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

Abstract

Analytical properties like existence, uniqueness, and asymptotic behavior of solutions are studied for the following singular initial value problem: 𝑔_𝑖(𝑡)𝑦^′_𝑖(𝑡)=𝑎_𝑖𝑦_𝑖(𝑡)(1+𝑓_𝑖(𝑡,𝐲(𝑡),∫^𝑡₀₊𝐾_𝑖(𝑡,𝑠,𝐲(𝑡),𝐲(𝑠))𝑑𝑠)),   𝑦_𝑖(0⁺)=0,   𝑡∈(0,𝑡₀], where 𝐲=(𝑦₁,…,𝑦_𝑛),   𝑎_𝑖>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.

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: 𝑔_𝑖(𝑡)𝑦^′_𝑖(𝑡)=𝑎_𝑖𝑦_𝑖(𝑡)1+𝑓_𝑖𝑡,𝐲(𝑡),^𝑡₀₊𝐾_𝑖((𝑡)(𝑠,𝑦𝑡,𝑠,𝐲,𝐲))𝑑𝑠_𝑖0⁺=0,𝑡∈0,𝑡₀, (1.1)(1.2) where 𝐲=(𝑦₁,…,𝑦_𝑛), 𝑎_𝑖>0 are constants, 𝑓_𝑖∈𝐶⁰(𝐽×ℝ^𝑛×ℝ,ℝ), 𝐾_𝑖∈𝐶⁰(𝐽×𝐽×ℝ^𝑛×ℝ^𝑛,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 lim_𝑡→0⁺𝑓(𝑡)/𝑔(𝑡)=0.(iii)𝑓(𝑡)∼𝑔(𝑡)as𝑡→0⁺if there is validlim_𝑡→0⁺𝑓(𝑡)/𝑔(𝑡)=1.

Definition 1.1. The sequence of functions (𝜙_𝑛(𝑡)) is called an asymptotic sequence as 𝑡→0⁺ if 𝜙_𝑛+1(𝑡)=𝑜𝜙_𝑛(𝑡)as𝑡→0⁺ (1.3) for all 𝑛.

Definition 1.2. The series ∑𝑐_𝑛𝜙_𝑛(𝑡), 𝑐_𝑛∈ℝ, is called an asymptotic expansion of the function 𝑓(𝑡) up to 𝑁th term as 𝑡→0⁺ if(a)(𝜙_𝑛(𝑡)) is an asymptotic sequence, (b)𝑓(𝑡)−^𝑁_𝑛=1𝑐_𝑛𝜙_𝑛(𝑡)=𝑜𝜙_𝑁(𝑡),as𝑡→0⁺. (1.4)
The functions 𝑔_𝑖,𝑓_𝑖,  and  𝐾_𝑖 will be assumed to satisfy the following:(i)𝑔_𝑖(𝑡)∈𝐶¹(𝐽), 𝑔_𝑖(𝑡)>0, 𝑔_𝑖(0⁺)=0, 𝑔^′_𝑖(𝑡)∼𝜓_𝑖(𝑡)𝑔^𝜆𝑖_𝑖(𝑡) as 𝑡→0⁺, 𝜆_𝑖>0, 𝜓_𝑖(𝑡)𝑔^𝜏_𝑖(𝑡)=𝑜(1) as 𝑡→0⁺ for each 𝜏>0, 𝑖=1,…,𝑛,(ii)|𝑓_𝑖(𝑡,𝑢,𝑣)|≤|𝑢|+|𝑣|, |∫^𝑡₀₊𝐾_𝑖(𝑡,𝑠,𝐲(𝑡),𝐲(𝑠))𝑑𝑠|≤𝑟_𝑖(𝑡)|𝐲|, 0<𝑟_𝑖(𝑡)∈𝐶(𝐽), 𝑟_𝑖(𝑡)=𝜑_𝑖(𝑡,𝐶_𝑖)𝑜(1) as 𝑡→0⁺ where 𝜑_𝑖(𝑡,𝐶_𝑖)=𝐶_𝑖exp(∫^𝑡_𝑡0(𝑎_𝑖/𝑔_𝑖(𝑠))𝑑𝑠) is the general solution of the equation 𝑔_𝑖(𝑡)𝑦^′_𝑖(𝑡)=𝑎_𝑖𝑦_𝑖(𝑡).
In the text, we will apply topological method of Ważewski and Schauder’s theorem. Therefore we give a short summary of them.
Let 𝑓(𝑡,𝐲) be a continuous function defined on an open (𝑡,𝐲) set Ω⊂ℝ×ℝ^𝑛, Ω⁰ an open set of Ω,𝜕Ω⁰ the boundary of Ω⁰, and Ω⁰ the closure of Ω⁰. Consider the following system of ordinary differential equations: 𝐲^′=𝑓(𝑡,𝐲). (1.5)

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))∉Ω⁰ on interval 𝑡₀<𝑡≤𝑡₀+𝜖₁(𝑡₀−𝜖₁≤𝑡<𝑡₀) for an 𝜖₁.

Definition 1.4 (see [18]). An open subset Ω⁰ 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 functions 𝑢_𝑖(𝑡,𝐲)∈𝐶¹(Ω,ℝ),  𝑖=1,…,𝑚 and 𝑣_𝑗(𝑡,𝐲)∈𝐶[Ω,ℝ]𝑗=1,…,𝑛,𝑚+𝑛>0 such that Ω₀=(𝑡,𝐲)∈Ω∶𝑢_𝑖(𝑡,𝐲)<0,𝑣_𝑗(𝑡,𝐲)<0∀𝑖,𝑗. (1.6)(2)𝑢̇_𝛼(𝑡,𝐲)<0 holds for the derivatives of the functions 𝑢_𝛼(𝑡,𝐲), 𝛼=1,…,𝑚 along trajectories of system (1.5) on the set 𝑈_𝛼=(𝑡,𝐲)∈Ω∶𝑢_𝛼(𝑡,𝐲)=0,𝑢_𝑖(𝑡,𝐲)≤0,𝑣_𝑗(𝑡,𝐲)≤0,∀𝑗and𝑖≠𝛼. (1.7)(3)𝑣̇_𝛽(𝑡,𝐲)>0 holds for the derivatives of the functions 𝑣_𝛽(𝑡,𝐲), 𝛽=1,…,𝑛 along trajectories of system (1.5) on the set 𝑉_𝛽=(𝑡,𝐲)∈Ω∶𝑢_𝛽(𝑡,𝐲)=0,𝑢_𝑖(𝑡,𝐲)≤0,𝑣_𝑗𝑡,𝑦≤0,∀𝑖and𝑗≠𝛽. (1.8)The set of all points of egress (strict egress) is denoted by Ω⁰_𝑒(Ω⁰_se).

