# Singular Initial Value Problem for a System of Integro-Differential Equations

Authors: Zdeněk Šmarda; Yasir Khan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Analytical properties like existence, uniqueness, and asymptotic behavior of solutions are studied for the following singular initial value problem:gi(t)yi′(t)=aiyi(t)(1+fi(t,y(t),∫0+tKi(t,s,y(t),y(s))ds)),
yi(0+)=0,
t∈(0,t0], where y=(y1,…,yn),
ai>0,
i=1,…,n are constants and t0>0. An approach which combines topological method of T. Ważewski and Schauder's fixed point theorem is used. Particular attention is paid to construction of asymptotic expansions of solutions for certain classes of systems of integrodifferential equations in a right-hand neighbourhood of a singular point.


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## Body

## 1. Introduction and Preliminaries

Singular initial value problem for ordinary differential and integro-differential equations is fairly well studied (see, e.g., [1–16]), but the asymptotic properties of the solutions of such equations are only partially understood. Although the singular initial value problems were widely considered using various methods (see, e.g., [1–13, 16]), our approach to this problem is essentially different from others known in the literature. In particular, we use a combination of the topological method of T. Ważewski [8] and Schauder’s fixed point theorem [11]. Our technique leads to the existence and uniqueness of solutions with asymptotic estimates in the right-hand neighbourhood of a singular point. Asymptotic expansions of solutions are constructed for certain classes of systems of integrodifferential equations as well.Consider the following problem:(1.1)gi(t)yi′(t)=aiyi(t)(1+fi(t,y(t),∫0+t‍Ki(t,s,y(t),y(s))ds)),(1.2)yi(0+)=0,t∈(0,t0],
where y=(y1,…,yn), ai>0 are constants, fi∈C0(J×ℝn×ℝ,ℝ), Ki∈C0(J×J×ℝn×ℝn,ℝ), J=(0,t0], t0>0, i=1,…,n.Denote(i)
f(t)=O(g(t)) as t→0+ if there is a right-hand neighbourhood 𝒰(0) and a constant K>0 such that (f(t)/g(t))≤K for t∈𝒰(0).(ii)
f(t)=o(g(t))ast→0+  if there is valid limt→0+f(t)/g(t)=0.(iii)
f(t)~g(t)ast→0+if there is validlimt→0+f(t)/g(t)=1.Definition 1.1.
The sequence of functions(ϕn(t)) is called an asymptotic sequence as t→0+ if
(1.3)ϕn+1(t)=o(ϕn(t))ast→0+
for all n.Definition 1.2.
The series∑‍cnϕn(t), cn∈ℝ, is called an asymptotic expansion of the function f(t) up to Nth term as t→0+ if(a)
(ϕn(t)) is an asymptotic sequence,(b)
(1.4)[f(t)-∑n=1N‍cnϕn(t)]=o(ϕN(t)),ast→0+.
The functionsgi,fi,  and  Ki will be assumed to satisfy the following:(i)
gi(t)∈C1(J), gi(t)>0, gi(0+)=0, gi′(t)~ψi(t)giλi(t) as t→0+, λi>0, ψi(t)giτ(t)=o(1) as t→0+ for each τ>0, i=1,…,n,(ii)
|fi(t,u,v)|≤|u|+|v|, |∫0+t‍Ki(t,s,y(t),y(s))ds|≤ri(t)|y|, 0<ri(t)∈C(J), ri(t)=φi(t,Ci)o(1) as t→0+ where φi(t,Ci)=Ciexp(∫t0t‍(ai/gi(s))ds) is the general solution of the equation gi(t)yi′(t)=aiyi(t).
In the text, we will apply topological method of Ważewski and Schauder’s theorem. Therefore we give a short summary of them.
Letf(t,y) be a continuous function defined on an open (t,y) set Ω⊂ℝ×ℝn, Ω0 an open set of Ω,∂Ω0 the boundary of Ω0, and Ω¯0 the closure of Ω0. Consider the following system of ordinary differential equations:
(1.5)y′=f(t,y).Definition 1.3 (see [17]).
The point(t0,y0)∈Ω∩∂Ω0 is called an egress (or an ingress point) of Ω0 with respect to system (1.5) if for every fixed solution of the problem y(t0)=y0, there exists an ϵ>0 such that (t,y(t))∈Ω0 for t0-ϵ≤t<t0(t0<t≤t0+ϵ). An egress point (ingress point) (t0,y0) of Ω0 is called a strict egress point (strict ingress point) of Ω0 if (t,y(t))∉Ω¯0 on interval t0<t≤t0+ϵ1(t0-ϵ1≤t<t0) for an ϵ1.Definition 1.4 (see [18]).
