Nonlinear Operator Equations Research Articles

SOLUTION OF A NONLINEAR OPERATOR EQUATION OF THE FIRST KIND WITH A CONTINUOUS POSITIVE LINEAR OPERATOR

In the space H, a nonlinear operator equation of the first kind is studied, when the linear, nonlinear operator and the right-hand side of the equation are given approximately. Based on the method of Lavrent'ev M.M. an approximate solution of the equation in Hilbert space is constructed. The dependence of the regularization parameter on errors was selected. The rate of convergence of the approximate solution to the exact solution of the original equation is obtained.

A method of solving a nonlinear boundary value problem for the Fredholm integro-differential equation

We propose a method to solve a nonlinear boundary value problem for a Fredholm integro-differential equation on a finite interval. By introducing additional parameters chosen as the values of the solution at the left-end points of the partition subintervals, the problem under consideration is transformed into an equivalent boundary value problem for a system of nonlinear integro-differential equations with parameters on the subintervals. For fixed parameters, we obtain a special Cauchy problem for this system, which is represented as a nonlinear operator equation and solved by an iterative method. By substitution of the solution to the special Cauchy problem with parameters into the boundary condition and the continuity conditions of the solution to the original problem at the interior partition points, we construct a system of nonlinear algebraic equations in parameters. It is proved that the solvability of this system provides the existence of a solution to the original boundary value problem. The algorithm for solving the special Cauchy problem includes two auxiliary problems: the Cauchy problems for ordinary differential equations and the evaluation of definite integrals. The accuracy of the method that we propose to solve the boundary value problem depends on the accuracy of methods applied to the auxiliary problems and does not depend on the number of the partition subintervals. Since iterative methods are used to solve both the constructed system of algebraic equations and the special Cauchy problem, we offer an approach to find an initial guess for the solutions to these problems.

The theorem of implicit operator in the sectorial neighborhoods

A nonlinear operator equation with parameter in a linear normed space is considered. The operator in the main part of the equation is not continuously invertible so, the standard theorem of implicit operator cannot be applied. We obtain sufficient conditions for the existence of a continuous solution. Method of constructing this solution is given.

PRP-like algorithm for monotone operator equations

Many studies have been devoted to develop and improve the iterative methods for solving convex constraint nonlinear equations problem (CCP). Based on the projection technique, we introduce a derivative-free method for approximating the solution of CCP. The proposed method is suitable for solving large-scale nonlinear equations due to its lower storage requirements. The directions generated by the proposed method at every iteration are bounded. Under some mild conditions, we establish the global convergence result of the proposed method. Numerical experiments are provided to show the efficiency of the method in solving CCP. Moreover, we tested the capability of the method in solving the monotone nonlinear operator equation equivalent to the $$\ell _1$$ -norm regularized minimization problem.

FR-type algorithm for finding approximate solutions to nonlinear monotone operator equations

This paper focuses on the problem of convex constraint nonlinear equations involving monotone operators in Euclidean space. A Fletcher and Reeves type derivative-free conjugate gradient method is proposed. The proposed method is designed to ensure the descent property of the search direction at each iteration. Furthermore, the convergence of the proposed method is proved under the assumption that the underlying operator is monotone and Lipschitz continuous. The numerical results show that the method is efficient for the given test problems.

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.

Solving nonlinear monotone operator equations via modified SR1 update

In this paper, we propose two algorithms for solving nonlinear monotone operator equations. The two algorithms are based on the conjugate gradient method. The corresponding search directions were obtained via a modified memoryless symmetric rank-one (SR1) update. Independent of the line search, the two directions were shown to be sufficiently descent and bounded. Moreover, the convergence of the algorithms were established under suitable assumptions on the operator under consideration. In addition, numerical experiments were conducted on some benchmark test problems to depict the efficiency and competitiveness of the algorithms compared with existing algorithms. From the results of the experiments, we can conclude that the proposed algorithms are more efficient and robust.

Inertial Derivative-Free Projection Method for Nonlinear Monotone Operator Equations With Convex Constraints

In this paper, we propose an inertial derivative-free projection method for solving convex constrained nonlinear monotone operator equations (CNME). The method incorporates the inertial step with an existing method called derivative-free projection (DFPI) method for solving CNME. The reason is to improve the convergence speed of DFPI as it has been shown and reported in several works that indeed the inertial step can speed up convergence. The global convergence of the proposed method is proved under some mild assumptions. Finally, numerical results reported clearly show that the proposed method is more efficient than the DFPI.

Strength of thin-walled elastic building structures

The existence theorem is proved within the framework of the shear model by S.P. Timoshenko. The stress-strain state of elastic inhomogeneous isotropic shallow thin-walled shell constructions is studied. The stress-strain state of shell constructions is described by a system of the five equilibrium equations and by the five static boundary conditions with respect to generalized displacements. The aim of the work is to find generalized displacements from a system of equilibrium equations that satisfy given static boundary conditions. The research is based on integral representations for generalized displacements containing arbitrary holomorphic functions. Holomorphic functions are found so that the generalized displacements should satisfy five static boundary conditions. The integral representations constructed this way allow to obtain a nonlinear operator equation. The solvability of the nonlinear equation is established with the use of contraction mappings principle.

Higher Order Analysis on the Existence of Periodic Solutions in Continuous Differential Equations via Degree Theory

Recently, the higher order averaging method for studying periodic solutions of both Lipschitz differential equations and discontinuous piecewise smooth differential equations was developed in terms of Brouwer degree theory. Between the Lipschitz and the discontinuous piecewise smooth differential equations, there is a huge class of differential equations lacking in a higher order analysis on the existence of periodic solutions, namely the class of continuous non-Lipschitz differential equations. In this paper, based on the coincidence degree theory for nonlinear operator equations, we perform a higher order analysis of continuous (non-Lipschitz) perturbed differential equations and derive sufficient conditions for the existence of periodic solutions for such systems. We apply our results to study continuous (non-Lipschitz) higher order perturbations of a harmonic oscillator.

