Consider an operator equation (*) B(u) − f = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator B (u) is not boundedly invertible, and well-posed otherwise. The DSM (dynamical systems method) for solving equation (*) consists of a construction of a Cauchy problem, which has the following properties: 1) it has a global solution for an arbitrary initial data, 2) this s...

We prove the existence of strong solutions to a class of inhomogeneous boundary-value problems for an elliptic-hyperbolic equation. The equation is an arbitrarily small lower-order perturbation of an equation that arises in the linearization of an approximate model for high-frequency waves near a caustic. The domain boundary is allowed to extend into both the elliptic and hyperbolic regions of ...

in this paper, we improve some recent coupled fixed point resultsfor single-valued operators in the framework of ordered $b$-metricspaces established by bota et al. [m-f. bota, a. petrusel, g.petrusel and b. samet, coupled fixed point theorems forsingle-valued operators in b-metric spaces, fixed point theoryappl. (2015) 2015:231]. also, we prove that perov-type fixed pointtheorem in ordered gen...

The dynamical systems method (DSM), for solving operator equations, especially nonlinear and ill-posed, is developed in this paper. Consider an operator equation F (u) = 0 in a Hilbert space H and assume that this equation is solvable. Let us call the problem of solving this equation illposed if the operator F ′(u) is not boundedly invertible, and well-posed otherwise. The DSM for solving linea...

Consider an operator equation F(u) = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator F ′(u) is not boundedly invertible, and well-posed otherwise. If F is monotone C2 loc(H) operator, then we construct a Cauchy problem, which has the following properties: (1) it has a global solution for an arbitrary initial data, (2) this solution tends to a limit as time tends t...

In this paper ill-posed linear inverse problems that arises in many applications is considered. The instability of special kind of these problems and it's relation to the kernel, is described. For finding a stable solution to these problems we need some kind of regularization that is presented. The results have been applied for a singular equation.

In this paper we address some ill-posed problems involving the heat or the wave equation in one dimension, in particular the backward heat equation and the heat/wave equation with lateral Cauchy data. The main objective is to introduce some variational mixed formulations of quasi-reversibility which enable us to solve these ill-posed problems by using some classical Lagrange finite elements. Th...