A Nonlinear Autonomous Boundary Value Problem for a Non-
                         Degenerate Differential-Algebraic System
                         Denys Khusainov 1, Sergey Chuiko 2,3,4, Olga Nesmelova 2,4 and Daria Diachenko 2
                         1
                                 Taras Shevchenko National University of Kyiv, Volodymyrska St, 60, Kyiv, Ukraine;
                         2
                                Donbass State Pedagogical University, Donetsk region, Slavyansk, st. General Batyuk, 19, Slaviansk, 84
                                116, Ukraine
                         3
                                Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstrasse, 1, Magdeburg, 39106,
                                Germany
                         4
                                Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, st. General Batyuk, 19, Slaviansk,
                                84 116, Ukraine

                                           Abstract
                                           We have found the constructive necessary and sufficient conditions for solvability and the
                                           scheme for constructing solutions of a nonlinear autonomous boundary value problem for a
                                           nondegenerate differential-algebraic system. The nonlinear boundary value problem for the
                                           autonomous system significantly differs from similar autonomous boundary value problems
                                           by its dependence on an arbitrary continuous vector function. We have also constructed a
                                           convergent iterative scheme for finding approximate solutions of the nonlinear autonomous
                                           boundary value problem for the nondegenerate differential-algebraic system in critical and
                                           noncritical cases. Proposed in this paper scheme for studying the nonlinear autonomous
                                           boundary value problem for the nondegenerate system of differential-algebraic equations can
                                           be transferred to degenerate systems of differential-algebraic equations in the same way.
                                           Keywords 1
                                           Differential-algebraic equations, nonlinear autonomous boundary value problems.

                         1. Statement of the problem

                              Let 𝐴 and 𝐵 are (𝑚 × 𝑛) dimensional matrices and 𝑍(𝑧, 𝜀) is a 𝑛 dimensional vector function. We
                         will call a weakly nonlinear autonomous periodic differential-algebraic boundary value problem the
                         problem of finding solution [1]
                                                   𝑧(𝑡, 𝜀): 𝑧(⋅, 𝜀) ∈ 𝐶 1 [𝑎, 𝑏(𝜀)], 𝑧(𝑡,⋅) ∈ 𝐶[0, 𝜀0 ], 𝑏(0): = 𝑏 ∗
                         of a differential-algebraic system
                                                                         𝐴𝑧 ′ = 𝐵𝑧 + 𝜀𝑍(𝑧, 𝜀),                                (1)
                         which satisfy the boundary condition
                                                                               ℓ𝑧(⋅, 𝜀) = 𝛼.                                  (2)
                         Here, ℓ𝑧(⋅, 𝜀) is a linear bounded vector functional:
                                                                      ℓ𝑧(⋅, 𝜀): 𝐶[𝑎, 𝑏(𝜀)] → 𝑅𝑞 .
                         The solution of the problem (1), (2) is found in a small neighbourhood of the solution 𝑧0 (𝑡) ∈
                         𝐶 1 [𝑎, 𝑏∗ ] of the Noether (𝑞 ≠ 𝑛) differential-algebraic generating boundary value problem
                                                                                        A z 0  B z0  z0 ()    R q                                     (3)
                         The vector function 𝑍(𝑧, 𝜀) we assume to be continuously differentiable with respect to the unknown
                         𝑧(𝑡, 𝜀) in a small neighbourhood of the solution of the generating problem and continuously
                         differentiable with respect to a small parameter 𝜀 in a small positive neighbourhood of zero. The