Lemma 1.5 (see [18]). Let the set Ω₀ be a (𝑢,𝑣) subset of the set Ω with respect to system (1.5). Then Ω⁰_se=Ω⁰_𝑒=^𝑚_𝛼=1𝑈_𝛼⧵^𝑛_𝛽=1𝑉_𝛽. (1.9)

Definition 1.6 (see [18]). Let 𝑋 be a topological space and 𝐵⊂𝑋. Let 𝐴⊂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 set 𝐴⊂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]). Let Ω⁰ be some (𝑢,𝑣) subset of Ω with respect to system (1.5). Let 𝑆 be a nonempty compact subset of Ω⁰∪Ω⁰_𝑒 such that the set 𝑆∩Ω⁰_𝑒 is not a retract of 𝑆 but is a retract Ω⁰_𝑒. 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]). Let 𝐸 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 each 𝐶_𝑖≠0 there is one solution 𝐲(𝑡,𝐂)=(𝑦₁(𝑡,𝐶₁),𝑦₂(𝑡,𝐶₂),…,𝑦_𝑛(𝑡,𝐶_𝑛)), 𝐂=(𝐶₁,…,𝐶_𝑛) of initial problem (1.1) and (1.2) such that 𝑦_𝑖^(𝑗)𝑡,𝐶_𝑖−𝜑_𝑖^(𝑗)𝑡,𝐶_𝑖≤𝛿𝜑²_𝑖(𝑡,𝐶_𝑖)^(𝑗),𝑗=0,1, (2.1) for 𝑡∈(0,𝑡^Δ], where 0<𝑡^Δ≤𝑡₀, 𝛿>1 is a constant, and 𝑡^Δ depends on 𝛿,𝐶_𝑖, 𝑖=1,…,𝑛.

