Fixed Point Theory

Johnson and Prince have classified all translation planes of order 81 that admit SL(2, 5), where the 3-elements are elations. In this article, it is shown that whenever SL(2, 5) acts as a collineation group on a translation plane of order 81, the problem can always be reduced to the elation case. Hence, there is a complete classification of all translation planes of order 81 admitting SL(2, 5); there are exactly 14 translation planes of order 81 admitting SL(2, 5).

Johnson and Prince have classified all translation planes of order 81 that admit SL(2, 5), where the 3-elements are elations. In this article, it is shown that whenever SL(2, 5) acts as a collineation group on a translation plane of order 81, the problem can always be reduced to the elation case. Hence, there is a complete classification of all translation planes of order 81 admitting SL(2, 5); there are exactly 14 translation planes of order 81 admitting SL(2, 5).

We formalize and complete the Knot Infinity Golden Set core claim in an abstract fixed-point setting. Let (I, ≤) be a complete lattice with a monotone refinement operator F : I → I, and let the coinductive seal K ∞ := gfp(F) be defined by Knaster-Tarski. For any Extensive Monotone Operator Ψ : I → I that satisfies the upper semi-commutation inequality (the "Stability Ratchet") Ψ • F ≤ F • Ψ, we prove the Generalized Golden Invariance Theorem: Ψ(K ∞) = K ∞. This removes any need for idempotence and isolates semi-commutation as the sufficient hypothesis ensuring invariance of the seal. Specializing to two admissibility constraints-Golden Stability Φ and J 3-Neutrality J-we show that if each individually semi-commutes with F (ΦF ≤ F Φ and JF ≤ F J), then their generated closure A ↑ := n<ω (Φ • J) n also semi-commutes with F. Consequently, K ∞ ∈ Fix(A ↑). The proof proceeds by (i) one-step semi-commutation for Φ • J, (ii) induction on iterates, and (iii) a pointwise join argument, requiring no continuity (join-preservation) assumptions on F [1]. To establish necessity, we give a finite four-element counterexample where F is monotone and Φ, J are closures, yet A ↑ • F F • A ↑ because one component (J) fails to semi-commute with F. Thus, monotonicity of F alone is insufficient; the individual semi-commutation hypotheses are required. This closes a gap in prior drafts and yields a concise, drop-in, machine-verifiable core. Setup Let (I,) be a complete lattice of invariants, and let the refinement operator F : I → I be monotone. By Knaster-Tarski, the coinductive seal is K ∞ := gfp(F).

Neural and gene networks are often modeled by differential equations. If the continuous threshold functions in the differential equations are replaced by step functions, the equations become piecewise linear (PL equations). The flow through the state space is represented schematically by paths and directed graphs on an n-dimensional hypercube. Closed pathways, called cycles, may/effect periodic orbits with associated fixed points in a chosen Poincar6 section. A return map in the Poincar6 section can be constructed by the composition of fractional linear maps. The stable and unstable manifolds of the fixed points can be determined analytically. These methods allow us to analyze the dynamics in higher-dimensional networks as exemplified by a four-dimensional network that displays chaotic behavior. The three-dimensional Poincar6 map is projected to a two-dimensional plane. This much simpler piecewise linear two-dimensional map conserves the important qualitative features of the flow.

In this paper, by using a resolvent operator technique of maximal monotone mappings and the property of a fixed-point set of multi-valued contractive mapping, we study the behavior and sensitivity analysis of a solution set for a parametric generalized mixed multi-valued implicit quasi-variational inclusion problem in Hilbert space. Further, under some suitable conditions, we discuss the Lipschitz continuity (or continuity) of the solution set with respect to the parameter. By exploiting the technique of this paper, one can generalize and improve many known results in the literature.

This appendix demonstrates an explicit mapping from the E₈ root system to the SU 3) gauge subgroup-illustrating how your E₈-based substrate naturally contains Standard Model symmetries. We provide step-by-step algebraic decompositions and explanatory notes. E₈ has rank 8 and 240 roots in an 8-dimensional Euclidean space. Standard construction: These 240 vectors satisfy. Explanatory Note: The root system defines the geometric structure underlying the E₈ Lie algebra. Each root corresponds to a generator in the algebra. The E₈ Dynkin diagram contains an A₂ subdiagram, corresponding to SU 3. One standard embedding removes six nodes to isolate two connected nodes: o-o-o-o-o-o-o | o Retain an A₂ chain: two adjacent nodes in the long branch. Explanatory Note: A₂ is the Dynkin type for SU 3. By selecting an A₂ subdiagram within E₈ʼs diagram, we identify the SU 3) subalgebra.

