New general solution of a family higher order diﬀerential equations and its application to solve multipoint-integral problems

The family multipoint-integral problems of higher order diﬀerential equations is considered. An eﬀective method for solving to family multipoint-integral problems for higher order diﬀerential equations is oﬀered. A domain is divided into m parts, the values of a solution at the beginning lines of the subdomains are considered as functional parameters, and the family higher order diﬀerential equations are reduced to the family Cauchy problems on the subdomains for system of diﬀerential equations with functional parameters. Using the solutions to these family problems, new general solutions to family higher order diﬀerential equations are introduced and their properties are established. Based on the general solution, family multipoint-integral problems, and continuity conditions of a solution at the interior lines of the partition, the linear system of functional equations with respect to parameters is composed. Algorithms for ﬁnding solutions to families of multipoint-integral problems for higher order diﬀerential equations are constructed and conditions for unique solvability are established in the terms of initial data.

Various processes of the theory of oscillations, the theory of impulse systems, and the theory of multi-support beams are considered as a family multipoint-integral problems for higher order differential equations [1-13, 16-18, 22-31]. The development of effective and constructive methods for solving the family multipoint-integral problems for higher order differential equations is important and interesting.
The goal of the present paper is to develop an effective method for solving the family multipoint-integral problems for higher order differential equations and to construct an algorithm for finding solution to the family multipoint-integral problems (1.1), (1.2).
The paper is organized as follows.
In Section 2, the family multipoint-integral problems for higher order differential equations (1.1), (1.2) is considered. Introducing a new unknown functions the family multipoint-integral problems for higher order differential equations (1.1), (1.2) is transferred to an equivalent family multipoint-integral problems for system of differential equations. Further, the Dzhumabaev's parametrization method [21] is applied for solving the equivalent family problems. Using the lines t = t j , j = 0, 1, ..., m, we make a partition ∆ m of domain Ω: The values of a solution at the lines t = t r−1 of the subdomains [t r−1 , t r ) × [0, ω] are considered as functional parameters λ r (y), r = 1, 2, ..., m. The family systems of differential equations is reduced to the family Cauchy problems on the subdomains for the system of differential equations with functional parameters. Using the solutions to these problems, ∆ m general solutions to the family systems of differential equations and initial the family higher order differential equations are introduced and their properties are established. The ∆ m general solution, denoted by x(t, ∆ m , λ), contains an arbitrary vector λ = (λ 1 , λ 2 , ..., λ m ) ∈ R nm .
Based on the ∆ m general solution, multipoint-integral conditions, and continuity conditions of a solution at the interior lines of the partition is composed the system of functional equations with respect to parameters in Section 3.
Using u(t, y, ∆ m , λ), we set solvability criteria of considered problem and propose an algorithm for finding its solution. Applying the new general solution reduces the solvability of family multipoint-integral problems for higher order differential equations (1.1), (1.2) to the solvability of the system of functional equations for parameters. An effective method for solving the family multipoint-integral problems for higher order differential equations (1.1), (1.2) is offered. This method includes to solve the system of functional equations and the family Cauchy problems for the system of differential equations.
The principal differences results of the paper from other existing analogues are included the following assertions: the development of algorithms for the parametrization method for solving family multipoint-integral problems for higher order differential equations; the construction a new general solution for family higher order differential equations and the establishment its properties; the creation an effective method for solving family multipoint-integral problems for higher order differential equations based on the composition and solution of systems of linear functional equations for arbitrary vectors of new general solutions; the establishment unique solvability conditions for family multipoint-integral problems for higher order differential equations.
2 The ∆ m general solution to a family higher order differential equations and its properties Problem (1.1), (1.2) by introducing new functions is reduced to a family multipoint-integral problems for a system of differential equations where x(t, y) = col(x 1 (t, y), x 2 (t, y), ..., x n (t, y)) is unknown vector function, the n × n matrices A(t, y), C(t, y) and n-vector function g(t, y) have the next form a n (t, y) a n−1 (t, y) a n−2 (t, y) a n−3 (t, y) ... a 2 (t, y) a 1 (t, y) , and are continuous on Ω; the n × n matrices B s (y) and n-vector function d(y) have the form ...
Using the lines t = t j , j = 0, 1, ..., m, we make a partition ∆ m of domain Ω: By ∆ 1 denote the case of no partitioning of the domain Ω.
: Ω r → R n are continuous and have a finite left-hand side limits lim Let vector function x(t, y) be the solution to family systems (2.1) and x (r) (t, y) is its restriction to sub-domain Ω r , r = 1, m. Then the system functions ., x (m) (t, y)) belongs to C(Ω, ∆ m , R nm ), and its elements x (r) (t, y), r = 1, ..., m, satisfy to family systems of differential equations in the following form Now, we introduce a functional parameters λ r (y) = x (r) (t r−1 , y), r = 1, ..., m.
The system functions v([t], y, λ) is called a solution to the family Cauchy problems with functional parameters (2.4), (2.5).
(2.6) where F (t, y) is a square matrix or a vector of dimension n, continuous on Ω. Denote by A r (t, y, F ) a unique solution to the family Cauchy problems (2.6) on each rth domain. The uniqueness of the solution to the family Cauchy problems for linear system of differential equations give D r (t, y, F ) = Φ r (t, y) t t r−1 Φ −1 r (τ, y)F (τ, y)dτ, (t, y) ∈ Ω r , r = 1, ..., m.
If x(t, y) is a solution to the family systems (2.1), and x([t], y) = (x (1) (t, y), x (2) (t, y), ..., x (m) (t, y)) is a system functions composed of its restrictions to the subdomains Ω r ), r = 1, ..., m, then the equations hold. These equations are the continuity conditions for the solution to the family systems (2.1) at the interior lines t = t p , p = 1, ..., m − 1, of the partition ∆ m .
From the Theorem 2.3 it follows that the function u * (t, y), given by the equalities u * (t, y) = x 1,(r) (t, y) for (t, y) ∈ Ω r , r = 1, ..., m, and u * (T, y) = lim has a partial derivatives up to the (n − 1)st order in t on Ω, defining by equalities has a partial derivatives of nth order in t on Ω, and satisfies the family higher order differential equations (1.1).
For any partition ∆ m , Theorems 2.1 and 2.3 provide the validity of the following statement.

Definition 3.2
The problem with non-separated multipoint-integral condition (2.1), (2.2) is called uniquely solvable if for any pair (g(t), d), with g(t) ∈ C([0, T ], R n ) and d ∈ R n , it has a unique solution.  is valid for all ζ ∈ Ker(Q * (∆ m )) , where (·, ·) is the inner product in R nm .
Theorem 3.7 The problem with non-separated multipoint-integral conditions for high-order differential equations (1.1), (1.2) is uniquely solvable if and only if the nm × nm matrix Q * (∆ m ) is invertible.

Conclusion
Dzhumabaev's parametrization method is applied to solving problems with non-separated multipoint-integral conditions for high-order differential equations. A new general solution of high-order differential equations is constructed and its properties is clarified. Using a new general solution, criteria for the unique solvability are established for the problems with non-separated multipoint-integral conditions for high-order differential equations. The basic idea behind the proposed method is to construct and solve system of algebraic equations with respect to arbitrary parameters of new general solutions. The Cauchy problems for system of differential equations on the subintervals are the main tool of composing this system. Methods and results will be apply to a various problems for differential equations of higher order [1-11, 17-18, 22-31].