                         Dynamical System Modeling and Stability Investigation (DSMSI-2023), December 19-21, 2023, Kyiv, Ukraine
                         EMAIL: d.y.khusainov@gmail.com (A. 1); chujko-slav@ukr.net (A. 2); star-o@ukr.net (A. 3); dyachenkodaria2016@gmail.com (A. 4)
                         ORCID: 0000-0001-5855-029X (A. 1); 0000-0001-7186-0129 (A. 2); 0000-0003-2542-5980 (A. 3); 0000-0001-6287-1838 (A. 4)
                                      ©️ 2023 Copyright for this paper by its authors.
                                      Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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matrix 𝐴 we assume, in general, to be rectangular: 𝑚 ≠ 𝑛, or square, but degenerate [2, 3, 4]. Under
the condition
                                                    𝑃𝐴∗ = 0                                             (4)
the generating system (3) is reduced to the traditional system of ordinary differential equations [5]
                                         𝑧′0 = 𝐴+ 𝐵𝑧0 + 𝑃𝐴𝜌0 𝜈0 (𝑡);                                    (5)
here
                                              rank 𝐴: = 𝑚 < 𝑛.
In addition, 𝐴+ is a pseudo-inverse (by Moore-Penrose) matrix, 𝑃𝐴∗ is a matrix-orthoprojector:
                                             𝑃𝐴∗ : 𝑅𝑚 → 𝑁(𝐴∗ ),
𝑃𝐴𝜌0 is a (𝑛 × 𝜌0 ) matrix formed from 𝜌0 linearly independent columns of (𝑛 × 𝑛) matrix-
orthoprojector
                                               𝑃𝐴 : 𝑅𝑛 → 𝑁(𝐴),
𝜈0 (𝑡) ∈ 𝑅𝜌0 is an arbitrary continuous vector function. Under the condition (4) the system (1) we will
call nondegenerate. In the critical case
                                           𝑃𝑄∗ ≠ 0, 𝑄: = ℓ𝑋0 (⋅)
for a fixed vector function 𝜈0 (𝑡) ∈ 𝐶[𝑎, 𝑏 ∗ ] under the condition
                                      𝑃𝑄𝑑∗ {𝛼 − ℓ𝐾[𝑃𝐴𝜌0 𝜈0 (𝑠)](⋅)} = 0                                  (6)
the generating problem (3) has 𝑟 parametric family of solutions [5]
                            𝑧0 (𝑡, 𝑐𝑟 ) = 𝑋𝑟 (𝑡) 𝑐𝑟 + 𝐺[𝑃𝐴𝜌0 𝜈0 (𝑠)](𝑡), 𝑐𝑟 ∈ 𝑅𝑟 .
Here 𝑋0 (𝑡) is a normal (𝑋0 (𝑎) = 𝐼𝑛 ) fundamental matrix of the homogeneous part of the differential
system (5), and 𝐺[𝑃𝐴𝜌0 𝜈0 (𝑠)](𝑡) is the generalized Green’s operator [3, 5] of the generating periodic
differential-algebraic boundary value problem (3), 𝐾[𝑃𝐴𝜌0 𝜈0 (𝑠)](𝑡) is the generalised Green’s
operator [3, 5] of the Cauchy problem 𝑧(𝑎) = 0 for the differential-algebraic system (3). The matrix
𝑃𝑄𝑑∗ is formed from 𝑑 linearly independent rows of the matrix-orthoprojector 𝑃𝑄∗ , and the matrix 𝑃𝑄𝑟
is formed from 𝑟 linearly independent columns of the matrix-orthoprojector 𝑃𝑄 . Under the condition
(4) the system (1) leads to the traditional system of ordinary differential equations
                                   𝑧 ′ = 𝐴+ 𝐵 𝑧 + 𝑃𝐴𝜌 𝜈0 (𝑡) + 𝜀 𝐴+ 𝑍(𝑧, 𝜀);
                                                     0
                                                                                                        (7)
The periodic boundary value problem for an autonomous system (6) significantly differs [1, 6] from
similar autonomous boundary value problems by its dependence on an arbitrary vector function
𝜈0 (𝑡) ∈ 𝐶[𝑎, 𝑏∗ ]. In addition, only in exceptional cases, the autonomous boundary value problem (1),
(2) is solvable on a segment of fixed length.
   Example 1. Let us find a solution to the autonomous nonlinear differential-algebraic boundary
value problem for equation
                            𝐴𝑧 ′ = 𝐵𝑧 + 𝜀 𝑍(𝑧, 𝜀), 𝑡 ∈ [0, 𝑇], ℓ𝑧(⋅, 𝜀) = 0;                     (8)
here,
                              1 0 0                  0 0 1                 0 0 0
                       𝐴: = (          ) , 𝐵: = (               ) , 𝛺: = (          ),
                              0 0 1                 −1 0 0                 0 0 1
and
             𝑍(𝑧(𝑡, 𝜀), 𝜀): = 𝛺 𝑧(𝑡, 𝜀)(1 − 𝑧 ∗ (𝑡, 𝜀) 𝑧(𝑡, 𝜀)), ℓ𝑧(⋅, 𝜀): = 𝑧(0, 𝜀) − 𝑧(𝑇, 𝜀).

