Oscillation and stability of first-order delay differential equations with retarded impulses

Oscillation and stability of first-order delay differential equations with retarded impulses

Ba¸sak Karpuz

^B

Department of Mathematics, Faculty of Science, Tınaztepe Campus, Dokuz Eylül University, Buca, 35160 ˙Izmir, Turkey

Received 1 March 2015, appeared 17 October 2015 Communicated by John R. Graef

Abstract. In this paper, we study both the oscillation and the stability of impulsive differential equations when not only the continuous argument but also the impulse condition involves delay. The results obtained in the present paper improve and gen- eralize the main results of the key references in this subject. An illustrative example is also provided.

Keywords: oscillation, stability, uniform stability, delay differential equations, impulse effects.

2010 Mathematics Subject Classification: 34K11, 34K20, 34K45.

1 Introduction

The theory of impulsive differential equations is an important area of scientific activity, since every nonimpulsive differential equation can be regarded as an impulsive differential equation with no impulse effect, i.e., the corresponding impulse factor is the unit. This fact makes it more interesting than the corresponding theory of nonimpulsive differential equations. More- over, such equations naturally appear in the modeling of several real-world phenomena in many areas such as physics, biology and engineering.

The first paper on oscillation of impulsive delay differential equations [7] was published in 1989 (see also [12], one of the first papers on the stability of impulsive differential equations).

From the publication of this paper up to the present time, impulsive delay differential equa- tions started receiving attention of many mathematicians and numerous papers have been published on this class of equations. Most of the publications are devoted to oscillation of first-order impulsive delay differential equations with instantaneous impulse conditions (see for instance [3–5,7,16–18,20–22]). However, to the best of our knowledge, there is not much done in the direction of oscillation and stability of impulsive delay differential equations when the impulse condition also involves a delay argument. Results dealing with retarded impulse conditions are relatively scarce, for instance, we can only find a few papers which only deal

BEmail: bkarpuz@gmail.com

with the stability property of the solutions (see for instance [1,2]). These results include de- lays in the impulse conditions since they study equations under distributed delays. Thus, their impulse conditions require some memory. Unfortunately, there seems to be nothing accom- plished in revealing the oscillation properties of equations with retarded impulse conditions in the absence of distributions. In this paper, we shall draw our attention to this untouched problem. More precisely, the aim of this paper is to deliver answers to the following ques- tions on the stability and oscillatory behaviour of the solutions to impulsive delay differential equations: “If delay differential equations without impulses are oscillatory, will their solu- tions continue to oscillate in the absence of retarded impulse perturbations?” and “If the zero solution of delay differential equations without impulses is stable, when exposed to retarded impulse effects, under what conditions the equation maintains the stability?” The motivation of this paper mainly originates from the work [18], where Yan and Zhao studied impulsive delay differential equations of the form

(x⁰(t) +p(t)x τ(t)=0 fort ∈[θ₀,∞)\{θ_k}_k∈N0 x(θ_k⁺) =λ_kx(θ_k) fork∈_N0

and established very important connections with the following nonimpulsive delay differential equation

y⁰(t) +

"

∏

τ(t)≤θ_k<t

1 λ_k

#

p(t)y τ(t)=0 fort∈[θ₀,∞)almost everywhere.

Practically, their method introduces a transform which glues the continuous pieces in the graph of a jump type discontinuous function endwise, i.e., sticks together the points at which the jump magnitude of the function is momentary. The method developed by the authors is very effective and it says that oscillation and stability of impulsive differential equation is equivalent to that of the nonimpulsive differential equation (see also [3]). Combining the con- nection in [18] with the results for nonimpulsive differential equations (see [8,11]) drops many superfluous restrictions in the papers [4,5,7,12,16,20–22]. Due to technical and theoretical ob- stacles, the same method is useless for studying impulsive delay differential equations when the impulse condition involves delays as well. We shall therefore introduce a new method which generalizes the method in [18] to such problems. Roughly speaking, the technique still relies on construction of continuous functions from a jump type discontinuous function but unlike the previous case, the jump magnitude at each given time is now allowed to depend on the magnitude of a prior time. The technique employed in the present paper allows us to study both oscillation and stability of such equations without putting any sign condition on the coefficient. In [9,13,19], the readers may find stability results for delay differential equations without impulses, which can be combined with our results. The results of this paper improve, generalize and extend the qualitative theory of differential equations to im- pulsive differential equations with retarded impulse conditions. We refer the readers to the books [10,14,15], which cover the fundamental results of the theory on impulsive differential equations.

