We establish constructive conditions for the existence of solutions of an autonomous Noetherian weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case and develop a modified iterative procedure for finding its solutions. Statement of the ProblemWe investigate the problem of the construction of a solutionof the system of ordinary differential equationswith boundary condition [1,2] z(·, ε) = α + εJ(z(·, ε), ε).We seek a solution of the Noetherian (m = n) boundary-value problem (1), (2) in a small neighborhood of the solutionof the generating problemHere, A is a constant n × n matrix, f is a constant column vector, Z(z, ε) is a nonlinear vector function continuously differentiable with respect to the unknown z in a small neighborhood of the solution of the generating problem and with respect to the small parameter ε on the segment [0, ε 0 ], z(·, ε) is a linear vector functional, J(z(·, ε), ε) is a nonlinear vector functional,
UDC 517.9We establish necessary and sufficient conditions for the existence of solutions of a weakly nonlinear Noetherian boundary-value problem for a system of ordinary differential equations in the critical case. Statement of the ProblemWe investigate the problem of construction of the solution z.t; "/W z.t; / 2 C 1 OEa; b; z.t; / 2 C OE0; " 0 ; of the boundary-value problem [1,2] dz dt D A.t /z C f .t / C "Z.z; t; "/;`z. ; "/ D˛C "J.z. ; "/; "/:We seek a solution of problem (1) in a small neighborhood of a solution of the generating problemHere, A.t / is an n n matrix, f .t / is an n-dimensional vector column whose elements are real functions continuous on the segment OEa; b; and`z. / is the linear bounded vector functional`z. /W C OEa; b ! R m : We assume that the nonlinearities Z.z; t; "/ and J.z. ; "/; "/ of the Noetherian .m ¤ n/ problem (1) are twice continuously differentiable with respect to the unknown z in a small neighborhood of the generating solution and with respect to the small parameter " in a small neighborhood of zero. We also assume that the vector function Z.z; t; "/ is continuous in the independent variable t on the segment OEa; b: We investigate the critical case .P Q 6 D 0/ and assume that the conditionis satisfied. In this case, the generating problem (2) has an .r D n n 1 /-parameter family of solutions z 0 .t; c r / D X r .t /c r C GOEf .s/I˛.t /; c r 2 R r : Here, X.t/ is the normal .X.0/ D I n / fundamental matrix of the homogeneous part of system (2), Q D`X. / is an m n matrix, rank Q D n 1 ; X d .t/ D X.t/P Q r ; P Q d is an n d matrix composed of d D m n 1 linearly independent columns of the n n matrix orthoprojector P Q W R n ! N.Q/; P Q d is the r d matrix composed of d D m n 1 linearly independent rows of the n n matrix orthoprojector P Q W R n ! N.Q /;
We establish necessary and sufficient conditions for the existence of solutions of an autonomous Noetherian boundary-value problem for a system of second-order ordinary differential equations in the critical case. For the construction of solutions of a nonlinear Noetherian boundary-value problem in the critical case, we propose an iterative scheme that combines the Newton-Kantorovich method and the leastsquares technique. The efficiency of the proposed method is demonstrated in the analysis of a periodic problem for a Liénard-type equation.
UDC 517.9We establish necessary and sufficient conditions for the existence of solutions and propose an iterative scheme for the determination of solutions of the Fredholm weakly nonlinear boundary-value problem for a system of ordinary differential equations in the case of multiple roots of the equation for generating constants. Statement of the ProblemWe study the problem of construction of the solution z.t; "/ W z.;We seek the solution of problem (1) in a small neighborhood of the solution of the generating problemHere, A.t/ is an .n n/ matrix and f .t / is an n -dimensional column vector whose elements are real functions continuous on the segment OEa; b and`z./ is a linear bounded vector functional`z./ W C OEa; b ! R m : Assume that the nonlinearities Z.z; t; "/ and J.z.; "/; "/ of the Fredholm .m ¤ n/ problem (1) are twice continuously differentiable with respect to the unknown z in a small neighborhood of the generating solution and with respect to the small parameter " in a small positive neighborhood of zero. In addition, we also assume that the vector function Z.z; t; "/ is continuous in the independent variable t on the segment OEa; b: We investigate the critical case .P Q 6 D 0/ under the assumption thatIn this case, the generating problem has an .r D n n 1 / -parameter family of solutionsHere, X.t/ is a normal .X.a/ D I n / fundamental matrix of the homogeneous part of the generating system, Q D`X./ is an .m n/ matrix, rank Q D n 1 ; X r .t / D X.t /P Q r ; P Q r is an .n r/ matrix formed by r
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