ON THE CORRECTNESS OF A NONLOCAL PROBLEM FOR THE SECOND ORDER MIXED TYPE EQUATION OF THE SECOND KIND IN A RECTANGLE

In this work, the correctness of a nonlocal problem from the Sobolev spaces is proven under some restrictions to the coefficients of the considered second order mixed type equation of the second kind. The proof is accomplished with multiple methods namely, "ε-regularization", priory estimates, and the Galerkin method. ABSTRAK: Dalam karya ini, ketepatan satu masalah bukan setempat dari ruang Sobolev telah dibuktikan di dalam beberapa sekatan kepada koefisien persamaan jenis bercampur perintah kedua daripada jenis yang kedua yang dipertimbangkan. Buktinya dicapai dengan pelbagai kaedah iaitu, "ε-regularization", anggaran priori dan kaedah Galerkin.


INTRODUCTION
The basic concepts of the theory of differential equations in partial derivatives were formed in the study of classical problems of mathematical physics and are currently well understood.However, the current problems of natural sciences lead to the necessity of setting and qualitatively studying new tasks, which is a striking example of a class of nonlocal problems.
As nonlocal problems we call such problems that determine relationships between the value of the solution or its derivatives at the boundary and interior points of a considered domain.In recent decades, nonlocal problems for partial differential equations are being actively studied by many mathematicians.
Among the first studies of non-local problems, we note a work by Bitsadze and Samarskii [4].In this work, spatial and non-local problems for a certain class of elliptic equations were formulated and studied, which led to the study of non-self-adjoint spectral problems.Subsequently, the problem formulated in [4], has been named a Bitsadze-Samarskii problem.
The investigation of nonlocal problems is caused by both theoretical interest and practical necessity.This is due to the fact that the mathematical models of various physical, chemical, biological, and ecological processes are often problems in which, instead of the classical boundary conditions, definite connection values of the unknown function (or its derivatives) on and within the boundary are given.Problems of this type may arise in the study of phenomena related to plasma physics, the spread of heat, the process of moisture transfer in capillary porous media issues, demographics, mathematical biology, and some technological processes.Nonlocal problems also have practical value in solving the problems of solid mechanics.They allow to control the stress-strain state and these are similar to the control tasks [4, 8-12, 16, 17, 19].
For the first time, non-local problems in certain weight and negative spaces for equations of mixed type were investigated by functional methods in the works [1,6,8,10,11,17].
Problem: To find a generalized solution of Eq. ( 1) from the Sobolev space 2 ( ) In the previous works [6,7,11,17] the correctness of problem Eqs. ( 1)-( 3) in the case where ( , 0) ( , ) 0 K x K x T   was proven under comparatively strong conditions to the coefficients of Eq. ( 1).In the present work, we consider the case where, ( , 0) 0 ( , ) Note that Eq. ( 1) is the second kind of mixed type equation, since there is no restriction inside the domain Q to the sign of the variable t of the function ( , ) K x t [5].First, we consider the case where 2 l  .Assuming the coefficients of Eq. (1) are smooth enough and ( , ), ( , ) x t c x t  are periodic in function with respect to variable t .

UNIQUENESS FOR PROBLEM Theorem 1:
Let aforementioned conditions to the coefficients of Eq. ( 1) are fulfilled, moreover let 1 2 2 0; 0  is an integer number and it is known that at 5,[13][14][15].Here, and further, by m we designate positive constants, the exact values of which are not of interest.

Proof:
For any function 2 2 ( ) u W Q  , one can easily get the following equality by using integration by parts: is a unit vector of the inner normal to the Q  .Conditions of Theorem 1 guarantee non-negativity of the integral over the domain Q .Let  2)-( 3), on condition that Theorem 1, with respect to function ( , ) K x t , must meet the condition of ( , 0) 0 ( , ) . With respect to function ( , ) c x t , the periodicity of variables t and 2 T e    satisfy conditions Eqs. ( 2)-( 3), thereby leading to positivity of the following boundary integral.

