On dual integral equations in the semiconservative case

A dual integral equations on the whole real axis with an unknown function f is considered. It is supposed that the kernel functions of the equations are even and representable as superposition of exponents. The equation is transferred to a system of integral equations on the positive semi-axis with two unknown functions: f (1)(x) = f(x) and f (2)(x) = f(-x). Applying a factorization method and using the solution of Ambartsumian equation, a system of Laplace transforms alpha (1), alpha (2) of functions f (1), f (2) is obtained. Under some conditions on the free term, the existence and uniqueness of the solution of that system is proved in the semi-conservative case. A construction of the functions alpha (1), alpha (2) is given by means of successive approximations, and a construction method of the solution (f (1), f (2)) by alpha (1), alpha (2) is described.