PARAMETRIZATION FOR NON-LINEAR PROBLEMS WITH INTEGRAL BOUNDARY CONDITIONS

We consider the integral boundary-value problem for a cer- tain class of non-linear systems of ordinary differential equations of the form x 0 (t) = f (t, x(t)) , t 2 (0, T), Ax(0) + Z T 0 P(s)x(s)ds + Cx(T) = d, where f : (0, T) × D ! R n is continuous vector function, DR n is a closed and bounded domain. By using an appropriate parametrization technique, the given prob- lem is reduced to an equivalent parametrized family of two-point boun- dary-value problems with linear boundary conditions without integral terms. To study the transformed problem, we use a method based upon a special type of successive approximations which are constructed ana- lytically. We establish sufficient conditions for the uniform convergence of that sequence and introduce a certain finite-dimensional determining system whose solutions give all the initial values of the solutions of the given boundary-value problem. Based upon properties of the functions of the constructed sequence and of the determining equations, we give efficient conditions for the solvability of the original integral boundary- value problem.


Introduction
Recently, boundary-value problems with integral conditions for non-linear differential equations have attracted much attention, see, e. g. [3,17].However, mainly scalar non-linear differential equations of special kinds have been studied.According our best knowledge, there are only a few works dealing with a constructive investigation of systems of non-linear differential equations of a general form with integral boundary restrictions (see, e. g., [2,6,15,16]).
The aim of this paper is to extend the numerical-analytic technique, which had been used earlier successfully in relation to different types of boundaryvalue problems with two-point and multipoint linear and non-linear boundary conditions [4,5,7,13], for a class of non-linear differential systems of the form x ′ (t) = f (t, x (t)) , t ∈ [0, T ], under the integral boundary conditions Ax(0) + T 0 P (s)x(s)ds + Cx(T ) = d.
We use an approach based on an appropriate parametrization technique [12,13], which allows us to reduce the given problem to an equivalent family of parametrized two-point boundary-value problems with linear boundary conditions without integral terms.To study the transformed problem, we use a method based upon a special type of successive approximations constructed analytically.We give conditions sufficient for the uniform convergence of this sequence and introduce a certain finite-dimensional "determining" system of algebraic or transcendental equations whose solutions give all the initial values of the solutions of the given boundary-value problem.Using properties of the functions of the sequence and determining equations and applying an argument based on the Brouwer degree, we give efficient conditions ensuring the solvability of the original integral boundary-value problem.

Notation
(1) In the sequel, the operations | • |, ≥, ≤, max, min between matrices and vectors are understood componentwise.(2) L (R n ) is the algebra of n-dimensional square matrices with real elements.(3) 1 m and 0 m stand, respectively, for the unit and zero matrix of dimension m ≤ n. (4) For any u ∈ R n and any non-negative vector r ∈ R n , we put B(u, r) := {ξ ∈ R n : |ξ − u| ≤ r} . (2.1) (5) r(K) is the spectral radius of a matrix K.

Problem setting
We consider the non-linear system of differential equations subjected to the integral boundary conditions Here, we suppose that the vector function f : [0, T ] × D → R n is continuous, where D ⊂ R n is a closed and bounded domain.
The problem is to find a solution of the system of differential equations (3.1) with property (3.2) in the class of continuously differentiable vector functions x : [0, T ] → D.