Proof. (1) Denote 𝐸 the Banach space of vector-valued continuous functions 𝐡(𝑡) on the interval [0,𝑡₀] with the norm ‖𝐡(𝑡)‖=max_{𝑡∈[0,𝑡0]}ℎ_𝑖(𝑡),𝑖=1,…,𝑛. (2.2) The subset 𝑆 of Banach space 𝐸 will be the set of all functions ℎ(𝑡) from 𝐸 satisfying the inequality ℎ_𝑖(𝑡)−𝜑_𝑖𝑡,𝐶_𝑖≤𝛿𝜑²_𝑖𝑡,𝐶_𝑖. (2.3) The set 𝑆 is nonempty, convex, and closed.
(2) Now we will construct the mapping 𝑃. 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: 𝑔_𝑖(𝑡)𝑦^′_𝑖(𝑡)=𝑎_𝑖𝑦_𝑖(𝑡)1+𝑓_𝑖𝑡,𝐲(𝑡),^𝑡₀₊𝐾_𝑖𝑡,𝑠,𝐡₀(𝑡),𝐡₀(𝑠)𝑑𝑠,𝑖=1,…,𝑛. (2.4) Put 𝑦_𝑖(𝑡)=𝜑_𝑖𝑡,𝐶_𝑖+𝜑_𝑖^(1−𝜇)𝑡,𝐶_𝑖𝑌_0𝑖(𝑡),𝑦^′_𝑖(𝑡)=𝜑^′_𝑖()+1𝑡,𝐶𝑔_𝑖(𝑡)𝜑_𝑖^(1−𝜇)𝑡,𝐶_𝑖𝑌_1𝑖(𝑡), (2.5)(2.6) where 0<𝜇<1 is a constant and new functions 𝑌_0𝑖(𝑡), 𝑌_1𝑖(𝑡) satisfy the differential equations as 𝑔_𝑖(𝑡)𝑌^′_0𝑖(𝑡)=𝜇−1𝑎_𝑖𝑌_0𝑖(𝑡)+𝑌_1𝑖(𝑡),𝑖=1,…,𝑛. (2.7) From (2.3), it follows ℎ_0𝑖(𝑡)=𝜑_𝑖𝑡,𝐶_𝑖+𝐻_0𝑖(𝑡),𝐻_0𝑖(𝑡)≤𝛿𝜑²_𝑖𝑡,𝐶_𝑖. (2.8) Substituting (2.5), (2.6), and (2.8) into (2.4), we get 𝑌_1𝑖(𝑡)=𝑎_𝑖𝑌_0𝑖(𝑡)+𝑎_𝑖𝜑^𝜇_𝑖𝑡,𝐶_𝑖+𝑎_𝑖𝑌_0𝑖(𝑡)×𝑓_𝑖𝑡,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑌₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛𝑌_0𝑛(𝑡),_𝑡0+𝐾_𝑖𝑡,𝑠,𝜑₁𝑡,𝐶₁+𝐻₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝐻_0𝑛(𝑡),𝜑₁𝑠,𝐶₁+𝐻₀₁(𝑠),…𝜑_𝑛𝑠,𝐶_𝑛+𝐻_0𝑛(𝑠)𝑑𝑠. (2.9) Substituting (2.9) into (2.7), we get 𝑔_𝑖(𝑡)Y^′_0𝑖(𝑡)=𝜇𝑎_𝑖𝑌_0𝑖(𝑡)+𝑎_𝑖𝜑^𝜇_𝑖𝑡,𝐶_𝑖+𝑎_𝑖𝑌_0𝑖(𝑡)×𝑓_𝑖𝑡,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑌₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛𝑌_0𝑛(𝑡),_𝑡0+𝐾_𝑖𝑡,𝑠,𝜑₁𝑡,𝐶₁+𝐻₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝐻_0𝑛(𝑡),𝜑₁𝑠,𝐶₁+𝐻₀₁(𝑠),…𝜑_𝑛𝑠,𝐶_𝑛+𝐻_0𝑛(𝑠)𝑑𝑠. (2.10) 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 set Ω⊂ℝ⁺×ℝ^𝑛. Denote 𝐘₀=(𝑌₀₁,…,𝑌_0𝑛). Define an open subset Ω₀⊂Ω as follows: Ω₀=𝑡,𝐘₀∶𝑢_𝑖𝑡,𝐘₀<0,𝑣𝑡,𝐘₀,𝑈<0,𝑖=1,…,𝑛_𝛼=𝑡,𝐘₀∶𝑢_𝛼𝑡,𝐘₀=0,𝑢_𝑖𝑡,𝐘₀≤0,𝑣𝑡,𝐘₀,𝑉≤0,𝑖=1,…,𝑛,𝑖≠𝛼_𝛽=𝑉=𝑡,𝐘₀∶𝑣𝑡,𝐘₀=0,𝑢_𝑗𝑡,𝐘₀≤0,𝑖=1,…,𝑛, (2.11) where 𝑢_𝑖𝑡,𝐘₀=𝑌²_0𝑖−𝛿𝜑_𝑖^(1+𝜇)𝑡,𝐶_𝑖²,𝑣𝑡,𝐘₀=𝑡−𝑡₀,𝑖=1,…,𝑛. (2.12) Calculating the derivatives 𝑢̇_𝛼(𝑡,𝐘₀), 𝑣(̇𝑡,𝐘₀) along the trajectories of (2.10) on the set 𝑈_𝛼, 𝑉, 𝛼=1,…,𝑛 we obtain 𝑢̇_𝛼𝑡,𝐘₀=2𝑎_𝛼𝑔_𝛼(𝑡)𝜇𝑌²_0𝛼(𝑡)+𝑌_0𝛼(𝑡)𝜑^𝜇_𝛼𝑡,𝐶_𝛼+𝑌²_0𝛼(𝑡)×𝑓_𝛼𝑡,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑌₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛𝑌_0𝑛(𝑡),_𝑡0+𝐾_𝑖𝑡,𝑠,𝜑₁𝑡,𝐶₁+𝐻₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝐻_0𝑛(𝑡),𝜑₁𝑠,𝐶₁+𝐻₀₁(𝑠),…𝜑_𝑛𝑠,𝐶_𝑛+𝐻_0𝑛(𝑠)𝑑𝑠.−𝛿²1+𝜇𝜑_𝛼^2(1+𝜇)𝑡,𝐶_𝛼. (2.13) Since lim_𝑡→+0𝜓_𝑖(𝑡)𝑔^𝜏_𝑖(𝑡)𝑔=0forany𝜏>0,𝑖=1,…,𝑛_′𝑖(𝑡)∼𝜓_𝑖(𝑡)𝑔^𝜆𝑖_𝑖(𝑡)as𝑡→0⁺,𝜆_𝑖>0,𝑖=1,…,𝑛, (2.14) then there exists a positive constant 𝑀_𝑖 such that 𝑔^′_𝑖(𝑡)<𝑀_𝑖,𝑡∈0,𝑡₀,𝑖=1,…,𝑛. (2.15) Consequently, ^𝑡_𝑡0𝑑𝑠𝑔_𝑖(𝑠)<1𝑀_𝑖^𝑡_𝑡0𝑔^′_𝑖(𝑠)𝑑𝑡𝑔_𝑖(𝑠)=1𝑀_𝑖𝑔ln^𝑖(𝑡)𝑔_𝑖𝑡₀⟶−∞as𝑡⟶0⁺,𝑖=1,…,𝑛. (2.16) From here lim_𝑡→0⁺𝜑_𝑖(𝑡,𝐶_𝑖)=0 and by L’Hospital’s rule 𝜑^𝜏_𝑖(𝑡,𝐶_𝑖)𝑔^𝜎_𝑖(𝑡)=𝑜(1), for 𝑡→0⁺, 𝑖=1,…,𝑛,𝜎 is an arbitrary real number. These both identities imply that the powers of 𝜑_𝑖(𝑡,𝐶_𝑖) affect the convergence to zero of the terms in (2.13), in a decisive way.
