In this chapter, we survey recent results on the numerical solutions of the Hammerstein equations. Hammerstein equations arise naturally in connection with the Laplace equation with a certain class of nonlinear boundary conditions. The Hammerstein equations with smooth as well as weakly singular kernels will be treated.

We define the concept of the generalized u0-concave operators, which generalize the definition of the u0-concave operators. By using the iterative method and the partial ordering method, we prove the existence and uniqueness of fixed points of this class of the operators. As an example of the application of our results, we show the existence and uniqueness of solutions to a class of the Hammers...

In this paper, we obtained a necessary and sufficient condition for the embedding H?q([0,1])? IBVqp([0,1]), where IBVq p denotes set of functions bounded q-integral p-variation. Additionally, conditions composition superposition operators were provided to map space H?q([0,1]) into itself, by which these bounded. Finally, applied results examine existence uniqueness solutions Hammerstein integra...

In this paper, an iterative method of successive approximations to the approximate solution nonlinear Hammerstein-Fredholm integral equations using optimal quadrature formula for classes functions Lipschitz types is provided. Also, convergence analysis and numerical stability proposed are proved. Finally, some examples verify theoretical results show accuracy method.

In this paper a numerical method for finding the solution of FredholmHammerstein integral equations is proposed. At first the properties of the Walsh-hybrid functions, which combination of block-pulse functions and Walsh functions are proposed. The properties of the hybrid functions with the operational matrix of integration together Newton-Cotes nodes are then utilized to reduce the solution o...