   The condition (4) is satisfied, so the system (8) is nondegenerate. In this case, the matrix 𝐴 is
rectangular, and
                                                                0
                                         𝜌0 = 1 ≠ 0, 𝑃𝐴𝜌0 = (1),
                                                                0
therefore, the homogeneous part of the system (8) has a solution that depends on an arbitrary
continuous function; we put 𝜈0 (𝑡): = 0:
                             𝑧0 (𝑡, 𝑐) = 𝑋0 (𝑡)𝑐 + 𝐾[𝑃𝐴𝜌0 𝜈0 (𝑠)](𝑡), 𝑐 ∈ 𝑅3 ,
where

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                                     𝑐𝑜𝑠 𝑡         0 𝑠𝑖𝑛 𝑡                        0
                          𝑋0 (𝑡) = ( 0             1   0 ) , 𝐾[𝑃𝐴𝜌0 𝜈0 (𝑠)](𝑡) = (0).
                                    − 𝑠𝑖𝑛 𝑡        0 𝑐𝑜𝑠 𝑡                        0
The generating solution
                                                                   0
                                    𝑧0 (𝑡, 𝑐𝑟 ) = 𝑧(𝑡, 0), 𝑐𝑟 : = (𝑐2 ) , 𝑐2 ∈ 𝑅1
                                                                   0
determines the partial solution
                                        𝑧(𝑡, 𝜀) = 𝑧0 (𝑡, 𝑐𝑟 ) = 𝑐𝑟
of nonlinear autonomous differential-algebraic boundary value problem (8) on the segment [0, 𝑇] of
fixed length.

2. Necessary condition of solvability

    An autonomous boundary value problem for the system (7) is significantly different from similar
nonautonomous boundary value problems; in that the length of the interval [𝑎, 𝑏(𝜀)], on which we
determined the solution of a nonlinear boundary value problem for the system (7), in general, is
unknown. We will use the technique [1, 6], which consists in representing the unknown function
                                        𝑏(𝜀) = 𝑏∗ + 𝜀 (𝑏∗ − 𝑎) 𝛽(𝜀)
through a new unknown
                                        𝛽(𝜀) ∈ 𝐶[0, 𝜀0 ], 𝛽(0): = 𝛽∗ .
Function 𝛽(𝜀) is to be determined in the process of finding a solution of the boundary value problem
for the system (7). The technique consists in replacing the of the independent variable
                                        𝑡 = 𝑎 + (𝜏 − 𝑎)(1 + 𝜀𝛽(𝜀))
and finding the solution of the nonlinear boundary value problem (2), (7) and the function 𝛽(𝜀), as a
function of a small parameter. In the critical case, under the condition (6) for a fixed function 𝜈0 (𝜏)
the condition of solvability of the nonlinear boundary value problem (2), (7) takes the form
                        𝑃𝑄∗ {(1 + 𝜀𝛽(𝜀)) 𝛼 − ℓ𝐾[𝛽(𝜀)(𝐴+ 𝐵𝑧(𝑠, 𝜀) + 𝑃𝐴𝜌 𝜈0 (𝑠) +
                               𝑑                                                        0
                                                                                                         (9)
                                                          +
                                +(1 + 𝜀 𝛽(𝜀)) 𝐴 𝑍(𝑧(𝑠, 𝜀), 𝜀)](⋅)} = 0.
Using the continuity of the nonlinear vector function 𝑍(𝑧(𝑡, 𝜀), 𝜀), on 𝜀 in a small positive
neighborhood of zero, we find the limit for 𝜀 → 0 in equality (9) and obtain the necessary condition
             F (č 0)  PQ {  K [   ( A B z0 ( s cr )  PA  0 ( s )  A Z ( z0 ( s cr ) 0)]()}  0    (10)
                           d                                           0