Our attention in this paper centers on the qualitative behaviour of solutions of the impul- sive delay differential equation

(x⁰(t) +p(t)x τ(t)=_{0 for}t ∈[θ₀,∞)\{θ_k}_k∈N0

x(θ_k) =λ_kx(θ⁻_k−`) fork∈_N0 ^(1.1)

under the following primary assumptions:

(A1) p: [θ₀,∞)→_Ris a Lebesgue measurable and a locally essentially bounded function;

(A2) τ: [θ₀,∞) → _R is a Lebesgue measurable function satisfying τ(t) ≤ t for all t ∈ [θ₀,∞)_{and limt→∞}τ(t) =_∞;

(A3) ` ∈ <sub>N0</sub> and {θ<sub>k</sub>}<sub>k∈N</sub>∪ {θ−`,θ−`+1, . . . ,θ₀} is an increasing divergent sequence of reals;

(A4) {λ_k}_k∈N0 is a sequence of reals which has no zero terms.

We define AC([ρ_θ0,θ₀],R), where ρ_t :=inf{τ(ξ) : ξ ∈ [t,∞)}for t ∈ [θ₀,∞), to be the set of functions absolutely continuous functions defined on [ρ_θ0,θ₀]_.

Definition 1.1 (Solution). Suppose that(A1)–(A4) hold. A functionx: [ρ_θ0,∞) → _Rdenoted byx(·_,θ₀,ϕ)is called asolutionof the initial value problem











x⁰(t) +p(t)x τ(t) =_{0 for}t ∈[θ₀,∞) x(θ_k) =λ_kx(θ⁻_k−`) fork ∈_N0

x(t) = ϕ(t) fort∈[ρ_θ0,θ0),

(1.2)

where ϕ∈ AC([ρ_θ0,θ₀],R)is given, if the following conditions are satisfied:

(i) for anyk ∈_N0, xis absolutely continuous on the interval[θ_k,θ_k+₁);

(ii) for anyk ∈_N0, both right-sided and left-sided limits ofxexist atθ_k withx(θ_k) =_x(θ_k⁺)_; (iii) x is equal to the initial function ϕ on the interval [ρ_θ0,θ₀], and satisfies the differential

equation

x⁰(t) +p(t)x τ(t) =0 for every t∈[θ₀,∞)\{θ_k}_k∈N0; (iv) x satisfies the impulse condition

x(θ_k) =λ_kx(θ⁻_k−`) for allk∈ _N0,

and it may have jump type discontinuity at the impulse points{θ_k}_k∈N0.

Definition 1.2 (Oscillation). A solution of (1.1) is said to be nonoscillatory if it is eventually either positive or negative. Otherwise, the solution is called oscillatory. In other words, a solution is said to be oscillatory if there exists an increasing divergent sequence {ξ_k}_k∈N ⊂ [θ₀,∞)such thatx(ξ⁺_k )x(ξ⁻_k )≤0 for allk∈_N.

For any givenϕ∈ AC([ρ_t,t]_,R), we definekϕk:=_sup{|ϕ(ξ)|:ξ ∈ [ρ_t,t]}. Definition 1.3 (Stability).

(i) The zero solution of (1.1) is said to be stable, if for every ε > 0 and every θ ∈ [θ₀,∞), there exists δ = δ(ε,θ) > 0 such that any ϕ ∈ AC([ρ_θ,θ],R) with kϕk < δ implies

|x(t,θ,ϕ)|<εfor allt ∈[_θ,∞)_.

(ii) The zero solution of (1.1) is said to be uniformly stable, if for every ε > 0 and every θ ∈ [θ₀,∞), there exists δ = δ(ε) > 0 such that any ϕ ∈ AC([ρ_θ,θ],R)with kϕk < δ implies|x(t,θ,ϕ)|<εfor allt ∈[_θ,∞)_.

(iii) The zero solution of (1.1) is said to be asymptotically stable, if it is stable, and for any θ ∈ [θ₀,∞)there existsδ = δ(θ)such that any ϕ∈ AC([ρ_θ,θ],R)with kϕk< δ implies lim_t→_∞x(t,θ,ϕ) =0.