An open subsetΩ0 of the set Ω is called an (u,v) subset of Ω with respect to system (1.5) if the following conditions are satisfied. (1)
There exist functionsui(t,y)∈C1(Ω,ℝ),  i=1,…,m and vj(t,y)∈C[Ω,ℝ]j=1,…,n,m+n>0 such that
(1.6)Ω0={(t,y)∈Ω:ui(t,y)<0,vj(t,y)<0∀i,j}.(2)
u˙α(t,y)<0 holds for the derivatives of the functions uα(t,y), α=1,…,m along trajectories of system (1.5) on the set
(1.7)Uα={(t,y)∈Ω:uα(t,y)=0,ui(t,y)≤0,vj(t,y)≤0,∀jandi≠α}.(3)
v˙β(t,y)>0 holds for the derivatives of the functions vβ(t,y), β=1,…,n along trajectories of system (1.5) on the set
(1.8)Vβ={(t,y)∈Ω:uβ(t,y)=0,ui(t,y)≤0,vj(t,y)≤0,∀iandj≠β}.The set of all points of egress (strict egress) is denoted by Ωe0(Ωse0).Lemma 1.5 (see [18]).
Let the setΩ0 be a (u,v) subset of the set Ω with respect to system (1.5). Then
(1.9)Ωse0=Ωe0=⋃α=1m‍Uα∖⋃β=1n‍Vβ.Definition 1.6 (see [18]).
LetX be a topological space and B⊂X.LetA⊂B. A function r∈C(B,A) such that r(a)=a for all a∈A is a retraction from B to A in X.The setA⊂B is a retract of B in X if there exists a retraction from B to A in X.Theorem 1.7 (Ważewski’s theorem [18]).
LetΩ0 be some (u,v) subset of Ω with respect to system (1.5). Let S be a nonempty compact subset of Ω0∪Ωe0 such that the set S∩Ωe0 is not a retract of S but is a retract Ωe0. Then there is at least one point (t0,y0)∈S∩Ω0 such that the graph of a solution y(t) of the Cauchy problem y(t0)=y0 for (1.5) lies on its right-hand maximal interval of existence.Theorem 1.8 (Schauder’s theorem [19]).
LetE be a Banach space and S its nonempty convex and closed subset. If P is a continuous mapping of S into itself and PS is relatively compact then the mapping P has at least one fixed point.


## 2. Main Results

Theorem 2.1.
Let assumptions (i) and (ii) hold, then for eachCi≠0 there is one solution y(t,C)=(y1(t,C1),y2(t,C2),…,yn(t,Cn)), C=(C1,…,Cn) of initial problem (1.1) and (1.2) such that
(2.1)|yi(j)(t,Ci)-φi(j)(t,Ci)|≤δ(φi2(t,Ci))(j),j=0,1,
for t∈(0,tΔ], where 0<tΔ≤t0, δ>1 is a constant, and tΔ depends on δ,Ci, i=1,…,n.Proof.
(1) DenoteE the Banach space of vector-valued continuous functions h(t) on the interval [0,t0] with the norm
(2.2)∥h(t)∥=maxt∈[0,t0]|hi(t)|,i=1,…,n.
The subset S of Banach space E will be the set of all functions h(t) from E satisfying the inequality
(2.3)|hi(t)-φi(t,Ci)|≤δφi2(t,Ci).
The set S is nonempty, convex, and closed.
(2) Now we will construct the mappingP. Let h0(t)∈S be an arbitrary function. Substituting h0(t), h0(s) instead of y(t), y(s) into (1.1), we obtain the following differential equation:
(2.4)gi(t)yi′(t)=aiyi(t)(1+fi(t,y(t),∫0+t‍Ki(t,s,h0(t),h0(s))ds)),i=1,…,n.
Put
(2.5)yi(t)=φi(t,Ci)+φi(1-μ)(t,Ci)Y0i(t),(2.6)yi′(t)=φi′(t,C)+1gi(t)φi(1-μ)(t,Ci)Y1i(t),
where 0<μ<1 is a constant and new functions Y0i(t), Y1i(t) satisfy the differential equations as
(2.7)gi(t)Y0i′(t)=(μ-1)aiY0i(t)+Y1i(t),i=1,…,n.