of existence of a solution to the boundary value problem (1), (2) in the critical case; here
                                                           
                                                          cr  r 1
                                                   č 0    R 
                                                      
                                                          
                                                               
Thus, the following lemma is proved.
    Lemma. Assume that the autonomous differential-algebraic boundary value problem (1), (2) for a
fixed constant 𝜈0 ∈ 𝑅𝜌0 under the conditions (4) and (6) represents the critical case 𝑃𝑄∗ ≠ 0 and has
the solution
                    𝑧(𝑡, 𝜀) = col(𝑧 (1) (𝑡, 𝜀), . . . , 𝑧 (𝑛) (𝑡, 𝜀)), 𝑧 (𝑖) (⋅, 𝜀) ∈ 𝐶 1 [𝑎, 𝑏(𝜀)],
                                     𝑧 (𝑖) (𝑡,⋅) ∈ 𝐶[0, 𝜀0 ], 𝑖 = 1, 2, . . . , 𝑛,
which for 𝜀 = 0 turns into the generating 𝑧(𝑡, 0) = 𝑧0 (𝑡, 𝑐𝑟∗ ). Then the vector č 0 satisfies equation
(10).
    The first 𝑟 components of 𝑐𝑟∗ ∈ 𝑅𝑟 of the root of the equation (10) determine the amplitude of the
generating solution 𝑧0 (𝑡, 𝑐𝑟∗ ), in small neighborhood of which the desired solution of the initial
problem (1), (2) can exist. In addition, from the equation (10) the value 𝛽∗ , which determines the first
approximation to the unknown function
                                             𝑏1 (𝜀) = 𝑏 ∗ + 𝜀(𝑏∗ − 𝑎)𝛽∗ ,

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can be found. If the equation (10) has no real roots, then the initial differential algebraic problem (1),
(2) has no desired solutions.
    The equation (10) will be further called the equation for the generating constants of the
autonomous nonlinear differential-algebraic boundary value problem (1), (2). The statement of the
lemma generalizes the corresponding results of [1, 6, 7, 9] on the case of an autonomous nonlinear
differential-algebraic boundary value problem (1), (2), namely, on the case 𝐴 ≠ 𝐼𝑛 . Similarly to [1, 6,
9] we demonstrate that the periodic problem for the system (7) is solvable provided that the roots of
the equation for the generating constants (10) are simple.

