On the solvability of a nonlocal problem for the system of Sobolev-type differential equations with integral condition

1 Introduction

In this paper, in the domain Ω=[0,T]×[0,ω], we consider the following nonlocal problem for a system of Sobolev-type differential equations with integral condition:

where u⁢(t,x)=col⁡(u1⁢(t,x),u2⁢(t,x),…,un⁢(t,x)) is an unknown function, the n×n matrices A⁢(t,x), B⁢(t,x), C⁢(t,x), D⁢(t,x) and the n vector function f⁢(t,x) are continuous on Ω, the n×n matrices K1⁢(t,x) and K2⁢(t,x) are continuous on Ω, the n×n matrices M2⁢(x), M1⁢(x), L2⁢(x), L1⁢(x) and the n vector function φ⁢(x) are continuous on [0,ω], 0≤a≤T, 0≤b≤T, the n vector functions ψ1⁢(t) and ψ2⁢(t) are continuously differentiable on [0,T].

Let C⁢(Ω,Rn) be a space of vector functions u⁢(t,x), continuous on Ω, with the norm

Let C⁢([0,ω],Rn) be a space of vector functions φ⁢(x), continuous on [0,ω], with the norm

Let C1⁢([0,T],Rn) be a space of vector functions ψ⁢(t), continuously differentiable on [0,T], with the norm

The function u⁢(t,x)∈C⁢(Ω,Rn) that has the partial derivatives

is called a classical solution to problem (1.1)–(1.4) if it satisfies system (1.1) for all (t,x)∈Ω and boundary conditions (1.2), (1.3) and (1.4).

Nonlocal problems for a system of Sobolev-type differential equations appear while studying a variety of physical problems [1, 2, 6, 32, 33, 34]. Some classes of nonlocal problems for partial differential equations of third order are investigated in the papers [10, 23, 28, 33]. Depending on the methods applied, conditions for the solvability of nonlocal problems are obtained in different terms. In [25], an initial-boundary value problem for linear hyperbolic equations of higher orders with two independent variables is studied. Conditions for the well-posedness of the considered problem in terms of the well-posedness of the boundary value problem for ordinary differential equations with parameter are established. In [14, 16, 15, 17, 19, 20, 25, 24, 21, 22, 29, 31], the solvability questions are investigated for initial-boundary value problems for linear and nonlinear hyperbolic equations of higher order with two variables. The study of the questions of solvability and construction of approximate solutions to nonlocal problems for Sobolev-type differential equations and systems with integral condition is very important and relevant in connection with numerous applications.

In the present paper, we study the existence of classical solutions to a nonlocal problem with an integral condition for the system of Sobolev-type differential equations (1.1)–(1.4) and propose the methods of constructing their approximate solutions. A nonlocal boundary value problem for a system of Sobolev-type differential equations has not previously been considered.

In [4], a linear boundary value problem with integral condition for a system of hyperbolic equations of second order was investigated for the case a=b=T. The sufficient conditions for unique solvability were obtained, and a method for finding solutions to the considered problem was proposed. Subsequently, the well-posedness criteria for that problem was established in terms of initial data. For this, some new unknown functions, which are related to the first-order derivatives of the sought functions, were introduced. Thus the problem was reduced to an equivalent problem involving a family of boundary value problems for ordinary differential equations and the integral relations. The equivalence of the well-posedness of the considered problem to the well-posedness of a family of boundary value problems is proved.

In this paper, the results and methods of [4] are extended to a new class of problems, namely, nonlocal problems with integral condition for a system of Sobolev-type differential equations. The nonlocal problem for the system of Sobolev-type differential equations with integral condition is reduced to a nonlocal problem with integral condition for a system of hyperbolic equations of second order and the integral relation. We establish the sufficient conditions for the unique solvability of a nonlocal boundary value problem with integral condition (1.1)–(1.4) in terms of the system’s right-hand side, boundary functions and integral kernels. Algorithms for finding a solution to the considered problem are constructed, and their convergence is shown. The results can be used for the numerical solution of applied problems.

2 Reduction to an equivalent problem and the algorithm

In this section, we introduce the new unknown function v⁢(t,x)=∂⁡u⁢(t,x)∂⁡x and reduce problem (1.1)–(1.4) to the equivalent problem

For problem (2.1)–(2.4), the condition u⁢(t,0)=ψ1⁢(t) is taken into account in relation (2.4).

A pair {v⁢(t,x),u⁢(t,x)} of functions, continuous on Ω, is called a solution to problem (2.1)–(2.4) if the function v⁢(t,x) belonging to C⁢(Ω,Rn) has the continuous partial derivatives ∂⁡v⁢(t,x)∂⁡x, ∂⁡v⁢(t,x)∂⁡t, ∂2⁡v⁢(t,x)∂⁡x⁢∂⁡t on Ω and satisfies the nonlocal problem with integral condition for a system of hyperbolic equations of second order (2.1)–(2.3), where the function u⁢(t,x) is connected with v⁢(t,x) by the functional relation (2.4).

