1. Introduction

The non-local problems are the problems wherein instead of giving the values of the solution or its derivatives on the fixed part of the boundary, the relation of these values with the values of the same functions on another inner or boundary manifolds is given. Theory of non-local boundary value problems is important in itself as a section of general theory of boundary value problems for partial equations and it is important as a section of mathematics that has numerous applications in mechanics, physics, biology and other natural science disciplines.

The more general time non-local conditions were considered were considered on the papers of A.A.Kerefov, J.Chabrowsky ^[8], V.V.Shelukhin ^[9], G.M.Liberman ^[10], A.I.Kozhanov ^[11], and others.

Yu.A.Mitropolsky and B.I.Moiseenkov ^[1], J.M.T. Thompson, H.B. Stewart ^[2], B.S.Bardin, S.D.Furta ^[3], D.V.Kostin ^[4] and others have situated oscillation and wave motions of an elastic bar on an elastic foundation.

The simplest non-linear model of motion of a homogeneous bar is described by the equation

where is bar’s deflection (after displacement of the middle line points of an elastic bar along the axis ). Note that the similar equation arises in the theory of crystals ^[5].

2. Problem Statement and ITS Reduction to an Integral Equation

For the equation ^[4]

(1)

in the domain we consider a problem with ordinary periodic boundary conditions

(2)

and subject to the non-local boundary conditions

(3)

where , , are the given numbers, moreover , , are the given functions, is a sought function.

Definition

Under the classic solution of problem (1)-(3) we understand the function , continuous in the closed domain together with its all derivatives involved in equation (1), and satisfying all the conditions (1)-(3) in ordinary sense.

It is known ^[11] that the system

,,, ... ,,,...

is a basis in, where .

Then it is obvious that each classical solution of problem (1)-(3) has the form:

(4)

where

Then, applying the formal scheme of the Fourier method, from (1) and (3) we have:

(5)

(6)

(7)

(8)

where

It is clear that . Let suppose that, . Then, by solving problem (5)-(8) we find:

(9)

(10)

where

It is known that

(11)

(12)

(13)

(14)

After substituting the expressions and into (4), we get:

(15)

Thus, the solution of problems (1)-(3) is reduced to the solution of integral equation (15) with respect to the unknown function .

Similarly to ^[12], it is possible to prove the following lemma.

Lemma

If is any classical solution of problem (1)-(3), the functions

satisfy the systems (9), (10) in .

From the lemma indicated above it follows that if

is the solution of systems (9) , (10) then the function

is the solution of (15).

From the above mentioned lemma follows

Corollary

Suppose that equation (15) has a unique solution. Then the problem (1)-(3) may have at most one solution, i.e. of the solution of problem (1)-(3) exists it is unique.

3. Existence and Uniqueness of the Classical Solution

Denote by ^[13] the set of all functions of the form

defined on , where each of the functions , are continuous on and

We define the norm in this set as follows:

It is known that is a Banach space.

Now in the space we consider the operator

where

Hence, we get:

(16)

where

Assume that the data of problem (1)-(3) satisfy the following conditions:

1. 

2. 

Then from (12) we have

(17)

Denote

and rewrite (17) in the form:

(18)

Theorem

Let conditions 1-2, , be fulfilled. Then for sufficiently small values of , problem (1)-(3) has a unique classical solution in the ball of the space .

Proof:

In the space consider the equation

(19)

where the operator is defined by the right hand side of equation (15). Consider the operator in the ball from .

Similarly, from (18) we get that for any , the following estimates are valid

(20)

(21)

It follows from the estimates (20), (21) that for rather small values of the operator acts in the ball and is contractive. Therefore in the ball it has unique fixed point , that is a unique solution of equation (19). Moreover, integral equation (15) also has a unique solution belonging to the ball .

The function , as an element of the space , is continuous and has continuous derivatives , , , on .

Now we’ll show that is continuous in . Allowing for (10), from (6) we have

Hence, we have:

or

It follows from the last relation that the function is continuous in .

It is easy to verify that equation (1) and conditions (2), (3) are satisfied in the ordinary sense. So, is the solution of the problem (1)-(3) in the ball from . Since equation (15) has a unique solution in the ball from . By the above mentioned corollary the problem (1)-(3) has a unique classical solution in the ball from . The theorem is thus proved.

4. Conclusion

The following results have been obtained:

1. The existence of the solution of a nonlocal boundary value problem for the equation of motion of a homogeneous bar is proved;

2. The uniqueness of the solution of a nonlocal boundary value problem for the equation of motion of a homogeneous bar is shown.

References

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	In article
[2]	Thompson J.M.T., Stewart H.B. Nonlinear Dynamics and Chaos. Chichester; Singapore: Wiley, 1986.
	In article
[3]	Bardin B.S., Furta S.D.Urgent problems of classical and celestial mechanics. M.: Elf, 1998. pp. 13-22. (in Russian).
	In article
[4]	Kostin D.V. On one scheme of analysis of two-mode deflections of weakly inhomogeneous elastic bar. Doklady Akademii Nauk, 2008, vol. 418, No.3, pp. 295-299. (in Russian).
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[5]	Izyumov Yu.A., Syromyatnikov V.I. Phase transitions and symmetry of crystals. M.: Nauka, 1984. 245 p. (in Russian).
	In article
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	In article
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	In article
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	In article
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[12]	Azizbayov E.I., Mehraliyev Y.T. A time-nonlocal boundary value problem for the equation of motion of a homogeneous bar. Bulletin of the Kyiv National University. Series: mathematics and mechanics, 2012, Issue 27, pp.114-121.
	In article
[13]	Mehraliyev Y.T. On the solvability of inverse boundary value problem for an elliptic equation of the second order. Bulletin of the Tver State University, Series: Applied math., 2011, No. 23, pp. 25-38. (in Russian).
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