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The theory of complete generalized Jordan sets is employed to reduce the PDE with the irreversible linear operator $B$ of finite index to the regular problems. It is demonstrated how the question of the choice of boundary conditions is connected with the $B$-Jordan structure of coefficients of PDE. The various approaches shows the combination alternative Lyapunov method, Jordan structure coefficients and skeleton decomposition of irreversible linear operator from the main part equation are among the most powerfull methods to attack such challenging problem. On this base the complex problem of the {\it correct choice of boundary conditions} for the wide class of the singular PDE can be solved. Aggregated existence and uniqueness theorems can be proved, solution may continuously depend on the function determied from the experiments. Such theory can be applied to the integral--differential equations with partial derivatives.

This paper presents a nonlinear dynamical model which consists the system of differential and operator equations. Here differential equation contains a nonlinear operator acting in Banach space, a nonlinear operator equation with respect to two elements from different Banach spaces. This system is assumed to enjoy the stationary state (rest points or equilibrium). The Cauchy problem with the initial condition with respect to one of the desired functions is formulated. The second function controls the corresponding nonlinear dynamic process, the initial conditions are not set. The sufficient conditions of the global classical solution's existence and stabilization at infinity to the rest point are formulated. It is demonstrated that a solution can be constructed by the method of successive approximations under the suitable sufficient conditions. If the conditions of the main theorem are not satisfied, then several solutions may exist. Some of solutions can blow-up in a finite time, while others stabilize to a rest point. The special case of considered dynamical models are nonlinear differential-algebraic equation (DAE) have successfully modeled various phenomena in circuit analysis, power systems, chemical process simulations and many other nonlinear processes. Three examples illustrate the constructed theory and the main theorem. Generalization on the non-autonomous dynamical systems concludes the article.

The linear PDE ${\mathbf B} {\mathbf L} (\frac{\partial}{\partial x}) u ={\mathbf L}_1(\frac{\partial}{\partial x})u +f(x)$ with nonclassic conditions on boundary $\partial \Omega$ is considered. Here ${\mathbf B}$ is linear noninvertible bounded operator acting from linear space $E$ into $E,$ $x=(t,x_1,\dots, x_m) \in \Omega, $ $\Omega \subset {\mathbb R}^{m+1}.$ It is assumed that ${\mathbf B}$ enjoys the skeleton decomposition ${\mathbf B}={\mathbf A}_1 {\mathbf A}_2,$ ${\mathbf A}_2 \in {\mathcal L}(E\rightarrow E_1),$ ${\mathbf A}_1 \in {\mathcal L}(E_1\rightarrow E)$ where $E_1$ is linear normed space. Differential operators ${\mathbf L}, \, {\mathbf L}_1$ are partial differential operators. In the concrete cases the domains of definition of operators ${\mathbf L}, {\mathbf L}_1$ consist of linear manifolds $E_{\partial}$ of sufficiently smooth abstract functions $u(x)$ with domain in $\Omega$ and their ranges in $E,$ which satisfy certain system of homogeneous boundary conditions. The abstract function $f: \Omega \subset {\mathbb R}^{m+1} \rightarrow E $ is assumed to be given. It is requested to find the solution $u: \Omega \subset {\mathbb R}^{m+1} \rightarrow E_{\partial},$ which satisfy certain condition on boundary $\partial \Omega.$ The concept of a skeleton chains is introduced as sequence of linear operators ${\mathbf B}_i \in {\mathcal L}(E_i \rightarrow E_i), \, i=1,2,\dots, p,$ where $E_i$ are linear spaces corresponding to the skeleton decomposition of operator ${\mathbf B}.$ It is assumed that irreversible operator ${\mathbf B}$ generates skeleton chain of the finite length $p.$ The problem is reduced to a regular split system with respect to higher order derivative terms with certain initial and boundary conditions.

One of the most common problems of scientific applications is computation of the derivative of a function specified by possibly noisy or imprecise experimental data. Application of conventional techniques for numerically calculating derivatives will amplify the noise making the result useless. We address this typical ill-posed problem by application of perturbation method to linear first kind equations $Ax=f$ with bounded operator $A.$ We assume that we know the operator $\tilde{A}$ and source function $\tilde{f}$ only such as $||\tilde{A} - A||\leq \delta_1,$ $||\tilde{f}-f||< \delta_2.$ The regularizing equation $\tilde{A}x + B(\alpha)x = \tilde{f}$ possesses the unique solution. Here $\alpha \in S,$ $S$ is assumed to be an open space in $\mathbb{R}^n,$ $0 \in \overline{S},$ $\alpha= \alpha(\delta).$ As result of proposed theory, we suggest a novel algorithm providing accurate results even in the presence of a large amount of noise.

The review of existence theorems of bifurcation points of solutions for nonlinear operator equation in Banach spaces is presented. The sufficient conditions of bifurcation of solutions of boundary-value problem for Vlasov-Maxwell system are considered. The analytical method of Lyapunov-Schmidt-Trenogin is employed.

The sufficient conditions for existence and uniqueness of continuous solutions of the Volterra operator equations of the first kind with piecewise continuous kernel are derived. The asymptotic approximation of the parametric family of solutions are constructed in case of non-unique solution. The algorithm for the solution's improvement is proposed using the successive approximations method., Comment: 17 pages