From (2.3), it follows
(2.8)h0i(t)=φi(t,Ci)+H0i(t),|H0i(t)|≤δφi2(t,Ci).
Substituting (2.5), (2.6), and (2.8) into (2.4), we get
(2.9)Y1i(t)=aiY0i(t)+(aiφiμ(t,Ci)+aiY0i(t))×fi(∫0+t‍t,φ1(t,C1)+φ1(1-μ)(t,C1)Y01(t),…,φn(t,Cn)+φn(1-μ)(t,Cn)Y0n(t),∫0+t‍Ki(t,s,φ1(t,C1)+H01(t),…,φn(t,Cn)+H0n(t),φ1(s,C1)+H01(s),…φn(s,Cn)+H0n(s))ds∫0+t‍).
Substituting (2.9) into (2.7), we get
(2.10)gi(t)Y0i′(t)=μaiY0i(t)+(aiφiμ(t,Ci)+aiY0i(t))×fi(∫0+t‍t,φ1(t,C1)+φ1(1-μ)(t,C1)Y01(t),…,φn(t,Cn)+φn(1-μ)(t,Cn)Y0n(t),∫0+t‍Ki(t,s,φ1(t,C1)+H01(t),…,φn(t,Cn)+H0n(t),φ1(s,C1)+H01(s),…φn(s,Cn)+H0n(s))ds∫0+t‍).
In view of (2.5) and (2.6), it is obvious that a solution of (2.10) determines a solution of (2.4).
Now we use Ważewski’s topological method. Consider an open setΩ⊂ℝ+×ℝn. Denote Y0=(Y01,…,Y0n). Define an open subset Ω0⊂Ω as follows:
(2.11)Ω0={(t,Y0):ui(t,Y0)<0,v(t,Y0)<0,i=1,…,n},Uα={(t,Y0):uα(t,Y0)=0,ui(t,Y0)≤0,v(t,Y0)≤0,i=1,…,n,i≠α},Vβ=V={(t,Y0):v(t,Y0)=0,uj(t,Y0)≤0,i=1,…,n},
where
(2.12)ui(t,Y0)=Y0i2-(δφi(1+μ)(t,Ci))2,v(t,Y0)=t-t0,i=1,…,n.
Calculating the derivatives u˙α(t,Y0), v˙(t,Y0) along the trajectories of (2.10) on the set Uα, V, α=1,…,n we obtain
(2.13)u˙α(t,Y0)=2aαgα(t)[∫0+t‍μY0α2(t)+(Y0α(t)φαμ(t,Cα)+Y0α2(t))×fα(∫0+t‍t,φ1(t,C1)+φ1(1-μ)(t,C1)Y01(t),…,φn(t,Cn)+φn(1-μ)(t,Cn)Y0n(t),∫0+t‍Ki(t,s,φ1(t,C1)+H01(t),…,φn(t,Cn)+H0n(t),φ1(s,C1)+H01(s),…φn(s,Cn)+H0n(s))ds∫0+t‍).-δ2(1+μ)φα2(1+μ)(t,Cα)∫0+t‍].
Since
(2.14)limt→+0ψi(t)giτ(t)=0foranyτ>0,i=1,…,ngi′(t)~ψi(t)giλi(t)ast→0+,λi>0,i=1,…,n,
then there exists a positive constant Mi such that
(2.15)gi′(t)<Mi,t∈(0,t0],i=1,…,n.
Consequently,
(2.16)∫t0tdsgi(s)<1Mi∫t0tgi′(s)dtgi(s)=1Milngi(t)gi(t0)⟶-∞ast⟶0+,i=1,…,n.
From here limt→0+φi(t,Ci)=0 and by L’Hospital’s rule φiτ(t,Ci)giσ(t)=o(1), for t→0+, i=1,…,n,σ is an arbitrary real number. These both identities imply that the powers of φi(t,Ci) affect the convergence to zero of the terms in (2.13), in a decisive way.
Using the assumptions of Theorem2.1 and the definition of Y0(t), φi(t,Ci), i=1,…,n, we get that the first term μY0α2(t,Cα) in (2.13) has the following form:
(2.17)μY0α2(t)=μδ2φα2(1+μ)(t,Cα),
and the second term
(2.18)(Y0α(t)φαμ(t,Cα)+Y0α2(t))×fα(t,φ1(t,C1)+φ1(1-μ)(t,C1)Y01(t),…,φn(t,Cn)+φn(1-μ)(t,Cn)Y0n(t),∫0+t‍Kα(t,s,φ1(t,C1)+H01(t),…,φn(t,Cn)+H0n(t),φ1(s,C1)+H01(s),…φn(s,Cn)+H0n(s))ds{t,φ1(t,C1)+φ1(1-μ)(t,C1)Y01(t)}).
is bounded by terms with exponents which are greater than φα2(1+μ)(t,Cα), α=1,…,n. From here, we obtain
(2.19)sgnu˙α(t,Y0)=-δ2(1+μ)φα2(1+μ)(t,Cα)=-1
for sufficiently small t*, depending on Cαα=1,…,n, δ, 0<t*≤t0.