Using the assumptions of Theorem 2.1 and the definition of 𝑌₀(𝑡), 𝜑_𝑖(𝑡,𝐶_𝑖), 𝑖=1,…,𝑛, we get that the first term 𝜇𝑌²_0𝛼(𝑡,𝐶_𝛼) in (2.13) has the following form: 𝜇𝑌²_0𝛼(𝑡)=𝜇𝛿²𝜑_𝛼^2(1+𝜇)𝑡,𝐶_𝛼, (2.17) and the second term 𝑌_0𝛼(𝑡)𝜑^𝜇_𝛼𝑡,𝐶_𝛼+𝑌²_0𝛼(𝑡)×𝑓_𝛼𝑡,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑌₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛𝑌_0𝑛(𝑡),^𝑡₀₊𝐾_𝛼𝑡,𝑠,𝜑₁𝑡,𝐶₁+𝐻₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝐻_0𝑛(𝑡),𝜑₁𝑠,𝐶₁+𝐻₀₁(𝑠),…𝜑_𝑛𝑠,𝐶_𝑛+𝐻_0𝑛(𝑠)𝑑𝑠. (2.18) is bounded by terms with exponents which are greater than 𝜑_𝛼^2(1+𝜇)(𝑡,𝐶_𝛼), 𝛼=1,…,𝑛. From here, we obtain sgn𝑢̇_𝛼𝑡,𝐘₀=−𝛿²1+𝜇𝜑_𝛼^2(1+𝜇)𝑡,𝐶_𝛼=−1 (2.19) for sufficiently small 𝑡^∗, depending on 𝐶_𝛼𝛼=1,…,𝑛, 𝛿, 0<𝑡^∗≤𝑡₀.
It is obvious that sgn𝑣(̇𝑡,𝐘₀)=1.
Change the orientation of the axis t into opposite. Then, with respect to the new system of coordinates, the set Ω₀ 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 𝐘₀(𝑡)=(𝑌₀₁(𝑡),…,𝑌_0𝑛(𝑡)) be also the solution of (2.10). Putting 𝑍_0𝑖=𝑌_0𝑖−𝑌_0𝑖,𝑖=1,…,𝑛 (2.20) and substituting into (2.10), we obtain 𝑔_𝑖(𝑡)𝑍^′_0𝑖(𝑡)=𝜇𝑎_𝑖𝑌_0𝑖(𝑡)+𝑎_𝑖𝜑^𝜇_𝑖𝑡,𝐶_𝑖+𝑎_𝑖𝑍_0𝑖(𝑡)×𝑓_𝑖𝑡,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑍_0𝑖(𝑡)+𝑌₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛𝑍_0𝑛(𝑡)+𝑌_0𝑛(𝑡)_𝑡0+,𝐾_𝑖𝑡,𝑠,𝜑₁𝑡,𝐶₁+𝐻₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝐻_0𝑛(𝑡),𝜑₁𝑠,𝐶₁+𝐻₀₁(𝑠),…𝜑_𝑛𝑠,𝐶_𝑛+𝐻_0𝑛(𝑠)𝑑𝑠. (2.21) Define Ω₁(𝛿)=𝑡,𝐙₀∶0<𝑡<𝑡^∗,𝑢_1𝑖𝑡,𝐙₀<0,𝑣₁𝑡,𝐙₀<0,0<𝑡<𝑡^∗𝑈,𝑖=1,…,𝑛_1𝛼=𝑡,𝐙₀∶𝑢_1𝛼𝑡,𝐙₀=0,𝑢_1𝑖𝑡,𝐙₀≤0,𝑣₁𝑡,𝐙₀,𝑉≤0,𝑖=1,…,𝑛,𝑖≠𝛼_1𝛽=𝑉=𝑡,𝐙₀∶𝑣₁𝑡,𝐙₀=0,𝑢_𝑗𝑡,𝐙₀≤0,𝑖=1,…,𝑛, (2.22) where 𝑢_1𝑖𝑡,𝐙₀=𝑍²_0𝑖−𝛿𝜑_𝑖^{(1+𝜇−𝛾)2},0<𝛾<𝜇,𝑣₁𝑡,𝐙₀=𝑡−𝑡^∗. (2.23) Using the same method as above, we have sgn𝑢̇_1𝑖𝑡,𝐙₀=−1,sgn𝑣̇₁𝑡,𝐙₀=1,𝑖=1,…,𝑛 (2.24) for sufficiently small 𝑡^⋄, 0<𝑡^⋄≤𝑡^∗. It is obvious that Ω₀⊂Ω₁(𝛿) for 𝑡∈(0,𝑡^⋄). Let 𝐙₀(𝑡)=(𝑍₀₁(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 𝑦_𝑖𝑡,𝐶_𝑖−𝜑_𝑖𝑡,𝐶_𝑖≤𝛿𝜑²_𝑖𝑡,𝐶_𝑖,𝑖=1,…,𝑛, (2.25) where (𝑦₁(𝑡,𝐶₁),…,𝑦_𝑛(𝑡,𝐶_𝑛)) is the solution of (2.4) for 𝑡∈(0,𝑡^⋄]. Similarly, from (2.6) and (2.9), we have 𝑦^′_𝑖𝑡,𝐶_𝑖−𝜑^′_𝑖𝑡,𝐶_𝑖=1𝑔_𝑖(𝑡)𝜑_𝑖^(1−𝜇)𝑡,𝐶_𝑖𝑌_1𝑖(𝑡≤1)𝑔_𝑖(𝑡)𝜑_𝑖^(1−𝜇)𝑡,𝐶_𝑖2𝑎_𝑖𝛿𝜑_𝑖^(1−𝜇)𝑡,𝐶_𝑖=𝛿𝜑²_𝑖𝑡,𝐶_𝑖^′. (2.26) 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 that 𝑃𝑆 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 {𝐡_𝑘(𝑡)} be an arbitrary sequence vector-valued functions in 𝑆 such that 𝐡_𝑘(𝑡)−𝐡₀(𝑡)=𝜖_𝑘,lim_𝑘→∞𝜖_𝑘=0,𝐡₀(𝑡)∈𝑆. (2.27) The solution 𝐘_𝑘(𝑡)=(𝑌_𝑘1,…,𝑌_𝑘𝑛) of the following