Seppo Heikkilä

Nonlinear differential equations in ordered spaces

OPERATOR EQUATIONS IN ORDERED SPACES AND FIRST APPLICATIONS Operator Equations in Ordered Spaces Applications to Differential Equations EXTREMALITY RESULTS FOR FIRST ORDER DIFFERENTIAL EQUATIONS Explicit Initial Value Problems Explicit Boundary Value Problems Explicit Functional Problems Implicit Functional Problems UNIQUENESS, COMPARISON AND WELL-POSEDNESS RESULTS FOR QUASILINEAR DIFFERENTIAL EQUATIONS First Order Boundary Value Problems First Order Initial Value Problems Well-Posedness Results Second Order Problems SECOND ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS Explicit Sturm-Liouville Boundary Value Problems Implicit Sturm-Liouville Boundary Value Problems Convergence of Successive Approximations Explicit Phi-Laplacian Problems Implicit Phi-Laplacian Problems EXTREMALITY RESULTS FOR QUASILINEAR PDE Quasilinear Elliptic Boundary Value Problems Quasilinear Parabolic Problems Discontinuous Quasilinear Problems DIFFERENTIAL INCLUSIONS OF HEMIVARIATIONAL INEQUALITY TYPE Quasilinear Elliptic Inclusions with State-Dependent Subdifferentials State-Dependent Subdifferentials Perturbed by Discontinuous Nonlinearities Elliptic Inclusions with Generalized Gradients Quasilinear Parabolic Inclusions with State-Dependent Subdifferentials DISCONTINUOUS IMPLICIT ELLIPTIC AND PARABOLIC PROBLEMS Statement of the Problem and Notations Preliminaries Main Results Implicit Elliptic Problems APPENDIX Analysis in Ordered Spaces Inequalities Sobolev Spaces Pseudomonotone and Quasilinear Elliptic Operators Nonlinear First Order Evolution Equations Nonsmooth Analysis

On fixed points of multifunctions in ordered spaces

Let P be an ordered topological space and a multivalued mapping for which and X 1≤x 2 imply y 1≤y 2 for some y 2∊Fx 2 Several fixed point theorems are derived for F under the above condition and some extra conditions imposed on P and/or F. The use of a generalized iteration method allows us to drop all the continuity properties of F, and even the topology of P from these conditions. Some of the results obtained are new also for single-valued mappings. Applications are given to mapping families and operator equations.

Elliptic problems with lack of compactness via a new fixed point theorem

Abstract This paper provides a new fixed point theorem for increasing self-mappings G : B → B of a closed ball B ⊂ X , where X is a Banach semilattice which is reflexive or has a weakly fully regular order cone X + . By means of this fixed point theorem, we are able to establish existence results of elliptic problems with lack of compactness.

Fixed point results in ordered normed spaces with applications to abstract and differential equations

In [S] fixed point results are derived for an increasing operator A: [u, U] -+ [u, 01, where [u, u] is a conical segment in an ordered Banach space with normal order cone P. As the main results, an existence a fixed point of A is proved by assuming that A( [u, u]) is weakly relatively compact and separable. If P is also minihedral, then A is shown to have least and greatest fixed points. No continuity or linearity hypotheses are imposed on A. In this paper we shall show, for instance, that if [u, u] is a conical segment in an ordered normed space E, and if A: [u, u] + [u, u] is increasing and weakly relatively order compact, then A has the least and the greatest fixed point, provided that the order cone of E is normal or A( [u, u]) is separable. Moreover, these fixed points are constructed by countable iteration processes. The so obtained fixed point results are then applied to derive results on existence of extremal or unique solutions of the operator equation

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.

Extremality and comparison results for discontinuous implicit third order functional initial-boundary value problems

Extremality results are derived for implicit third order functional initial-boundary value problems. Differential equations, initial conditions and boundary conditions of problems may involve discontinuous nonlinearities.

Existence of solutions of third-order functional problems with nonlinear boundary conditions

In this paper some existence results for third-order differential equations with nonlinear boundary value conditions are derived. Functional dependence in the data is allowed. In the proofs we use the method of upper and lower solutions, Schauders fixed point theorem and results from Cabada and Heikkila on third-order differential equations with linear and nonfunctional initial-boundary value conditions.

Existence and comparison results for operator and differential equations in abstract spaces

In this paper we apply the chain methods developed in (Heikkila and Lakshmikantham, 1994) to obtain new fixed point theorems and new existence and comparison results for operator equations in partially ordered sets. These results are then applied to discontinuous implicit functional differential equations in ordered Banach spaces.

On operator and integral equations with discontinuous right-hand side

On considere des equations operateurs dans C(A,E) ou A est un espace topologique et E est un espace topologique partiellement ordonne ou un groupe

On summability, integrability and impulsive differential equations in Banach spaces

MSC: 26A06, 26A18, 26A39, 26B12, 26E20, 34A36, 34A37, 34G20, 40A05, 40D05, 40F05, 47H07, 47H10.PurposeTo study summability of families indexed by well-ordered sets of R∪{∞} in normed spaces. To derive integrability criteria for step mappings and for right regulated mappings from an interval of R∪{∞} to a Banach space. To study solvability of impulsive differential equations.Main methodsA generalized iteration method presented, e.g., (Heikkilä and Lakshmikantham in Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, 1994). Summability of families in normed spaces indexed with well-ordered subsets of R∪{∞}.ResultsNecessary and sufficient conditions for global and local HK, HL, Bochner and Riemann integrability of step mappings and right regulated mappings defined on an interval of R∪{∞}, and having values in a Banach space. Applications to impulsive differential equations are also presented. Families indexed with well-ordered subsets of R∪{∞} are used to represent impulsive parts of considered equations and to approximate their solutions.