It is obvious thatsgnv˙(t,Y0)=1.
Change the orientation of the axis t into opposite. Then, with respect to the new system of coordinates, the setΩ0 is the (u,v) subset with respect to system (2.10). By Ważewski’s topological method, we state that there exists at least one integral curve of (2.10) lying in Ω0 for t∈(0,t*). It is obvious that this assertion remains true for an arbitrary function h0(t)∈S.
Now we prove the uniqueness of a solution of (2.10). Let Y¯0(t)=(Y¯01(t),…,Y¯0n(t)) be also the solution of (2.10). Putting
(2.20)Z0i=Y0i-Y¯0i,i=1,…,n
and substituting into (2.10), we obtain
(2.21)gi(t)Z0i′(t)=μaiY0i(t)+(aiφiμ(t,Ci)+aiZ0i(t))×fi(∫0+t‍t,φ1(t,C1)+φ1(1-μ)(t,C1)(Z0i(t)+Y¯01(t)),…,φn(t,Cn)+φn(1-μ)(t,Cn)(Z0n(t)+Y¯0n(t)),∫0+t‍Ki(t,s,φ1(t,C1)+H01(t),…,φn(t,Cn)+H0n(t),φ1(s,C1)+H01(s),…φn(s,Cn)+H0n(s))ds∫0+t‍).
Define
(2.22)Ω1(δ)={(t,Z0):0<t<t*,u1i(t,Z0)<0,v1(t,Z0)<0,0<t<t*,i=1,…,n}U1α={(t,Z0):u1α(t,Z0)=0,u1i(t,Z0)≤0,v1(t,Z0)≤0,i=1,…,n,i≠α},V1β=V={(t,Z0):v1(t,Z0)=0,uj(t,Z0)≤0,i=1,…,n},
where
(2.23)u1i(t,Z0)=Z0i2-(δφi(1+μ-γ))2,0<γ<μ,v1(t,Z0)=t-t*.
Using the same method as above, we have
(2.24)sgnu˙1i(t,Z0)=-1,sgnv˙1(t,Z0)=1,i=1,…,n
for sufficiently small t♢, 0<t♢≤t*. It is obvious that Ω0⊂Ω1(δ) for t∈(0,t♢). Let Z¯0(t)=(Z¯01(t),…,Z¯0n(t)) be any nonzero solution of (2.10) such that (t1,Z0¯(t1))∈Ω1 for 0<t1<t♢. Let δ¯∈(0,δ) be such a constant that (t1,Z0¯(t1))∈∂Ω1(δ¯). If the curve Z0¯(t) lay in Ω1(δ¯) for 0<t<t1, then (t1,Z0¯(t1)) would have to be a strict egress point of ∂Ω1(δ¯) with respect to the original system of coordinates. This contradicts the relation (2.24). Therefore there exists only the trivial solution Z0(t)≡0 of (2.21), so Y0=Y0¯(t) is the unique solution of (2.10).
From (2.5) we obtain
(2.25)|yi(t,Ci)-φi(t,Ci)|≤δφi2(t,Ci),i=1,…,n,
where (y1(t,C1),…,yn(t,Cn)) is the solution of (2.4) for t∈(0,t♢]. Similarly, from (2.6) and (2.9), we have
(2.26)|yi′(t,Ci)-φi′(t,Ci)|=|1gi(t)φi(1-μ)(t,Ci)Y1i(t)|≤|1gi(t)φi(1-μ)(t,Ci)2aiδφi(1-μ)(t,Ci)|=δ(φi2(t,Ci))′.
It is obvious (after a continuous extension of y(t,C) for t=0, y(0+)=0) that P:h0→y maps S into itself and PS⊂S.
(3) We will prove thatPS is relatively compact and P is a continuous mapping.
It is easy to see, by (2.25) and (2.26), that PS is the set of uniformly bounded and equicontinuous functions for t∈[0,t♢]. By Ascoli’s theorem, PS is relatively compact.
