Class Of Nonlinear Operator Equations Research Articles

Two-sided inequalities with nonmonotone sublinear operators

The theorems on solutions and their two-sided estimates for one class of nonlinear operator equations $x=Fx$ with nonmonotone operators.

Positive fixed points for a class of nonlinear operators and applications

In this paper, we study the existence and uniqueness of positive solutions for a class of nonlinear operator equations on ordered Banach spaces. Various applications are also considered to illustrate our obtained results (existence of solutions to quadratic integral equations with a linear modification of the argument, positive solution of second-order Neumann boundary value problem, and positive definite solutions of a class of nonlinear matrix equations).

Perturbed algorithms for solving nonlinear relaxed cocoercive operator equations with general A-monotone operators in Banach spaces

Based on the notion of general A-monotonicity, the new proximal mapping technique and Alber’s inequalities, a new class of nonlinear relaxed cocoercive operator equations with general A-monotone operators in Banach spaces is introduced and studied. Further, we also discuss the convergence and stability of a new perturbed iterative algorithm with errors for solving this class of nonlinear operator equations in Banach spaces. Since general A-monotonicity generalizes general H-monotonicity (and in turn, generalizes A-monotonicity, H-monotonicity and maximal monotonicity), our results improve and generalize the corresponding results of recent works.

THE EXISTENCE RESULTS OF COUPLED QUASI-SOLUTIONS FOR A CLASS OF OPERATOR EQUATIONS

In this paper, by using the semi-order method, two new exis- tence theorems of coupled quasi-solutions for a class of nonlinear operator equations in Banach spaces are proved under some suitable conditions.

Projection-algebraic approximation of linear and nonlinear operator differential equations in Banach spaces

We develop a Calogero-type projection-algebraic method of discrete approximations for linear differential equations in Banach spaces and analyze the convergence of finite-dimensional approximations based on the functional-analytic approach to discrete approximations and methods of operator theory in Banach spaces. Applications of the obtained results to the functional-interpolation scheme of the projection-algebraic method of discrete approximations are considered. Based on a generalized Leray-Schauder-type theorem, we consider the projection-algebraic scheme of discrete approximations and analyze its solvability and convergence for a special class of nonlinear operator equations.

Implicit function theorem via the DSM

Sufficient conditions are given for an implicit function theorem to hold. The result is established by an application of the Dynamical Systems Method (DSM). It allows one to solve a class of nonlinear operator equations in the case when the Fréchet derivative of the nonlinear operator is a smoothing operator, so that its inverse is an unbounded operator.

Fixed point theorems for a class of nonlinear operators in Banach spaces and applications

In this paper, we study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using the cone theory and Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions. The results in this paper are applied to a class of abstract semilinear evolution equations with noncompact semigroup in Banach spaces and the initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. The results obtained here improve and generalize many known results.

New Variational Principle and Duality for a Certain Class of Nonlinear Operator Equations

We provide a duality theory and existence results for a superlinear operator equation We use the abstract version of the so called dual variational method. The solution is obtained as a limit of a minimizng sequence whose existence and convergence is proved.

The exact solutions of a kind of nonlinear operator equations

This paper discusses the method how to solve a class of nonlinear operator equations in the reproducing kernel space W 2 1 [ a , b ] . Under certain conditions, we give the exact solution’s expression of this class of equations. Finally, we show through two numerical experiments our result is valid.

Solvability and Galerkin Approximations of a Class of Nonlinear Operator Equations

Solvability and Galerkin Approximations of a Class of Nonlinear Operator Equations

Mesh independence principle for nonlinear equations on hilbert spaces by preconditioning

In the paper [2] the author prove that, under certain assumptions, the Mesh Independence Principle is satisfied for a class of nonlinear operator equations (on Banach spaces) and their discretizations. In this paper we show that, under few natural assumptions, the same results can be obtained for a class of nonlinear operator equations on Hilbert spaces and a family of preconditioned finite element discretizations of them. We also obtain (using a different proof technique) a generalization of some similar results from the paper [1]. In the last section of the paper we present numerical examples which confirm our theoretical results from the previous ones.

Nonlocal problems for one class of nonlinear operator equations that arise in the theory of sobolev type equations

Let Open image in new window , be Hilbert spaces forming a sequence of compact imbeddingsH 3 ⊂H 2 ⊂H 1 ⊂H 0. Consider the nonlinear equation Open image in new window in H2 and suppose that conditions (8)–(12) and (15) hold for the operators A and K(u) and the external force F(t). Four nonlocal problems (cf. 1–4) are studied for such equations. Examples are given [cf. (2)–(6)] for nonlinear dissipative equations of Sobolev type that are reducible to an abstract non-linear equation [cf. (7)–(12), (15)].

Solvability of nonlinear perturbations of linear operator equations with parameters

The solvability of a class of nonlinear operator equations is discussed in the reflexive Banac space possessing a normalized Schauder basis.

One class of nonlinear operator equations with nonselfadjoint right sides

One class of nonlinear operator equations with nonselfadjoint right sides

On Adomian's decomposition method and some comparisons with Picard's iterative scheme

We consider the general mathematical framework of Adomian's decomposition method (G. Adomian, “Stochastic Systems,” Academic Press, New York, 1983) for a large class of nonlinear operator equations. Picard's iterative scheme is considered for the same class for equations and compared with the decomposition method. The paper identifies carefully all substantial differences between the two methods and shows that various advantages exist for the decomposition method.

Solutions of nonlinear boundary value problems by the decomposition method

Boundary value problems for a class of nonlinear operator equations are studied and a method for determining their solutions in approximated form is proposed, based on the Adomian decomposition procedure. The theory is applied with satisfactory results to the study of a classical boundary value problem in continuum mechanics.

Exact bounds for the radially symmetric shape of confined plasma in the unit circle

AbstractThe method of “cone iteration”; is used to obtain exact bounds for the solutions of a class of nonlinear operator equations. The general theory is applied to the problem of finding radially symmetric solutions of the plasma problem in the unit circle. Existence and uniqueness results are obtained at the same time. Some numerical examples demonstrate the method's high degree of accuracy.

Approximate methods for solving a class of non-linear operator equations

Approximate methods for solving a class of non-linear operator equations

Existence theorems for a class of nonlinear operator equations

Existence theorems for a class of nonlinear operator equations

Minimax Principles for the Solution of Semilinear Gradient Operator Equations in Hilbert Space

A new variational characterization of solutions for an important class of nonlinear operator equations is obtained. The result obtained is used to derive sharp necessary and sufficient conditions for the solvability of such operator equations. Examples of the applicability of the results obtained to nonlinear Dirichlet problems and global differential geometry are discussed.