3. Sufficient condition of solvability
    Assume that the equation for the generating constants (10) has real roots. Fixing one of the
solutions č 0 ∈ 𝑅𝑟+1 of the equation (10), we come to the problem of finding solutions of the problem
(1), (2)
                                         𝑧(𝜏, 𝜀) = 𝑧0 (𝜏, 𝑐𝑟∗ ) + 𝑥(𝜏, 𝜀)
in the neighbourhood of the generating solution
                                 𝑧0 (𝜏, 𝑐𝑟∗ ) = 𝑋𝑟 (𝜏)𝑐𝑟∗ + 𝐺[𝑃𝐴𝜌0 𝜈0 (𝑠)](𝜏).
The deviation of 𝑥(𝑡, 𝜀) from the generating solution is determined by the boundary value problem
              𝑥 ′ = 𝐴+ 𝐵𝑥 + 𝜀{𝛽𝐴+ 𝐵𝑧 + 𝛽𝑃𝐴𝜌0 𝜈0 (𝑠) + (1 + 𝜀𝛽)𝐴+ 𝑍(𝑧, 𝜀)}, ℓ𝑥(⋅, 𝜀) = 0.                                  (11)
Using the continuous differentiability of the function 𝑍(𝑧, 𝜀) with respect to both the first and second
arguments in the neighbourhood of the generating solution 𝑧0 (𝜏, 𝑐𝑟∗ ) and the point 𝜀 = 0, we get the
expansion of this function
                                      𝑍(𝑧0 (𝜏, 𝑐𝑟∗ ) + 𝑥(𝜏, 𝜀), 𝜀) = 𝑍(𝑧0 (𝜏, 𝑐𝑟∗ ),0) +
                                   +𝐴1 (𝜏)𝑥(𝜏, 𝜀) + 𝜀𝐴2 (𝜏) + 𝑅(𝑧0 (𝜏, 𝑐𝑟∗ ) + 𝑥(𝜏, 𝜀), 𝜀),
where
                                          𝜕𝑍(𝑧, 𝜀)                      𝜕𝑍(𝑧, 𝜀)
                         A1 (𝜏) =                  |         , A2 (𝜏) =          |
                                            ∂z 𝑧=𝑧0 (𝜏,𝑐𝑟∗),              ∂𝜀       𝑧=𝑧0 (𝜏,𝑐𝑟∗ ),
                                                         𝜀=0,                                  𝜀=0,
The residual 𝑅(𝑧(𝜏, 𝜀), 𝜀) of the expansion of the function 𝑍(𝑧(𝜏, 𝜀), 𝜀) of higher order of smallness
on 𝑥 and 𝜀 in the neighbourhood of the points 𝑥 = 0 and 𝜀 = 0, than the first three terms of the
expansion, so
                                                     𝜕𝑅(𝑧, 𝜀)                           𝜕𝑅(𝑧, 𝜀)
                  𝑅(𝑧, 𝜀)|                    ≡ 0,     𝜕𝑧       |                ≡ 0,     𝜕𝜀 |                      ≡0
                             𝑧=𝑧0 (𝜏,𝑐𝑟∗ ),                     𝑧=𝑧0 (𝜏,𝑐𝑟∗ ),                     𝑧=𝑧0 (𝜏,𝑐𝑟∗ ),
                                𝜀=0,                               𝜀=0,                               𝜀=0,

Let us denote by (𝑑 × (𝑟 + 1)) the dimensional matrix
                 𝐵0 = −𝑃𝑄𝑑∗ ℓ𝐾{𝐴+ [𝛽∗ 𝐵 + 𝐴1 (𝑠)]𝑋𝑟 (𝑠); 𝑃𝐴𝜌0 𝜈0 (𝑠) + 𝐴+ 𝐵𝑧0 (𝑠, 𝑐𝑟∗ )}(⋅).
In the critical case, under the condition (6) for a fixed function 𝜈0 (𝜏) and the solution č 0 of equation
(10) the solution of the nonlinear boundary value problem (11) has the form
                                               𝑥(𝜏, 𝜀) = 𝑋𝑟 (𝜏)𝑐𝑟 (𝜀) + 𝑥 (1) (𝜏, 𝜀);
here,
                    𝑥 (1) (𝜏, 𝜀): = 𝜀 𝐺[𝛽𝐴+ 𝐵𝑧 + 𝛽𝑃𝐴𝜌0 𝜈0 (𝑠) + (1 + 𝜀𝛽) 𝐴+ 𝑍(𝑧, 𝜀)](𝜏).
and
                         𝛽(𝜀): = 𝛽∗ + 𝛾(𝜀), 𝑐(𝜀): = col(𝑐𝑟 (𝜀), 𝛽(𝜀)) ∈ 𝐶[0, 𝜀0 ],
while the condition of solvability of the nonlinear boundary value problem (11) leads to the equation
                 𝐵0 𝑐(𝜀) = −𝑃𝑄𝑑∗ ℓ𝐾{𝛽∗ 𝐴 + 𝐵𝑥 (1) (𝑠, 𝜀) + 𝐴+ [𝜀𝐴2 (𝑠) + 𝑅(𝑧(𝑠, 𝜀), 𝜀)]}(⋅),
which solvable under the condition [1, 10, 11]
                   𝑃𝐵0∗ 𝑃𝑄𝑑∗ ℓ𝐾{𝛽∗ 𝐴 + 𝐵𝑥 (1) (𝑠, 𝜀) + 𝐴+ [𝜀𝐴2 (𝑠) + 𝑅(𝑧(𝑠, 𝜀), 𝜀)]}(⋅) = 0.