Let{hk(t)} be an arbitrary sequence vector-valued functions in S such that
(2.27)∥hk(t)-h0(t)∥=ϵk,limk→∞ϵk=0,h0(t)∈S.
The solution Y¯k(t)=(Y¯k1,…,Y¯kn) of the following equation:
(2.28)gi(t)Y0i′(t)=μaiY0i(t)+(aiφiμ(t,Ci)+aiY0i(t))×fi(∫0+t‍t,φ1(t,C1)+φ1(1-μ)(t,C1)Y01(t),…,φn(t,Cn)+φn(1-μ)(t,Cn)Y0n(t),∫0+t‍Ki(t,s,φ1(t,C1)+H01(t),…,φn(t,Cn)+H0n(t),φ1(s,C1)+Hk1(s),…φn(s,Cn)+Hkn(s))ds∫0+t‍).
corresponds to the function hk(t) and Yk¯(t)∈Ω0 for t∈(0,t♢). Similarly, the solution Y0¯(t) of (2.10) corresponds to the function h0(t). We will show that |Yk¯(t)-Y0¯(t)|→0 uniformly on [0,tΔ], where 0<tΔ≤t♢, tΔ is a sufficiently small constant which will be specified later. Consider the following region:
(2.29)Ω0k={(t,Y0):0<t<t♢,u0ki(t,Y0)<0,v0(t,Y0)<0,i=1,…,n},
where
(2.30)u0ki(t,Y0)=(Y0i(t)-Y¯0i(t))2-(ϵkφi(1+μ-ν)(t,Ci))2,0<ν<α,i=1,…,n,k≥1,v0(t,Y0)=t-t♢.
There exists sufficiently small constant tΔ≤t♢ such that Ω0⊂Ω0k for any k,t∈(0,tΔ). Investigate the behaviour of integral curves of (2.28) with respect to the boundary ∂Ω0k, t∈(0,tΔ]. Using the same method as above, we obtain the following trajectory derivatives:
(2.31)sgnu˙0k(t,Y0)=-1,sgnv˙0(t,Y0)=1
for t∈(0,tΔ] and any k. By Ważewski’s topological method, there exists at least one solution Y¯k(t) lying in Ω0k, 0<t<tΔ. Hence, it follows that
(2.32)|Y¯ki(t)-Y¯0i(t)|≤ϵkφi1+μ-ν≤Niϵk,Ni>0, i=1,…,n are constants depending on Ci, tΔ. From (2.5), we obtain
(2.33)|yki(t)-y0i(t)|=φi(1-μ)(t,Ci)|Y¯ki(t)-Y¯0i(t)|≤niϵk,
where ni>0, i=1,…,n are constants depending on tΔ, Ci, Ni. This estimate implies that P is continuous.
We have thus proved that the mappingP satisfies the assumptions of Schauder’s fixed point theorem and hence there exists a function h(t)∈S with h(t)=P(h(t)). The proof of existence of a solution of (1.1) is complete.
Now we will prove the uniqueness of a solution of (1.1). Substituting (2.5) and (2.6) into (1.1), we get
(2.34)Y1i(t)=aiY0i(t)+(aiφiμ(t,Ci)+aiY0i(t))×fi(∫0+t‍t,φ1(t,C1)+φ1(1-μ)(t,C1)Y01(t),…,φn(t,Cn)+φn(1-μ)(t,Cn)Y0n(t),∫0+t‍Ki(t,s,φ1(t,C1)+φ1(1-μ)(t,C1)Y01(t),…,φn(t,Cn)+φn(1-μ)(t,Cn)Y0n(t),φ1(s,C1)+φ1(1-μ)(s,C1)Y01(s),…φn(s,Cn)+φn(1-μ)(s,Cn)Y0n(s))ds∫0+t‍).
Equation (2.7) may be written in the following form:
(2.35)gi(t)Y0i′(t)=aiY0i(t)+(aiφiμ(t,Ci)+aiY0i(t))×fi(∫0+t‍t,φ1(t,C1)+φ1(1-μ)(t,C1)Y01(t),…,φn(t,Cn)+φn(1-μ)(t,Cn)Y0n(t),∫0+t‍Ki(t,s,φ1(t,C1)+φ1(1-μ)(t,C1)Y01(t),…,φn(t,Cn)+φn(1-μ)(t,Cn)×Y0n(t),φ1(s,C1)+φ1(1-μ)(s,C1)Y01(s),…φn(s,Cn)+φn(1-μ)(s,Cn)Y0n(s))ds∫0+t‍).
Now we know that there exists the solutiony0(t)=(y01(t,C1),…,y0n(t,Cn)) of (1.1) satisfying (1.2) such that
(2.36)y0i(t,Ci)=φi(t,Ci)+φi(1-μ)(t,Ci)T0i(t),i=1,…,n,
where T0(t)=(T01(t),…,T0n(t)) is the solution of (2.35).
DenoteWi0(t)=Y0i(t)-T0i(t), i=1,…,n. Substituting Wi0(t) into (2.35), we obtain
(2.37)gi(t)W0i′(t)=aiW0i(t)+(aiφiμ(t,Ci)+aiW0i(t))×[∫0+t‍fi(∫0+t‍t,φ1(t,C1)+φ1(1-μ)(t,C1)(W01(t)+T01(t)),…,φn(t,Cn)+φn(1-μ)(t,Cn)×(W0n(t)+T0n(t)),∫0+t‍Ki(t,s,φ1(t,C1)+φ1(1-μ)(t,C1)(W01(t)+T01(t)),…,φn(t,Cn)+φn(1-μ)(t,Cn)×(W0n(t)+T0n(t)),φ1(s,C1)+φ1(1-μ)(s,C1)×(W01(s)+T01(s)),…φn(s,Cn)+φn(1-μ)(s,Cn)(W0n(s)+T0n(s)))ds∫0+t‍)-fi(∫0+t‍t,φ1(t,C1)+φ1(1-μ)(t,C1)T01(t),…,φn(t,Cn)+φn(1-μ)(t,Cn),∫0+t‍Ki(t,s,φ1(t,C1)+φ1(1-μ)(t,C1)T01(t),…,φn(t,Cn)T0n(t)+φn(1-μ)(t,Cn)T0n(t),φ1(s,C1)+φ1(1-μ)(s,C1)×T01(s),…φn(s,Cn)+φn(1-μ)(s,Cn)T0n(s))ds∫0+t‍)∫0+t‍].
Let
(2.38)1Ω0={(t,W0):0<t<tΔ,u1i(t,W0)<0,v1(t,W0)<0},
where
(2.39)u1i(t,W0)=W1i2-(δφi(1+μ-ρ)(t,Ci))2,0<ρ<μ,v1(t,W0)=t-tΔ,i=1,…,n.
If (2.37) had only the trivial solution lying in1Ω0, then Y0(t)=T0(t) would be only one solution of (2.37) and from here, by (2.35), y0(t) would be only one solution of (1.1) satisfying (1.2) for t∈(0,tΔ].
We will suppose that there exists nontrivial solutionW¯0(t) of (2.37) lying in 1Ω0. Substituting W¯0i(s) instead of W0i(s), i=1,…,n into (2.37), we obtain the following differential equation:
(2.40)gi(t)W0i′(t)=aiW0i(t)+(aiφiμ(t,Ci)+aiW0i(t))×[∫0+t‍fi(∫0+t‍t,φ1(t,C1)+φ1(1-μ)(t,C1)(W01(t)+T01(t)),…,φn(t,Cn)+φn(1-μ)(t,Cn)×(W0n(t)+T0n(t)),∫0+t‍Ki(t,s,φ1(t,C1)+φ1(1-μ)(t,C1)(W01(t)+T01(t)),…,φn(t,Cn)+φn(1-μ)(t,Cn)×(W0n(t)+T0n(t)),φ1(s,C1)+φ1(1-μ)(s,C1)×(W¯01(s)+T01(s)),…φn(s,Cn)+φn(1-μ)(s,Cn)(W¯0n(s)+T0n(s)))ds∫0+t‍)-fi(∫0+t‍t,φ1(t,C1)+φ1(1-μ)(t,C1)T01(t),…,φn(t,Cn)+φn(1-μ)(t,Cn),∫0+t‍Ki(t,s,φ1(t,C1)+φ1(1-μ)(t,C1)T01(t),…,φn(t,Cn)T0n(t)+φn(1-μ)(t,Cn)T0n(t),φ1(s,C1)+φ1(1-μ)(s,C1)×T01(s),…φn(s,Cn)+φn(1-μ)(s,Cn)T0n(s))ds∫0+t‍)∫0+t‍].
Calculating the derivativeu˙1i(t,W0) along the trajectories of (2.40) on the set ∂1Ω0, we get sgnu˙1i(t,W0)=-1 for t∈(0,tΔ], i=1,…,n.