The paper is organized as follows: in Section 2, we construct the major equipments of the paper which all the results in the sequel will depend on; in Section 3, we present our main results, which combine qualitative theory of delay differential equations and qualitative theory of delay differential equations in the absence of retardations in the impulse conditions;

in Section4, to conclude the paper, we make our final comments and give a simple example to mention the significance and applicability of the main results. In the sequel, we always assume without further mentioning that∏_∅ :=1 and∑_∅:=0.

2 Preparatory results

In this section, we shall introduce several tools required for our main purpose. For simplicity of notation, we letθ<sub>−(`+1)</sub>:=$<sub>θ0</sub>. Fori∈ {0, 1, . . . ,`}, we define

ϑⁱ_k :=











θ₀−θ_{−(`+1)+i+1</sub>−θ<sub>−(`+1)+i}

, k= −1

θ₀, k=0

ϑ_kⁱ₋₁+θ_k(`+<sub>1</sub>)+<sub>i</sub>+<sub>1</sub>−θ<sub>k</sub>(`+₁)+_i, k∈ _N, which explicitly yields

ϑⁱ_k =θ₀+

∑

k ν=1

θ_{ν(`+1)+i+1</sub>−θ<sub>ν(`+1)+i}

fork∈_N.

Note that [

ν∈_N0

ϑⁱ_ν−1,ϑⁱ_ν

=θ₀−(θ_{−(`+1)+i+1</sub>−θ<sub>−(`+1)+i}),∞

for eachi∈ {0, 1, . . . ,`}.

Fori∈ {0, 1, . . . ,`}, define

α_i: [ϑ₋ⁱ ₁,∞)→ ^[

ν∈_N0

θ_{(ν−1)(`+1)+i</sub>,θ<sub>(ν−1)(`+1)+i+1}

by

α_i(t)_:=











t+θ_i−θ₀

+

∑

θ0<ϑⁱ_j≤t j∈_N

θ_{j(`+1)+i</sub>−θ<sub>(j−1)(`+1)+i+1}

, t∈[θ₀,∞)

t−θ₀−θ_−(`+1)+i+1

, t∈[ϑⁱ₋₁,θ₀).

Then α_i (i ∈ {0, 1, . . . ,`}) maps the interval [ϑ<sup>i</sup><sub>k−1</sub>,ϑ<sup>i</sup><sub>k</sub>) onto [θ<sub>(k−1)(`+1)+i</sub>,θ_{(k−1)(`+1)+i+1}) for eachk ∈_N0.

Figure2.1is an illustration of the functionsα_i (i=0, 1, . . . ,`).

In addition to the primary assumptions introduced in the previous section, below, we list two more primary assumptions.

Θ₀= J₀⁰

J_-1⁰ J₁⁰ J₂⁰ J₃⁰t Θ_-3

Θ_-2 Θ_-1 Θ₁ Θ₂ Θ₃ Θ₄ Θ₅ Θ₆ Θ₇ Θ₈ Θ₉ Α₀

(a) The graph of the function α0

Θ₀= J₀¹

J_-1¹ J₁¹ J₂¹ J₃¹t Θ_-3

Θ_-2 Θ_-1 Θ₁ Θ₂ Θ₃ Θ₄ Θ₅ Θ₆ Θ₇ Θ₈ Θ₉ Α₁

(b) The graph of the function α₁

Θ₀= J₀²

J_-1² J₁² J₂² J₃²t Θ_-3

Θ_-2 Θ_-1 Θ₁ Θ₂ Θ₃ Θ₄ Θ₅ Θ₆ Θ₇ Θ₈ Θ₉ Α₂

(c) The graph of the function α2

Figure 2.1: An illustration of the functionsα_i (i=0, 1, 2) with`=2 (A5) For eachi∈ {0, 1, . . . ,`},

t ∈ ^[

ν∈_N0

θ_{ν(`+1)+i</sub>,θ<sub>ν(`+1)+i+1}

if and only if τ(t)∈ ^[

ν∈_N0

θ_{(ν−1)(`+1)+i</sub>,θ<sub>(ν−1)(`+1)+i+1} .

(A6) There exist functionsσ_i: [θ₀,∞)→[ϑⁱ₋₁,∞)(i∈ {0, 1, . . . ,`}) such that α_i σ_i(t) =τ α_i(t) for allt∈ [θ₀,∞).

