A technique, based on the investigations of the images of maps, for obtaining fixed-point and coincidence results in a new class of maps and domains is described. In particular, we show that the problem concerning the existence of fixed points of expansive set-valued maps and inner set-valued maps on not necessarily convex or compact sets in Hausdorff topological vector spaces has a solution. As a consequence, we prove a new intersection theorem concerning not necessarily convex or compact sets and its applications. We also give new coincidence and section theorems for maps defined on not necessarily convex sets in Hausdorff topological vector spaces. Examples and counterexamples show a fundamental difference between our results and the well-known ones.

The notions of relaxed submonotone and relaxed monotone mappings in Banach spaces are introduced and many of their properties are investigated. For example, the Clarke subdifferential of a locally Lipschitz function in a separable Banach space is relaxed submonotone on a residual subset. For example, it is shown that this property need not be valid on the whole space. We prove, under certain hypotheses, the surjectivity of the relaxed monotone mappings.

The significance of fixed-point theory stems from the fact that it furnishes a unified approach and constitutes an important tool in solving equations which are not necessarily linear. On the other hand, if the fixed-point equation $Tx=x$ does not possess a solution, it is contemplated to resolve a problem of finding an element $x$ such that $x$ is in proximity to $Tx$ in some sense. Best proximity pair theorems analyze the conditions under which the optimization problem, namely ${\text{min}}_{x\in A}d\left(x,Tx\right)$ has a solution. In this paper, we discuss the difference between best approximation theorems and best proximity pair theorems. We also discuss an application of a best proximity pair theorem to the theory of games.

We consider the modulus of $u$-convexity of a Banach space introduced by Ji Gao (1996) and we improve a sufficient condition for the fixed-point property (FPP) given by this author. We also give a sufficient condition for normal structure in terms of the modulus of $u$-convexity.

For a scalar nonlinear delay differential equation$\dot{N}\left(t\right)=r\left(t\right)N\left(t\right)\left(K-N\left(h\left(t\right)\right)\right)/\left(K+s\left(t\right)N\left(g\left(t\right)\right)\right)$, $r\left(t\right)\ge 0$, $k>0$, $h\left(t\right)\le \mathrm{t}$, $g\left(t\right)\le \mathrm{t}$ and some generalizations of this equation, we establish explicit oscillation and nonoscillation conditions. Coefficient $r\left(t\right)$ and delays are not assumed to be continuous.