Let $E$ be a topological vector space and $C$ a pointed closed convex cone. For a set $Q$ in $E$ we prove arcwise connectedness of the efficient point set $Max(Q|C)$ between any two points of a closed set $M \subset Max(Q|C)$ with a compact closed convex hull and having certain additional property. An application to a class of non-convex in general sets is given. The method generalizes the one from [6] concerning compact convex sets and allows also for such sets to obtain a more general result.

In this work the Hyers--Ulam type stability of the functional equation $f(x+y+xy)=f(x)+f(y+xy)$ is proved.

In [10] we reviewed Gödel's incompleteness theorem and gave a new proof along with an application which leads to a contradiction when applying the Gödel's discussion to the set theory ZFC itself. We stated a possible solution to avoid contradiction by removing the self-reference by appealing to the axiomatic formulation of a theory with referring to its validity in no explicit ways. We will in this paper give a more specific possible solution that one can avoid the Gödel type self-contradiction by preventing oneself from telling anything definite about the validity of oneself's assertion.

An algebraic approach for factorizing nonlinear partial differential equations (PDEs) and systems of PDEs is provided. In the particular case of second order linear and nonlinear PDEs and systems of PDEs, necessary and sufficient conditions of factorization are given.

We consider the invertibility of parabolic pseudodifferential operators in exponential weighted Sobolev spaces. We suppose that the symbol $a$ of the operator $Op(a)$ is analytically extended with respect to the impulse variable in an unbounded tube domain $\mathbb{R}^{n}+iD$ and satisfies conditions of uniform parabolicity . We prove that under these conditions the pseudodifferential operator $Op(a)$ is invertible in admissible weighted Sobolev spaces with weights connected with the domain $D.$ As an application we obtain exponential estimates of solutions (including estimates of the fundamental solution) for parabolic differential operators.

We consider a differential inclusion $$\dot{x} \in A(t)x + f(t,x) + g(t,x,X_1)$$ in an arbitrary Banach space $X$ with a general exponential dichotomy, where $X_1$ is the closed unit ball of $X.$ The right-hand side is strongly measurable in the time variable and Lipschitz continuous in the others. We prove the existence and uniqueness of quasibounded solutions corresponding to suitable selectors. The stable and unstable sets of these quasibounded solutions are characterised as graphs of certain multifunctions. Exponential dichotomy criteria are also presented.

We introduce new definitions of semi-continuity for multifunctions, combining the topological and the ordered structure of a Banach space induced by a closed convex cone. We prove two types Nash equilibrium theorems for multifunctions using scalarization and the Ky Fan’s inequality. As corollaries we obtain saddle point theorems for convex-concave multifunctions, which can be considered as generalization to the vector-valued set-valued case of the Von Neumann minimax theorem.