In his study of Cantor-type intersection theorems and fixed points of almost affine mappings, Jachymski posed four open questions. In this note, we provide an affirmative answer to the fourth question.

We introduce and study a new class of contractive maps on metric spaces defined by a contraction condition on the second elementary symmetric polynomial of the three pairwise distances of any triple of distinct points. A map satisfying this condition is called an e2-contraction; the corresponding strict version is a strict e2-contraction. For complete metric spaces containing at least three points, every e2-contraction is continuous and has a fixed point if and only if it admits no periodic point of prime period 2, with at most two fixed points in total. For arbitrary metric spaces containing at least three points, a strict e2-contraction with no period-2 point for which some orbit admits a convergent subsequence necessarily has a fixed point. As consequences, we recover the classical Banach and Edelstein fixed point theorems, and show that on spaces where every point is an accumulation point every e2-contraction is a Banach contraction. Explicit finite examples show that e2-contractions (resp. strict e2-contractions) strictly extend the Banach (resp. Edelstein) and perimeter-contracting classes of [11] (resp. [2]).

In this paper, we establish the existence and uniqueness of fixed points for alpha beta-phi contractive mappings in sb-suprametric spaces using a new polygonal inequality. We then derive fixed-point theorems for contractions, focusing on points that are either binary-related or partially ordered. Additionally, we explore several qualitative properties of the fixed point problem, such as its Ulam-Hyers stability, well-posedness, and the limit shadowing property. As an application, we prove the existence of a unique solution to a periodic boundary value problem.

We introduce the concept of rectangular b-suprametric spaces, in which we prove fixed point theorems of Banach, Kannan, Reich and Chatterjea types. We also provide an error estimate of the distance between the fixed point and the terms of its approximate sequence. Moreover, we present simple proofs of some existing theorems in rectangular b-metric and b-suprametric spaces. Further, we derive the Jungck’s common fixed point theorem. Our results generalize and improve several results from fixed point theory.

Let $C$ be a subset of a Hilbert space, and let $f$ and $g$ be self-maps on $C$ such that the range of $f$ is a convex, closed, and bounded subset of the range of $g$. If $f$ does not increase distances more than $g$, we demonstrate that $f$ and $g$ have coincidence points. This result generalizes a fixed point theorem of Browder-Petryshyn. As applications, we establish the existence of solutions to both matrix and integral equations.

We introduce the sets of periodic, recurrent, ?-limit and nonwandering points for a selfmap defined on a nonempty set endowed with an Eilenberg-Jachymski collection. Then, under some appropriate conditions, we show that these sets all coincide. Moreover, we establish fixed and periodic point theorems for a new class of ?-contractive mappings in ?-dislocated metric spaces, and generalize some results obtained by Edelstein, Matkowski and Bessenyei-P?les.

In this paper, we present a Caristi-type coincidence and common-fixed point theorem and its dual in Hausdorff spaces. We also extend the Caristi-Jachymski and Caristi-Kirk-Saliga fixed point theorems. Moreover, we give a positive answer to a question of Kirk and Shahzad without assuming the standard distance axioms.

Let C be a bounded closed convex subset of a uniformly convex Banach space, and let f and g be selfmaps of C such that f is expansive relative to g . Without assuming compactness of C , we show that f and g have coincidence points, and they have common fixed points if they commute. As a consequence, we derive the fixed point theorem of Browder-G & ouml;hde-Kirk.

In this paper, we introduce the strong b-suprametric spaces in which we prove the fixed point principles of Banach and Edelstein. Moreover, we prove a variational principle of Ekeland and deduce a Caristi fixed point theorem. Furthermore, we introduce the strong b-supranormed linear spaces in which we establish the fixed point principles of Brouwer and Schauder. As applications, we study the existence of solutions to an integral equation and to a third-order boundary value problem.

We introduce new concepts of triangular and rectangular multiplicative metric spaces, and establish fixed-point theorems in such spaces. Then, we derive Kannan-type fixed-point theorems in triangular and rectangular multiplicative metric spaces.

We introduce the concept of $b$-suprametric spaces and establish a fixed point result for mappings satisfying a nonlinear contraction in such spaces. The obtained result generalizes a fixed point theorem of Czerwik and a recent result of the author.

