In this paper, the following Fuchsian equation with four singularities is studied: $$L_z[y(z)]= p_0(z) y''(z)+ p_1(z) y'(z)+ p_2(z) y(z)= \lambda y(z),\tag 1$$ with $p_0(z)= z(z- 1)(z+ s)$, $p_1(z)= c(z- 1)(z+ s)+ dz(z+ s)+ ez(z- 1)$, $p_2(z)= abz$ and the parameters $a$, $b$, $c$, $d$, $e$ satisfy the Fuchs identity $a+ b+ 1- c- d- e=0$ and the additional conditions $c\ge 1$, $d\ge 1$, $e\ge 1$, $a\ge b$. The other parameter $s$ is a small positive one.
A boundary problem on the interval $[0,1]$ is posed for equation (1) with the boundary conditions $|y(0)|< \infty$, $|y(1)|< \infty$. The eigenvalues $\lambda_n$ of this problem are functions of the parameter $s$.
The authors study the behaviour of the eigenvalue curves $\lambda_n= \lambda_n(s)$ in the vicinity of $s= 0$, i.e. for $s\to 0+$. The qualitative behaviour of the eigenfunctions $y_n(z)$ is also studied.
A nice application of the obtained results is demonstrated on an important physical model -- the plastic deformation of crystalline materials under stress.