System Of Nonlinear Operator Equations Research Articles

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

Abstract 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.

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.

Existence and uniqueness of solutions for systems of fractional differential equations with Riemann\u2013Stieltjes integral boundary condition

In this article, we first establish an existence and uniqueness result for a class of systems of nonlinear operator equations under more general conditions by means of the cone theory and monotone iterative technique. Furthermore, the iterative sequence of the solution and the error estimation of the system are given. Then we use this new result to study the existence and uniqueness of the solution for boundary value problems of systems of fractional differential equations with a Riemann–Stieltjes integral boundary condition in real Banach spaces. The results obtained in this paper are more general than many previous results and complement them.

A Fixed Point Theorem for Systems of Nonlinear Operator Equations and Applications to $$(p_1, p_2)$$ ( p 1 , p 2 ) -Laplacian System

A new fixed point theorem for systems of nonlinear operator equations is established by means of topological degree theory and positively 1-homogeneous operator, where the components has a positively 1-homogeneous majorant or minorant. As applications, the existence of positive solutions for $$(p_1, p_2)$$ -Laplacian system is considered under some conditions concerning the positive eigenvalues corresponding to the relevant positively 1-homogeneous operators.

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.

Simulation of a Moving Boundary in Plasma Electron Sources

The numerical algorithms and the software package to model the electro-optical systems with a moving plasma emitter are developed. The proposed algorithms are based on the method of the computational region decomposition. The main point of the method is a computational area divided into two subregions: near-cathode subregion and main subregion. The solution in the near-cathode subregion is found analytically whereas its determination in the main subregion is via numerical methods. The problem to find the position and shape of a plasma boundary is defined as the problem of solving the nonlinear Poincare–Steklov equations. These equations are obtained by the conjugation conditions for an electric field potential and its derivatives on the boundary between the near-cathode and main subregions. These equations are approximated by the system of nonlinear operator equations that is solved by the efficient version of the descent method. This paper provides examples of the methodological and practical problem solutions.

Strong Probabilistic Contractor Couple and Nonlinear Operator Equations in Menger PN Space

The purpose of the paper is to introduce the concept of strong probabilistic contractor couple in Menger PN spaces and to study the existence and uniqueness of solutions for a system of nonlinear operator equations with probabilistic contractor couples in Menger PN spaces non-Archimedean Menger PN spaces.

A System of Nonlinear Operator Equations for a Mixed Family of Fuzzy and Crisp Operators in Probabilistic Normed Spaces

By using a random version of the theory of contractor introduced by Altman, we introduce and study a system of nonlinear operator equations for a mixed family of fuzzy and crisp operators in probabilistic normed spaces. We construct some new iterative algorithms for solving this kind of nonlinear operator equations. We also prove some new existence theorems of solutions of a new system of nonlinear operator equations for a mixed family of fuzzy and crisp operators and some new convergence results of sequences generated by iterative algorithms under joint orbitally complete conditions.

DIFFRACTION OF ELECTROMAGNETIC WAVES BY A LAYER FILLED WITH A KERR-TYPE NONLINEAR MEDIUM

The diffraction of a plane wave by a transversely inhomogeneous isotropic nonmagnetic linearly polarized dielectric layer filled with a Kerr-type nonlinear medium is considered. The analytical and numerical solution techniques are developed. The diffraction problem is reduced to a singular boundary value problem for a semilinear second-order ordinary differential equation with a cubic nonlinearity and then to a cubic-nonlinear integral equation (IE) of the second kind and to a system of nonlinear operator equations of the second kind solved using iterations. Sufficient conditions of the unique solvability are obtained using the contraction principle.

Non-resonance and existence for systems of non-linear operator equations

An existence theory for systems of two non-linear operator equations in Hilbert spaces is presented under non-resonance conditions with respect to two spectra and in terms of matrices convergent to zero. The theory is then applied to elliptic systems.

Diffraction by a Kerr-type Nonlinear Dielectric Layer

The difiraction of a plane wave by a transversely inhomogeneous isotropic nonmag- netic linearly polarized dielectric layer fllled with a Kerr-type nonlinear medium is considered. The difiraction problem is reduced to a cubic-nonlinear integral equation (IE) of the second kind and to a system of nonlinear operator equations of the second kind solved using iterations. Su-cient conditions of the IE unique solvability are obtained using the contraction principle.

A regularized Newton method for one-dimensional profile inversion of a lossy cylinder

A regularized Newton iterative method is presented for the determination of one-dimensional profile of an inhomogeneous circularly cylindrical region. The region is assumed to be filled with a lossy dielectric material of which electromagnetic parameters have an arbitrary variation in the radial direction. By using the finite Fourier transformation of field expressions, the problem is first reduced to a system of non-linear operator equations which can be solved through regularized Newton method. The method requires to solve the scattered field and the Fréchet derivatives in the inhomogeneous region in each iteration step. Numerical simulations are carried out to test the applicability and the effectiveness of the method.

ITERATIVE SOLUTIONS FOR SYSTEMS OF NONLINEAR OPERATOR EQUATIONS IN BANACH SPACE

By using partial order method, the existence, uniqueness and iterative approximation of solutions for a class of systems of nonlinear operator equations in Banach space are discussed. The results obtained in this paper extend and improve recent results.

A note on probabilistic contractor couple and solutions for a system of nonlinear equations in n a menger pn-spaces

The concept of (ϕ , δ)-type probabilistic contractor couple was introduced which simplifies and weakens the definition of probabilistic contractor couple given by Zhang Shisheng. The existence and uniqueness of the solutions for a system of nonlinear operator equations with this kind of probabilistic contractor couple in N. A. Menger PN-spaces were studied. The works improve and extend the corresponding results by M. Altman, A. C. Lee, W. J. Padgett et at.

Probabilistic contractor couple and solutions for a system of nonlinear equations in non-archimedean menger probabilistic normed spaces

The Purpose of this paper is to introduce the concept of probabilistic contractor couple in non-Archimedean probabilistic normed spaces and to study the existence and uniqueness of solutions for a system of nonlinear operator equations with probabilistic contractor couples in non-Archimedean probabilistic normed spaces. The results presented in this paper improve and extend the corresponding results in [1–5].

SOLUTIONS FOR A SYSTEM OF RANDOM OPERATOR EQUATIONS AND SOME APPLICATIONS

The present paper is a continuation of [1]. By using more general type of random contractor than that defined by Lee and Padgett, we show several existence and uniqueness theorems for solutions to a system of nonlinear random operator equations with stochastic domain. Next, we obtain a random coincidence theorem and a random implicit function theorem, which improves and generalizes the corresponding results of Ding, Reddy and Subrahmanyam, Altman, Eng, Krasnoselskii and others. Finally, some applications of our results are given for the solutions of a system of nonlinear random integral equations.