Existence results for a system of nonlinear operator equations and block operator matrices in locally convex spaces

The purpose of this paper is to prove some fixed point results dealing with a system of nonlinear equations defined in an angelic Hausdorff locally convex space ( X , { | ⋅ | p } p ∈ Λ ) (X,\{\lvert\,{\cdot}\,\rvert_{p}\}_{p\in\Lambda}) having the 𝜏-Krein–Šmulian property, where 𝜏 is a weaker Hausdorff locally convex topology of 𝑋. The method applied in our study is connected with a family Φ Λ τ \Phi_{\Lambda}^{\tau} -MNC of measures of weak noncompactness and with the concept of 𝜏-sequential continuity. As a special case, we discuss the existence of solutions for a 2 × 2 2\times 2 block operator matrix with nonlinear inputs. Furthermore, we give an illustrative example for a system of nonlinear integral equations in the space C ⁢ ( R + ) × C ⁢ ( R + ) C(\mathbb{R}^{+})\times C(\mathbb{R}^{+}) to verify the effectiveness and applicability of our main result.

Approximate methods for solving amplitude-phase problem for discrete signals

Methods for solving amplitude and phase problems for one and two-dimensional discrete signals are proposed. Methods are based on using nonlinear singular integral equations. In the one-dimensional case amplitude and phase problems are modeled by corresponding linear and nonlinear singular integral equations. In the two-dimensional case amplitude and phase problems are modeled by corresponding linear and nonlinear bisingular integral equations. Several approaches are presented for modeling two-dimensional problems: 1) reduction of amplitude and phase problems to systems of linear and nonlinear singular integral equations; 2) using methods of the theory of functions of many complex variables, problems are reduced to linear and nonlinear bisingular integral equations. To solve the constructed nonlinear singular integral equations, methods of collocation and mechanical quadrature are used. These methods lead to systems of nonlinear algebraic equations, which are solved by the continuous method for solution of nonlinear operator equations. The choice of this method is due to the fact that it is stable against perturbations of coefficients in the right-hand side of the system of equations. In addition, the method is realizable even in cases where the Frechet and Gateaux derivatives degenerate at a finite number of steps in the iterative process. Some model examples have shown effectiveness of proposed methods and numerical algorithms.

Optimal Control for a Nonlocal Model of Non-Newtonian Fluid Flows

This paper deals with an optimal control problem for a nonlocal model of the steady-state flow of a differential type fluid of complexity 2 with variable viscosity. We assume that the fluid occupies a bounded three-dimensional (or two-dimensional) domain with the impermeable boundary. The control parameter is the external force. We discuss both strong and weak solutions. Using one result on the solvability of nonlinear operator equations with weak-to-weak and weak-to-strong continuous mappings in Sobolev spaces, we construct a weak solution that minimizes a given cost functional subject to natural conditions on the model data. Moreover, a necessary condition for the existence of strong solutions is derived. Simultaneously, we introduce the concept of the marginal function and study its properties. In particular, it is shown that the marginal function of this control system is lower semicontinuous with respect to the directed Hausdorff distance.

Existence of Three Solutions for Nonlinear Operator Equations and Applications to Second-Order Differential Equations

The existence of three solutions for nonlinear operator equations is established via index theory for linear self-adjoint operator equations, critical point reduction method, and three critical points theorems obtained by Brezis-Nirenberg, Ricceri, and Averna-Bonanno. Applying the results to second-order Hamiltonian systems satisfying generalized periodic boundary conditions or Sturm-Liouville boundary conditions and elliptic partial differential equations satisfying Dirichlet boundary value conditions, we obtain some new theorems concerning the existence of three solutions.

Approximate methods of solving amplitude-phase problem for continuous signals

Amplitude and phase problems in physical research are considered. The construction of methods and algorithms for solving phase and amplitude problems is analyzed without involving additional information about the signal and its spectrum. Mathematical models of the amplitude and phase problems in the case of one-dimensional and two-dimensional continuous signals are proposed and approximate methods for their solution are constructed. The models are based on the use of nonlinear singular and bisingular integral equations. The amplitude and phase problems are modeled by corresponding nonlinear singular and bisingular integral equations defined on the numerical axis (in the one-dimensional case) and on the plane (in the two-dimensional case). To solve the constructed nonlinear singular and bisingular integral equations, spline-collocation methods and the method of mechanical quadratures are used. Systems of nonlinear algebraic equations that arise during the application of these methods are solved by the continuous method of solving nonlinear operator equations. A model example shows the effectiveness of the proposed method for solving the phase problem in the two-dimensional case.

Two-step Ulm-type method for solving nonlinear operator equations

In this paper, we present a two-step Ulm-type method to solve systems of nonlinear equations without computing Jacobian matrices and solving Jacobian equations. we prove that the two-step Ulm-type method converges locally to the solution with R-convergence rate 3. Numerical implementations demonstrate the effectiveness of the new method.

Finite-dimensional iteratively regularized processes with an a posteriori stopping for solving irregular nonlinear operator equations

We construct and study a class of numerically implementable iteratively regularized Gauss–Newton type methods for approximate solution of irregular nonlinear operator equations in Hilbert space. The methods include a general finite-dimensional approximation for equations under consideration and cover the projection, collocation and quadrature discretization schemes. Using an a posteriori stopping rule for the iterative processes and the standard source condition on the solution, we establish accuracy estimates for the approximations generated by the methods. We also investigate projected versions of the processes which take into account a priori information about a convex compact containing the solution. An iteratively regularized quadrature process is applied to an inverse 2D problem of gravimetry.

Solutions of Nonlinear Operator Equations by Viscosity Iterative Methods

Finding the solutions of nonlinear operator equations has been a subject of research for decades but has recently attracted much attention. This paper studies the convergence of a newly introduced viscosity implicit iterative algorithm to a fixed point of a nonexpansive mapping in Banach spaces. Our technique is indispensable in terms of explicitly clarifying the associated concepts and analysis. The scheme is effective for obtaining the solutions of various nonlinear operator equations as it involves the generalized contraction. The results are applied to obtain a fixed point of λ-strictly pseudocontractive mappings, solution of α-inverse-strongly monotone mappings, and solution of integral equations of Fredholm type.

On Global Dynamics in Duffing Equation with Quasiperiodic Perturbation

An iterative method for solution of Cauchy problem for one-dimensional nonlinear hyperbolic differential equation is proposed in this paper. The method is based on continuous method for solution of nonlinear operator equations. The keystone idea of the method consists in transition from the original problem to a nonlinear integral equation and its successive solution via construction of an auxiliary system of nonlinear differential equations that can be solved with the help of different numerical methods. The result is presented as a mesh function that consists of approximate values of the solution of stated problem and is constructed on a uniform mesh in a bounded domain of two-dimensional space. The advantages of the method are its simplicity and also its universality in the sense that the method can be applied for solving problems with a wide range of nonlinearities. Finally it should be mentioned that one of the important advantages of the proposed method is its stability to perturbations of initial data that is substantiated by methods for analysis of stability of solutions of systems of ordinary differential equations. Solving several model problems shows effectiveness of the proposed method.