Construction of the successive approximations
Assume that the function f in the right hand-side of (3.1) satisfies the Lipschitz condition of the form Let us put Furthermore, introduce the vector and assume that the set D * defined according to the formula Recalling notation (2.1), we see that the inclusion z ∈ D * holds if and only if the vector [(1 Let us associate with the parametrized boundary-value problem (3.1), (4.4) the sequence of functions defined recurrently by the formula EJQTDE, 2012 No. 99, p. 4 for t ∈ [0, T ], m = 1, 2, 3, . . ., where and the vectors z, λ, and η are considered as parameters.
It is easy to check that the functions x m (•, z, λ, η) satisfy linear parametrized boundary conditions (4.4) for all m ≥ 1, z, η, λ ∈ R n .The following statement establishes the convergence of sequence (5.4).
obtained for the sequence of functions (5.20).By virtue of the estimate (5.21), from (5.19) we get for all t ∈ [0, T ] and m = 0, 1, 2, . .., where the matrix Q is given by (5.13).Therefore, in view of (5.22), Since, due to the condition (5.6), the maximal eigenvalue of the matrix Q of the form (5.13) does not exceed 1, we have and lim m→∞ Q m = 0 n , where 0 n is the n × n zero matrix.Therefore, we conclude from (5.23) that, according to the Cauchy criterion, the sequence {x m (•, z, λ, η) : m ≥ 1} of the form (5.4) uniformly converges in the domain [0, T ] × D * × P × D to a limit function x * (•, z, λ, η).Since all the functions x m (•, z, λ, η) of the sequence (5.4) satisfy the boundary conditions (4.4) for all values of the introduced parameters, we conclude that the limit function x * (•, z, λ, η) also satisfies these conditions.Passing to the limit as m → ∞ in equality (5.4), we show that the limit function satisfies both the integral equation (5.8) and the Cauchy problem (5.9), (5.10),where ∆ (z, λ, η) is given by (5.11).
Theorem 5.2.Let z ∈ D * , λ ∈ P, η ∈ D and µ ∈ R n be fixed.Suppose that for the system of differential equations (3.1) all conditions of Theorem 5.1 hold.
Then, for the solution x(•, z, λ, η, µ) of the initial-value problem (5.24), (5.25) to satisfy the parametrized boundary conditions (4.4), it is necessary and sufficient that µ be given by the formula µ = µ z,λ,η , where In that case, (5.28) in the right-hand side of the system of differential equations (5.24).By virtue of Theorem 5.1, the limit function (5.7) of the sequence (5.4) is the unique solution of the problem (5.24), (4.4) for the fixed values of parameters z, λ and η and µ of form (5.28).Furthermore, the function x * (•, z, λ, η) satisfies the initial conditions (5.25), i. e., it is a solution of the Cauchy problem (5.24), (5.25) for that µ.Thus, we have found the value of the parameter µ given by (5.26), for which (5.27) holds.
Let us find out the relation of the limit function x * (•, z, λ, η) of the sequence (5.4)  V (z, λ, η) = 0, (5.47) where Proof.It suffices to apply Theorem 5.2 and notice that the differential equation (5.9) coincides with (3.The next statement proves that the system of determining equations (5.46)-(5.48)defines all possible solutions of the original non-linear boundary-value problem (3.1) with integral boundary restrictions (3.2).Lemma 5.1.Let all conditions of Theorem 5.1 be satisfied.Furthermore there exist some vectors z ∈ D * , λ ∈ P and η ∈ D that satisfy the system of determining equations (5.46)-(5.48). Then: (1) The non-linear boundary-value problem (3.1), (3.2) with integral boundary conditions has a solution x(•) such that Moreover, this solution is given by the formula (5.50) Proof.We will apply Theorems 5. where the vectors λ, η are defined by (5.50).However, µ z,λ,η is given by formula (5.26), and hence the first equation (5.46) of the determining system is satisfied, if z, λ, and η are given by (5.50).Using (4.4), we obtain that the other two equations (5.47), (5.48) of the determining system also hold.So, we have specified values (z, λ, η) that satisfy the system of determining equations (5.46)-(5.48),which proves the lemma.

Remarks on the constructive applications of the method
Although Theorem 5.3 gives sufficient and necessary conditions for the solvability and construction of the solution of the given problem, its application faces with difficulties due the fact that the explicit form of the functions ∆ : 5.48) is usually unknown.This complication can be overcome by using the properties of the function x m (•, z, λ, η) of the form (5.4) for a fixed m, which will lead one, instead of the exact determining system (5.46)-(5.48), to the mth approximate system of determining equations ∆ m (z, λ, η) = 0, (6.1) V m (z, λ, η) = 0, (6.2) where ∆ m : D * × P × D → R n and V m : D * × P × D → R n are given by the formulas and x m (•, z, λ, η) is the vector-function defined according to relation (5.4).
In the next section we will show how, under certain natural assumptions, the approximate determining system can be used in solvability analysis.

Existence of solutions of the integral boundary-value problem
In the sequel, we need a lemma providing an estimate for functions (5.11) and (6.4).
where Q, δ D (f ) are given by (5.13), (5.Proof.Let us fix arbitrary z, λ, η of the form (4.1).By virtue of the estimate (5.12), we have: The last estimate completes the proof.

Definition 7.1 ([10]
).Let H ⊂ R 3n be an arbitrary non-empty set.For any pair of functions f j = col (f j1 , . . ., f j,3n ) : H → R 3n , j = 1, 2, we write if and only if there exist a function EJQTDE, 2012 No. 99, p. 15 Remark 7.1.Relation (7.5) means that at every point x ∈ H at least one of the components of the vector f 1 (x) is greater then the corresponding component of the vector f 2 (x).

