On the construction of the approximate solution of a special type integral boundary value problem

We consider the integral boundary value problem (BVP) for a certain class of non-linear system of ordinary differential equations of the form dx (t) dt = f (t, x (t)) , Ax(0) + ∫ T 0 P(s)k(s, x(s))ds + Cx(T) = d, where t ∈ [0, T], x ∈ Rn, f : [0, T]× D → Rn and k : [0, T]× D → Rn are continuous vector functions, D ⊂ Rn is a closed and bounded domain, A, C and d are arbitrary matrices and vector with real components, det C 6= 0. We give a new approach for studying this problem, namely by using an appropriate parametrization technique the original BVP is reduced to the equivalent parametrized two-point one with linear restrictions without integral term. To study the transformed problem we use a method based upon a special type of successive approximations constructed analytically.


Notations
• Operations =, <, >, ≤, ≥, max, min between matrices and vectors mean componentwise; • L(R n ) is an algebra of n-dimensional matrices with real components; • I n and O n are unit and zero n-dimensional matrices, respectively; • for any vector u ∈ R n and non-negative vector r ∈ R n we write B(u, r) := {ξ ∈ R n : |ξ − u| ≤ r} as an r-neighborhood of u ∈ R n ; • r(K) -spectral radius of matrix K. Email:katya_marinets@ukr.net

Problem setting and parametrization of the integral boundary conditions
Let us investigate the solutions of the system of nonlinear differential equations subjected to the special type integral boundary conditions of the form: where t ∈ [0, T], f are some given matrices and vector and P is a continuous n-dimensional matrix function.
Suppose that the vector function f in the right-hand side of the system of differential equations is continuous, where D ⊂ R n is a closed and bounded domain, and let us put The problem is to find the continuously differentiable solution x : [0, T] → D of the system of differential equations (2.1) satisfying integral boundary restrictions (2.2).
Remark 2.1.The set of the solutions of the non-linear BVP with integral boundary conditions (2.1), (2.2) coincides with the set of the solutions of the parametrized problem (2.1) with linear boundary restrictions (2.5), satisfying additional conditions (2.3).

Construction of the successive approximations and their uniform convergence
Let us introduce the vector and assume that the BVP (2.1), (2.5) satisfies following conditions: A) there exists a set D β ⊂ D such that for all λ ∈ D 0 , t ∈ [0, T]; B) function f in the right-hand side of (2.1) satisfies Lipschitz condition of the form for all t ∈ [0, T] , {u, v} ⊂ D with some non-negative constant matrix C) the spectral radius r(Q) satisfies the inequality where Let us connect with the parametrized BVP (2.1), (2.5) the sequence of functions: where m ∈ N, x m (t, z, λ) = col (x m,1 (t, z, λ) , x m,2 (t, z, λ) , . . . ,x m,n (t, z, λ)) and z, λ are considered as param- eters.
Note that the functions x m of the sequence (3.6) were built from the linear parametrized boundary conditions (2.5), so they satisfy them for all m ∈ N, z ∈ D β , λ ∈ D 0 .
Similarly to [4], let us establish the uniform convergence of the sequence (3.6).
Theorem 3.1.Assume that for the system of differential equations (2.1) and the parametrized boundary restrictions (2.5) conditions A)-C) are satisfied.Then for all fixed z ∈ D β , λ ∈ D 0 the following hold.
1.The functions of the sequence (3.6) are continuously differentiable and satisfy the parametrized boundary conditions (2.5): 2. The sequence of functions (3.6) for t ∈ [0, T] converges uniformly as m → ∞ to the limit function (3.7) 3. The limit function x ∞ satisfies the parametrized linear two-point boundary conditions: 4. The limit function (3.7) is a unique continuously differentiable solution of the integral equation i.e., it is the unique solution on [0, T] of the Cauchy problem for the modified system of differential equations: where 5. The following error estimation holds: where δ D ( f ) and Q are given by (3.1), (3.5).
Proof.Let us prove that the sequence of functions (3.6) is a Cauchy sequence in the Banach space C([0, T], R n ).First we show that x m (t, z, λ) ∈ D, for all (t, z, λ) ∈ [0, T] × D β × D 0 , m ∈ N. Let us note, that the function x 0 (t, z, λ) ∈ D for all (t, z, λ) ∈ [0, T] × D β × D 0 .Then, using the estimation from [5]: where Therefore, by virtue of (3.15), we conclude that By induction we can easily establish that all functions (3.6) are also contained in the domain Now, consider the difference of functions: and introduce the notation: By virtue of the estimation (3.13) and of the Lipschitz condition (3.3), we have: According to (3.15), Using the inequality from [5] α m+1 (t) ≤ 10 9 obtained for the sequence of functions from (3.16) for m = 1 follows: By induction using (3.18), we can easily obtain that where α m+1 is calculated according to (3.18), and δ D ( f ) is given by (3.1).By virtue of the estimate (3.17) from (3.19) we get: ∀m ∈ N, where matrix Q is given by (3.5).Therefore, in view of (3.20): Since, due to the condition (3.4), the maximum eigenvalue of the matrix Q of the form (3.5) does not exceed the unity, we have Therefore, we conclude from (3.21) that, according to the Cauchy criterion, the sequence {x m } of the form (3.6) uniformly converges in the domain (t, z, λ) ∈ [0, T] × D β × D 0 to the limit function x ∞ .Since all functions x m of the sequence (3.6) satisfy the boundary conditions (2.5) for all values of the artificially introduced parameters, the limit function x ∞ also satisfies these conditions.Passing to the limit as m → ∞ in equality (3.6) we show that for all z ∈ D β and λ ∈ D 0 the limit function x ∞ (•, z, λ) is a solution of both integral equation (3.8) and the Cauchy problem (3.9), (3.10) with ∆ given by (3.11).The uniqueness of x ∞ (•, z, λ) follows from the Lipschitz condition imposed on the function f .