By the same method as in the case of the existence of a solution of (1.1), we obtain that in 1Ω0 there is only the trivial solution of (2.40). The proof is complete.


## 3. Asymptotic Expansions of Solutions

Diblík [3] investigated a singular initial problem for implicit ordinary differential equations and constructed asymptotic expansions of solutions in a right-hand neighbourhood of a singular point. Some results about asymptotic expansions of solutions for integrodifferential equations with separable kernels are given in [3, 10, 12].The aim of this section is to show that results of paper [2] for ordinary differential equations are possible to extend on certain classes systems integrodifferential equations with a separable kernel in the following form:
(3.1)g(t)yi′=yi+∫0+t‍(∑|σi|+|ωi|=2Ni‍uσiωi(t)vσiωi(s)yσi(t)yωi(s))ds,
where Ni∈ℕ, σi=(li1,…,lin), ωi=(ji1,…,jin), lik,jik∈ℕ∪{0}, k=1,…,n,
(3.2)|σi|=∑k=1n‍lik,|ωi|=∑k=1n‍jik,yσi(t)=∏k=1n‍yklik(t),yωi(s)=∏k=1n‍ykjik(s),uσiωi(t),vσiωi(t)∈C0(J),J=(0,t0],i=1,…,n.We will construct the solution of (3.1) in the form of one parametric asymptotic expansions as
(3.3)yi(t,C)=∑h=1∞‍fih(t)ϕh(t,C),i=1,…,n,
where ϕ(t,C) is the general solution of the differential equation g(t)y′=y so that
(3.4)ϕ(t,C)=Cexp[∫t0t‍dτg(τ)],fi1(t)≡1, fih(t), h≥2, i=1,…,n are unknown functions, C≠0 is a constant.Consider the following differential equation:(3.5)g(t)y′=qy+p(t).
Diblík [3] proved asymptotic estimates of the solution of (3.5) which we can be formulated as follows.Theorem 3.1.
Assume that(I)
Letq be a constant, q<0, g(t)∈C1(J), g(t)>0, limt→t0+g(t)=0, g′(t)~ψ1(t)gλ1(t) as t→t0+, λ1>0, limt→t0+ψ1(t)gτ(t)=0,τ is any positive number.(II)
p(t)∈C(J), p(t)=b0(t)gλ(t)+O(b1(t)gλ+ϵ(t)), ϵ>0,limt→t0+bm(t)g′(t)=0, m=0,1,b0(t)∈C(J),b0(t)≠0, b0′(t)~ψ2(t)gλ2(t) as t→t0+,   λ2+1>0, limt→t0+ψ2(t)gτ(t)=0, limt→t0+gτ(t)(b0(t))-1=0.
Then (3.5) has a unique solution on (0,t-],0<t-≤t0, satisfying asymptotic estimates
(3.6)y(x)=-1qb0(x)gλ(x)+O(gν(x)),y′(x)=O(gν-1(x)),
where ν∈(λ,λ+min{λ1,λ2+1,ϵ}).Now we will show the results of Theorem3.1. regarding only differential equation (3.5) we can apply to system of integrodifferential equations (3.1).Substituting (3.3) into (3.1) and comparing the terms with the same powers of ϕ(t,C), we obtain the following system of recurrence equations:
(3.7)g(t)fih′=(1-h)fih+ϕ-h(t,C)∫0+t‍Rih(t,s)ds,h≥2,i=1,…,n and
(3.8)Rih(t,s)=Rih[f11(t),…,fih-1(t),…,fn1(t),…,fnh-1(t),f11(s),…,f1h-1(s),…,fn1(s),…,fnh-1(s)].
Denote
(3.9)pih(t)=ϕ-h(t,C)∫0+t‍Rih(t,s)ds,
then it is obvious that the recurrence equations
(3.10)g(t)fih′=(1-h)fih+pih(t)h≥2, i=1,…,n have the same form as (3.5) with the constant q=1-h. Hence we can apply Theorem 3.1, after the modification of assumption (II) of Theorem 3.1 for indices h≥2, i=1,…,n, to recurrence (3.10) which we will demonstrate with the following example.Example 3.2.
Consider the following system of integrodifferential equations:(3.11)t2y1′=y1+∫0+t‍1t3y1(s)y2(s)ds,t2y2′=y2+∫0+t‍ty1(t)y2(s)ds.
System (3.11) has the form of system (3.1) for
(3.12)σ1ω1=(0,0,1,1),uσ1ω1(t)=1t3,vσ1ω1(s)=1,N1=2σ2ω2=(1,0,0,1),uσ2ω2(t)=t,vσ2ω2(s)=1,N2=2.