Notes on proving the solvability
According to the approach developed here, the proof of the solvability of the original boundary-value problem (3.1), (3.2) is based on Theorems 5.1 and 7.1.Theorem 5.1 ensures the convergence of the iteration method and, in particular, justifies the further argument that involves functions of sequence (5.4) and their limit (5.7).On the other hand, applying Theorem 7.1, one can use properties of finitely many functions of sequence (5.4) to establish that the solution of (3.1), (3.2) exists.
Remark 8.1.In order to apply Theorem 7.1, one has to: • compute the vector δ D (f ) according to (5.2) (or estimate it from above) • construct the function x m (•, z, λ, η) analytically for a certain fixed value m = m 0 , keeping z, λ, and η as parameters • select a suitable set Ω and verify conditions (7.7), (7.8) for m = m 0 .Remark 8.2.To verify condition (7.7) of Theorem 7.1 in concrete cases, one has to use the recurrence formula (5.4) to compute the function x m (•, z, λ, η) depending on z ∈ D * , λ ∈ P, η ∈ D as parameters and verify whether at least one of the components of the vector |Φ m (z, λ, η)| is strictly greater than the corresponding component of the appropriate vector in the righthand side at every point (z, λ, η) of ∂Ω.
After that, we need verify in (7.8) whether the topological degree of Φ m is not zero.This is rather difficult problem in general.However, there are sufficient conditions applicable in a number of important cases.In particular, when Φ m is an odd mapping, i. e., Φ m (−z, −λ, −η) = −Φ m (z, λ, η) for all (z, λ, η), then, according to the Borsuk theorem (see [1, Theorem A2.12]), its Brouwer degree is an odd number and therefore, is not equal to zero.
The vector δ D (f ) can be estmated as The role of D * is played by the domain defined by inequalities: The domain P is such that One can verify that, for the parametrized boundary-value problem (9.1), (9.4), 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 computation shows that the approximate solutions of the determining system (9.9)-(9.12)for m = 1 are The error of the first approximation is max t∈[0,

0 P
(s)x(s)ds + Cx(T ) = d, (3.2) where A is arbitrary and C is a given singular n × n matrix of the form C = C 11 C 12 C 21 0 n−p , where C 11 is a p × p matrix, det C 11 = 0, C 12 is a p × (n − p) matrix, C 21 is a (n − p) × p matrix, and P : [0, T ] → L (R n ) is a continuous n × n EJQTDE, 2012 No. 99, p. 2 matrix-valued function.We also assume that det(1 n−p − C 21 C −1 11 C 12 ) = 0. (3.3)

Remark 4 . 1 . 3 Remark 4 . 2 .
matrix, and λ and η are the parameters with meaning (4.1).In view of assumption (3.3), the matrix C 1 is non-singular in condition (4.4).Let us put d(λ, η) := d − λ + η. (4.3) Taking (4.3) into account, one can rewrite the parametrized boundary conditions (4.2) in the form Ax(0) + C 1 x(T ) = d(λ, η).(4.4)The parametrization technique that we are going to use suggests that, instead of the original boundary-value problem with the integral boundary conditions (3.1), (3.2), we study the family of parametrized boundary value problems (3.1), (4.4), where the boundary restrictions are linear.We then go back to the original problem by choosing the values of the parameters appropriately.EJQTDE, 2012 No. 99, p.The set of the solutions of the non-linear boundary-value problem with integral boundary conditions (3.1), (3.2) coincides with the set of the solutions of the parametrized problem (3.1) with linear boundary restrictions (4.4), satisfying additional conditions (4.1).
η) belongs to D together with its "vector" T 2 δ D (f )-neighbourhood for any λ ∈ P, η ∈ D, and t ∈ [0, T ].Remark 5.1.The technical assumption (5.3) means that the domain D, where the right-hand side of the differential equation is assumed to satisfy the Lipschitz condition, is wide enough.

Remark 5 . 2 .
It follows from the definition of the set D * that the values of function (5.5) do not escape from D for any z ∈ D * , λ ∈ P, and η ∈ D.

Lemma 7 . 1 .
Let conditions of Theorem 5.1 be satisfied.Then, for an arbitrary m ≥ 1, the exact and approximate determining functions ∆ : D * × P × D → R n and ∆ m : D * × P × D → R n defined by (5.11) and (6.4) satisfy the estimate