This work introduces a novel theoretical framework for analyzing topological structures, centered on two core concepts: Knot Infinity (K∞ ) and the Golden Set (Gϕ). We define K∞ coinductively as the final coalgebra of an iterative refinement operator, F. This reframes the problem of knot classification from one of static invariants to one of behavioral equivalence, where K∞ represents the stable, scale-invariant information that survives a dynamical process analogous to a Renormalization Group (RG) flow. Concurrently, we define the Golden Set, Gϕ, as the greatest fixed point of an admissibility operator, representing a locus of maximally stable states. These states are characterized by Discrete Scale Invariance (DSI) with the golden ratio (ϕ) as the preferred scaling factor. The privileged role of ϕ is justified through an appeal to the Kolmogorov-Arnold-Moser (KAM) theorem, which suggests that ϕ-based systems exhibit maximal resilience to resonant perturbations. The framework's central thesis is the synthetic inclusion K∞ ⊆G ϕ, which posits a profound connection between the process of topological simplification and the principles of maximal dynamical stability. This synthesis bridges concepts from theoretical computer science (coinduction), theoretical physics (RG flow), and dynamical systems theory (KAM stability). We further propose that this framework offers a new paradigm for Topological Data Analysis (TDA), where the iterative refinement operator F can serve as a "topology-aware" denoising filter, providing a dynamical alternative to persistent homology.

A generalization of a viscosity generalized Halpern iteration scheme is analyzed. It is proven that the solution converges asymptotically strongly to a unique fixed point of an asymptotically nonexpansive mapping which drives the iteration together with a contractive self-mapping, a viscosity term and two driving external forcing terms.

A generalization of Halpern&#39;s iteration is investigated on a compact convex subset of a smooth Banach space. The modified iteration process consists of a combination of a viscosity term, an external sequence, and a continuous nondecreasing function of a distance of points of an external sequence, which is not necessarily related to the solution of Halpern&#39;s iteration, a contractive mapping, and a nonexpansive one. The sum of the real coefficient sequences of four of the above terms is not required to be unity at each sample but it is assumed to converge asymptotically to unity. Halpern&#39;s iteration solution is proven to converge strongly to a unique fixed point of the asymptotically nonexpansive mapping.

In this paper we introduce the notions of ϕ-contractive parentchild infinite iterated function system (pcIIFS) and orbital ϕ-contractive infinite iterated function system (oIIFS) and we prove that the corresponding fractal operator is weakly Picard. The corresponding notions of shift space, canonical projection and their properties are also treated.

In this paper we introduce the notions of φ-contractive parentchild infinite iterated function system (pcIIFS) and orbital φ-contractive infinite iterated function system (oIIFS) and we prove that the corresponding fractal operator is weakly Picard. The corresponding notions of shift space, canonical projection and their properties are also treated.

In this paper we construct a new interval method for the inclusion of one simple or multiple complex polynomial zero in circular complex arithmetic. We present the convergence analysis starting from the computationally verifiable initial condition that guarantees the convergence of this inclusion method. We also give two numerical examples in order to demonstrate convergence behavior of the proposed method.

Ultrafast manipulations of magnetic phases are eliciting increasing attention from the scientific community, because potentially relevant to the understanding of nonequilibrium phase transitions and to novel technologies. Here, we focus on manipulations applied to magnetic impurities in metallic hosts. By considering small nanoring geometries, we show how currents can induce a dynamical switching between different types of exchange interactions in these systems. Our work thus opens a study window on nonequilibrium Doniach&#39;s magnetic phase diagrams, and time-dependent Kondo-vs-RKKY scenarios.

In this article we define multivalued R-weakly contractive multi-valued mappings in ordered cone metric spaces without assumption of normality on cone, and generalize many results existing in the literature. We provide applications to solutions of integral inclusions and give nontrivial examples to support our main theorem.

In this article, we introduce the notion of a Chatterjea-type cyclic weakly contraction and derive the existence of a fixed point for such mappings in the setup of complete metric spaces. Our result extends and improves some fixed point theorems in the literature. Example is given to support the usability of the result.

The existence of minimal and maximal fixed points for monotone operators defined on probabilistic Banach spaces is proved. We obtained sufficient conditions for the existence of coupled fixed point for mixed monotone condensing multivalued operators.