Stress state of hinged supported thin-wall elastic structures

The paper studies the stress-strain state of flat elastic isotropic thin-walled shell structures in the framework of the S. P. Timoshenko shear model with pivotally supported edges. The stress-strain state of shell structures is described by a system of five second-order nonlinear partial differential equations under given static boundary conditions with respect to generalized displacements. The system of equations under study is linear in terms of tangential displacements, rotation angles, and nonlinear in terms of normal displacement. To find a solution to the system that satisfies the given static boundary conditions, integral representations for generalized displacements containing arbitrary holomorphic functions are used. Finding holomorphic functions is one of the main and difficult points in the proposed study. The integral representations constructed in this way allow us to reduce the original problem to a single nonlinear operator equation with respect to the deflection, the solvability of which is established using the principle of compressed maps.

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.

Extended convergence analysis of Newton-Potra solver for equations

In the paper a local and a semi-local convergence of combined iterative process for solving nonlinear operator equations is investigated. This solver is built based on Newton solver and has R-convergence order 1.839.... The radius of the convergence ball and convergence order of the investigated solver are determined in an earlier paper. Modifications of previous conditions leads to extended convergence domain. These advantages are obtained under the same computational effort. Numerical experiments are carried out on the test examples with nondifferentiable operator.

A New Algorithm for the Common Solutions of a Generalized Variational Inequality System and a Nonlinear Operator Equation in Banach Spaces

We study a new algorithm for the common solutions of a generalized variational inequality system and the fixed points of an asymptotically non-expansive mapping in Banach spaces. Under some specific assumptions imposed on the control parameters, some strong convergence theorems for the sequence generated by our new viscosity iterative scheme to approximate their common solutions are proved. As an application of our main results, we solve the standard constrained convex optimization problem. The results here generalize and improve some other authors’ recently corresponding results.

Застосування колокаційно-ітеративного методу до нелінійних інтегро-функціональних рівнянь

The article investigates the application of the collocation-iterative method to one type of nonlinear integro-functional equation. The conditions that guarantee the existence of a single solution of such an equation are given. The basic algorithm of the method is given and the conditions under which this method will be convergent are indicated. In substantiating this method, the fact is used that the original nonlinear integro-functional equation can be reduced to a nonlinear integral equation, the kernels of integral operators of which are written explicitly. Another important point in the justification is that the collocation-iterative method can be interpreted as a partial case of the projection-iterative method. The application of the latter to different types of nonlinear operator equations has been systematically investigated in the works of A. Y. Luchkа and his students. A significant difference between the method studied in this work is that at each step of the iteration it is necessary to solve systems of nonlinear algebraic or transcendental equations, which is the main technical complexity of this process. But the problem of finding solutions of such systems is simpler than solving the original nonlinear integro-functional equation. The article shows that the method of successive approximations and the method of collocation of the solution of the initial nonlinear equation can be considered as partial cases of the collocation-iterative method. In addition to the basic algorithm of the method, its computational algorithm is given, which is more convenient for direct calculations than the method itself and which, if necessary, can be successfully implemented on a computer by creating an appropriate program.

High-resolution computer-generated hologram for creating 2D images with kinematic effects of motion

A new method is proposed for computing the microrelief of phase diffractive optical elements (DOEs) forming 2D images with kinematic effects of motion. The DOE is subdivided into elementary areas with a size of approximately 50 microns. The synthesis of optical security elements includes the computation of the beam pattern in each elementary area. The beam pattern in each elementary area uniquely defines the phase function of the optical element. The computation of the phase function involves solving a nonlinear operator equation. Efficient methods are developed for computing the microrelief of DOEs forming 2D images with kinematic effects of motion. Sample optical elements are made using electron-beam technology. The microrelief is shaped with a precision of ∼10 nm, and the resolution is equal to ∼0.1 micron. The synthesis method developed is universal in nature. Given different sets of frames, one can achieve different behaviours of the motion of image fragments. The new optical elements can be used to protect banknotes, passports, IDs, and brands against counterfeiting.

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

Abstract 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.

An accelerated Kaczmarz type method for nonlinear inverse problems in Banach spaces with uniformly convex penalty

In this paper, we propose and analyze a novel Kaczmarz type method for solving inverse problems which can be written as systems of nonlinear operator equations in Banach spaces. The proposed method is formulated by combining homotopy perturbation iteration and Kaczmarz approach with uniformly convex penalty terms. The penalty term is allowed to be non-smooth, including the L1 and the total variation like penalty functionals, to reconstruct special features of solutions such as sparsity and piecewise constancy. To accelerate the iteration, we introduce a sophisticated rule to determine the step sizes per iteration. Under certain conditions, we present the convergence result of the proposed method in the exact data case. When the data is given approximately, together with a suitable stopping rule, we establish the stability and regularization properties of the method. Finally, some numerical experiments on parameter identification in partial differential equations by boundary as well as interior measurements are provided to validate the effectiveness of the proposed method.

Controllability for abstract semilinear control systems with homogeneous properties

This paper considers the approximate controllability for a class of abstract semilinear control systems using a method called surjectivity theorems for nonlinear operator equations under restrictive assumptions. The sufficient conditions for approximate controllability are derived under the natural assumptions on the nonlinear terms, which is an odd homogeneous and the nonlinear inverse considered as a multivalued mapping is bounded. An example is given to illustrate our main results.