Nonlocal inverse problem for a pseudohyperbolic-pseudoelliptic type differential equation

Abstract

The problems of solvability of a nonlocal inverse boundary value problem for a mixed type pseudohyperbolic-pseudoelliptic differential equation with a spectral parameter and integral conditions are considered. The equation is characterized by the fact that it has three unknown functions: with respect to the main unknown function the equation is a mixed type differential equation; and with respect to redefinition functions it is Fredholm integral equations of the second kind. The cases of regular and irregular values of the spectral parameter are studied. By the aid of the method of the Fourier series, a nonlinear system of two countable systems of Fredholm ordinary integral equations of the second kind is obtained. From this system we prove the existence and uniqueness of the Fourier coefficients of redefinition functions. In the proof of the unique solvability of this system of two countable systems of Fredholm ordinary integral equations of the second kind the method of compressive mappings is applied. In this proof we use also the Cauchy-Schwartz inequality and the Bessel inequality. For regular values of the spectral parameter, a criterion of unique solvability of the inverse boundary value problem is established. Solutions of the inverse boundary value problem are obtained in the form of Fourier series. The convergence of the obtained Fourier series and the possibility of term-by-term differentiation of the main Fourier series are shown. In the case of irregular values of spectral parameter, it is constructed an infinite number of solutions of the inverse problem in the view of Fourier series. The results are formulated as a theorem.