Existence, uniqueness and approximate solutions of problems with a parameter. (English) Zbl 0893.34062

The systems of functional equations with a parameter (vector) and the systems of functional-differential equations of neutral type including equations with deviated arguments and also integro-differential and integral equations of Volterra and Fredholm types are the subject of research. The purpose of this paper is to present a general theory of existence and uniqueness of solutions, convergence of successive approximations and also convergence of numerical algorithms constructed for finding approximate solutions of the discussed problems.
Existence, uniqueness and continuous dependence theorems are formulated by using nonlinear comparison operators. The convergence of successive approximations, including methods of Seidel type, is proved. An extensive study concerns the case when these operators have linear form and the obtained results are better than the corresponding ones. When our problem satisfies the Volterra condition then a shooting method may also be applied to find the solution. Some examples illustrate the results.
The problem of approximate solutions of neutral functional-differential equations with a parameter is discussed, too. A combination of one-step (or multi-step) methods with the corresponding iterative formulas gives useful numerical methods for this problem, and such methods, including the shooting algorithm with Newton’s method are discussed.


34K05 General theory of functional-differential equations
34K40 Neutral functional-differential equations
45B05 Fredholm integral equations
45D05 Volterra integral equations
45J05 Integro-ordinary differential equations
65Q05 Numerical methods for functional equations (MSC2000)