                                                                                                                         93
In particular, the condition of solvability of the nonlinear boundary value problem (11) is satisfied in
the case of
                                                    𝑃𝐵0∗ 𝑃𝑄𝑑∗ = 0.                                      (12)
Thus, under the condition (12) we get an operator system that is equivalent to the problem of finding
solutions of the boundary value problem (1), (2)
                      𝑧(𝜏, 𝜀) = 𝑧0 (𝜏, 𝑐𝑟∗ ) + 𝑥(𝜏, 𝜀), 𝑥(𝜏, 𝜀) = 𝑋𝑟 (𝜏)𝑐𝑟 (𝜀) + 𝑥 (1) (𝜏, 𝜀),
                    𝑥 (1) (𝜏, 𝜀): = 𝜀𝐺[𝛽𝐴+ 𝐵𝑧 + 𝛽𝑃𝐴𝜌0 𝜈0 (𝑠) + (1 + 𝜀𝛽)𝐴+ 𝑍(𝑧, 𝜀)](𝜏),
                𝑐(𝜀) = −𝐵0+ 𝑃𝑄𝑑∗ ℓ𝐾{𝛽∗ 𝐴 + 𝐵𝑥 (1) (𝑠, 𝜀) + 𝐴+ [𝜀𝐴2 (𝑠) + 𝑅(𝑧(𝑠, 𝜀), 𝜀)]}(⋅).                     (13)
The operator system (13) belongs to the class of systems for which the method of simple iterations is
applicable [1]. Thus, the following theorem is proved.
    Theorem. Assume that the autonomous differential-algebraic boundary value problem (1), (2) for
a fixed constant 𝜈0 ∈ 𝑅𝜌0 under the conditions (4) and (6) represents the critical case 𝑃𝑄∗ ≠ 0.
Suppose also that the equation for the generating constants (10) has real roots. Under the conditions
(4), (6) and (12) for the fixed function 𝜈0 (𝜏) and for the solution č 0 of equation (10) the operator
system (13) is equivalent to the problem of finding solutions of the boundary value problem (1), (2)
and has at least one solution. To find the solution to the operator system (13) the method of simple
iteration is applicable.
    For finding the solution of the autonomous boundary value problem (1), (2) in the neighbourhood
of the generating solution, the Newton-Kantorovich method can also be used [8].
   Example 2. Let’s find a solution to the autonomous nonlinear differential-algebraic boundary
value problem for the equation
                            𝐴𝑧 ′ = 𝐵𝑧 + 𝜀 𝑍(𝑧, 𝜀), 𝑡 ∈ [0, 𝑇], ℓ𝑧(⋅, 𝜀) = 0;                   (14)
here
                                                                               𝑢(𝑡, 𝜀)
                         1 0 0)          0 1 1)          1 0 0)
                𝐴: = (          , 𝐵: = (        , 𝛺: = (        , 𝑧(𝑡, 𝜀): = ( 𝑣(𝑡, 𝜀) ),
                         0 0 1          −1 0 0           0 0 1
                                                                              𝑤(𝑡, 𝜀)
and
                                         𝑢(𝑡, 𝜀)𝑣(𝑡, 𝜀)
                         𝑍(𝑧(𝑡, 𝜀), 𝜀): = (            ) , ℓ𝑧(⋅, 𝜀): = 𝑧(0, 𝜀) − 𝑧(𝑇, 𝜀).
                                         𝑢(𝑡, 𝜀)𝑣(𝑡, 𝜀)
   Since the condition (4) is satisfied, the system (14) is nondegenerate. In this case, the matrix 𝐴 is
rectangular, and
                                                                  0
                                              𝜌0 = 1 ≠ 0, 𝑃𝐴𝜌0 = (1),
                                                                  0
so the homogeneous part of the system (14) has a solution that depends on an arbitrary continuous
function; we put 𝜈0 (𝑡): = 𝑐𝑜𝑠 3 𝑡:
