Theory Of Nonlinear Integral Equations Research Articles

Existence of infinitely many weak solutions to Kirchhoff–Schrödinger–Poisson systems and related models

In this article we investigate the existence of infinitely many weak solutions for Kirchhoff–Schrödinger–Poisson systems via the critical point theory. We obtain infinitely many large energy solutions and negative energy solutions of the system with superlinear nonlinearities by using the fountain theorem and the dual fountain theorem, respectively. Moreover, we establish the existence of infinitely many weak solutions of the problem with sublinear nonlinearities by applying the genus theory introduced by Krasnolsel’skii [Topological Methods in the Theory of Nonlinear Integral Equations (The Macmillan Company, New York, 1964)]. In particular, we do not use the classical Ambrosetti–Rabinowitz condition.

On the development of nonlinear integral equations at the early stage and the contribution of domestic mathematics

The paper considers the preconditions and the origin of the theory of nonlinear integral equations. The appearance of this theory was a natural consequence of the development of all mathematics of the XVIII-XIX cc. At the same time, the growing interest in nonlinear problems in the late XIX and early XX centuries had a strong motivating effect. The direct investigation of specific nonlinear integral equations was triggered by an urgent applied problem on the equilibrium figures of rotating liquid masses, which has attracted a significant number of major mathematicians since Newton. In the first decades of the development of the theory of nonlinear integral equations, traditional approaches were cultivated, which were used to study differential and algebraic equations, according to the equation-solution scheme. That is, the foreground was the calculation and assessment of its accuracy. The complexity and originality of nonlinear problems immediately revealed the relevance of questions of the existence and uniqueness of their solutions, which made it necessary to involve other, just emerging areas of mathematics. The theory of integral equations in general was one of the origins of functional analysis. Moreover, both theories were closely intertwined and mutually stimulated each other in their evolution. This fully applies to nonlinear integral equations, for which qualitative methods have become of paramount importance. At the stage considered in this work, there was a parallel development and mixing of traditional methods for studying equations and new approaches of a qualitative nature. In the next phase, new approaches came to the fore, merging with functional analysis and topology.

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.

Indentation of a round punch into a rough elastic half-space

We consider the axisymmetric contact problem on the indentation of a round punch with plane base into an elastic half-space. Roughness is taken into account in the framework of the well-known model in which the local surface displacements are proportional to some power of the contact pressures at the same point. To study the problem, we use the theory of nonlinear integral equations of Hammerstein type together with the mathematical technique of orthogonal Legendre polynomials and the contraction mapping principle. We obtain an approximate analytic solution of the problem, numerically analyze the results, and reveal typical laws of variation of the main mechanical variables.

Existence and multiplicity of solutions for fourth-order boundary value problems with parameters

In this paper we study the existence and multiplicity of the solutions for the fourth-order boundary value problem (BVP) u ( 4 ) ( t ) + η u ″ ( t ) − ζ u ( t ) = λ f ( t , u ( t ) ) , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , where f : [ 0 , 1 ] × R → R is continuous, ζ , η ∈ R and λ ∈ R + are parameters. By means of the idea of the decomposition of operators shown by Chen [W.Y. Chen, A decomposition problem for operators, Xuebao of Dongbei Renmin University 1 (1957) 95–98], see also [M. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Gostehizdat, Moscow, 1956], and the critical point theory, we obtain that if the pair ( η , ζ ) is on the curve ζ = − η 2 / 4 satisfying η < 2 π 2 , then the above BVP has at least one, two, three, and infinitely many solutions for λ being in different interval, respectively.

DISCRETE REGULARIZATION FOR OPERATOR EQUATION OF HAMMERSTEIN'S TYPE

The purpose of this note is to study a convergence for a method in form of combination of discrete approximations with regularization for solving operator equations of Hammerstein's type in Banach spaces. For illustration, an example in the theory of nonlinear integral equations is given.

Weak compactness of Fréchet-derivatives: application to composition operators

AbstractWe prove a result on compactness properties of Fréchet-derivatives which implies that the Fréchet-derivative of a weakly compact map between Banach spaces is weakly compact. This result is applied to characterize certain weakly compact composition operators on Sobolev spaces which have application in the theory of nonlinear integral equations and in the calculus of variations.

Topological Methods in the Theory of Nonlinear Integral Equations.

Topological Methods in the Theory of Nonlinear Integral Equations.

TOPOLOGICAL METHODS IN THE THEORY OF NON-LINEAR INTEGRAL EQUATIONS

TOPOLOGICAL METHODS IN THE THEORY OF NON-LINEAR INTEGRAL EQUATIONS

Critical points and gradient fields of scalars in Hilbert space

The paper is concerned with certain aspects of a theory of critical points of a scalar (i.e., a real valued funct ion) i(5) defined in a Hilbert space H, especially with the relation of the cr i t ica l points to the gradient field of i(~). Moreover, applications to the theory of non-linear integral equations are made. Le t V be a bounded open convex set of H, and S its boundary. We suppose t ha t i(~) is defined in an open region V' conta ining V + S in its interior. I f i(5) is wri t ten in the form

On the theory of non-linear integral equations

On the theory of non-linear integral equations