equation: 𝑔_𝑖(𝑡)𝑌^′_0𝑖(𝑡)=𝜇𝑎_𝑖𝑌_0𝑖(𝑡)+𝑎_𝑖𝜑^𝜇_𝑖𝑡,𝐶_𝑖+𝑎_𝑖𝑌_0𝑖(𝑡)×𝑓_𝑖𝑡,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑌₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛𝑌_0𝑛(𝑡),_𝑡0+𝐾_𝑖𝑡,𝑠,𝜑₁𝑡,𝐶₁+𝐻₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝐻_0𝑛(𝑡),𝜑₁𝑠,𝐶₁+𝐻_𝑘1(𝑠),…𝜑_𝑛𝑠,𝐶_𝑛+𝐻_𝑘𝑛(𝑠)𝑑𝑠. (2.28) corresponds to the function 𝐡_𝑘(𝑡) and 𝐘_𝑘(𝑡)∈Ω₀ for 𝑡∈(0,𝑡^⋄). Similarly, the solution 𝐘₀(𝑡) of (2.10) corresponds to the function ℎ₀(𝑡). We will show that |𝐘_𝑘(𝑡)−𝐘₀(𝑡)|→0 uniformly on [0,𝑡^Δ], where 0<𝑡^Δ≤𝑡^⋄, 𝑡^Δ is a sufficiently small constant which will be specified later. Consider the following region: Ω_0𝑘=𝑡,𝐘₀∶0<𝑡<𝑡^⋄,𝑢_0𝑘𝑖𝑡,𝐘₀<0,𝑣₀𝑡,𝐘₀<0,𝑖=1,…,𝑛, (2.29) where 𝑢_0𝑘𝑖𝑡,𝐘₀=𝑌_0𝑖(𝑡)−𝑌_0𝑖(𝑡)²−𝜖_𝑘𝜑_𝑖^{(1+𝜇−𝜈)}𝑡,𝐶_𝑖²𝑣,0<𝜈<𝛼,𝑖=1,…,𝑛,𝑘≥1,₀𝑡,𝐘₀=𝑡−𝑡^⋄. (2.30) There exists sufficiently small constant 𝑡^Δ≤𝑡^⋄ such that Ω₀⊂Ω_0𝑘 for any 𝑘,𝑡∈(0,𝑡^Δ). Investigate the behaviour of integral curves of (2.28) with respect to the boundary 𝜕Ω_0𝑘, 𝑡∈(0,𝑡^Δ]. Using the same method as above, we obtain the following trajectory derivatives: sgn𝑢̇_0𝑘𝑡,𝐘₀=−1,sgn𝑣̇₀𝑡,𝐘₀=1 (2.31) for 𝑡∈(0,𝑡^Δ] and any 𝑘. By Ważewski’s topological method, there exists at least one solution 𝐘_𝑘(𝑡) lying in Ω_0𝑘, 0<𝑡<𝑡^Δ. Hence, it follows that 𝑌_𝑘𝑖(𝑡)−𝑌_0𝑖(𝑡)≤𝜖_𝑘𝜑_𝑖^{1+𝜇−𝜈}≤𝑁_𝑖𝜖_𝑘, (2.32)𝑁_𝑖>0, 𝑖=1,…,𝑛 are constants depending on 𝐶_𝑖, 𝑡^Δ. From (2.5), we obtain 𝑦_𝑘𝑖(𝑡)−𝑦_0𝑖(𝑡)=𝜑_𝑖^(1−𝜇)𝑡,𝐶_𝑖𝑌_𝑘𝑖(𝑡)−𝑌_0𝑖(𝑡)≤𝑛_𝑖𝜖_𝑘, (2.33) where 𝑛_𝑖>0, 𝑖=1,…,𝑛 are constants depending on 𝑡^Δ, 𝐶_𝑖, 𝑁_𝑖. This estimate implies that 𝑃 is continuous.
We have thus proved that the mapping 𝑃 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 𝑌_1𝑖(𝑡)=𝑎_𝑖𝑌_0𝑖(𝑡)+𝑎_𝑖𝜑^𝜇_𝑖𝑡,𝐶_𝑖+𝑎_𝑖𝑌_0𝑖(𝑡)×𝑓_𝑖𝑡,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑌₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛𝑌_0𝑛(𝑡),_𝑡0+𝐾_𝑖𝑡,𝑠,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑌₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛𝑌_0𝑛(𝑡),𝜑₁𝑠,𝐶₁+𝜑₁^(1−𝜇)𝑠,𝐶₁𝑌₀₁(𝑠),…𝜑_𝑛𝑠,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑠,𝐶_𝑛𝑌_0𝑛(𝑠)𝑑𝑠. (2.34) Equation (2.7) may be written in the following form: 𝑔_𝑖(𝑡)𝑌^′_0𝑖(𝑡)=𝑎_𝑖𝑌_0𝑖(𝑡)+𝑎_𝑖𝜑^𝜇_𝑖𝑡,𝐶_𝑖+𝑎_𝑖𝑌_0𝑖(𝑡)×𝑓_𝑖𝑡,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑌₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛𝑌_0𝑛(𝑡),_𝑡0+𝐾_𝑖𝑡,𝑠,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑌₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛×𝑌_0𝑛(𝑡),𝜑₁𝑠,𝐶₁+𝜑₁^(1−𝜇)𝑠,𝐶₁𝑌₀₁(𝑠),…𝜑_𝑛𝑠,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑠,𝐶_𝑛𝑌_0𝑛(𝑠)𝑑𝑠. (2.35)
Now we know that there exists the solution 𝐲₀(𝑡)=(𝑦₀₁(𝑡,𝐶₁),…,𝑦_0𝑛(𝑡,𝐶_𝑛)) of (1.1) satisfying (1.2) such that 𝑦_0𝑖𝑡,𝐶_𝑖=𝜑_𝑖𝑡,𝐶_𝑖+𝜑_𝑖^(1−𝜇)𝑡,𝐶_𝑖𝑇_0𝑖(𝑡),𝑖=1,…,𝑛, (2.36) where 𝐓₀(𝑡)=(𝑇₀₁(𝑡),…,𝑇_0𝑛(𝑡)) is the solution of (2.35).
