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

Definition 1.1. The sequence of functions (ϕ_n(t)) is called an asymptotic sequence as t → 0⁺ if

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)
  $$ ()

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, $$ as t → 0⁺, λ_i > 0, $$ as t → 0⁺ for each τ > 0, i = 1, …, n,

• (ii)

  |f_i(t, u, v)| ≤ |u| + |v|, $$, 0 < r_i(t) ∈ C(J), r_i(t) = φ_i(t, C_i)o(1) as t → 0⁺ where $$ is the general solution of the equation $$.

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 × ℝⁿ, Ω⁰ an open set of Ω, ∂Ω⁰ the boundary of Ω⁰, and $$ the closure of Ω⁰. Consider the following system of ordinary differential equations:

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 $$ 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

  $$ ()

• (2)

  $$ holds for the derivatives of the functions u_α(t, y), α = 1, …, m along trajectories of system (1.5) on the set

  $$ ()

• (3)

  $$ holds for the derivatives of the functions v_β(t, y), β = 1, …, n along trajectories of system (1.5) on the set

  $$ ()

The set of all points of egress (strict egress) is denoted by $$.

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

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 $$ such that the set $$ is not a retract of S but is a retract $$. 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.

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

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

The subset S of Banach space E will be the set of all functions h(t) from E satisfying the inequality 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:

Put where 0 < μ < 1 is a constant and new functions Y_0i(t), Y_1i(t) satisfy the differential equations as From (2.3), it follows Substituting (2.5), (2.6), and (2.8) into (2.4), we get Substituting (2.9) into (2.7), we get 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⁺ × ℝⁿ. Denote Y₀ = (Y₀₁, …, Y_0n). Define an open subset Ω₀ ⊂ Ω as follows:

where Calculating the derivatives $$, $$ along the trajectories of (2.10) on the set U_α, V, α = 1, …, n we obtain Since then there exists a positive constant M_i such that Consequently, From here $$ and by L’Hospital’s rule $$, 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 $$ in (2.13) has the following form:

and the second term is bounded by terms with exponents which are greater than $$, α = 1, …, n. From here, we obtain for sufficiently small t^*, depending on C_α  α = 1, …, n, δ, 0 < t^* ≤ t₀.

It is obvious that $$.

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 $$ be also the solution of (2.10). Putting

and substituting into (2.10), we obtain Define where Using the same method as above, we have for sufficiently small t^⋄, 0 < t^⋄ ≤ t^*. It is obvious that Ω₀ ⊂ Ω₁(δ) for t ∈ (0, t^⋄). Let $$ be any nonzero solution of (2.10) such that $$ for 0 < t₁ < t^⋄. Let $$ be such a constant that $$. If the curve $$ lay in $$ for 0 < t < t₁, then $$ would have to be a strict egress point of $$ 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 $$ is the unique solution of (2.10).

From (2.5) we obtain

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 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

The solution $$ of the following equation: corresponds to the function h_k(t) and $$ for t ∈ (0, t^⋄). Similarly, the solution $$ of (2.10) corresponds to the function h₀(t). We will show that $$ 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: where 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: for t ∈ (0, t^Δ] and any k. By Ważewski’s topological method, there exists at least one solution $$ lying in Ω_0k, 0 < t < t^Δ. Hence, it follows that N_i > 0, i = 1, …, n are constants depending on C_i, t^Δ. From (2.5), we obtain 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

Equation (2.7) may be written in the following form:

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

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

Let where If (2.37) had only the trivial solution lying in  ¹Ω₀, 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 $$ of (2.37) lying in   ¹Ω₀. Substituting $$ instead of W_0i(s), i = 1, …, n into (2.37), we obtain the following differential equation:

Calculating the derivative $$ along the trajectories of (2.40) on the set ∂¹Ω₀, we get $$ 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.

Theorem 3.1. Assume that

• (I)

  Let q be a constant, q < 0, g(t) ∈ C¹(J), g(t) > 0, $$, $$ as $$, λ₁ > 0, $$ is any positive number.

• (II)

  p(t) ∈ C(J), p(t) = b₀(t)g^λ(t) + O(b₁(t)g^λ+ϵ(t)), $$, m = 0,1,   b₀(t) ∈ C(J),  b₀(t) ≠ 0,    $$ as $$,   λ₂ + 1 > 0, $$, $$.

Then (3.5) has a unique solution on $$, satisfying asymptotic estimates

where ν ∈ (λ, λ + min {λ₁, λ₂ + 1, ϵ}).

Example 3.2. Consider the following system of integrodifferential equations:

System (3.11) has the form of system (3.1) for We will construct a solution of system (3.11) in the following form: 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: Put Differentiating both equations (3.17), we obtain the following differential equations: Equation (3.18) satisfies assumptions of Theorem 3.1. with following functions and coefficients:

Hence we can choose a constant λ₂ + 1 > 1/2 and similarly ϵ > 1/2. By Theorem 3.1., we have

Second equation (3.19) is different from (3.18) only in the constant a = −1. Thus 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: For (3.23), we can put

Then we can choose a constant λ₂ + 1 > 1/2. By Theorem 3.1., we get

Similarly for (3.24), we can put a = −1, b₀(t) = 1, $$, λ = 5/4, b₁(t) = 1, ϵ = ν₂ − 1,

Then we can choose a constant λ₂ + 1 > 1/2. By Theorem 3.1., we have

Substituting coefficients f₁₂, f₂₂ into (3.16) and using the same method as in the calculation of coefficients f₁₂, f₂₂, we have Thus the solution of system (3.11) has for h = 3 the following asymptotic expansions: