On the 21st of July, 2022, a remarkable mathematician and a former rector of the Maria Curie-Skłodowska University in Lublin (UMCS) – prof. Kazimierz Goebel left us

In 1988 N.A. Assad showed that a nonself nonlinear contraction taking a closed subset of a complete metrically convex space into the space so that the boundary of this subset is mapped back into the subset itself has a unique fixed point. In the present paper we extend this result by replacing the complete metrically convex space with a complete metric space which is a length space.

Assume that $(\Omega, \Sigma, \mu)$ is a $\sigma$-finite measure space and $p\colon\Omega\to [1,\infty]$ a variable exponent. In the case of a purely atomic measure, we prove that the w-FPP for mappings of asymptotically nonexpansive type in the Nakano space $\ell^{p(k)}$, where $p(k)$ is a sequence in $[1,\infty]$, is equivalent to several geometric properties of the space, as weak normal structure, the w-FPP for nonexpansive mappings and the impossibility of containing isometrically $L^1([0,1])$. In the case of an arbitrary $\sigma$-finite measure, we prove that this characterization also holds for pointwise eventually nonexpansive mappings. To determine if the w-FPP for nonexpansive mappings and for mappings of asymptotically nonexpansive type are equivalent is a long standing open question [19]. According to our results, this is the case, at least, for pointwise eventually nonexpansive mappings in Lebesgue spaces with variable exponents.

We provide a concrete isometric description of all the preduals of $\ell_1$ for which the standard basis in $\ell_1$ has a finite number of $w^*$-limit points. Then, we apply this result to give an example of an $\ell_1$-predual $X$ such that its dual $X^*$ lacks the weak$^*$ fixed point property for nonexpansive mappings (briefly, $w^*$-FPP), but $X$ does not contain an isometric copy of any hyperplane $W_{\alpha}$ of the space $c$ of convergent sequences such that $W_\alpha$ is a predual of $\ell_1$ and $W_\alpha^*$ lacks the $w^*$-FPP. This answers a question left open in the 2017 paper of the present authors.

The paper is devoted to present a result on the existence of solutions of an infinite system of quadratic integral equations of the Urysohn type considered on the real half-axis. Our investigations are conducted in the Banach space consisting of bounded and continuous functions defined on the real half-axis with values in the space of real sequences converging to zero. That space is equipped with the standard supremum norm. The main tools used in our study is the technique of measures of noncompactness and the Schauder fixed point principle. We illustrate our result by a suitable example.

We define some classes of generalized nonexpansive mappings under assumptions concerning the convex combinations of two consecutive points in their orbits. For these mappings, in the setting of Banach spaces that enjoy normal structure, we provide several fixed point results.

In this paper we prove a fixed point theorem for asymptotically nonexpansive mappings acting in modular spaces. This result generalises the 1972 fixed point theorem by K. Goebel and W.A. Kirk. In the process, we extend several other results (including the Milman-Pettis theorem) from the class of Banach spaces to the larger class of regular modular spaces.

In this article, a notion and classification of spherical caps in the sequence space $\ell_1$ are introduced, and the least Lipschitz constant of Lipschitz retractions from the unit ball onto a spherical cap is defined. In addition, an approximation of this value for the specific spherical cap, the simple spherical cap, is calculated. This approximation reveals a rough relation between these values, denoted by $\kappa(\alpha)$, and the answer of the optimal retraction problem for the space $\ell_1$, denoted by $k_0(\ell_1)$. To be precise, there exists $-1< \mu< 0$ such that $k_0(\ell_1)\leq\kappa(\alpha)\leq2+k_0(\ell_1)$ whenever $-1< \alpha< \mu$; here $\alpha$ is the level of spherical cap.

The purpose of this paper is to study the existence of mild solutions to a class of second order nonlinear evolution equations of the form\begin{equation*}\begin{cases} u''(t)+A(u'(t))+B(u(t))\ni f(t), &t\in(0,T),\\u(0)=u_0, \quad u'(0)=g(u')\end{cases}\end{equation*} where $A\colon D(A)\subseteq X\rightarrow 2^{X}$ is an $m$-accretive operator on a Banach space $X,$ $B\colon X\rightarrow X$ is a Lipschitz mapping, $g\colon C([0,T];X)\to X$ is a function and $f\in L^1(0,T,X)$. We obtain sufficient conditions for this problem to have at least a mild solution.

In this paper we will extend the concept of multi-valued Feng-Liu contraction, by imposing a nonlinear assumption on the operator. Then, fixed point, strict fixed point and stability theorems for the fixed point inclusion with multi-valued nonlinear Feng-Liu contractions are given. An application illustrates the main theoretical results.

Let $C$ be a bounded, closed and convex subset of a reflexive metric space with a digraph $G$ such that $G$-intervals along walks are closed and convex. In the main theorem we show that if $T\colon C\rightarrow C$ is a monotone $G$-nonexpansive mapping and there exists $c\in C$ such that $Tc\in [c,\rightarrow )_{G}$, then $T$ has a fixed point provided for each $a\in C$, $[a,a]_{G}$ has the fixed point property for nonexpansive mappings. In particular, it gives an essential generalization of the Dehaish-Khamsi theorem concerning partial orders in complete uniformly convex hyperbolic metric spaces. Some counterparts of this result for modular spaces, and for commutative families of mappings are given too.

We study the existence of positive solutions for a fractional differential equation with integral boundary conditions. Our solutions are placed in the space of Hölder functions and the main tools used in the proof of the results are the classical Schauder fixed point theorem and a sufficient condition about the relative compactness in Hölder spaces. Moreover, some examples are shown illustrating the results.

We prove that the countable product of lines contains a Haar-null Haar-meager Borel linear subspace $L$ that cannot be covered by countably many closed Haar-meager sets. This example is applied to studying the interplay between various classes of "large" sets and Kuczma-Ger classes in the topological vector spaces ${\mathbb R}^n$ for $n\le \omega$.

The goal of this paper is to establish a general fixed point theorem for compact single-valued continuous mappings in Hausdorff $p$-vector spaces, and a fixed point theorem for upper semicontinuous set-valued mappings in locally $p$-convex spaces for $p\in (0, 1]$. These results not only provide a solution to Schauder conjecture in the affirmative under the setting of $p$-vector spaces for compact single-valued continuous mappings, but also show the existence of fixed points for upper semicontinuous set-valued mappings defined on $s$-convex subsets in Hausdorff locally $p$-convex spaces, which would be fundamental for nonlinear functional analysis, where $s, p \in (0, 1]$.

Let $C(\Delta)$ denote the Banach space of all continuous real-valued functions on the Cantor set $\Delta$ and $C_0(\Delta)=\lbrace f\in C(\Delta): f(1)=0\rbrace$. From the 1966 theorem of Cambern, it is well-known that the Banach-Mazur distance $d(C(\Delta), C_0(\Delta))\geq 2$. We prove that, in fact, $d(C(\Delta), C_0(\Delta))= 2$. As a consequence, we answer a question left open in the 2012 paper of Candido and Galego.

It is known that a strict contraction on complete metric spaces with graphs possesses a fixed point. In the present paper we show that this property holds for mappings on complete metric spaces with graphs which are merely contractive. We also obtain some stability results.

In this paper we study some results on common fixed points of families of mappings on metric spaces by imposing orbit Lipschitzian conditions on them. These orbit Lipschitzian conditions are weaker than asking the mappings to be Lipschitzian in the traditional way. We provide new results under the two classic approaches in the theory of fixed points for uniformly Lipschitzian mappings: the one under the normal structure property of the space (which can be regarded as the Cassini-Maluta's approach) and the one after the Lifschitz characteristic of the metric space (Lifschitz's approach). Although we focus on the case of semigroup of mappings, our results are new even when a mapping is considered by itself.

In this paper, the solvability of an infinite system of integral equations of the Volterra-Hammerstein type in Banach space $BC(\mathbb{R_+}\times \mathbb{R_+},c_0$) is examined. Technique associated with the measure of noncompactness plays the most important role in adopted analysis and authors present an example to validate the applicability of the result.

We consider relations between normal structure of a Banach space and the fixed point property for various classes of generalized nonexpansive mappings under additional assumptions, such as that of continuity. In this way we answer some open questions about the behaviour of such maps.

We explore the behavior of convex functions defined on locally polyhedral type spaces. As a relevant example of such spaces we show that locally compact $\mathbb{R}$-trees with the geodesic extension property around any point are locally polyhedral spaces.

We characterize the family of iterated nonexpansive mappings that are stable under every renorming. The family of iterated nonexpansive mappings contains the family of nonexpansive mappings, it also contains quasi-nonexpansive and Suzuki's (C)-type mappings with fixed points, among others. We also give the corresponding characterizations for quasi-nonexpansive and some Suzuki's (C)-type mappings with fixed points.