                              𝑧0 (𝑡, 𝑐) = 𝑋0 (𝑡) 𝑐 + 𝐾[𝑃𝐴𝜌0 𝜈0 (𝑠)](𝑡), 𝑐 ∈ 𝑅3 ,
where
                        𝑐𝑜𝑠 𝑡        𝑠𝑖𝑛 𝑡             𝑠𝑖𝑛 𝑡                           3 𝑐𝑜𝑠 𝑡 − 3 𝑐𝑜𝑠 3 𝑡
                                                                                   1
             𝑋0 (𝑡) = ( 0              1                 0 ) , 𝐾[𝑃𝐴𝜌0 𝜈0 (𝑠)](𝑡) = 24 (     8 𝑠𝑖𝑛 3 𝑡      ).
                       − 𝑠𝑖𝑛 𝑡     −1 + 𝑐𝑜𝑠 𝑡          𝑐𝑜𝑠 𝑡                            −3 𝑠𝑖𝑛 𝑡 + 𝑠𝑖𝑛 3 𝑡
In this case, the condition (6) is satisfied, so the linear part of the problem (14) has a solution that
depends on the continuous function 𝜈0 (𝑡): = 𝑐𝑜𝑠 3 𝑡:
                               𝑧0 (𝑡, 𝑐𝑟 ) = 𝑋𝑟 (𝑡)𝑐 + 𝐺[𝑃𝐴𝜌0 𝜈0 (𝑠)](𝑡), 𝑋𝑟 (𝑡): = 𝑋0 (𝑡);
here
                                   𝐺[𝑃𝐴𝜌 𝜈0 (𝑠)](𝑡) = 𝐾[𝑃𝐴𝜌 𝜈0 (𝑠)](𝑡), 𝑐𝑟 ∈ 𝑅3 .
                                          0                         0

The equation for the generating constants (10) has a real root
                                                            1
                                              
                                                  cr     
                                         č 0 
                                                  R  c   0   R3     0
                                                       
                                                            4   

                                                
                                                              r
                                                      
                                                          1
                                                             

                                                                                                                94
which corresponds to the full rank matrix
                                             𝜋    0 1 0 16
                                       𝐵0 = 8 (                    ),
                                                  0 17 0 −16
thus, the condition of solvability (12) of the nonlinear boundary value problem (14) is satisfied.
    Proposed in this paper scheme for studying a nonlinear autonomous boundary value problem for a
nondegenerate system of differential-algebraic equations [12,13,14,15,16,17,18] can be, analogously
[5], transferred to degenerate systems of differential-algebraic equations in the same way. For finding
the solution of an autonomous boundary value problem for a nondegenerate system of differential-
algebraic equations (1), (2) the main requirement is the solvability requirement (12), which is
equivalent to the condition of simplicity of the roots of the equation for the generating constants (10)
[1, 6]. The scheme for studying a nonlinear autonomous boundary value problem for a nondegenerate
system of differential-algebraic equations proposed in this paper can be similarly transferred to
systems of differential-algebraic equations with a variable rank of derivative matrix.

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