Solutions Of Nonlinear Operator Equations Research Articles

Minimal Branches of Solutions of Nonlinear Operator Equations in Banach Spaces

Minimal Branches of Solutions of Nonlinear Operator Equations in Banach Spaces

On the solution of nonlinear operator equations and the invariant subspace

On the solution of nonlinear operator equations and the invariant subspace

On small solutions of nonlinear operator equations with noninvertible operator in the principal term

On small solutions of nonlinear operator equations with noninvertible operator in the principal term

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.

The majorant function modelling to solve nonlinear algebraic system

The majorize modelling of the modified Newton method (MNM) is an effective tool for concluding the existence and uniqueness of the solution of nonlinear operator equations. In this paper, we consider MNM together with a new majorant function (MF) that needed weak conditions to solve a nonlinear algebraic system (NLAS). The main advantage of this method is that you only require to establish the inverse of the Jacobian matrix (JM) at the initial guess (IG) and there is no need to compute it at the other iterations. The proposed strategy in this paper is logical and easy to execute. The theorems of MF and convergence are established for the method. Numerical results show that the strategy is efficient and promising.

Oscillation analysis of numerical solution of Nonlinear Operator Equation

There is a problem of low accuracy in the analysis of the vibration of the numerical solution of the nonlinear operator equation. In this work, the vibration analysis equation is constructed by the step-by-step search method, and the vibration quadrant of the equation is divided by the dichotomy method. The vibration spectrum is determined by the iteration method, and the vibration analysis model of the numerical solution of the nonlinear operator equation is constructed. The vibration analysis of the numerical solution of the nonlinear operator equation is completed based on the solution of the model and the numerical calculation and display of the step-by-step Fourier. The experimental results show that the proposed method has higher accuracy than the traditional vibration analysis method, which meets the requirements of the vibration analysis of the numerical solution of nonlinear operator equation.

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.

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.

On an iterative method for solution of direct problem for nonlinear hyperbolic differential equations

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.

On the local convergence of Weerakoon–Fernando method with $$\omega $$ continuity condition in Banach spaces

In this manuscript, the study of local convergence analysis for the cubically convergent Weerakoon–Fernando method is presented to obtain solutions of nonlinear operator equations in Banach spaces. The novelty of our paper is that our analysis only requires the $$\omega $$ continuity of the first Frechet derivative and extends the applicability of the algorithm when both Lipschitz and Holder conditions fail without engaging derivatives of the higher order which do not appear on this method. The local convergence offers radii of balls of convergence, the error bounds and uniqueness of the solution. Furthermore, the convergence region for the scheme to approximate the zeros of various polynomials is studied using basins of attraction tool. Several numerical tests are performed to show the usefulness of our theoretical results.

Some iterative algorithms for approximating solutions of nonlinear operator equations in Banach spaces

In this paper, iterative algorithms of Krasnoselkii-type and of Halpern-type for approximating an element of the set of common zeros of a countable family of inverse strongly monotone maps, common fixed points of a countable family of totally quasi- $$\phi $$ -asymptotically nonexpansive nonself multi-valued maps, and a solution of a system of generalized mixed equilibrium problems are constructed and studied. Strong convergence of the sequences generated by these algorithms is established in uniformly smooth and 2-uniformly convex real Banach spaces. The theorems obtained extend, improve and generalize several recent important results.

Existence of solutions for nonlinear operator equations

We investigate solutions for nonlinear operator equations and obtain some abstract existence results by linking methods. Some well-known theorems about periodic solutions for second-order Hamiltonian systems by M. Schechter are special cases of these results.

A Third Order Newton-Like Method and Its Applications

In this paper, we design a new third order Newton-like method and establish its convergence theory for finding the approximate solutions of nonlinear operator equations in the setting of Banach spaces. First, we discuss the convergence analysis of our third order Newton-like method under the ω -continuity condition. Then we apply our approach to solve nonlinear fixed point problems and Fredholm integral equations, where the first derivative of an involved operator does not necessarily satisfy the Hölder and Lipschitz continuity conditions. Several numerical examples are given, which compare the applicability of our convergence theory with the ones in the literature.

On a Three-Step Method with the Order of Convergence 1 + 2 $$ \sqrt{2} $$ for the Solution of Systems of Nonlinear Operator Equations

We propose a three-step modification of a method with an order of convergence 1+ $$ \sqrt{2} $$ aimed at the solution of nonlinear operator equations. We prove that the method is convergent and estimate its error. We also perform the numerical investigation of this modification on test examples, compare the results with the base method, and make conclusions on the basis of these results. The results of verification of the method confirm the theoretical predictions.

An iterative method for solution of nonlinear operator equation

In the note, for finding a solution of nonlinear operator equation of Hammerstein’s type an iterative process in infinite-dimentional Hilbert space is shown, where a new iteration is constructed basing on two last steps. An example in the theory of nonlinear integral equations is given for illustration.

Ball convergence theorems for eighth-order variants of Newton’s method under weak conditions

We present a local convergence analysis for eighth-order variants of Newton’s method in order to approximate a solution of a nonlinear equation. We use hypotheses up to the first derivative in contrast to earlier studies such as Amat et al. (Appl Math Comput 206(1):164–174, 2008), Amat et al. (Aequationes Math 69:212–213, 2005), Chun et al. (Appl Math Comput. 227:567–592, 2014), Petkovic et al. (Multipoint methods for solving nonlinear equations. Elsevier, Amsterdam, 2013), Potra and Ptak (Nondiscrete induction and iterative processes. Pitman Publ, Boston, 1984), Rall (Computational solution of nonlinear operator equations. Robert E. Krieger, New York, 1979), Ren et al. (Numer Algorithms 52(4):585–603, 2009), Rheinboldt (An adaptive continuation process for solving systems of nonlinear equations. Banach Center, Warsaw, 1975), Traub (Iterative methods for the solution of equations. Prentice Hall, Englewood Cliffs, 1964), Weerakoon and Fernando (Appl Math Lett 13:87–93, 2000), Wang and Kou (J Differ Equ Appl 19(9):1483–1500, 2013) using hypotheses up to the seventh derivative. This way the applicability of these methods is extended under weaker hypotheses. Moreover, the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

Sign-Changing Solutions for Nonlinear Operator Equations

The existence of six solutions for nonlinear operator equations is obtained by using the topological degree and fixed point index theory. These six solutions are all nonzero. Two of them are positive, the other two are negative, and the fifth and sixth ones are both sign-changing solutions. Furthermore, the theoretical results are applied to elliptic partial differential equations.

Positive solutions of nonlinear operator equations with sign-changing kernel and its applications

In this paper, the existence of positive solutions of nonlinear operator equation u=λTu are studied, where λ>0 is a parameter, T is a compact operator with sign-changing kernel. By using the Leray–Schauder degree theory, the existence of positive solutions of nonlinear operator equations with sign-changing kernel are obtained. The obtained abstract theorem can be applied easily, to illustrate this point the paper also introduces some applications.

A twist condition and multiple solutions of unbounded self-adjoint operator equation with symmetries

By using the index theory for unbounded self-adjoint operator equations and the symmetric mountain pass theorem, we investigate the existence of multiple solutions for nonlinear operator equations with twist conditions. We prove an abstract theorem, and give some applications to first order Hamiltonian systems with Sturm–Liouville boundary conditions and delay differential equations.