Theorem 2:
Let the following conditions holds true in Q : Then for any function 2 , ( ) there exists a unique regular solution of problem in Eqs. ( 6)-( 8) and the following inequalities are true: The proof of inequality I) will be done similarly to the proof of Theorem 1, from which the uniqueness of the regular solution of problem in Eqs. ( 6)-( 8) follows.Now, the second prior estimate is proven.
Using these sequences of functions, a solution of the auxiliary problem is constructed 2 , (12) Obviously, problem in Eqs. ( 12)-( 13) is uniquely solvable and its solution has the form for some set of sequences of functions 1 2 , ,..., m    , then acting on this sum by the operator  results in . Note, from the construction of function ( , ) j x t  the following conditions to the functions ( , ) ( ) ) 0 (15) Now an approximate solution of Eqs. ( 6)-( 8) is searched for in the form The unique solvability of the algebraic system in Eq. ( 16) is proven.Multiplying every equation of ( 16) by 2 j c and summarizing with respect to j from 1 to N , considering problem in Eqs. ( 9)-( 13) results in exp( ) exp( ) .( 17) From which, by virtue of Theorem 2, by integrating the identity in Eq. ( 17) an approximate solution of problem in Eqs. ( 6)-( 8) is obtained that estimates I) i.e.
Now the second priory estimate II) must be proven.Thanks to Eqs. ( 9)-( 13), from the identity in Eq. ( 16) comes Multiplying each equation of ( 18) by 2 2 j j c  and summing j from 1 to N , taking Eqs. ( 13),( 14) into account from Eq. ( 18), the following is derived Integrating Eq. ( 19) according to the conditions of Theorem 2 and the boundary conditions in Eqs. ( 14)-( 15), we obtain the following inequality where 1 J is integral along the domain and i J , 2,3 i  are integral along the boundary.Considering the condition of Theorem 2, using Young's inequality, results in: ) Based on boundary conditions in Eqs. ( 14), (15) and the condition of Theorem 2, 0, 1, 2. (22 does not depend on N , hence from Eqs. ( 17)-( 22) the second estimate for the approximate solution of Eqs. ( 6)- (8) follows.An estimate of Eq. ( 17) together with Eq. (21) allows to pass to limit at N   and conclude that a subsequence   k N u  converges in view of the uniqueness (Theorem 1) in 2 ( ) L Q together with a first-and second-order to the desired regular solution ( , ) u x t  of Eqs. ( 6)-( 8) possessing the properties stated in Theorem 2 [6,8,10,13,14].For ( , ) u x t  by virtue of Eq. ( 21), the inequality holds true Theorem 2 is thus proven.Now using the method of "ε-regularization" the solvability of Eqs.(1)-( 3) is proven.

Theorem 3:
Let all conditions of Theorem 2 be fulfilled.Then the generalized solution of the problem from 2  2 ( ) W Q exists and is unique.

Proof:
The uniqueness of the solution of problem in Eqs. ( 1)-(3) from 2 2 ( ) W Q is proven in Theorem 1.Now the existence of a generalized solution of Eqs. ( 1)-(3) from 2 2 ( ) W Q is proven.For this, consider Eq. ( 6) in domain Q with boundary conditions in Eqs. ( 7), (8) at 0   .Since all conditions of Theorem 2 are fulfilled, then there exists a unique regular solution of problem Eqs. ( 6)-( 8) at 0   and for this solution estimates I), II) are true.
From here follows, under the known theorem of weak compactness, that from a set of functions  , From here, the following inequality will be performed From (25) approaching a limit at 0 i   we get the unique solution of Eqs. ( 1)-(3).Thus, theorem 3 is proven.

SMOOTHNESS OF THE SOLUTION
Now, the more general case of 3 l… must be proven.Further, the coefficients of Eq. ( 1) are assumed to be infinitely differentiable in a closed domain Q .

Theorem 4:
Let the conditions of Theorem 3 be fulfilled and let 2( ) 0 there exists a unique generalized solution to problem (1)-( 3) from the space From the smoothness of the solution to problem in Eqs. ( 9)-( 13), the following conditions for the approximate solution of Eqs. ( 6 From Theorem 2, it follows that the set of functions   Further, from the conditions of Theorem 3, one can easily see that operators ,( 0) P    satisfy the conditions of Theorem 4. From here, based on estimates of I) and II) for a function   v  , the following analogical estimates are obtained , (29) . The set of functions     is uniformly bounded in 1 2 ( ) W Q i.e.

CONCLUSION
Solvability of the problem considered can be formulated in the terms of smoothness of K (x, t) in the case where, ( , 0) 0 ( , ) K x K x T   . In this the second kind of mixed type equation studied, since there is no restriction inside the domain.Under the assumption that the coefficients of the equation are smooth enough and ( , ), ( , ) x t c x t  are periodic with respect to variable t .