Function Theorem Research Articles

Existence of solutions and well-posedness for set optimization problems with an application in the robust optimization

In this paper, first a generalization of the Weierstrass theorem for set-valued mappings in the absence of usual continuity and compactness assumptions is derived. Then, new notions of well-posedness for set-valued optimization problems are presented and some necessary and sufficient conditions for well-posedness of these problems are obtained. Finally, as an application, we provide some results for a robust set-valued optimization problem.

Quasi-Periodic Orbits of a Difference Equation with Varied Coefficients

In this paper, we study the stability of the equilibrium point and investigate the periodicity properties of a class of difference equations [Formula: see text], [Formula: see text] whose parameters and initial values are positive. In particular, [Formula: see text] is a two-periodic coefficient with respect to n. Moreover, we depict the phase transition of orbits and analyze the deviation of the quasi-periodic orbits from some periodic orbits related to the arithmetic property of [Formula: see text], including Diophantine and Liouvillean numbers. We associate the response solution problem of differential equations with the fixed point problem of the area-preserving mapping related to multiperiodic difference equations, and pose some future work. The techniques are based on Moser’s twist mapping theorem, reversibility and some invariant structures embedded in the system.

Bivariate density estimation under marginal unimodality constraints

Empirical analyses of observed pairs of variables often suggest that the underlying joint distribution of these variables is highly dependent and each one of them is marginally unimodal. Exploiting marginal shape information can improve the bivariate density estimation. We develop a nonparametric density estimation method where the joint density of the bivariate outcome variables is estimated using a flexible class of mixtures of scaled Beta distributions and the mixing weights are suitably constrained to ensure marginal unimodality. Large sample consistency of the proposed density estimator is established using the method of sieves and Argmax Continuous Mapping Theorem. The proposed method enlarges the scope of previous studies in two important aspects: (i) joint density provides a simultaneous estimation of higher-order moments beyond mean and variance functions, and (ii) marginal shape constraints provide more efficient estimates. The proposed procedures are illustrated using simulated and real case study datasets on gestational age and infant birth weight.

Chatterjea-Type Contraction Mapping Theorem for Four Self-Mappings in Cone Pentagonal Metric Space

In this paper we obtain a common fixed point theorem for four self mappings under Chatterjea contractive conditions in cone pentagonal metric space. We present an example in support of the main result. Some Corollaries conclude the paper.

S-Contractive Mappings on Vector-Valued White Noise Functional Space and Their Applications

In this paper, we propose a new notion, which we name S-contractive mapping, in a framework of vector-valued white noise functionals Wω⊗N⊂Γ(H)⊗K⊂(Wω⊗N)*. And we give concrete definitions of S-contractive mappings for vector-valued white noise functionals. We establish the fixed-point theorems of S-contractive mappings. As applications, by applying the fixed-point theorems of generalized S-contractive mappings, we prove the existence and uniqueness of a generalized form of differential equations of vector-valued white noise functionals with weak conditions and investigate Wick-type differential equations of vector-valued white noise functionals with generalized conditions.

Oka-1 manifolds: New examples and properties

In this paper we investigate Oka-1 manifolds and Oka-1 maps, a class of complex manifolds and holomorphic maps recently introduced by Alarcón and Forstnerič. Oka-1 manifolds are characterised by the property that holomorphic maps from any open Riemann surface to the manifold satisfy the Runge approximation and Weierstrass interpolation conditions, while Oka-1 maps enjoy similar properties for liftings of maps from open Riemann surfaces in the absence of topological obstructions. We also formulate and study the algebraic version of the Oka-1 condition, called aOka-1. We show that it is a birational invariant for compact algebraic manifolds and holds for all rational manifolds. This gives a Runge approximation theorem for maps from compact Riemann surfaces to uniformly rational projective manifolds. Finally, we study a class of complex manifolds with an approximation property for holomorphic sprays of discs. This class lies between the smaller class of Oka manifolds and the bigger class of Oka-1 manifolds and has interesting functorial properties.

Bloch-Type Theorems for Harmonic Mappings

Bloch-Type Theorems for Harmonic Mappings

A class of free boundary problems describing the local-nonlocal cross-diffusion models with two species in ecology

We study a class of free boundary problems describing the local-nonlocal cross-diffusion models with two species in ecology. The first is a native species with nonlocal cross-diffusion, and the second is an invasion species with local cross-diffusion. Since the local and nonlocal cross-diffusions are taken into account, the models are described by the strongly coupled systems of integral and differential equations. In addition, the position of each free boundary is determined by the Stefan condition. By using the transformations of function and variable, various estimates and contraction mapping theorem, we show the global existence, uniqueness and regularity of solutions for the free boundary problems. Besides, the long time behavior of the solution for a special type of local-nonlocal cross-diffusion competition model is also obtained.

A Note on the Immovable Point Theorem for Nonlinear Mappings

In this paper we discussed the generalization of the findings on fixed-point theorem in a complete convex metric space with uniformly normal structure of Mukherjee and Som as well as Gillespie and Williams.

