Perturbed Hammerstein Integral Equations Research Articles

Nontrivial Solutions of Systems of Perturbed Hammerstein Integral Equations with Functional Terms

We discuss the solvability of a fairly general class of systems of perturbed Hammerstein integral equations with functional terms that depend on several parameters. The nonlinearities and the functionals are allowed to depend on the components of the system and their derivatives. The results are applicable to systems of nonlocal second order ordinary differential equations subject to functional boundary conditions, this is illustrated in an example. Our approach is based on the classical fixed point index.

Positive solutions of systems of perturbed Hammerstein integral equations with arbitrary order dependence.

Motivated by the study of systems of higher-order boundary value problems with functional boundary conditions, we discuss, by topological methods, the solvability of a fairly general class of systems of perturbed Hammerstein integral equations, where the nonlinearities and the functionals involved depend on some derivatives. We improve and complement earlier results in the literature. We also provide some examples in order to illustrate the applicability of the theoretical results. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.

Pointwise conditions for perturbed Hammerstein integral equations with monotone nonlinear, nonlocal elements

We consider perturbed Hammerstein integral equations of the form $$\begin{aligned} y(t)= & {} \gamma _1(t)F_1(\varphi _1(y),\varphi _2(y),\dots ,\varphi _{n_1} (y))+\gamma _2(t)F_2(\psi _1(y),\psi _2(y),\dots ,\psi _{n_2}(y))\\&+\,\lambda \int _0^1G(t,s)f(s,y(s)) ds, \end{aligned}$$ where the \(\varphi _i\)’s and \(\psi _i\)’s are linear functionals realized as Stieltjes integrals with associated signed Stieltjes measures. Positive solutions of the integral equation can be related to positive solutions of boundary value problems, and we demonstrate that such problems have solutions under relatively mild hypotheses on the functions \(F_1\), \(F_2\), and f. We provide examples to illustrate the applicability of the results and their relationship to existing results in the literature, and we mention applications to radially symmetric solutions of PDEs as well as beam deflection modeling. Our results are achieved by using a nonstandard order cone together with an associated nonstandard open set.

Positive and increasing solutions of perturbed Hammerstein integral equations with derivative dependence

We discuss the existence and non-existence of non-negative, non-decreasing solutions of certain perturbed Hammerstein integral equations with derivative dependence. We present some applications to nonlinear, second order boundary value problems subject to fairly general functional boundary conditions. The approach relies on classical fixed point index theory.

Solvability of Hammerstein Integral Equations with Applications to Boundary Value Problems

In this paper we present some new results regarding the solvability of nonlinear Hammerstein integral equations in a special cone of continuous functions. The proofs are based on a certain fixed point theorem of Leggett and Williams type. We give an application of the abstract result to prove the existence of nontrivial solutions of a periodic boundary value problem. We also investigate, via a version of Krasnosel'skiĭ's theorem for the sum of two operators, the solvability of perturbed Hammerstein integral equations in the space of continuous functions of bounded variation in the sense of Jordan. As an application of these results, we study the solvability of a boundary value problem subject to integral boundary conditions of Riemann–Stieltjes type. Some examples are presented in order to illustrate the obtained results.

Solutions of perturbed Hammerstein integral equations with applications

By means of topological methods, we provide new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for systems of perturbed Hammerstein integral equations. In order to illustrate our theoretical results, we study some problems that occur in applied mathematics, namely models of chemical reactors, beams and thermostats. We also apply our theory in order to prove the existence of nontrivial radial solutions of systems of elliptic boundary value problems subject to nonlocal, nonlinear boundary conditions.

Perturbed Hammerstein integral equations with sign-changing kernels and applications to nonlocal boundary value problems and elliptic PDEs

We demonstrate the existence of at least one positive solution to the perturbed Hammerstein integral equation \[ y(t)=\gamma _1(t)H_1\big (\varphi _1(y)\big )+\gamma _2(t)H_2\big (\varphi _2(y)\big )\] \[\qquad \qquad \qquad \quad +\lambda \int _0^1G(t,s)f\big (s,y(s)\big )\, ds,\] where certain asymptotic growth properties are imposed on the functions $f$, $H_1$ and $H_2$. Moreover, the functionals $\varphi _1$ and $\varphi _2$ are realizable as Stieltjes integrals with signed measures, which means that the nonlocal elements in the Hammerstein equation are possibly of a very general, sign-changing form. We focus here on the case where the kernel $(t,s)\mapsto G(t,s)$ is allowed to change sign and demonstrate the existence of at least one positive solution to the integral equation. As applications, we demonstrate that, by choosing $\gamma _1$ and $\gamma _2$ in particular ways, we obtain positive solutions to boundary value problems, both in the ODEs and elliptic PDEs setting, even when the Green's function is sign-changing, and, moreover, we are able to localize the range of admissible values of the parameter~$\lambda $. Finally, we also provide a result that for each $\lambda >0$ yields the existence of at least one positive solution.

Non-trivial solutions of local and non-local Neumann boundary-value problems

We prove new results on the existence, non-existence, localization and multiplicity of non-trivial solutions for perturbed Hammerstein integral equations. Our approach is topological and relies on the classical fixed-point index. Some of the criteria involve a comparison with the spectral radius of some related linear operators. We apply our results to some boundary-value problems with local and non-local boundary conditions of Neumann type. We illustrate in some examples the methodologies used.

Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications

We provide a theory to establish the existence of nonzero solutions of perturbed Hammerstein integral equations with deviated arguments, being our main ingredient the theory of fixed point index. Our approach is fairly general and covers a variety of cases. We apply our results to a periodic boundary value problem with reflections and to a thermostat problem. In the case of reflections we also discuss the optimality of some constants that occur in our theory. Some examples are presented to illustrate the theory.

Nontrivial solutions of boundary value problems for second-order functional differential equations

In this paper we present a theory for the existence of multiple nontrivial solutions for a class of perturbed Hammerstein integral equations. Our methodology, rather than to work directly in cones, is to utilize the theory of fixed point index on affine cones. This approach is fairly general and covers a class of nonlocal boundary value problems for functional differential equations. Some examples are given in order to illustrate our theoretical results.

Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations

We study the existence of positive solutions for a system of perturbed Hammerstein integral equations. Our setting is quite general and covers a wide class of systems of boundary value problems. The main ingredient in our theory is the classical fixed point index theory for compact maps.

Constructive approximation for a class of perturbed Hammerstein integral equations

Constructive approximation for a class of perturbed Hammerstein integral equations

Existence of positive continuous solutions of perturbed hammerstein equations

We improve previous existence results for a class of perturbed Hammerstein integral equations, where the relevant Hammerstein operator is decreasing on positive functions. We strongly weaken the assumptions on the nonlinearities involved, and obtain existence of positive continuous solutions, even on noncompact domains, applying the Schauder-Tychonoff fixed-point theorem, via the generalized Ascoli theorem and the regularizing effect of the integral operator. An application to perturbed quadratic integral equations of interest in transport theory is also given