Hammerstein Integral Equations with Nontrivial Solutions

By means of critical point theory, existence theorems for nontrivial solutions to the Hammerstein equation x = KFx are given, where K is a compact linear integral operator and F is a nonlinear superposition operator. To this end, appropriate conditions on the spectrum of the linear parte are combined with growth and representation conditions on the nonlinear part to ensure the applicability of the mountain — pass lemma. The abstract existence theorems are applied to nonlinear elliptic equations and systems subject to Dirichlet boundary conditions.     

Stability in the almost everywhere sense: A linear transfer operator approach

The problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework. The infinitesimal generator of a linear transfer operator (Perron-Frobenius) is used to provide stability conditions of an autonomous ordinary differential equation. It is shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of a non-negative solution for a steady state advection type linear partial differential equation. We refer to this non-negative solution, verifying almost everywhere global stability, as Lyapunov density. A numerical method using finite element techniques is used for the computation of Lyapunov density.     

On the convergence region for decomposition solutions

The author's decomposition method using his A_n polynomials for the nonlinearities has been shown to apply to wide classes of nonlinear (or nonlinear stochastic) operator equations providing a computable, accurate solution which converges rapidly. In computation the above is sufficient for a rapid test of convergence region.     

On a class of essentially nonlinear elliptic differential-difference equations

An essentially nonlinear differential-difference equation containing the product of the p-Laplacian and a difference operator is considered. Sufficient conditions are obtained for the corresponding nonlinear differential-difference operator to be coercive and pseudomonotone in the case of nonvariational statement of the differential equation. The existence of a generalized solution to the Dirichlet problem for the nonlinear equation is proved.     

Approximation methods and alternative problems

Projection methods applied to abstract problems of the form Ax = Nx, where A is a linear operator with a nontrivial null space and N is a nonlinear operator, both on a normed space, are studied. Convergence results are obtained and then are applied to periodic two-point boundary value problems using splines as approximations.     

Asymptotic error estimates of a linearized projection-difference method for a differential equation with a monotone operator

We study a projection-difference method of solving the Cauchy problem for an operatordifferential equation with a selfadjoint leading operator A(t) and a nonlinear monotone subordinate operator K(·) in a Hilbert space. This method leads to a solution of a system of linear algebraic equations at each time level. Error estimates are derived for approximate solutions as well as for fractional powers of the operator A(t). The method is applied to a model parabolic problem.     

Nonlinear boundary value problems and operators TT1

We consider nonlinear boundary value problems of the type $\text{L? + N? = 0}$ for the existence of solutions. It is assumed that L is a 2nth-order linear differential operator in the real Hilbert space S = L²[a, b] which admits a decomposition of the form $\text{L =}{}^{\mathrm{TT1}}$ where T is an nth-order linear differential operator and N is a nonlinear operator defined on a subspace of S. The decomposition of L induces a natural decomposition of the generalized inverse of L. Using the method of “alternative problems,” we split the boundary value problem into an equivalent system of two equations. The theory of monotone operators and the theory of nonlinear Hammerstein equations are then utilized to consider the solvability of the equivalent system.     

New approximation-solvability of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators

In this paper, we consider and study a class of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators in Hilbert spaces. We also construct a new perturbed iterative algorithm framework with errors and investigate variational graph convergence analysis for this algorithm framework in the context of solving the nonlinear operator inclusion couple along with some results on the resolvent operator corresponding to (A, η, m)-maximal monotonicity. The obtained results improve and generalize some well known results in recent literatures.     

Nonuniformly nonlinear elliptic equations of p-biharmonic type

This paper deals with the existence and multiplicity of weak solutions to nonlinear differential equations involving a general p-biharmonic operator (in particular, p-biharmonic operator) under Dirichlet boundary conditions or Navier boundary conditions. Our method is mainly based on variational arguments.     

Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part I. Finite-dimensional discretization

We consider discretized Hamiltonian PDEs associated with a Hamiltonian function that can be split into a linear unbounded operator and a regular nonlinear part. We consider splitting methods associated with this decomposition. Using a finite-dimensional Birkhoff normal form result, we show the almost preservation of the actions of the numerical solution associated with the splitting method over arbitrary long time and for asymptotically large level of space approximation, provided the Sobolev norm of the initial data is small enough. This result holds under generic non-resonance conditions on the frequencies of the linear operator and on the step size. We apply these results to nonlinear Schrödinger equations as well as the nonlinear wave equation.     

