Dirichlet boundary value problem related to the p(x)-Laplacian with discontinuous nonlinearity

In this paper, we prove the existence of a weak solution for the Dirichlet boundary value problem related to a certain p(x)-Laplacian, by using the degree theory after turning the problem into a Hammerstein equation. The right hand side is a possibly discontinuous function in the second variable satisfying some non-standard growth conditions.

Approximation of solutions of Hammerstein equations with monotone mappings in real Banach spaces

Let E be a uniformly convex and uniformly smooth real Banach space with dual space, E∗. Let F : E → E∗, K : E∗ → E be maximal monotone mappings. An iterative algorithm is constructed and the sequence of the algorithm is proved to converge strongly to a solution of the Hammerstein equation u+KF u = 0. This theorem is a significant improvement of some important recent results which were proved in Lp spaces, 1 < p ≤ 2 under the assumption that F and K are bounded. This restriction on K and F have been dispensed with even in the more general setting considered here. Finally, a numerical experiment is presented to illustrate the convergence of the sequence of the algorithm which is found to be much faster, in terms of the number of iterations and the computational time than the convergence obtained with existing algorithms.

Algorithm for Hammerstein equations with monotone mappings in certain Banach spaces

For q > 1 and p > 1, let E be a 2-uniformly convex and q-uniformly smooth or p- uniformly convex and 2-uniformly smooth real Banach space and F : E → E∗, K : E∗ → E be bounded and strongly monotone maps with D(K) = R(F) = E∗. We construct a coupled iterative process and prove its strong convergence to a solution of the Hammerstein equation u + KF u = 0. Futhermore, our technique of proof is of independent of interest.

Algorithm for Hammerstein equations with monotone mappings in certain Banach spaces

For q > 1 and p > 1, let E be a 2-uniformly convex and q-uniformly smooth or p- uniformly convex and 2-uniformly smooth real Banach space and F : E → E∗, K : E∗ → E be bounded and strongly monotone maps with D(K) = R(F) = E∗. We construct a coupled iterative process and prove its strong convergence to a solution of the Hammerstein equation u + KF u = 0. Futhermore, our technique of proof is of independent of interest.

A note on perturbed Hammerstein equations with applications to nonlocal boundary value problems

We consider the perturbed Hammerstein equation

A New Method for Proving Existence Theorems for Abstract Hammerstein Equations

An abstract Hammerstein equation is an equation of the formu+KFu=0. Anew methodis introduced to prove the existence of a solution of this equation whereKandFare nonlinear accretive (monotone) operators. The method does not involve the complicated technique of factorizing a linear map via a Hilbert space and does not involve the use of deep variational techniques.

ASYMPTOTIC EXPANSIONS FOR APPROXIMATE SOLUTIONS OF HAMMERSTEIN INTEGRAL EQUATIONS WITH GREEN'S FUNCTION TYPE KERNELS

We consider approximation of a nonlinear Hammerstein equation with a kernel of the type of Green's function using the Nyström method based on the composite midpoint and the composite modified Simpson rules associated with a uniform partition. We obtain asymptotic expansions for the approximate solution unat the node points as well as at the partition points and use Richardson extrapolation to obtain approximate solutions with higher orders of convergence. Numerical results are presented which confirm the theoretical orders of convergence.