Shouchuan Hu

Differential equations with discontinuous right-hand sides☆

Many phenomena from physics, control theory, and economics have been successfully described by differential equations with discontinuous functions. This in turn leads to a great need to develope a theory for such differential equations. However, as far as existence and qualitative behavior are concerned, such a theory is far from being complete. Even existence results are available only for very special discontinuous function f (as the discontinuous part in equation), e.g., see [S] for piecewise continuous f; [9] for f having one discontinuous point, [S, 121 for f of bounded variation. In this paper, we are going to establish some existence results for differential equations with discontinuous right-hand sides with great generality. Some of the results in this paper can be stated and proved for high order ordinary and partial differential equations. For example, we may consider the following problem

Handbook of multivalued analysis

Preface. I. Evolution Inclusions Involving Monotone Coercive Operators. II. Evolution Inclusions of the Subdifferential Type. III. Special Topics in Differential and Evolution Inclusions. IV. Optimal Control. V. Calculus of Variations. VI. Mathematical Economics. VII. Stochastic Games. VIII. Special Topics in Mathematical Economics and Optimization. Appendix. References. Symbol. Index. Errata of Volume A.

On the stability of double homoclinic and heteroclinic cycles

In this paper we give a criterion for the stability of planar double homoclinic and heteroclinic cycles with one or two saddles in some degenerate case.

Periodic solutions for nonconvex differential inclusions

In this paper we prove the existence of periodic solutions for differential inclusions with nonconvex-valued orientor field. Our proof is based on degree theoretic arguments.

Nontrivial solutions for superquadratic nonautonomous periodic systems

We consider a nonautonomous second order periodic system with an indefinite linear part. We assume that the potential function is superquadratic, but it may not satisfy the Ambrosetti-Rabinowitz condition. Using an existence result for

Nonlinear elliptic problems of Neumann-type

C^1

Elliptic equations with indefinite and unbounded potential and a nonlinear concave boundary condition

-functionals having a local linking at the origin, we show that the system has at least one nontrivial solution.

Evolution Inclusions of the Subdifferential Type

In this paper we study a nonlinear elliptic differential equation driven by thep-Laplacian with a multivalued boundary condition of the Neumann type. Using techniques from the theory of maximal monotone operators and a theorem of the range of the sum of monotone operators, we prove the existence of a (strong) solution.

Evolution Inclusions Involving Monotone Coercive Operators

We consider an elliptic problem driven by the negative Laplacian plus an indefinite and unbounded potential and a superlinear reaction. The boundary condition is parametric, nonlinear and superlinear near zero. Thus, the problem is a new version of the classical “convex–concave” problem (problem with competing nonlinearities). First, we prove a bifurcation-type result describing the set of positive solutions as the parameter λ > 0 varies. We also show the existence of a smallest positive solution ūλ and investigate the properties of the map λ →ūλ. Finally, by imposing bilateral conditions on the reaction we generate two more solutions, one of which is nodal.

Special Topics in Mathematical Economics and Optimization

In this section we study a second class of evolution inclusions, which are driven by time-dependent subdifferential operators. The t-dependence of the convex function φ(t, ·) is an important feature of our inclusions and general enough to incorporate cases where domφ(t, ·) ⋂ domφ(s, ·) = O for t ≠ s. So the abstract framework of this chapter is suitable for the analysis of problems with time-varying obstacles. The structure of this chapter basically parallels that of chapter I.