Denote 𝑊_𝑖0(𝑡)=𝑌_0𝑖(𝑡)−𝑇_0𝑖(𝑡), 𝑖=1,…,𝑛. Substituting 𝑊_𝑖0(𝑡) into (2.35), we obtain 𝑔_𝑖(𝑡)𝑊^′_0𝑖(𝑡)=𝑎_𝑖𝑊_0𝑖(𝑡)+𝑎_𝑖𝜑^𝜇_𝑖𝑡,𝐶_𝑖+𝑎_𝑖𝑊_0𝑖(𝑡)×𝑓_𝑖𝑡,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑊₀₁(𝑡)+𝑇₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛×𝑊_0𝑛(𝑡)+𝑇_0𝑛(𝑡)_𝑡0+,𝐾_𝑖𝑡,𝑠,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑊₀₁(𝑡)+𝑇₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛×𝑊_0𝑛(𝑡)+𝑇_0𝑛(𝑡),𝜑₁𝑠,𝐶₁+𝜑₁^(1−𝜇)𝑠,𝐶₁×𝑊₀₁(𝑠)+𝑇₀₁(𝑠),…𝜑_𝑛𝑠,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑠,𝐶_𝑛𝑊_0𝑛(𝑠)+𝑇_0𝑛(𝑠)𝑑𝑠−𝑓_𝑖𝑡,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑇₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛,_𝑡0+𝐾_𝑖𝑡,𝑠,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑇₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛𝑇_0𝑛(𝑡)+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛𝑇_0𝑛(𝑡),𝜑₁𝑠,𝐶₁+𝜑₁^(1−𝜇)𝑠,𝐶₁×𝑇₀₁(𝑠),…𝜑_𝑛𝑠,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑠,𝐶_𝑛𝑇_0𝑛(𝑠)𝑑𝑠. (2.37) Let ¹Ω₀=𝑡,𝐖₀∶0<𝑡<𝑡^Δ,𝑢_1𝑖𝑡,𝐖₀<0,𝑣₁𝑡,𝐖₀<0, (2.38) where 𝑢_1𝑖𝑡,𝐖₀=𝑊²_1𝑖−𝛿𝜑_𝑖^{(1+𝜇−𝜌)}(𝑡,𝐶_𝑖)²,0<𝜌<𝜇,𝑣₁𝑡,𝐖₀=𝑡−𝑡^Δ,𝑖=1,…,𝑛. (2.39) 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,𝑡^Δ].
We will suppose that there exists nontrivial solution 𝐖₀(𝑡) of (2.37) lying in ¹Ω₀. Substituting 𝑊_0𝑖(𝑠) instead of 𝑊_0𝑖(𝑠), 𝑖=1,…,𝑛 into (2.37), we obtain the following differential equation: 𝑔_𝑖(𝑡)𝑊^′_0𝑖(𝑡)=𝑎_𝑖𝑊_0𝑖(𝑡)+𝑎_𝑖𝜑^𝜇_𝑖𝑡,𝐶_𝑖+𝑎_𝑖𝑊_0𝑖(𝑡)×𝑓_𝑖𝑡,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑊₀₁(𝑡)+𝑇₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛×𝑊_0𝑛(𝑡)+𝑇_0𝑛(𝑡)_𝑡0+,𝐾_𝑖𝑡,𝑠,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑊₀₁(𝑡)+𝑇₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛×𝑊_0𝑛(𝑡)+𝑇_0𝑛(𝑡),𝜑₁𝑠,𝐶₁+𝜑₁^(1−𝜇)𝑠,𝐶₁×𝑊₀₁(𝑠)+𝑇₀₁(𝑠),…𝜑_𝑛𝑠,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑠,𝐶_𝑛𝑊_0𝑛(𝑠)+𝑇_0𝑛(𝑠)𝑑𝑠−𝑓_𝑖𝑡,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑇₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛,_𝑡0+𝐾_𝑖𝑡,𝑠,𝜑₁𝑡,𝐶₁+𝜑₁^(1−𝜇)𝑡,𝐶₁𝑇₀₁(𝑡),…,𝜑_𝑛𝑡,𝐶_𝑛𝑇_0𝑛(𝑡)+𝜑_𝑛^(1−𝜇)𝑡,𝐶_𝑛𝑇_0𝑛(𝑡),𝜑₁𝑠,𝐶₁+𝜑₁^(1−𝜇)𝑠,𝐶₁×𝑇₀₁(𝑠),…𝜑_𝑛𝑠,𝐶_𝑛+𝜑_𝑛^(1−𝜇)𝑠,𝐶_𝑛𝑇_0𝑛(𝑠)𝑑𝑠. (2.40)
Calculating the derivative 𝑢̇_1𝑖(𝑡,𝐖₀) along the trajectories of (2.40) on the set 𝜕¹Ω₀, we get sgn𝑢̇_1𝑖(𝑡,𝐖₀)=−1 for 𝑡∈(0,𝑡^Δ], 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: 𝑔(𝑡)𝑦^′_𝑖=𝑦_𝑖+^𝑡₀₊^𝑁𝑖_{|𝜎𝑖|+|𝜔𝑖|=2}𝑢_{𝜎𝑖𝜔𝑖}(𝑡)𝑣_{𝜎𝑖𝜔𝑖}(𝑠)𝐲^𝜎𝑖(𝑡)𝐲^𝜔𝑖(𝑠)𝑑𝑠, (3.1) where 𝑁_𝑖∈ℕ, 𝜎_𝑖=(𝑙_𝑖1,…,𝑙_𝑖𝑛), 𝜔_𝑖=(𝑗_𝑖1,…,𝑗_𝑖𝑛), 𝑙_𝑖𝑘,𝑗_𝑖𝑘∈ℕ∪{0}, 𝑘=1,…,𝑛, 𝜎_𝑖=^𝑛_𝑘=1𝑙_𝑖𝑘,𝜔_𝑖=^𝑛_𝑘=1𝑗_𝑖𝑘,𝐲^𝜎𝑖(𝑡)=^𝑛_𝑘=1𝑦^𝑙𝑖𝑘_𝑘(𝑡),𝐲^𝜔𝑖(𝑠)=^𝑛_𝑘=1𝑦^𝑗𝑖𝑘_𝑘(𝑠),𝑢_{𝜎𝑖𝜔𝑖}(𝑡),𝑣_{𝜎𝑖𝜔𝑖}(𝑡)∈𝐶⁰(𝐽),𝐽=0,𝑡₀,𝑖=1,…,𝑛. (3.2)

