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Celebrating 100 years of the Banach contraction principle, we prove some fixed point theorems having all ingredients of the principle, but dealing with common fixed points of a contractive semigroup of nonlinear mappings acting in a modulated topological vector space. This research follows the ideas of the author’s recent papers [‘On modulated topological vector spaces and applications’, Bull. Aust. Math. Soc. 101 (2020), 325–332, and ‘Normal structure in modulated topological vector spaces’, Comment. Math. 60 (2020), 1–11]. Modulated topological vector spaces generalise, among others, Banach spaces and modular function spaces. The interest in modulars reflects the fact that the notions of ‘norm like’ but ‘noneuclidean’ (and not even necessarily convex) constructs to measure a level of proximity between complex objects are frequently used in science and technology. To prove our fixed point results in this setting, we introduce a new concept of Opial sets using analogies with the norm-weak and modular versions of the Opial property. As an example, the results of this work can be applied to spaces like 
$L^p$ for 
$p> 0 $, variable Lebesgue spaces 
$L^{p(\cdot )}$ where 
$1 \leq p(t) < + \infty $, Orlicz and Musielak–Orlicz spaces.
We prove the existence of exponentially and superexponentially localized breather solutions for discrete nonlinear Klein–Gordon systems. Our approach considers $d$
-dimensional infinite lattice models with general on-site potentials and interaction potentials being bounded by an arbitrary power law, as well as, systems with purely anharmonic forces, cases which are much less studied particularly in a higher-dimensional set-up. The existence problem is formulated in terms of a fixed-point equation considered in weighted sequence spaces, which is solved by means of Schauder's Fixed-Point Theorem. The proofs provide energy bounds for the solutions depending on the lattice parameters and its dimension under physically relevant non-resonance conditions.
We show how some Ulam stability issues can be approached for functions taking values in 2-Banach spaces. We use the example of the well-known Cauchy equation 
$f(x+y)=f(x)+f(y)$, but we believe that this method can be applied for many other equations. In particular we provide an extension of an earlier stability result that has been motivated by a problem of Th. M. Rassias. The main tool is a recent fixed point theorem in some spaces of functions with values in 2-Banach spaces.
Given two (real) normed (linear) spaces 
$X$ and 
$Y$, let 
$X\otimes _{1}Y=(X\otimes Y,\Vert \cdot \Vert )$, where 
$\Vert (x,y)\Vert =\Vert x\Vert +\Vert y\Vert$. It is known that 
$X\otimes _{1}Y$ is 
$2$-UR if and only if both 
$X$ and 
$Y$ are UR (where we use UR as an abbreviation for uniformly rotund). We prove that if 
$X$ is 
$m$-dimensional and 
$Y$ is 
$k$-UR, then 
$X\otimes _{1}Y$ is 
$(m+k)$-UR. In the other direction, we observe that if 
$X\otimes _{1}Y$ is 
$k$-UR, then both 
$X$ and 
$Y$ are 
$(k-1)$-UR. Given a monotone norm 
$\Vert \cdot \Vert _{E}$ on 
$\mathbb{R}^{2}$, we let 
$X\otimes _{E}Y=(X\otimes Y,\Vert \cdot \Vert )$ where 
$\Vert (x,y)\Vert =\Vert (\Vert x\Vert _{X},\Vert y\Vert _{Y})\Vert _{E}$. It is known that if 
$X$ is uniformly rotund in every direction, 
$Y$ has the weak fixed point property for nonexpansive maps (WFPP) and 
$\Vert \cdot \Vert _{E}$ is strictly monotone, then 
$X\otimes _{E}Y$ has WFPP. Using the notion of 
$k$-uniform rotundity relative to every 
$k$-dimensional subspace we show that this result holds with a weaker condition on 
$X$.
Let 
$X$ and 
$Y$ be two normed spaces over fields 
$\mathbb{F}$ and 
$\mathbb{K}$, respectively. We prove new generalised hyperstability results for the general linear equation of the form 
$g(ax+by)=Ag(x)+Bg(y)$, where 
$g:X\rightarrow Y$ is a mapping and 
$a,b\in \mathbb{F}$, 
$A,B\in \mathbb{K}\backslash \{0\}$, using a modification of the method of Brzdęk [‘Stability of additivity and fixed point methods’, Fixed Point Theory Appl.2013 (2013), Art. ID 285, 9 pages]. The hyperstability results of Piszczek [‘Hyperstability of the general linear functional equation’, Bull. Korean Math. Soc.52 (2015), 1827–1838] can be derived from our main result.
We extend the results of Schu [‘Iterative construction of fixed points of asymptotically nonexpansive mappings’, J. Math. Anal. Appl.158 (1991), 407–413] to monotone asymptotically nonexpansive mappings by means of the Fibonacci–Mann iteration process where 
$$\begin{eqnarray}x_{n+1}=t_{n}T^{f(n)}(x_{n})+(1-t_{n})x_{n},\quad n\in \mathbb{N},\end{eqnarray}$$
$T$ is a monotone asymptotically nonexpansive self-mapping defined on a closed bounded and nonempty convex subset of a uniformly convex Banach space and 
$\{f(n)\}$ is the Fibonacci integer sequence. We obtain a weak convergence result in 
$L_{p}([0,1])$, with 
$1<p<+\infty$, using a property similar to the weak Opial condition satisfied by monotone sequences.
Let 
$X$ be a nonempty subset of a normed space such that 
$0\notin X$ and 
$X$ is symmetric with respect to 
$0$ and let 
$Y$ be a Banach space. We study the generalised hyperstability of the Drygas functional equation where 
$$\begin{eqnarray}f(x+y)+f(x-y)=2f(x)+f(y)+f(-y),\end{eqnarray}$$
$f$ maps 
$X$ into 
$Y$ and 
$x,y\in X$ with 
$x+y,x-y\in X$. Our first main result improves the results of Piszczek and Szczawińska [‘Hyperstability of the Drygas functional equation’, J. Funct. Space Appl.2013 (2013), Article ID 912718, 4 pages]. Hyperstability results for the inhomogeneous Drygas functional equation can be derived from our results.
