Fredholm integral-differential equation with integral conditions and spectral parameters

Integro-differential equations are of great interest in terms of applications. Different problems are posed and studied for ordinary integro-differential equations. In cases where the boundary of the flow domain of a physical process is not available for measurements nonlocal conditions in integral form can serve as additional information sufficiently for one-valued solvability of the problem. Nonlocal problems with integral conditions for differential and integro-differential equations were considered in the works of many mathematicians.
On segment $[0;T]$ we consider the following integro-differential equation
$$ u''(t)+\lambda^2\, u(t)+\nu \int\limits_{0}^{T} K(t,s) u(s)\, ds = 0 (1) $$
under the following integral conditions
$$ u(T)+\int\limits_{0}^{T}u(t)\, dt =\alpha, \qquad u'(T)+\int\limits_{0}^{T}u'(t)\, t\, dt = \beta, (2) $$
where $T>0$ is a given real number, $\lambda>0$ is a real spectral parameter, $\alpha=$const, $\beta=$const, $\nu$ is a real nonzero spectral parameter,
$$ K(s,t)=\sum\limits_{i=1}^k a_i(t)\,b_i(s), \qquad a_i(t),\,\, b_i(s)\in C[0;T]. $$
In this paper we assume that functions $\{a_i(t)\}_{i=1}^k$ and $\{b_i(t)\}_{i=1}^k$ are linearly independent.
The article considers the issues of solvability and construction of solutions of the nonlocal boundary-value problem for the second-order Fredholm integro-differential equation with the degenerate kernel, integral conditions, and spectral parameters are considered. We calculate the values of spectral parameters and construct the solutions corresponding to these values. This paper studies the singularities arising in the course of integration and establishes the criteria for solvability of the problem.