Remark 2.1. Note that for the case` =0, we haveα₀(t) = tfor t ∈[θ₀,∞), and thus we may letσ0(t) =τ(t)fort∈[θ0,∞)to have(A5)and(A6)satisfied.

Lemma 2.2. Assume that (A1)–(A6) hold. Let x = x(·,θ₀,ϕ) be a solution of (1.2). Then, for i∈ {0, 1, . . . ,`}, the function y_i: [θ₀,∞)→_Rdefined by

y_i(t):=











∏

ϑⁱ_j≤t j∈_N0

1 λ_j(`+1)+i











x α_i(t) for t∈ [θ₀,∞) (2.1)

is absolutely continuous on [θ₀,∞). Moreover, i ∈ {0, 1, . . . ,`}, the function y_i is the solution of the initial value problem













 y⁰_i(t) +











∏

σ_i(t)<ϑⁱ_j≤t j∈_N0

1 λ_j(`+1)+i











p α_i(t)y_i σ_i(t) =0 for t∈ [θ0,∞)almost everywhere

y_i(t) =ϕ α_i(t) for t∈[ϑⁱ₋₁,θ0].

(2.2)

Proof. Let i ∈ {0, 1, . . . ,`}, it is easy to see that y_i is absolutely continuous on each of the intervals[ϑ_kⁱ₋₁,ϑⁱ_k)for eachk∈_N0. Note that

α_i(ϑⁱ_k⁺) =θ_kⁱ⁺_{(`+1)+i</sub> and α<sub>i</sub>(ϑ<sup>i</sup><sub>k</sub><sup>−</sup>) =θ<sup>−</sup><sub>(k−1)(`+1)+i+1}= θ_k⁻_(`+1)+i−` fork ∈_N0.

We now show thaty_i(ϑⁱ_k⁺) =y_i(ϑⁱ_k⁻)for allk ∈N. Indeed, fork∈ _{N, we have}

y_i(ϑⁱ_k) =











∏

ϑⁱ_j≤ϑⁱ_k j∈_N0

1 λ_j(`+1)+i











x θ_k(`+1)+i

=











∏

ϑⁱ_j≤ϑⁱ_k j∈_N0

1 λ_j(`+1)+i











λ_{k(`+1)+i</sub>x θ<sub>k</sub><sup>−</sup><sub>(`+1)+i−`}

=











∏

ϑⁱ_j<ϑⁱ_k j∈_N0

1 λ_j(`+1)+i











x θ⁻_k(`+1)+i−`

=











∏

ϑⁱ_j<ϑⁱ_k j∈_N0

1 λ_j(`+1)+i











x θ₍⁻_{k−1)(`+1)+i+1}

=y_i(ϑⁱ_k⁻),

which proves thaty_i(ϑ_kⁱ⁺) = y_i(ϑ_kⁱ⁻), and hence y_i is absolutely continuous on [ϑⁱ₋₁,∞). On the other hand, we have

y⁰_i(t) +











∏

σi(t)<ϑⁱ_j≤t j∈_N0

1 λ_j(`+1)+i











p α_i(t)y_i σ_i(t)

=











∏

ϑⁱ_j≤t j∈_N0

1 λ_j(`+1)+i











x⁰ α_i(t)

+











∏

σ_i(t)<ϑⁱ_j≤t j∈_N0

1 λ_j(`+1)+i











p α_i(t)











∏

ϑⁱ_j≤σ_i(t) j∈_N0

1 λ_j(`+1)+i











x α_i σ_i(t)

=











∏

ϑⁱ_j≤t j∈_N0

1 λ_j(`+1)+i









 n

x⁰ α_i(t)+p α_i(t)x τ(α_i(t))^o=0

(2.3)

for allt∈ [θ₀,∞)\{θ_k}_k∈N0. Also it is not hard to see that the initial function associated with this equation is ϕ◦α_i on [ϑⁱ₋₁,θ₀). The proof is therefore completed.

Figure 2.2 is a graphical illustration of the main idea in the construction of the functions y_i (i=0, 1, . . . ,`) from the solutionx of (1.1).

