ABSTRACT
In this section we illustrate the applicability of topological methods, monotonicity methods, and variational methods to the solution of various types of nonlinear integra-differential equations of Barbashin type and nonlinear partial integral equations. We show first how to solve initial and boundary value problems for nonlinear Barbashin equations by means of the fixed point theorems considered in Subsection 7.1 and Subsection 9.4. An important step consists here in transforming the boundary value problem into an equivalent operator equation involving Uryson-type integral operators. Finally, we show how to use Minty's monotonicity principle to prove (unique) solvability of a Barbashin equation containing Hammerstein-type integral operators. For studying the boundary value problem (17.13)/(17.14) we apply a scheme which has recently been proposed in ApPELL-KALITVIN-ZABREJKO [1994, 1996], see also KALITVIN [1993a].
