Investigation of the solution to the Cauchy problem for a singularly perturbed differential equation (Q1603085)

The author considers the Cauchy problem for the singularly perturbed differential equation \[ \varepsilon^h C(\varepsilon){dx\over dt}= [A- B(\varepsilon)] x(t,\varepsilon,\;t\in [0,\infty),\;\varepsilon\in (0,\varepsilon_0],\quad h\in Q,\tag{1} \] where \(A\) is a closed, linear, Fredholm (generally speaking, nonbounded) operator acting from the Banach space \(E_1\) into the Banach space \(E_2\), with the domain dense in \(E_1\) \(\dim\ker A= \dim co\ker A= 1\); \[ B(\varepsilon)= \sum^\infty_{k=1} \varepsilon^k B_k,\qquad C(\varepsilon)= \sum^\infty_{k=0} \varepsilon^k C_k; \] \(B_k\), \(C_k\) are linear operators acting from \(E_1\) into \(E_2\), \(\varepsilon^{-1} B(\varepsilon)\) and \(C(\varepsilon)\) are uniformly bounded operators on \((0,\varepsilon_0]\). The operator \(C(\varepsilon)\) can be irreversible for all sufficiently small \(\varepsilon\), therefore the Cauchy problem for equation (1) can possess a solution not for all values of \(x(0,\varepsilon)\); a solution, if it exists, can be nonunique. Here, the author constructs a subspace in \(E_1\), where the Cauchy problem is uniquely solvable. He investigates the structure of the solution.