We obtain common fixed points and points of coincidence of a pair of mappings satisfying a generalized contractive type condition in cone metric spaces. Our results generalize some well-known recent results in the literature. generalize some well-known recent results in the literature. A large variety of the problems of analysis and applied mathematics reduce to finding solutions of non-linear functional equations which can be formulated in terms of finding the fixed points of a nonlinear mapping. In fact, fixed point theorems are very important tools for proving the existence and uniqueness of the solutions to various mathematical models (differential, integral and partial differential equations and variational inequalities etc.) representing phenomena arising in different fields, such as steady state temperature distribution, chemical equations, neutron transport theory, economic theories, financial analysis, epidemics, biomedical research and flow of fluids. They are also used to study the problems of optimal control related to these systems . Fixed point theory concerned with ordered Banach spaces helps us in finding exact or approximate solutions of boundary value problems . In 1963, S. Ghaler, generalized the idea of metric space and introduced 2-metric space which was followed by a number of papers dealing with this generalized space. A lot of materials are available in other generalized metric spaces, such as, semi metric spaces, quasi semi metric spaces and Dmetric spaces. Huang and Zhang [6] introduced the concept of cone metric space and established some fixed point theorems for contractive type mappings in a cone metric space. Subsequently, some other authors studied the existence of fixed points, points of coincidence and common fixed points of mappings Manuscript

Some statistical and dynamical properties for the problem of relativistic charged particles in a wave packet are studied. We show that the introduction of dissipation change the structure of the phase space and attractors appear. Additionally, by changing at least one of the control parameters, the unstable manifold touches the stable manifold of the same saddle fixed point and a boundary crisis occurs. We show that the chaotic attractor is destroyed given place to a transient which follows a power law with exponent À1 when varying the control parameters near the criticalities. On the other hand, by changing at least two control parameters and by using the Lyapunov exponents to classify orbits with chaotic and periodic behaviour, we show the existence of infinite shrimp-shaped domains, which correspond to the periodic attractors, embedded in a region with chaotic behaviour. Finally, we show the first indication of a shrimp in a three dimension parameter space.

Here, f is of countable weight [18] if there exists a map h: X → I∞ such that f × g embeds X into Y × I∞. The C-space property was introduced by Haver [10] for compact metric spaces and then extended by Addis and Gresham [1] for general spaces. The source limitation topology (or ...

A B S T RA CT In this paper we prove a unique common fixed point theorem in 2-metric space .the existence of fixed point for two weakly compatible maps into 2-metric space is established under new contractive condition of integral type by using another functions φ and. ψ

The aim of this paper is to introduce convex structure G-metric spaces and extended Mann's iteration algorithm to these spaces. By using Mann's iteration scheme, a series of fixed point results and Mann's iteration algorithm was generalized. Also, strong convergence theorems for contraction mappings in convex G-metric spaces were developed. Furthermore, the problem of T-stability of the Mann's iteration procedure for the mappings in complete convex Gmetric spaces was considered.

In recent years, there has been a significant rise in the application of bmetric spaces in mathematical and analytical studies, particularly regarding fixed-point theory. Nonetheless, these spaces present certain challenges. A key issue stems from their fundamental two-dimensional characteristics, which complicate the direct application or extension of results-especially fixed-point theorems-to higher-dimensional frameworks, such as threedimensional spaces. Previous research has largely concentrated on fixedpoint theorems within the realm of b-metric spaces, particularly those associated with the Sehgal-Guseman contraction principle. While these investigations have contributed valuable insights, they are limited in their breadth. To address this constraint, the current study introduces a new approach by presenting an alternative metric framework-specifically, the G-metric space. The main aim of this research is to generalize and expand the Sehgal-Guseman-type contraction results from the b-metric context to the more extensive and adaptable G-metric environment. This study explores the existence, uniqueness, and characteristics of fixed points under these generalized conditions using complete G-metric spaces.

We analyse the spectral phase diagram of Schrödinger operators T + λV on regular tree graphs, with T the graph adjacency operator and V a random potential given by iid random variables. The main result is a criterion for the emergence of absolutely continuous (ac) spectrum due to fluctuation-enabled resonances between distant sites. Using it we prove that for unbounded random potentials ac spectrum appears at arbitrarily weak disorder (λ 1) in an energy regime which extends beyond the spectrum of T . Incorporating considerations of the Green function's large deviations we obtain an extension of the criterion which indicates that, under a yet unproven regularity condition of the large deviations' 'free energy function', the regime of pure ac spectrum is complementary to that of previously proven localization. For bounded potentials we disprove the existence at weak disorder of a mobility edge beyond which the spectrum is localized.