Connection of the limit function x ∞ with the solution of the BVP (2.1), (2.2)
Consider the Cauchy problem where µ ∈ R n is a control parameter and z ∈ D β .By analogy to [4] let us prove the control parameter theorem.
Theorem 4.1.Let z ∈ D β , λ ∈ D 0 and µ ∈ R n be some given vectors.Suppose that for the system of differential equations (2.1) all conditions of Theorem 3.1 hold.Then for the solution x = x(•, z, µ) of the initial value problem (4.1), (4.2) to be defined on [0, T] and to satisfy boundary conditions (2.5), it is necessary and sufficient that µ satisfies where and x ∞ (•, z, λ) is a function from the assertion 2. of Theorem 3.1.
In that case Proof.Sufficiency.Let us suppose that µ = µ z,λ on the right-hand side of the system of differential equations (4.1) is given by (4.4).By virtue of Theorem 3.1, the limit function (3.7) of the sequence (3.6) is the unique solution of the initial value problem (4.1), (4.2).Furthermore, the limit function x ∞ satisfies (2.5).Thus we have found the value of the parameter µ given by (4.4), for which (4.5) holds.Necessity.Now we show that the parameter value (4.4) is unique, i.e., that for any µ = μ = µ z,λ the solution x(t, z, μ) of the initial value problem (4.6), (4.2),where does not satisfy boundary condition (2.5).Indeed, assume the contrary.Then there exists μ ∈ R n such that µ z,λ = μ and the solution x(•) := x(•, z, μ) of the Cauchy problem (4.1)-(4.2) is defined on [0, T] and satisfies boundary condition (2.5).
It is obviously that the functions x z,λ and x satisfy the following integral equations and By assumption, the functions x z,λ , x satisfy parametrized boundary conditions (2.5) and the initial conditions (4.2).That is why we have x z,λ (0) = z, (4.10) x(0) = z.(4.12) Taking into account (4.9)-(4.12)we get By virtue of (4.7), (4.8) for t = T, µ z,λ and µ can be written as Using (4.17), (4.18) it is obvious that On the basis of the Lipschitz condition (3.3), from the relation (4.19) we get that the function satisfies integral inequalities: where α 1 is given by (3.14).Using (4.19) recursively, we come to an inequality: where m ∈ N and functions α m are given by the formula (3.18).Taking into account (3.17), from (4.22) for each m ∈ N we get an estimation: By passing to the limit as m → ∞ in the last inequality and by virtue of (3.4), we come to the conclusion that max It means, according to (4.20), that the function x z,λ coincides with x.Starting with (4.15) and (4.16), we get that µ z,λ = μ.The contradiction we received proves the necessity part of the theorem.
Let us find out the relation of the limit function x = x ∞ (•, z, λ) of the sequence (3.6) to the solution of the parametrized two-point BVP (2.1) with linear boundary conditions (2.5) or the equivalent non-linear problem (2.1) with integral conditions (2.2) [4].

Theorem 4.2. Let the conditions A)-C) hold for the original BVP (2.1), (2.2).
Then x ∞ (•, z * , λ * ) is a solution of the integral BVP (2.1), (2.2) if and only if the pair (z * , λ * ) is a solution of the determining system of algebraic or transcendental equations: where  Moreover, this solution is given by formula where x ∞ is the limit function of the sequence (3.6).
where µ z,λ is given by (4.4).Therefore, the first equation (4.23) of the determining system is satisfied.
Taking into account the above-proved equality (4.27), it follows from (4.28) that the second equation of the determining system also holds.

Remarks on the constructive applications of the method
Although Theorem gives sufficient and necessary conditions for the solvability and construction of the solution of the given BVP, its application faces with difficulties due the fact that the explicit forms of the functions in (4.23), (4.24) are usually unknown.
This complication can be overcome by using the properties of the function x m (•, z, λ) of the form (3.6) for a fixed m, which will lead one instead of the exact determining system (4.23),(4.24) to the m-th approximate system of determining equations of the form: V m (z, λ) = 0, ( where ∆ m , V m : D β × D 0 → R n are defined by the determining function given by formulae and x m (•, z, λ) is a vector function, that is defined by the recursive relation (3.6).
It is important to note that, unlike to system (4.23),(4.24), the m-th approximate determining system (5.1),(5.2) contains only terms involving the function x m and, thus known explicitly.

An illustrative example
Let us apply the numerical-analytic scheme described above to the system of differential equations x 1 (t) = x 2 , considered for t ∈ [0, 1] with the two-point integral boundary conditions where 13/30 , and It is easy to check that the pair of functions is an exact solution of the problem (6.1), (6.2).Suppose that the BVP (6.1), (6.2) is considered in the domain Following (2.3), introduce the parameters: T 0 P(s) f (s, x(s))ds =: col (λ 1 , λ 2 ) .(6.4) The formal substitution (6.3) transforms the boundary restrictions (6.2) to the linear conditions where d(λ Then (6.1) takes the form (2.1) with T = 1, n = 2, and it is then easy to check that the matrix K from the Lipschitz condition (3.3) can be taken as Calculations show that matrix Q = 0 0.3 0.3 0.375 and r (Q) < 0.55 < 1.
The vector δ D ( f ) can be estimated as The role of D β is played by the domain defined by inequalities: The domain D 0 is such that One can verify that, for the parametrized BVP (6.1), (6.5), all the needed conditions are fulfilled, and we can proceed with application of the numerical-analytic scheme described above.As a result, we construct the sequence of approximate solutions.
The graphs of the first approximation and the exact solution of the original BVP are shown on Figure 6