Generalized inverse operators and Noether boundary value problems. (Q2785558)

The main purpose of this book is an investigation of boundary value problems for systems of functional-differential equations. NEWLINENEWLINENEWLINEIn Chapter 1-3 the problem of construction of generalized inverse operators to linear Fredholm operators which act in Banach or Hilbert spaces is considered. Next, using the Lyapunov-Schmidt method, the authors develop a general scheme of investigation of weakly nonlinear boundary value problems with Fredholm linear part: NEWLINE\[NEWLINE(Lz)(t)= \varphi(t)+ \varepsilon Z(z,t,\varepsilon), \qquad lz= \alpha +\varepsilon J(z(\cdot,\varepsilon),\cdot,\varepsilon),NEWLINE\]NEWLINE where \(z: [a,b] \to \mathbb R^n\), \(\varphi: [a,b]\to\mathbb R^n\), \(-\infty \leq t \leq b < + \infty\), \(L: B_1\to B_2\) is a linear bounded operator acting in Banach spaces \(B_i\), \(l: B_1\to\mathbb R^m\) is a linear bounded vector functional. The classification of critical (resonant) and noncritical (nonresonant) cases for different classes of functional-differential equations is suggested as well as convergent iterative algorithms for construction of solutions. A coefficient criterion for the existence of solutions is presented. NEWLINENEWLINENEWLINEThis approach is applied to the analysis of traditional operator systems: systems of ordinary differential equations with deviating arguments (Chapter 5) and without it (Chapter 6), and impulsive systems (Chapter 7). The book contains a lot of examples.