Let u∗⁢(t,x) be a classical solution to problem (1.1)–(1.4). Then the pair {v∗(t,x), u∗(t,x)}, where v∗⁢(t,x)=∂⁡u∗⁢(t,x)∂⁡x, is a solution to problem (2.1)–(2.4). Conversely, if the pair {v~⁢(t,x),u~⁢(t,x)} is a solution to problem (2.1)–(2.4), then u~⁢(t,x) is a classical solution to problem (1.1)–(1.4).

For fixed u⁢(t,x), problem (2.1)–(2.3) is a nonlocal problem with integral condition for a system of hyperbolic equations of second order. Nonlocal problems with and without integral condition for the system of hyperbolic equations (2.1) were studied by many authors [4, 3, 5, 7, 8, 9, 12, 18, 26, 27, 28, 29, 30, 35, 36, 37]. A general linear boundary value problem with integral condition for a system of hyperbolic equations was investigated in [4] for a=b=T.

System (2.1) depends on the unknown function u⁢(t,x). For fixed v⁢(t,x), the function u⁢(t,x) is determined from the integral relation (2.4).

If we know v⁢(t,x)∈C⁢(Ω,Rn), then we find u⁢(t,x)∈C⁢(Ω,Rn) from the integral relation (2.4). Conversely, if we know u⁢(t,x)∈C⁢(Ω,Rn), then we can find v⁢(t,x)∈C⁢(Ω,Rn) from problem (2.1)–(2.3). Since v⁢(t,x) and u⁢(t,x) are unknown, to find a solution to problem (2.1)–(2.4), we use an iterative method. Determine a pair {v∗(t,x), u∗(t,x)}∈C(Ω,Rn)×C(Ω,Rn) as a limit of the sequence {v(m)⁢(t,x),u(m)⁢(t,x)}∈C⁢(Ω,Rn)×C⁢(Ω,Rn), m=0,1,2,…, by the following algorithm:

1. Solving the nonlocal problem with integral condition for a hyperbolic equation of second order (2.1)–(2.3) for u⁢(t,x)=ψ1⁢(t), we find the initial approximation v(0)⁢(t,x)∈C⁢(Ω,Rn). Further, from the integral relation (2.4), under v⁢(t,x)=v(0)⁢(t,x), we determine u(0)⁢(t,x)∈C⁢(Ω,Rn).

2. From the nonlocal problem (2.1)–(2.3) for u⁢(t,x)=u(0)⁢(t,x) on Ω, we find v(1)⁢(t,x)∈C⁢(Ω,Rn). Further, from the integral relation (2.4), under v⁢(t,x)=v(1)⁢(t,x), we determine u(1)⁢(t,x)∈C⁢(Ω,Rn).

   And so on.

3. From the nonlocal problem (2.1)–(2.3), for u⁢(t,x)=u(m-1)⁢(t,x) on Ω, we find v(m)⁢(t,x)∈C⁢(Ω,Rn). Further, from the integral relation (2.4), under v⁢(t,x)=v(m)⁢(t,x), we determine u(m)⁢(t,x)∈C⁢(Ω,Rn), m=1,2,….

This algorithm divides the process of finding unknown functions into two parts:

1. From the nonlocal problem with integral condition for a system of hyperbolic equations of second order (2.1)–(2.3), we find the introduced function v⁢(t,x).

2. From the integral relation (2.4), we find the unknown function u⁢(t,x).

3 Solvability conditions for problem (1.1)–(1.4)

In this section, we use an auxiliary problem for establishing conditions of unique solvability to problem (1.1)–(1.4). Consider the family of boundary value problems with integral condition for the system of ordinary differential equations

where V⁢(t,x) is an unknown function, the n vector function F⁢(t,x) is continuous on Ω, the n vector function Φ⁢(x) is continuous on [0,ω], x∈[0,ω].

A continuous function V:Ω→ℝn that has a continuous derivative with respect to t on Ω is called a solution to the family of boundary value problems with integral condition (3.1), (3.2) if it satisfies system (3.1) and condition (3.2) for all (t,x)∈Ω and x∈[0,ω], respectively.

For fixed x∈[0,ω], problem (3.1), (3.2) is a linear boundary value problem with integral condition for the system of ordinary differential equations. Suppose a variable x takes values on the interval [0,ω]; then we obtain a family of boundary value problems with an integral condition for ordinary differential equations.

The following theorem provides the conditions of unique solvability to problem (1.1)–(1.4) in terms of the solvability of family of problems (3.1), (3.2).

Now let us state the main theorem on the feasibility and convergence of the proposed algorithm. The assertion also provides the sufficient conditions for the unique solvability of problem (1.1)–(1.4).

Let Ps⁢(t,x) be a matrix of the form

where s∈ℕ. It is clear that Ps⁢(t,x)⇉Z⁢(t,x) as s→∞ on Ω (see [11, p. 145]).

Introduce the notations