t, x(t),^RLI^α1x(t),^RLI^α2x(t)) =

0

−3e^−4tRLI²³x₂(t)−5e^−4th

RLI^1/2x₂(t)i² .

! , Obviously,

kf(t, x, u₁, u₂)k ≤3e^−4tku1k+ 5e^−4tku2k². (4.2) The system has the form

˙

x(t) =Ax(t) +

0

−3e^−4tRLI²³x2(t)−5e^−4th

RLI^1/2x2(t)i²

!

. (4.3)

Now, let us find the constantsa >0 andK >0 from condition (A1). The Jordan block of the matrixAhas the form

α β

−β α,

,

whereα=−1,β= 1. One can prove by using the Putzer’s method (see [24]) that e^Atx=e^αt

cos(βt)I+1

β sin(βt)(A−I)

x, (4.4)

whereI is the unit matrix. This yields the estimate ke^Atxk ≤e^−t kIk+kA−Ik

kxk, (4.5)

where we use the norm kCk = max{|c11|+|c12|,|c21|+|c22|} of a 2×2 matrix C= (cij). For this norm the inequalitykCyk ≤ kCkkykis valid for anyy= (y1, y2) with the normkyk = max{|y1|,|y2|}. Since kIk = 1,kA−Ik = 7, from (4.4) we obtain

ke^Atxk ≤8e^−tkxk for allx∈R², (4.6)