We will construct a solution of system (3.11) in the following form:
(3.13)y1=∑k=1∞‍f1k(t)ϕk(t,C),y2=∑k=1∞‍f2k(t)ϕk(t,C),
where ϕ(t,C) is the general solution of the equation t2y′=y. We will demonstrate the calculation of coefficients fih for h=3. Substituting (3.13) in (3.11) and comparing the terms with the same powers of ϕ(t,C), we obtain the following system of recurrence equations:
(3.14)ϕ1(t,C):1=1,1=1.(3.15)ϕ2(t,C):t2f12′=-f12+ϕ-2(t,C)∫0+t‍1t3ϕ2(s,C)ds,t2f22′=-f22+ϕ-2(t,C)∫0+t‍tϕ(t,C)ϕ(s,C)ds.(3.16)ϕ3(t,C):t2f13′=-2f13+ϕ-3(t,C)∫0+t‍1t3[f12(s)+f22(s)]ϕ3(s,C)ds,t2f23′=-2f23+ϕ-3(t,C)∫0+t‍t[f12(t)ϕ2(t,C)ϕ(s,C)+f22(s)ϕ2(s,C)ϕ(t,C)]ds.
Put
(3.17)u1=ϕ-2(t,C)∫0+t‍ϕ2(s,C)ds,u2=ϕ-1(t,C)∫0+t‍ϕ(s,C)ds.
Differentiating both equations (3.17), we obtain the following differential equations:
(3.18)t2u1′=-2u1+t2,(3.19)t2u2′=-u2+t2.
Equation (3.18) satisfies assumptions of Theorem 3.1. with following functions and coefficients:
(3.20)a=-2,b0(t)=1,gλ(t)=(t2)1⇒λ=1,b1(t)=0,g′(t)=(t2)′=2(g(t))1/2⇒λ1=12,b0′(t)=0·gλ2(t).
Hence we can choose a constantλ2+1>1/2 and similarly ϵ>1/2. By Theorem 3.1., we have
(3.21)u1=12t2+O(t2ν1),ν1∈(1,32).
Second equation (3.19) is different from (3.18) only in the constant a=-1. Thus
(3.22)u2=t2+O(t2ν2),ν2∈(1,32).
Substituting solutions (3.21) and (3.22) into (3.15) instead of integral terms, we obtain for unknown coefficients f12,f22 the following differential equations:
(3.23)t2f12′=-f12+12t+O(t2ν1-3),(3.24)t2f22′=-f22+t5/2+O(t2ν2+1/2).
For (3.23), we can put
(3.25)a=-1,b0(t)=12,gλ(t)=(t2)-1/2,λ=-12,b1(t)=1,ϵ=ν1-1,g′(t)=(t2)′=2(g(t))1/2⇒λ1=12,b0′(t)=0·gλ2(t).
Then we can choose a constantλ2+1>1/2. By Theorem 3.1., we get
(3.26)f12(t)=12t+O(t2ν12),f12′(t)=O(t2ν12-2),ν12∈(-12,0).
Similarly for (3.24), we can put a=-1, b0(t)=1, gλ(t)=(t2)5/4, λ=5/4,b1(t)=1, ϵ=ν2-1,
(3.27)g′(t)=(t2)′=2(g(t))1/2⇒λ1=12,b0′(t)=0·gλ2(t).
Then we can choose a constantλ2+1>1/2. By Theorem 3.1., we have
(3.28)f22(t)=t5/2+O(t2ν22),f22′(t)=O(t2ν22-2),ν22∈(54,74).
Substituting coefficients f12, f22 into (3.16) and using the same method as in the calculation of coefficients f12, f22, we have
(3.29)f13(t)=112t2+O(t2ν13),f13′(t)=O(t2ν13-1),ν13∈(-1,-12),f23(t)=14t3/2+O(t2ν23),f23′(t)=O(t2ν23-1),ν23∈(34,54).
Thus the solution of system (3.11) has for h=3 the following asymptotic expansions:
(3.30)y1≈ϕ(t,C)+[12t+O(t2ν12)]ϕ2(t,C)+[112t2+O(t2ν13)]ϕ3(t,C),y2≈ϕ(t,C)+[t5/2+O(t2ν22)]ϕ2(t,C)+[14t3/2+O(t2ν23)]ϕ3(t,C).



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*Source: 918281-2012-12-17.xml*