On Solvability of an Inverse Boundary Value Problem for Pseudo Hyperbolic Equation of the Fourth Order

Abstract We analyze the solvability of the inverse boundary problem with an unknown coefficient depended on time for the pseudo hyperbolic equation of fourth order with periodic and integral conditions.The initial problem is reduced to an equivalent problem. With the help of the Fourier method, the equivalent problem is reduced to a system of integral equations. The existence and uniqueness of the solution of the integral equations is proved. The obtained solution of the integral equations is also the only solution to the equivalent problem. Basing on the equivalence of the problems, the theorem of the existence and uniqueness of the classical solutions of the original problem is proved.


Introduction
There are many cases where the needs of the practice bring about the problems of determining coefficients or the right hand side of differential equations from some knowledge of its solutions.Such problems are called inverse boundary value problems of mathematical physics.Inverse boundary value problems arise in various areas of human activity such as seismology, mineral exploration, biology, medicine, quality control in industry etc., which makes them an active field of contemporary mathematics.
The inverse problems are favorably developing section of up-to-date mathematics.Recently, the inverse problems are widely applied in various fields of science.
In searching of local and non-local boundary value problems for pseudohyperbolic equations practical and theoretical interests assume great importance and is more actively studied now days.
In this paper, due to the ( Mehraliyev, 2011)-( Mehraliyev, 2012), we proved the existence and uniqueness of the solution of the inverse boundary value problem for the pseudohyperbolic equation of fourth order with periodic and integral conditions.

Problem Statement and Its Reduction to Equivalent Problem
Lets consider for the equation (Gabov & Orazov,1986) and with additional condition where x 0 ∈ (0, 1) are the given number, f (x, t),φ(x),ψ(x),h(t) are the given functions, and u(x, t) , a(t) are the required functions.
The condition ( 4) is a non-local integral condition of first kind, i.e. the one not involving values of unknown functions at the domains boundary points.
Definition.The classic solution of problem (1) -( 5) is the pair{u (x, t) , a (t)} of the functions u(x, t) and a(t) with the following properties: 1)the function u(x, t) is continuous in D T together with all its derivatives contained in equation ( 1); 2)the function a(t) is continuous on [0, T ]; 3) all the conditions of (1) -(5) are satisfied in the ordinary sense.
The following lemma is valid.
Then the problem on finding the classic solution of problem (1) -( 5) is equivalent to the problem on defining of the function u(x, t) and a(t), possessing the properties 1) and 2) of definition of the classic solution of problem (1) -(5), from relations (1) -(3) and satisfying Proof.Let {u (x, t) , a (t)} be a classical solution to the problem (1) -( 5).Integrating equation (1) with respect to x from 0 to 1, we have Taking into account that ∫ 1 0 f (x, t)dx = 0 (0 ≤ t ≤ T ) and ( 4), we find that Since problem ( 9), ( 10)has only a trivial solution, we have u x (1, t) − u x (0, t) = 0, i.e. the condition ( 6) is fulfilled.
Similarly to (Mehraliyev, 2012) it is possible to prove the following lemma.
Remark 1 It follows from lemma 2 that to prove the uniqueness of the solution to the problem (1) -( 3), ( 6), ( 7), it suffices to prove the uniqueness of the solution to the system ( 26), ( 29).
It is easy to see that 1 Taking into account these relations, by means of simple transformations we find Then the problem (1) -(3), ( 6), ( 7) has a unique solution in the ball K = K R (∥z∥ T .Remark 2. Inequality (36) is satisfied for sufficiently small values at Proof.In the space E 3 T consider the equation where z = {u, a} and the components Φ i (u, a) (i = 1, 2) of the operator Φ(u, a) are given by the right hand sides of the equations ( 26), ( 29).Consider the operator Φ(u, a) in the ball K = K R from E 3 T .Similar to (35), we see that for any z, z 1 , z 2 ∈ K R the following estimates hold: Then, it follows from (36) together with the estimates ( 38) and ( 39) that the operator Φ acts in the ball K = K R and is contractive.Therefore, in the ball K = K R the operator Φ has a unique fixed point {u, a}, that is a unique solution of equation ( 37) in the ball K = K R , i.e. it is a unique solution of system ( 26), (29) in the ball K = K R .
The function u(x, t), as an element of the space B 3 2,T is continuous and has continuous derivatives u x (x, t) and u xx (x, t) in D T .
By lemma 1 the unique solvability of the initial problem (1)-( 5) follows from the theorem.