Now, we defineχ_i (i∈ {0, 1, . . . ,`}) to be the characteristic function of the interval [

ν∈_N0

θ_{ν(`+1)+i</sub>,θ<sub>ν(`+1)+i+1} ,

i.e.,

χ_i(t)_:=







1, t ∈ ^[

ν∈_N0

θ_{ν(`+1)+i</sub>,θ<sub>ν(`+1)+i+1} 0 otherwise.

Θ₀

Θ_-3 Θ_-2 Θ_-1 Θ₁ Θ₂ Θ₃ Θ₄ Θ₅ Θ₆ Θ₇ Θ₈ Θ₉ t x

(a) The graph of a solutionxof (1.1)

Θ₀

Θ_-3 Θ_-2 Θ_-1 Θ₁ Θ₂ Θ₃ Θ₄ Θ₅ Θ₆ Θ₇ Θ₈ Θ₉ t x

(b) A colored version of the solutionx

J₀⁰

J_-1⁰ J₁⁰ J₂⁰ J₃⁰ t y0

(c) The graph of the function y₀

J_k¹

J_-1¹ J₁¹ J₂¹ J₃¹t y1

(d) The graph of the function y₁

J_k²

J_-1² J₁² J₂² J₃²t y2

(e) The graph of the function y₂

Figure 2.2: An illustration of the functionsy_i (i=0, 1, 2) with`=2 For i∈ {0, 1, . . . ,`}, we define the function

β_i: ^[

ν∈_N0

θ_{(ν−1)(`+1)+i</sub>,θ<sub>(ν−1)(`+1)+i+1}

→ϑⁱ₋₁,∞ by

β_i(t):=













 t−











θ_i−θ₀

+

∑

θ_j(`+1)+i≤t j∈_N

θ_{j(`+1)+i</sub>−θ<sub>(j−1)(`+1)+i+1}











, t ∈ ^[

ν∈_N0

θ_{ν(`+1)+i</sub>,θ<sub>ν(`+1)+i+1}

t+θ0−θ_−(`+1)+i+₁

, t ∈θ_{−(`+1</sub>)+i,θ<sub>−(`+1})+i+₁

. It is not hard to see that for each i ∈ {0, 1, . . . ,`}, α_i◦β_i and β_i ◦α_i are the identity mappings on the sets^S_ν∈N0

θ_{(ν−1)(`+1)+i</sub>,θ<sub>(ν−1)(`+1)+i+1} and

ϑⁱ₋₁,∞

, respectively.

Lemma 2.3. Assume that(A1)–(A4)hold. Let x be a solution of (1.1)and the functions y_i: [θ0,∞)→ R(i∈ {0, 1, . . . ,`}) be defined by(2.1). Then

x(t) =

∑

` µ=0

χ_µ(t)









∏

θ_j(`+1)+µ≤t j∈_N0

λ_j(`+1)+µ









y_µ(β_µ(t)) for t∈[θ₀,∞). (2.4)

Proof. Leti∈ {0, 1, . . . ,`}andk∈_N0. From (2.1), we have

x α_i(t) =











∏

ϑⁱ_j≤t j∈_N0

λ_j(`+₁)+_i











y_i(t) for allt ∈θ0,∞),

which yields

x(t) =











∏

ϑⁱ_j≤βi(t) j∈_N0

λ_j(`+1)+i











y_i β_i(t) =









∏

θ_j(`+1)+i≤t j∈_N0

λ_j(`+1)+i









y_i β_i(t) for allt ∈[θ₀,∞).

Therefore, we learn that (2.4) is true.

To give a converse analogue of Lemma 2.2, we need the following additional primary assumption.

(A7) There exist functionsσ_i: [θ₀,∞)→[ϑ₋ⁱ ₁,∞)(i∈ {0, 1, . . . ,`}) such that σ_i β_i(t)= β_i τ(t) for allt ∈[θ₀,∞).

Lemma 2.4. Assume that(A1)–(A4),(A5)and(A7)hold. Let y_i =y_i(·,θ₀,ϕ_i)(i ∈ {0, 1, . . . ,`}) be solutions of

(y⁰_i(t) +q_i(t)y_i σ_i(t) =0 for t ∈[θ₀,∞)

y_i(t) = ϕ_i(t) for t ∈[ϑⁱ₋₁,θ₀]. (2.5) Then, x defined by(2.4)is a solution of the initial value problem

























 x⁰(_t) +

∑

` µ=0

χ_µ(_t)









∏

τ(t)<θ_j(`+1)+µ≤t j∈_N0

λ_j(`+1)+µ









qµ β_µ(_t)_x τ(_t)=_{0 for t}∈ [θ₀,∞)

x(θ_k) =λ_kx(θ_k⁻_−`) for k∈ _N0 x(t) =

∑

` µ=0

χµ(t)ϕµ βµ(t) for t ∈[θ_−(`+1),θ0).

