An approximation—Iteration method for solving nonlinear operator equations

Convergence of the two-step combined method and uniqueness of the solution of nonlinear operator equations

Recurrence relations for a Newton-like method in Banach spaces

The Numerical Solution of Integral Equations on the Half-Line

We can compute with bounding sets of numbers, vectors, or functions using the techniques of interval mathematics. The techniques can be used to computationally verify sufficient conditions for existence, uniqueness, and convergence and to construct upper and lower bounds on solutions of nonlinear operator equations. Rounding errors are taken into account. We can compute with set-valued functions and operators on them. The techniques are also useful in search procedures for finding safe starting regions for iterative methods and for constructing natural stopping criteria.

Fixed points and approximate solutions for nonlinear operator equations

The purpose of Chap. 7 is to give a natural definition of generalized solution of non-solvable non-linear equation. In Sect. 7.1 we give the definition of generalized solution of equation with operator which acts in metric spaces. In Sect. 7.2 we give the definition of new concept of near-solution of non-linear operator equation. In Sect. 7.3 and 7.4 the existence, uniqueness, and correctness of generalized solution are proved. In Sect. 7.5 and 7.6 we investigate issues of embedding of metric spaces and extension of operators which are important for the theory of generalized solutions. Sections 7.7 and 7.8 contain examples of operators and describe numerical aspects of computation of generalized solutions. Section 7.9 is devoted to the development of the theory of generalized solutions of equations in uniform spaces and proximity spaces.KeywordsGeneralize SolutionOperator EquationBasic SpaceContinuous OperatorCauchy SequenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

Two theorems on an iteration method of order three -- usually referred to as Chebyshev's method -- for the approximate solution of nonlinear operator equations in Banach spaces are given. Very simple and weak assumptions are obtained as well as error estimates which are much better compared with known results. Some worked examples of operator equations of different types are given.

Iteratively regularized Gauss–Newton type methods for approximating quasi–solutions of irregular nonlinear operator equations in Hilbert space with an application to COVID–19 epidemic dynamics

In the present paper some Newton-like iteration methods are developed to enclose solutions of nonlinear operator equations of the kindF(x)=0. HereF maps a certain subset of a partially ordered vector space into another partially ordered vector space. The obtained results are proved without any special properties of the orderings by taking use of a new kind of a generalized divided difference operator, so that they even hold for nonconvex operators. Furthermore a method for constructing including starting points is presented and two examples are given.

Some theorems about the existence of solutions of nonlinear operator equation and the global homeomorphism in abstract space by the global existence theorem of an initial value problem are given.They are generalizations of some well-known results in these areas.

In this paper, we study ill-posedness concepts of nonlinear and linear operator equations in a Hilbert space setting. Such ill-posedness information may help to select appropriate optimization approaches for the stable approximate solution of inverse problems, which are formulated by the operator equations. We define local ill-posedness of a nonlinear operator equation F(x) = y 0 in a solution point x 0:and consider the interplay between the nonlinear problem and its linearization using the Fréchet derivative F′(x 0). To find a corresponding ill-posedness concept for the linearized equation we define intrinsic ill-posedness for linear operator equations A x = y and compare this approach with the ill-posedness definitions due to Hadamard and Nashed

A method of deletion of dimensions for a mathematical model which gives the number of variables less than that prescribed by the π-theorem was proposed. In a number of cases, it is possible to exclude from consideration all similarity criteria or, in other words, achieve self-similarity in them. In the framework of deletion of dimensions, this is expressed in a transition from criteria to similarity numbers. Thus, information is reduced without its loss. The limit reduction of the number of variables in the mathematical model makes it possible to use analytical approximation dependences as approximate solutions. Such dependences are obtained from the initial solutions by using the coefficients of stretch, which corresponds to the group methods for solving equations. It was proposed to use solutions of the linearized forms of the original nonlinear equations as initial solutions. Such approach makes it possible to take into account physical character of the change in the studied quantities in the approximation dependences when solution of nonlinear equations cannot be realized using standard functions. Efficiency of the method was illustrated by the example of study of the pendulum motion, which is a counterpart of inertial link of the second order in the theory of automatic control. Solutions were obtained for the cases of presence and absence of environmental resistance. The last variant is interesting by the feasibility of comparing the proposed and available analytical solutions in terms of elliptic integrals.

In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 𝒟 ⁢ ( F ) ⊂ 𝖴 → 𝖸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 𝖴 {\mathsf{U}} and 𝖸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 𝒟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions.

In a Banach space with a strictly convex norm we consider a nonlinear equation u' + A(t)u = 0 of general form. Suppose that a monotonicity condition is satisfied: for any two solutions u1(t) and u2(t) the function g(t) = ||u1(t) – u2(t)|| is nonincreasing with respect to t; suppose A(t) is almost periodic (in some sense) with respect to t. The basic theorem reads as follows: given strong (weak) continuity of the solutions with respect to the initial conditions and the coefficients, there exists at least one almost periodic solution if there exists a compact (weakly compact) solution on t≥0. Bibliography: 26 items.