On a class of nonlinear dynamical systems: The structure of a differential operator in the application of the decomposition method

A suitable expresion of a differential operator is deduced, allowing the explicit calculation of the general term f_h in the power series expansion of a function f(x, t), to be utilized for solving nonlinear differential equations.     

On small solutions of nonlinear equations with vector parameter in sectorial neighborhoods

We consider the nonlinear operator equation B(λ)x + R(x, λ) = 0 with parameter λ, which is an element of a linear normed space Λ. The linear operator B(λ) has no bounded inverse for λ = 0. The range of the operator B(0) can be nonclosed. The nonlinear operator R(x, λ) is continuous in a neighborhood of zero and R(0, 0) = 0. We obtain sufficient conditions for the existence of a continuous solution x(λ) → 0 as λ → 0 with maximal order of smallness in an open set S of the space Λ. The zero of the space Λ belongs to the boundary of the set S. The solutions are constructed by the method of successive approximations.     

Numerical study of fisher's equation by Adomian's method

The effect of the linear operator L, used in the Adomian's method for solving nonlinear partial differential equations, on the convergence is studied on the Fisher's equation, which describes a balance between linear diffusion and nonlinear reaction. The results show that the convergence of this method is not influenced by the choice of the operator L in the equation to be solved. Furthermore, under some conditions, these results are close to those obtained by using other numerical techniques.     

<Emphasis Type="Italic">p</Emphasis>-Adic Analogue of the Porous Medium Equation

We consider a nonlinear evolution equation for complex-valued functions of a real positive time variable and a p-adic spatial variable. This equation is a non-Archimedean counterpart of the fractional porous medium equation. Developing, as a tool, an \(L^1\)-theory of Vladimirov’s p-adic fractional differentiation operator, we prove m-accretivity of the appropriate nonlinear operator, thus obtaining the existence and uniqueness of a mild solution. We give also an example of an explicit solution of the p-adic porous medium equation.     

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.     

Pointwise estimation of the difference of the solutions of a controlled functional operator equation in Lebesgue spaces

For a functional operator equation in Lebesgue space, we prove a statement on the pointwise estimate of the modulus of the increment of its global (on a fixed set Π ? ? ⁿ ) solution under the variation of the control function appearing in this equation. As an auxiliary statement, we prove a generalization of Gronwall’s lemma to the case of a nonlinear operator acting in Lebesgue space. The approach used here is based onmethods from the theory of stability of existence of global solutions to Volterra operator equations.     

The Kramers–Moyal expansion of the master equation that describes human migration in a bounded domain

It is known in quantitative sociodynamics that human migration in a bounded domain can be described by a nonlinear integro-partial differential equation, which is called the master equation. This equation has its origin in statistical physics. At a physical level of rigor we can formally expand the nonlinear integral operator contained in the master equation into an infinite series whose terms are nonlinear partial differential operators. The infinite series thus obtained is called the Kramers–Moyal expansion. The purpose of this paper is to give a mathematical justification of this formal expansion.     

Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators

We study the scattering theory of a conservative nonlinear one-parameter group of operators on a Hilbert space X relative to a group of linear unitary operators. Under certain hypotheses, the scattering operator carries a neighborhood of 0 in X into X. The theory is designed to apply to the semilinear Schrödinger and Klein-Gordon equations.     

A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations

This paper is concerned with an operator equation Ax+Bx+Cx=x on ordered Banach spaces, where A is an increasing α-concave operator, B is an increasing sub-homogeneous operator and C is a homogeneous operator. The existence and uniqueness of its positive solutions is obtained by using the properties of cones and a fixed point theorem for increasing general β-concave operators. As applications, we utilize the fixed point theorems obtained in this paper to study the existence and uniqueness of positive solutions for two classes nonlinear problems which include fourth-order two-point boundary value problems for elastic beam equations and elliptic value problems for Lane-Emden-Fowler equations.     

A third order method for fixed points in Banach spaces

This paper deals with a third order Stirling-like method used for finding fixed points of nonlinear operator equations in Banach spaces. The semilocal convergence of the method is established by using recurrence relations under the assumption that the first Fréchet derivative of the involved operator satisfies the Hölder continuity condition. A theorem is given to establish the error bounds and the existence and uniqueness regions for fixed points. The R-order of the method is also shown to be equal to at least (2p+1) for p∈(0,1]. The efficacy of our approach is shown by solving three nonlinear elementary scalar functions and two nonlinear integral equations by using both Stirling-like method and Newton-like method. It is observed that our convergence analysis is more effective and give better results.