(2.6)

Proof. We shall first show that x defined by (2.4) satisfies the impulse condition in (2.6). Let k∈_N0, then we may findr,i∈_N0such thatk=r(`+1) +iwandi≤`, and thus

x(θ⁺_k ) =x θ_r⁺_(`+1)+i

=

∑

` µ=0

χµ θ_r(`+1)+i









∏

θ_{j(`+1)+µ</sub>≤θ<sub>r(`+1)+i} j∈_N0

λ_j(`+1)+µ









yµ βµ(θ⁺_r(`+1)+i)

=









∏

θ_{j(`+1)+i</sub>≤θ<sub>r(`+1)+i} j∈_N0

λ_j(`+1)+i









y_i β_i(θ_r⁺_(`+1)+i)

=λ_r(`+1)+i









∏

θ_{j(`+1)+i</sub><θ<sub>r(`+1)+i} j∈_N0

λ_j(`+1)+i







 y_i(ϑⁱ_r)

=λ_r(`+1)+i









∏

θ_{j(`+1)+i</sub><θ<sub>r(`+1)+i} j∈_N0

λ_j(`+1)+i









y_i β_i(θ₍⁻_{r−1)(`+1)+i+1})

=λ_r(`+1)+i

∑

` µ=0

χ_µ θ₍⁻_{r−1)(`+1)+i+1}









∏

θ_j(`+1)+µ<θ<sub>r</sub>(`+1)+i

j∈_N0

λ_j(`+1)+µ









y_µ β_µ(θ₍⁻_{r−1)(`+1)+i+1})

=λ_{r(`+1)+i</sub>x θ<sub>(</sub><sup>−</sup><sub>r−1)(`+1)+i+1}

=λ_{r(`+1)+i</sub>x θ<sub>r</sub><sup>−</sup><sub>(`+1)+i−`}

=λ_kx(θ⁻_k−`).

Now, we show that x defined by (2.4) satisfies the differential equation condition in (2.6) but first note that

x⁰(t) =

∑

` µ=0

χµ(t)









∏

θ_j(`+1)+i≤t j∈_N0

λ_j(`+₁)+µ









y⁰_µ(βµ(t))

for all t ∈ (θ_k,θ_k+1) and all k ∈ _N0 since the factor of y_i is a step function (its derivative is therefore 0) and β_i is a combination of lines of slope 1. Now, we can compute that

x⁰(t) +

∑

` µ=₀

χ_µ(t)









∏

τ(t)<θ_j(`+1)+µ≤t j∈_N0

λ_j(`+1)+µ









q_µ β_µ(t)x τ(t)

=

∑

` µ=0

χ_µ(t)









∏

θ_j(`+1)+µ≤t j∈_N0

λ_j(`+1)+µ









y⁰_µ(β_µ(t))

+











∑

` µ=0

χ_µ(t)









∏

τ(t)<θ_j(`+1)+µ≤t j∈_N0

λ_j(`+1)+µ









q_µ β_µ(t)











×











∑

` µ=0

χ_µ τ(t)









∏

θ_j(`+1)+µ≤τ(t) j∈_N0

λ_j(`+1)+µ









y_µ β_µ(τ(t))











=

∑

` µ=0

χµ(t)









∏

θ_j(`+1)+µ≤t j∈_N0

λ_j(`+1)+µ









y⁰_µ(βµ(t))

+











∑

` µ=0

χµ(t)









∏

τ(t)<θ_j(`+₁)+_µ≤t j∈_N0

λ_j(`+1)+µ









qµ βµ(t)











×











∑

` µ=0

χµ(t)









∏

θ_j(`+1)+_µ≤τ(t) j∈_N0

λ_j(`+1)+µ









yµ σµ βµ(t)











=

∑

` µ=0

χµ(t)









∏

θ_j(`+₁)+_µ≤t j∈_N0

λ_j(`+1)+µ







 n

y⁰_µ βµ(t)+qµ βµ(t)yµ σµ βµ(t)^o=0,