We will construct the solution of (3.1) in the form of one parametric asymptotic expansions as 𝑦_𝑖(𝑡,𝐶)=^∞_ℎ=1𝑓_𝑖ℎ(𝑡)𝜙^ℎ(𝑡,𝐶),𝑖=1,…,𝑛, (3.3) where 𝜙(𝑡,𝐶) is the general solution of the differential equation 𝑔(𝑡)𝑦^′=𝑦 so that 𝜙(𝑡,𝐶)=𝐶exp^𝑡_𝑡0𝑑𝜏𝑔(𝜏), (3.4)𝑓_𝑖1(𝑡)≡1, 𝑓_𝑖ℎ(𝑡), ℎ≥2, 𝑖=1,…,𝑛 are unknown functions, 𝐶≠0 is a constant.

Consider the following differential equation: 𝑔(𝑡)𝑦^′=𝑞𝑦+𝑝(𝑡). (3.5) Diblík [3] proved asymptotic estimates of the solution of (3.5) which we can be formulated as follows.

Theorem 3.1. Assume that (I)Let 𝑞 be a constant, 𝑞<0, 𝑔(𝑡)∈𝐶¹(𝐽), 𝑔(𝑡)>0, lim_𝑡→𝑡⁺₀𝑔(𝑡)=0, 𝑔^′(𝑡)∼𝜓₁(𝑡)𝑔^𝜆1(𝑡) as 𝑡→𝑡⁺₀, 𝜆₁>0, lim_𝑡→𝑡⁺₀𝜓₁(𝑡)𝑔^𝜏(𝑡)=0,𝜏 is any positive number. (II)𝑝(𝑡)∈𝐶(𝐽), 𝑝(𝑡)=𝑏₀(𝑡)𝑔^𝜆(𝑡)+𝑂(𝑏₁(𝑡)𝑔^𝜆+𝜖(𝑡)), 𝜖>0,lim_𝑡→𝑡⁺₀𝑏_𝑚(𝑡)𝑔^′(𝑡)=0, 𝑚=0,1,𝑏₀(𝑡)∈𝐶(𝐽),𝑏₀(𝑡)≠0, 𝑏^′₀(𝑡)∼𝜓₂(𝑡)𝑔^𝜆2(𝑡) as 𝑡→𝑡⁺₀,   𝜆₂+1>0, lim_𝑡→𝑡⁺₀𝜓₂(𝑡)𝑔^𝜏(𝑡)=0, lim_𝑡→𝑡⁺₀𝑔^𝜏(𝑡)(𝑏₀(𝑡))⁻¹=0.
Then (3.5) has a unique solution on (0,𝑡],0<𝑡≤𝑡₀, satisfying asymptotic estimates 𝑦(𝑥)=−1𝑞𝑏₀(𝑥)𝑔^𝜆(𝑥)+𝑂𝑔^𝜈(𝑥),𝑦^′(𝑥)=𝑂𝑔^𝜈−1(𝑥), (3.6) where 𝜈∈(𝜆,𝜆+min{𝜆₁,𝜆₂+1,𝜖}).

