Cauchy problem for a sedond-order degeneracy differential equation in a Banach space

This article is devoted to the study of the Cauchy problem for a second-order differential equation given in Banach spaces $E_1\to E_2$ with closed linear operator coefficients that are everywhere dense in the $E_1$ domain of definition. The operator $A$ is degenerate, which is why the solution of the Cauchy problem does not exist for every value of the initial data. This operator is Fredholm with zero index (hereinafter, Fredholm). Its kernel is assumed to be $n$-dimensional. The Fredholm property allows one to split the equation and conditions into the corresponding equations and conditions in subspaces of decreasing dimensions. On the right hand side, the operator coefficients are variable, which is different from other works. We study the case $\Delta(t)\ne0$ for each $t\in[0;T]$, where $\Delta(t)$ is some matrix constructed using operator coefficients. Conditions obtained the conditions under which the solution of the problem exists is unique; this solution is found in analytical form. An illustrative example is given.