Artificial General Intelligence (AGI) represents the aspiration to create systems that can learn, reason, and adapt across diverse domains with human-like flexibility. The foundations of AGI are deeply mathematical: they involve structures from logic, topology, algebra, geometry, probability, and category theory. In this paper I explore the mathematics underpinning AGI, with a focus on fibre bundles, category theory, optimization, algebraic geometry, information theory etc showing how abstract mathematics can unify cognitive architectures.

This paper mainly focuses on multiple current controller methods for a grid-connected inverter-based distributed generation. PI, PR, DQ, and Hysteresis controllers are the different control methods used for the analysis. Switching pulses for the conventional H-bridge inverter are generated, and the output total harmonic distortion analysis is performed using these controllers. Disadvantages of a PI current control include its steady-state errors as well as its extremely low rejection capability. As a solution to the issues of PI controller, PR controller was invented. In addition to the PI and PR controllers, Direct-Quadrature and hysteresis controllers are also studied and compared in terms of total harmonic distortion. Analytical results of the hysteresis current controller are compared with different bandwidths. The performance of these controllers has been examined using MATLAB/Simulink by evaluating the switching frequency and current THD of the inverter.

We present in this paper a study on highly resistive SiC nanowires in a singly clamped geometry. We demonstrate that these field emission nanoelectromechanical systems (NEMS) can be synchronized ton an external AC signal and act as an amplifier.

In this manuscript, a two-fluid magnetized plasma oscillator subjected to parametric excitation with square delayed feedback is studied both analytically and numerically. The Hamilton systems of triple-well and narrow single-well are discussed in detail. The scenarios of phase portraits and equilibria are given. Homoclinic and heteroclinic orbits are strictly derived. With the Melnikov method, the critical value of chaos arising from homoclinic or heteroclinic intersections is derived analytically. We have discovered some interesting dynamic phenomena, such as uncontrollable time delays with which chaos always occurs for this system. The influence of time-delay on the chaotic property is also studied rigorously. On the basis of theoretical analysis, some numerical simulations including time histories, phase portraits, bifurcation diagrams, Poincaré sections, Lyapunov exponential spectrums and attractor basins are given. Numerical simulations are consistent with theoretical results.

Abstract: The whole complex process to obtain a protein encoded by a gene is difficult to include in a mathematical model. There are many models for describing different aspects of a genetic network. Finding a better model is one of the most important and interesting ...

In this paper, we obtain some fixed point theorems for dominated mappings satisfying locally contractive conditions on a closed ball in a left K-sequentially O-complete ordered quasi-partial metric space and in a right K-sequentially O-complete ordered quasi-partial metric space, respectively. Our results improve several well known results.

Sufficient conditions for existence of random fixed point of a nonexpansive rotative random operator are obtained and existence of random periodic points of a random operator is proved. We also derive random periodic point theorem for ǫ-expansive random operator.

Let (X, d) be a metric space and F : X ; X be a set valued mapping. We obtain sufficient conditions for the existence of a fixed point of the mapping F in the metric space X endowed with a graph G such that the set V (G) of vertices of G coincides with X and the set of edges of G is E(G) = {(x, y) : (x, y) ∈ X × X }.

We obtain necessary conditions for the existence of fixed point and approximate fixed point of nonexpansive and quasi nonexpansive maps defined on a compact convex subset of a uniformly convex complete metric space. We obtain results on best approximation as a fixed point in a strictly convex metric space.

We are interested in addressing the problem of coordinating a large number of simple agents in order to achieve a given task. Stated in this way, the question leads naturally to the Swarm Intelligence field. In this paper we use a new type of model, directly inspired by Kaneko's coupled map gas model which we have adapted to the multi-agent system paradigm, so as to tackle this generic objective. This model is called a logistic multi-agent system (LMAS): it is composed of reactive situated agents whose individual behavior is governed by a logistic map or more generally a quadratic map. The collective behavior results from couplings between agents and local controls on agents adjusted by local environmental conditions. This way of modelling reveals to enable a wide range of pattern formations and various forms of adaptation to the environment. This paper focuses on the way to design the constitutive mechanisms of LMAS -particularly the perception and action processesand on the way a self-adaptation process may result from these mechanisms. This study is illustrated with experiments on the predators-prey pursuit problem, in which a set of agents (predators) has to encircle a moving prey. We show that coupling the internal states of agents leads to amplifying the predator aggregation around the prey, whereas altering the internal control variable in each agent through environment perceptions modifies the predator sensitivity to the prey. We finally complete this study by relating the concept of adaptation with concepts of the dynamical system theory: a qualitative dynamical analysis of the capturing process leads to view the prey as a dynamical fixed point of the system.