Now we will show the results of Theorem 3.1. 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: 𝑔(𝑡)𝑓^′_𝑖ℎ=(1−ℎ)𝑓_𝑖ℎ+𝜙^−ℎ(𝑡,𝐶)^𝑡₀₊𝑅_𝑖ℎ(𝑡,𝑠)𝑑𝑠, (3.7)ℎ≥2,𝑖=1,…,𝑛 and 𝑅_𝑖ℎ(𝑡,𝑠)=𝑅_𝑖ℎ𝑓₁₁(𝑡),…,𝑓_𝑖ℎ−1(𝑡),…,𝑓_𝑛1(𝑡),…,𝑓_𝑛ℎ−1(𝑡),𝑓₁₁(𝑠),…,𝑓_1ℎ−1(𝑠),…,𝑓_𝑛1(𝑠),…,𝑓_𝑛ℎ−1(𝑠). (3.8) Denote 𝑝_𝑖ℎ(𝑡)=𝜙^−ℎ(𝑡,𝐶)^𝑡₀₊𝑅_𝑖ℎ(𝑡,𝑠)𝑑𝑠, (3.9) then it is obvious that the recurrence equations 𝑔(𝑡)𝑓^′_𝑖ℎ=(1−ℎ)𝑓_𝑖ℎ+𝑝_𝑖ℎ(𝑡) (3.10)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: 𝑡²𝑦^′₁=𝑦₁+^𝑡₀₊1𝑡³𝑦₁(𝑠)𝑦₂(𝑠)𝑡𝑑𝑠,₂𝑦^′₂=𝑦₂+^𝑡₀₊√𝑡𝑦₁(𝑡)𝑦₂(𝑠)𝑑𝑠. (3.11) System (3.11) has the form of system (3.1) for 𝜎₁𝜔₁=(0,0,1,1),𝑢_𝜎1𝜔1(𝑡)=1𝑡³,𝑣_𝜎1𝜔1(𝑠)=1,𝑁₁𝜎=2₂𝜔₂=(1,0,0,1),𝑢_𝜎2𝜔2(𝑡)=√𝑡,𝑣_𝜎2𝜔2(𝑠)=1,𝑁₂=2. (3.12) We will construct a solution of system (3.11) in the following form: 𝑦₁=^∞_𝑘=1𝑓_1𝑘(𝑡)𝜙^𝑘(𝑡,𝐶),𝑦₂=^∞_𝑘=1𝑓_2𝑘(𝑡)𝜙^𝑘(𝑡,𝐶), (3.13) where 𝜙(𝑡,𝐶) is the general solution of the equation 𝑡²𝑦^′=𝑦. We will demonstrate the calculation of coefficients 𝑓_𝑖ℎ 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: 𝜙¹()𝜙𝑡,𝐶:1=1,1=1.₂(𝑡,𝐶):𝑡²𝑓^′₁₂=−𝑓₁₂+𝜙⁻²(𝑡,𝐶)^𝑡₀₊1𝑡³𝜙²()𝑡𝑠,𝐶𝑑𝑠,₂𝑓^′₂₂=−𝑓₂₂+𝜙⁻²(𝑡,𝐶)^𝑡₀₊√()𝜙()𝜙𝑡𝜙𝑡,𝐶𝑠,𝐶𝑑𝑠.₃(𝑡,𝐶):𝑡²𝑓^′₁₃=−2𝑓₁₃+𝜙⁻³(𝑡,𝐶)^𝑡₀₊1𝑡³𝑓₁₂(𝑠)+𝑓₂₂(𝑠)𝜙³()𝑡𝑠,𝐶𝑑𝑠,₂𝑓^′₂₃=−2𝑓₂₃+𝜙⁻³(𝑡,𝐶)^𝑡₀₊√𝑡𝑓₁₂(𝑡)𝜙²(𝑡,𝐶)𝜙(𝑠,𝐶)+𝑓₂₂(𝑠)𝜙²(𝑠,𝐶)𝜙(𝑡,𝐶)𝑑𝑠. (3.14)(3.15)(3.16) Put 𝑢₁=𝜙⁻²(𝑡,𝐶)^𝑡₀₊𝜙²(𝑠,𝐶)𝑑𝑠,𝑢₂=𝜙⁻¹(𝑡,𝐶)^𝑡₀₊𝜙(𝑠,𝐶)𝑑𝑠. (3.17) Differentiating both equations (3.17), we obtain the following differential equations: 𝑡²𝑢^′₁=−2𝑢₁+𝑡²,𝑡²𝑢^′₂=−𝑢₂+𝑡². (3.18)(3.19) Equation (3.18) satisfies assumptions of Theorem 3.1. with following functions and coefficients: 𝑎=−2,𝑏₀(𝑡)=1,𝑔^𝜆(𝑡)=𝑡²¹⇒𝜆=1,𝑏₁(𝑡)𝑔=0,_′(𝑡)=𝑡^2′=2𝑔(𝑡)^1/2⇒𝜆₁=12,𝑏^′₀(𝑡)=0⋅𝑔^𝜆2(𝑡). (3.20)
Hence we can choose a constant 𝜆₂+1>1/2 and similarly 𝜖>1/2. By Theorem 3.1., we have 𝑢₁=12𝑡²+𝑂𝑡^2𝜈1,𝜈₁∈31,2. (3.21) Second equation (3.19) is different from (3.18) only in the constant 𝑎=−1. Thus 𝑢₂=𝑡²+𝑂𝑡^2𝜈2,𝜈₂∈31,2. (3.22) 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: 𝑡²𝑓^′₁₂=−𝑓₁₂+1𝑡2𝑡+𝑂^2𝜈1−3,𝑡²𝑓^′₂₂=−𝑓₂₂+𝑡^5/2+𝑂𝑡^2𝜈2+1/2. (3.23)(3.24) For (3.23), we can put 𝑎=−1,𝑏₀(𝑡)=12,𝑔^𝜆(𝑡)=𝑡^2−1/21,𝜆=−2,𝑏₁(𝑡)=1,𝜖=𝜈₁−1,𝑔^′(𝑡)=𝑡^2′=2𝑔(𝑡)^1/2⇒𝜆₁=12,𝑏^′₀(𝑡)=0⋅𝑔^𝜆2(𝑡). (3.25)
Then we can choose a constant 𝜆₂+1>1/2. By Theorem 3.1., we get 𝑓₁₂(𝑡)=1𝑡2𝑡+𝑂^2𝜈12,𝑓^′₁₂(𝑡)=𝑂𝑡^2𝜈12−2,𝜈₁₂∈−12,0. (3.26) Similarly for (3.24), we can put 𝑎=−1, 𝑏₀(𝑡)=1, 𝑔^𝜆(𝑡)=(𝑡²)^5/4, 𝜆=5/4,𝑏₁(𝑡)=1, 𝜖=𝜈₂−1, 𝑔^′(𝑡)=𝑡^2′=2𝑔(𝑡)^1/2⇒𝜆₁=12,𝑏^′₀(𝑡)=0⋅𝑔^𝜆2(𝑡). (3.27)
Then we can choose a constant 𝜆₂+1>1/2. By Theorem 3.1., we have 𝑓₂₂(𝑡)=𝑡^5/2+𝑂𝑡^2𝜈22,𝑓^′₂₂(𝑡)=𝑂𝑡^2𝜈22−2,𝜈₂₂∈54,74. (3.28) Substituting coefficients 𝑓₁₂, 𝑓₂₂ into (3.16) and using the same method as in the calculation of coefficients 𝑓₁₂, 𝑓₂₂, we have 𝑓₁₃(𝑡)=112𝑡²+𝑂𝑡^2𝜈13,𝑓^′₁₃(𝑡)=𝑂𝑡^2𝜈13−1,𝜈₁₃∈1−1,−2,𝑓₂₃(𝑡)=14𝑡^3/2+𝑂𝑡^2𝜈23,𝑓^′₂₃(𝑡)=𝑂𝑡^2𝜈23−1,𝜈₂₃∈34,54. (3.29) Thus the solution of system (3.11) has for ℎ=3 the following asymptotic expansions: 𝑦₁()+1≈𝜙𝑡,𝐶𝑡2𝑡+𝑂^2𝜈12𝜙²()+1𝑡,𝐶12𝑡²+𝑂𝑡^2𝜈13𝜙³(),𝑦𝑡,𝐶₂≈𝜙(𝑡,𝐶)+𝑡^5/2+𝑂𝑡^2𝜈22𝜙²()+1𝑡,𝐶4𝑡^3/2+𝑂𝑡^2𝜈23𝜙³(𝑡,𝐶). (3.30)

Acknowledgments

The first author is supported by Grant FEKT-S-11-2-921 of the Faculty of Electrical Engineering and Communication, Brno University of Technology and Grant P201/11/0768 of the Czech Grant Agency (Prague).