Seppo Seikkala

On the fuzzy initial value problem

Abstract The initial value problem x ′( t ) = f ( t , x ( t )), x (0)= x 0 , with fuzzy initial value and with deterministic or fuzzy function f is considered. Two different approaches, viz. the extension principle and the use of extremal solutions of deterministic initial value problems, are applied. Generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated.

On fuzzy metric spaces

Abstract In this paper we introduce the concept of a fuzzy metric space. The distance between two points in a fuzzy metric space is a non-negative, upper semicontinuous, normal and convex fuzzy number. Properties of fuzzy metric spaces are studied and some fixed point theorems are proved.

Towards the theory of fuzzy differential equations

Comparative analysis of two ideologically distinct approaches in the theory of fuzzy differential equations is given. Domains of applications, advantages and shortcomings of those approaches are described. New definitions of solutions to fuzzy differential equations are proposed.

On the estimation of successive approximations in abstract spaces

The comparison principle in the estimation of solutions to equations in abstract spaces usually yields inequalities possessing maximal solutions of comparison equations as upper estimates (I, 71. One purpose of this paper is to show that some of these maximal solution estimates can be replaced by the corresponding minimal ones, if the solutions of the original equations are limits of successive approximations. First, comparison results are derived for cluster values (subsequential limits) of iterative sequences. The accuracy of these results is illustrated by an example which, together with the applications, describes their applicability to the theory of differential and integral equations.

Solution of a nonlinear two-point boundary value problem with Neumann-type boundary data

Abstract In this article we propose a new method for solution of nonlinear two-point boundary value problems of Neumann type. Our approach is based on a generalized integral equation representation of the solution together with an additional control variable. The method is constructive using Picard iterations. Some numerical examples are given.

On a classical Nicoletti boundary value problem

We shall derive existence, uniqueness and comparison results for the functional differential equationx′(t)=f(t, x), a. e.t∈I, with classical Nicoletti boundary conditionsxi(ti)=yi∈X, i∈A, whereI is a real interval,A is a nonempty set andX is a Banach space.

Existence, Uniqueness and Comparison Results for Nonlinear Boundary Value Problems Involving a Deviating Argument

In this paper we present existence, uniqueness and solution estimates for the differential equation with deviating argument Wt) =f(b m x(4(G)) (1) defined on the interior i of the interval I = [to, tl]. We assume that all functions herein are real valued, f: I x F?

A Resonance Problem for a Second-Order Vector Differential Equation

We shall consider the boundary value problem (BVP)

Integral Equations Arising in Boundary Value Problems at Resonance

\left\{ \begin{gathered} \left( {\begin{array}{*{20}c} {x_1 ^{\prime \prime } } \\ {x_2 ^{\prime \prime } } \\ \end{array} } \right) + A\left( {\begin{array}{*{20}c} {x_1 } \\ {x_2 } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {f_1 (ax_1 + bx_2 )} \\ {f_2 (cx_1 + dx_2 )} \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} {b_1 (t)} \\ {b_2 (t)} \\ \end{array} } \right) \hfill \\ x_1 (0) = x_2 (0) = x_1 (\pi ) = x_2 (\pi ) = 0. \hfill \\ \end{gathered} \right.,