We introduce the concept of generalized suprametric spaces, which subsumes some existing abstract metric spaces. Then, we show the existence of fixed points for maps satisfying nonlinear contractions involving either extended comparison or ρ\rho -subhomogeneous functions. This study was carried out in generalized suprametric spaces as well as partially ordered generalized suprametric spaces. Some related results in JS-metric spaces and in bb-suprametric spaces are improved or extended.

We establish a new fixed point theorem in abstract spaces. We then derive two main consequences in topological spaces for mappings admitting precompact images or leading to a nonempty omega-limit set. The study is carried out by introducing a cone of special functions which enables us to extend, unify and improve fixed point results due to Bailey, Ciric, Dass-Gupta, Edelstein, Hardy-Rogers, Jaggi, Karapinar, Liepins, Nemytskii, Popa, Popescu, Reich, Suzuki and Wardowski. Finally, we introduce the notion of xi-Lipschitz property and we investigate the existence of solutions to a class of Cauchy problems.

We introduce the concept of suprametric space and study some basic properties of its topology. Then we show that certain contraction maps in suprametric spaces have a unique fixed point either the space is complete or it contains a nonempty $$\omega $$ -limit set. Next, we construct three suprametrics in partially ordered vector spaces, and utilize them to derive several fixed point theorems. Finally, we apply the obtained results to investigate the existence of solutions to some nonlinear integral and matrix equations.

Existence of certain minimal and maximal solutions to a class of operator equations in a Banach space ordered by a cone P are established. Precisely, we show the existence of these solutions as coincidence points of two operators A and B in an ordered interval D if some suitable conditions are satisfied at the extremities of D, B-1(D) is a subset of D, A is B-nondecreasing and one of the following cases holds: (i) P is normal and A is B -condensing; (ii) P is regular and A is B-demicontinuous; (iii) P is strongly minihedral; (iv) P is regular. In case where P is an abstract cone and A(B-1) preserves chains, two supplementary cases are investigated when any subset of A(B-1(D)) is either (v) relatively compact, or (vi) relatively weakly compact. As applications, we discuss the existence of solutions to some nonlinear functional equations and to various classes of nonlinear matrix equations. Several examples are presented to demonstrate the applicability of the results.

We introduce the concept of Maia α-ψ contractive type mappings and establish new fixed point theorems. We derive some results for comparable mappings in partially ordered metric spaces, which extend and generalize various known theorems on the topic. As applications, we study the existence and uniqueness of solutions to inhomogeneous Fredholm integral equations of the second kind, and we apply this study to a nonlinear third order two–point boundary value problem.

In this paper, we introduce the concept of Eilenberg–Jachymski collection on a nonempty set. Then, we establish three results equivalent to Bourbaki–Kneser’s fixed point theorem, and, therefore, to the axiom of choice. As consequences, we present new fixed point theorems in compact topological spaces, which extend and unify those of Nemytskii–Edelstein, Liepinš and Suzuki.

We establish a coincidence point theorem in complete metric spaces. As a consequence, we show the existence of solutions to a system of initial value problem of fractional differential equations involving Riemann–Liouville fractional derivatives. Next, we derive several coincidence point theorems for new classes of sublinear and superlinear operators, in the context of ordered Banach spaces. Finally, we apply these results to discuss the existence of positive solutions to a class of initial value problem of fractional differential equations.

Consider an ordered Banach space and f,gf,g two self-operators defined on the interior of its positive cone. In this article, we prove that the equation f(X)=g(X)f(X)=g(X) has a positive solution, whenever f is strictly α\alpha -concave g-monotone or strictly (−α)(-\alpha )-convex g-antitone with g super-homogeneous and surjective. As applications, we show the existence of positive definite solutions to new classes of nonlinear matrix equations.

In this paper, we study the existence of positive solutions for classes of nonlinear operator equations in Banach spaces ordered by cones. We establish new coincidence point theorems via a generalized monotone iterative method. As applications, we discuss the existence of positive solutions for systems of nonlinear matrix equations and a system of integral equations of Volterra type.