The existence of anti-periodic solutions is studied for a second order difference inclusion associated with a maximal monotone operator in Hilbert spaces. It is the discrete analogue of a well-studied class of differential equations.

Let (X,T) be a paracompact space, Y a complete metric space, $F:X\to 2^Y$ a lower semicontinuous multifunction with nonempty closed values. We prove that if $T^{+}$ is a (stronger than T) topology on X satisfying a compatibility property, then F admits a $T^{+}$-continuous selection. If Y is separable, then there exists a sequence $\left(f_n\right)$ of $T^{+}$-continuous selections such that $F\left(x\right)=\overline{f_n\left(x\right);n\ge 1}$ for all x ∈ X. Given a Banach space E, the above result is then used to construct directionally continuous selections on arbitrary subsets of ℝ × E.

We study the Hamilton-Jacobi equation of the minimal time function in a domain which contains the target set. We generalize the results of Clarke and Nour [J. Convex Anal., 2004], where the target set is taken to be a single point. As an application, we give necessary and sufficient conditions for the existence of solutions to eikonal equations.

We study the Hamilton-Jacobi equation of the minimal time function in a domain which contains the target set. We generalize the results of Clarke and Nour [J. Convex Anal., 2004], where the target set is taken to be a single point. As an application, we give necessary and sufficient conditions for the existence of solutions to eikonal equations.

We investigate the regularity of semipermeable surfaces along barrier solutions without the assumption of smoothness of the right-hand side of the differential inclusion. We check what can be said if the assumptions concern not the right-hand side itself but the cones it generates. We examine also the properties of families of sets with semipermeable boundaries.

We consider the problem of the existence of solutions of the random set-valued equation: (I) $D_H{X}_t=F(t,X_t)P.1$, t ∈ [0,T] -a.e.; X₀ = U p.1 where F and U are given random set-valued mappings with values in the space $K_c\left(E\right)$, of all nonempty, compact and convex subsets of the separable Banach space E. Under certain restrictions on F we obtain existence of solutions of the problem (I). The connections between solutions of (I) and solutions of random differential inclusions are investigated.

We present the concepts of set-valued stochastic integrals in a plane and prove the existence of a solution to stochastic integral inclusions of the form $z_{s,t}\in {\phi }_{s,t}+{\int }_0^s{\int }_0^t{F}_{u,v}\left(z_{u,v}\right)dudv+{\int }_0^s{\int }_0^t{G}_{u,v}\left(z_{u,v}\right)d{w}_{u,v}$

We study a system of two differential inclusions such that there is a singular perturbation in the second one. We state new convergence results of solutions under assumptions concerning contingent derivative of the perturbed inclusion. These results state that there exists at least one family of solutions which converges to some solution of the reduced system. We extend this result to perturbed systems with state constraints.

We give some results about the topological structure of solution sets of multivalued Sturm-Liouville problems in Banach spaces.

In this paper, some algebraic fixed point theorems for multi-valued discontinuous operators on ordered spaces are proved. These theorems improve the earlier fixed point theorems of Dhage (1988, 1991) Dhage and Regan (2002) and Heikkilä and Hu (1993) under weaker conditions. The main fixed point theorems are applied to the first order discontinuous differential inclusions for proving the existence of the solutions under certain monotonicity condition of multi-functions.