^{1, 2}

^{1, 2}

^{1}

^{2}

By method of integral equations, unique solvability is proved for the solution of boundary value problems of loaded third-order integrodifferential equations with Riemann-Liouville operators.

The theory of mixed type equations is one of the principal parts of the general theory of partial differential equations. The interest for these kinds of equations arises intensively because of both theoretical and practical uses of their applications. Many mathematical models of applied problems require investigations of this type of equations. The first fundamental results in this direction were obtained in 1920–1930 by Tricomi [

In the recent years, in connection with intensive research on problems of optimal control of the agroeconomical system, long-term forecasting, and regulating the level of ground waters and soil moisture, it has become necessary to investigate a new class of equations called “loaded equations.” Such equations were investigated for the first time in works of N. N. Nazarov and N. Kochin. However, they did not use the term “loaded equation.” For the first time, the term has been used in works of Nakhushev [

Basic questions of the theory of boundary value problems for partial differential equations are the same for the boundary value problems for the loaded differential equations. However, existence of the loaded operator does not always make it possible to apply directly the known theory of boundary value problems.

Works of Nakhushev, M. Kh. Shkhankov, A. B. Borodin, V. M. Kaziev, A. Kh. Attaev, C. C. Pomraning, E. W. Larsen, V. A. Eleev, M. T. Dzhenaliev, J. Wiener, B. Islomov and D. M. Kuriazov, K. U. Khubiev, and M. I. Ramazanov et al. are devoted to loaded second-order partial differential equations.

It should be noted that boundary value problems for loaded equations of a hyperbolic, parabolic-hyperbolic, and elliptic-hyperbolic types of the third order are not well understood. We indicate only the works of V. A. Eleev, Islomov, D. M. Kur’yazov, and A. V. Dzarakhokhov.

The present paper is devoted to formulation and investigation of the analogue of the Cauchy-Goursat problem for the loaded equation of a hyperbolic type

Let

Let us consider the following analogue of the Cauchy-Goursat problem for the loaded equation (

If

An important role in proving Theorem

Lemma 2.

Then, vice versa, let

Let us prove the validity of relation (

It follows from the latter representation that

Lemma 2 is proved.

Invoking that the function

Let us solve the Cauchy problem for (

The solution to the Cauchy problem for (

The last equality with respect to designation and after some transformation becomes

And with recurring index

By virtue of representation (

Similarly to [

Assuming that

We can assume without loss of generality that

Let

Let

Hence, we conclude that the integral equations (

Thus, it is proved that problem A is uniquely solvable. Theorem

Let us introduce the following notation:

Let us term the function

the sewing conditions

We note the unique solvability of problem C and Gellerstedt problems for a loaded differential equation (

If

The following theorem holds.

Lemma 4.

The lemma is proved similarly to Lemma 2.

Invoking that the function

Solution of the Cauchy problem for (

By virtue of representation (

Assuming that

Denoting

Due to the property of Problem C and in view of (

The equality (

Omitting the function

Assuming that

Hence, (

In view of (

Hence, we conclude that (

Hence, by virtue of the condition

Thus, the solution of problem C in the domain

Thus, problem C is uniquely solvable. Theorem