New common fixed point theorems for quartet mappings on orthogonal S -metric spaces with applications

New common fixed point theorems for quartet mappings on orthogonal S -metric spaces with applications

A new nonlocal impulsive fractional differential hemivariational inclusions with an application to a frictional contact problem

This paper is addressed to the study of a novel impulsive fractional differential hemivariational inclusions (IFDHI) with a nonlocal condition, comprising an impulsive fractional differential inclusion (IFDI) with a nonlocal condition and a hemivariational inequality (HVI), within separable reflexive Banach spaces. Initially, we establish the unique solvability of the HVI by adopting the surjectivity theorem for set-valued mappings. Subsequently, we demonstrate that there exist mild solutions for the new IFDHI by utilizing the theory of measure of noncompactness (MNC) and fixed point theorem (FPT) for condensing set-valued mappings. Additionally, we employ our principal findings to establish the solvability of a new frictional contact problem (FCP) concerning an elastic body interacting with a foundation within a finite time interval, considering the temperature effect.

Integral Operators in b-Metric and Generalized b-Metric Spaces and Boundary Value Problems

We study fixed-point theorems of contractive mappings in b-metric space, cone b-metric space, and the newly introduced extended b-metric space. To generalize an existence and uniqueness result for the so-called Φs functions in the b-metric space to the extended b-metric space and the cone b-metric space, we introduce the class of ΦM functions and apply the Hölder continuous condition in the extended b-metric space. The obtained results are applied to prove the existence and uniqueness of solutions and positive solutions for nonlinear integral equations and fractional boundary value problems. Examples and numerical simulation are given to illustrate the applications.

On structural proof theory of the modal logic K+ extended with infinitary derivations

Abstract We consider an extension of the modal logic of transitive closure $\textsf{K}^{+}$ with certain infinitary derivations and present a sequent calculus for this extension, which allows non-well-founded proofs. We establish continuous cut-elimination for the given calculus using fixed-point theorems for contractive mappings. The infinitary derivations mentioned above are well founded and countably branching, while the non-well-founded proofs of the sequent calculus can only be finitely branching. The ordinary derivations in $\textsf{K}^{+}$, as we show additionally, correspond to the non-well-founded proofs of the calculus that are regular and cut-free. Therefore, in this article, we explore the relationship between deductive systems for $\textsf{K}^{+}$ with well-founded infinitely branching derivations and (regular) non-well-founded finitely branching proofs.

Some Geraghty Type Inequalities in b-fuzzy Metric Spaces with an Application

In this note, we introduce novel Geraghty-type inequalities within the framework of a b-fuzzy metric space and develop new fixed point theorems for such mappings in a G-complete b-fuzzy metric space. To substantiate our findings, we present several illustrative examples using graphical methods. Additionally, we demonstrate the application of our introduced theorems by solving a non-linear integral equation, showing the practical utility of our results.

ROBUST STABILITY OF UNCERTAIN HOPFIELD NEURAL NETWORKS WITH PROPORTIONAL DELAYS

The problem of robust stability is investigated for a class of uncertain Hopfield-type neural networks with proportional delays. The existence and uniqueness of an equilibrium is first established using the homeomorphic mapping theorem. Then, by employing a modified Lyapunov–Krasovskii functional, a new criterion for the global asymptotic stability of an equilibrium point of the system is formulated.

The Bałaban variational problem in the non-linear sigma model

The minimization of the action of a QFT with a constraint dictated by the block averaging procedure is an important part of Bałaban’s approach to renormalization. It is particularly interesting for QFTs with non-trivial target spaces, such as gauge theories or non-linear sigma models on a lattice. We analyze this step for the [Formula: see text] non-linear sigma model in two dimensions and demonstrate, in this case, how various ingredients of Bałaban’s approach play together. First, using variational calculus on Lie groups, the equation for the critical point is derived. Then, this non-linear equation is solved by the Banach contraction mapping theorem. This step requires detailed control of lattice Green functions and their integral kernels via random walk expansions.

Existence and uniqueness of the solution to initial and inverse problems for integro-differential heat equations with fractional load

This work is devoted to the unique solvability of the direct and inverse problems for a multidimensional heat equation with a fractional load in Holder spaces. In the problem under consideration, the loaded term is in the form of a fractional integral operator for the time variable. We prove the existence and uniqueness of the solution to these problems by the contraction mapping theorem and the theory of integral equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/64/abstr.html

A note on an invariant distance of the bidisk

In this short paper, we discuss relation between an invariant distance of the bidisk and Kreĭn space geometry. In particular, an interpolation theorem for rational maps with respect to our invariant distance is proven.

Controls insensitizing the norm of solution of a Schrödinger type system with mixed dispersion

The main goal of this manuscript is to prove the existence of insensitizing controls for the fourth-order dispersive nonlinear Schrödinger equation with cubic nonlinearity. To obtain the main result we prove a null controllability property for a coupled fourth-order Schrödinger cascade type system with zero-order coupling which is equivalent to the insensitizing control problem. Precisely, employing a new Carleman estimates, we first obtain a null controllability result for the linearized system around zero, and then the null controllability for the nonlinear case is extended using an inverse mapping theorem.

The Bowen-Franks type theorems for multivalued maps and impulsive differential equations

The aim of the present paper is to extend the well known Bowen-Franks type theorems to interval and circle multivalued maps. In this way, a positive topological entropy, including its lower estimate, can be implied by the existence of periodic orbits whose orders differ from a power of 2, or provided the topological degree of given circle maps is absolutely greater than 1. Simple applications are also given to impulsive differential equations.