ABSTRACT. We prove a strong duality result between a convex optimization problem with both cone and equality constraints and its Lagrange dual formulation, provided that a constraint qualification condition related to the notion of quasi-relative interior holds true. ...

The purpose of this article is to introduce the concept of residual ν-metric space as a synthesis of a type of generalization of metric space and its extensions namely, b-metric space, extended b-metric space, strong b-metric space, strong controlled b-metric space, double controlled metric type space, suprametric space, b-suprametric space, rectangular metric space, rectangular b-metric space, extended rectangular b-metric space, homothetic rectangular metric space, controlled rectangular b-metric space, ν-generalized metric space and b_ν(s)metric space. Moreover we give a general form of the notion of contraction in a residual ν-metric space and we prove the analogue of the Banach Contraction Principle in this new framework.

Efficient allocation of network bandwidth is a critical challenge in optimizing network performance. In this study, we compare different fixed-point iterative schemes to determine a most effective approach for solving the bandwidth allocation problem. Our findings indicate that the three-step fixed-point iterative method significantly outperforms the one-step scheme
in terms of convergence speed and efficiency. The results suggest that increasing the number of iteration steps beyond four may further enhance the convergence of the allocation point, leading to improved optimization. Building on this insight, we propose future research to develop and analyze a fixed-point iteration method with more than four steps. Such an approach has the potential to offer even greater efficiency in solving network bandwidth allocation problems. The findings of this study provide a foundation for further advancements in iterative optimization techniques for network resource management.

The main aim of these lectures is to study the connection between symplectic symmetries of K3 surfaces and the Mathieu group M24, and its Enriques analogy, that is, a conjectural connection between semi-symplectic symmetries of Enriques surfaces and another Mathieu group M12. Algebraic varieties are the subject of study of algebraic geometers. The place of K3 and Enriques surfaces among them can be depicted in the following way:

Dance performances in which dancers wield sections of PVC pipe to form a number of polyhedral designs led to the question: what is the largest graph which edge decomposes the “skeletons” of all five Platonic solids? We show that a certain 6-edge tree is the largest such graph, and investigate related questions. PVC Pipe Polyhedra. The tetrahedron in Figure 1 is composed of a folded octagon, while the cube and octahedron are each composed of six folded paths of length 2. These constructions have been used in dances created by the author and his collaborators. The photo from the author’s dance “Pipe Dreams” shows an octahedron in which each dancer wields three lengths of PVC pipe held together by cord at the two internal vertices labeled a in the diagram on the right. The white pipe tends to stand out in bright stage lighting, though in some dances the pipes are painted with fluorescent paint and black lights make the pipes “pop” even more. The shapes created by the dancers, which mig...

We consider a SU (N )×SU (M ) generalization of the multichannel single-impurity Kondo model which we solve analytically in the limit N → ∞, M → ∞, with γ = M/N fixed. Non-Fermi liquid behavior of the single electron Green function and of the local spin and flavor susceptibilities occurs in both regimes, N ≤ M and N > M , with leading critical exponents identical to those found in the conformal field theory solution for all N and M (with M ≥ 2). We explain this remarkable agreement and connect it to "spin-flavor separation", the essential feature of the non-Fermi-liquid fixed point of the multichannel Kondo problem.

Cyclic mappings describe fixed paths for which each point is sequentially transmitted from one set to another. Cyclic mappings satisfying certain cyclic contraction conditions have been used to obtain the best proximity points, which constitute a suitable framework for the mirror reflection model. Alternative contraction mappings introduced by Chen (Symmetry 11:750, 2019) built a new model containing several mirrors in which the light reflected from a mirror does not go to the next mirror sequentially, and its path may diverge to any other mirror. The aim of this paper is to present a new variant of alternative contraction called alternative p-contraction and study its properties. The best proximity point result for such contractions under the alternative UC property is proved. An example to support the result proved herein is provided.