The mathematical description of numerous phenomena in many areas of
electronics, theory of nonlinear oscillation, mechanics, biology and radio
engineering leads to the necessity of investigating nonlinear partial
differential equations. In a particular case, the latter equation leads to
nonlinear differential equations unsolved with respect to the derivative. A
linearization of such equations, leads to a linear differential-algebraic
equation of the form
$$A(t)z_0(t) = B(t)z(t) + f(t)$$
The monographs of A.M. Samoilenko, M.O. Perestiuk, V.P. Yakovets, O.O.
Boichuk, as well as numerous works by foreign authors S. Campbell, J.R.
Magnus, V.F. Chistyakov and others are devoted to the study of linear
differential-algebraic equations using the central canonical form and perfect
pairs and triples of matrices The difference of this report is that the
finding of constructive conditions for existence and construction of
solutions of nonlinear differential-algebraic boundary-value problems was
done without using the central canonical form. This made it possible to study
solutions of differential-algebraic boundary-value problems that depend on
arbitrary continuous functions. The relevance of the studying nonlinear
boundary-value problems unsolved with respect to the derivative is due to the
fact that the study of a traditional problem resolved with respect to the
derivative is sometimes difficult, in the case of obtaining nonlinearities
not integrable in elementary functions. Thus this report is devoted to the
study of the problems of finding constructive conditions for existence and
construction of solutions of nonlinear boundary-value problems problems
unsolved with respect to the derivative. The report will also deal with
linear and nonlinear differential-algebraic boundaryvalue problems. The cases
of degeneracy and nondegeneracy of differential-algebraic system will be
investigated. The classification of nonlinear differential-algebraic
boundary-value problems has been improved. The constructive solvability
conditions and schemes for constructing solutions of nonlinear
differential-algebraic boundary-value problems in critical and noncritical
cases are found. Convergent iteration schemes for finding approximations to
solutions of nonlinear differential-algebraic boundary value problems are
constructed.