Single-valued Mappings Research Articles

Some Results on Multivalued Proximal Contractions with Application to Integral Equation

In this manuscript, for the purpose of investigating the coincidence best proximity point, best proximity point, and fixed point results via alternating distance ϕ, we discuss some multivalued (ϕ−Fτ)CP and (ϕ−Fτ)BP−proximal contractions in the context of rectangular metric spaces. To ascertain the coincidence best proximity point, best proximity point, and the fixed point for single-valued mappings, we reduce these findings using (Fτ)CP and (Fτ)BP−proximal contractions. To make our work more understandable, examples of both single- and multivalued mappings are provided. These examples support our core findings, which rely on coincidence points, as well as the corollaries that address fixed point conclusions. In the final phase of our study, we use the obtained results to verify that a solution to a Fredholm integral equation exists. This application highlights the theoretical framework we built throughout our study.

A variational approach to stability relative to a set of single-valued and set-valued mappings

Based on the limiting normal cone relative to a set, we present in this paper the novel versions of the limiting coderivative relative to a set and subdifferentials relative to a set of multifunctions and singleton mappings, respectively. In addition to giving the necessary and sufficient conditions for the Aubin property relative to a set of multifunctions, the limiting coderivative relative to a set also provides a coderivative criterion for the metric regularity relative to a set of multifunctions. Besides, our study establishes sudifferential characteristics of the metric regularity and the locally Lipschitz continuity relative to a set for single-valued mappings. In finite dimensional spaces, our results are more general than the previous results. Furthermore, we also give examples to illustrate our results.

An existence result for integro-differential degenerate sweeping process with convex sets.

In this paper, we study the well-posedness in the sense of existence and uniqueness of a solution of integrally perturbed degenerate sweeping processes, involving convex sets in Hilbert spaces. The degenerate sweeping process is perturbed by a sum of a singlevalued map satisfying a Lipschitz condition and an integral forcing term. The integral perturbation depends on two time-variables, by using a semi-discretization method. Unlike the previous works, the Cauchy’s criterion of the approximate solutions is obtained without any new Gronwall’s like inequality.

Analysis of Caputo Sequential Fractional Differential Equations with Generalized Riemann–Liouville Boundary Conditions

This paper delves into a novel category of nonlocal boundary value problems concerning nonlinear sequential fractional differential equations, coupled with a unique form of generalized Riemann–Liouville fractional differential integral boundary conditions. For single-valued maps, we employ a transformation technique to convert the provided system into an equivalent fixed-point problem, which we then address using standard fixed-point theorems. Following this, we evaluate the stability of these solutions utilizing the Ulam–Hyres stability method. To elucidate the derived findings, we present constructed examples.

Motivic coaction and single-valued map of polylogarithms from zeta generators

We introduce a new Lie-algebraic approach to explicitly construct the motivic coaction and single-valued map of multiple polylogarithms in any number of variables. In both cases, the appearance of multiple zeta values is controlled by conjugating generating series of polylogarithms with Lie-algebra generators associated with odd zeta values. Our reformulation of earlier constructions of coactions and single-valued polylogarithms preserves choices of fibration bases, exposes the correlation between multiple zeta values of different depths and paves the way for generalizations beyond genus zero.

Noncoercive convex sweeping processes with velocity constraints

In this paper, we investigate noncoercive monotone convex sweeping processes with velocity constraints, a topic not previously investigated. Using some fundamental results on Bochner integration, the Tikhonov regularization method, a solution existence result for coercive convex sweeping processes with velocity constraints and a useful fact on measurable single-valued mappings, we prove the solution existence and properties of the solution set of noncoercive convex sweeping processes with velocity constraints under suitable conditions. These results represent a significant contribution, addressing a specific aspect of an open question posed in our recent work [Adly et al., Convex and nonconvex sweeping processes with velocity constraints: well-posedness and insights. Appl Math Optim. 2023;88(Paper No. 45)]. Additionally, we resolve two other open questions from the same paper concerning the behaviour of the regularized trajectories, relying on the Dominated Convergence Theorem for Bochner integration.

Mordukhovich derivatives of the set-valued metric projection operator in general Banach spaces

ABSTRACT In this paper, we investigate the properties and the precise solutions of the Mordukhovich derivatives of the set-valued metric projection operator onto some closed balls in some general Banach spaces. In the Banach space c, we find the properties of Mordukhovich derivatives of the set-valued metric projection operator onto the closed subspace c 0. We show that the metric projection from C[0, 1] to polynomial with degree less than or equal to n is a single-valued mapping. We investigate its Mordukhovich derivatives and G a ˆ teaux directional derivatives.

Fixed point theorem for composition of multivalued and single-valued mappings in Banach spaces

n this paper, we present a new characterization for finding a solution of common fixed points problem involving multivalued and single-valued mappings in Banach spaces by applying the properties of composition without commuting assumption. We equally obtain a convergence result to a solution of system of inclusion problems in Banach spaces. Our results unify, generalize and complement various known comparable results from the current literatur

Fixed point theorems for continuous single-valued and upper semicontinuous set-valued mappings in $p$-vector and locally $p$-convex spaces

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 $s, p \in (0, 1]$.

Generalized fixed points for fuzzy and nonfuzzy mappings in strong b-metric spaces

The main purpose of this research article is to generalize Kannan-type fixed-point (FP) theorems for single-valued mappings and Chatterjea-type FP result for fuzzy mappings (FMs) in the context of complete strong b-metric spaces (MSs). Moreover, fuzzy FPs are established for Suzuki-type fuzzy contraction in the setting of complete strong b-MSs. The conclusions are supported by nontrivial examples to enhance the validity of the results obtained in this study. In addition, previous findings have been made as corollaries from the relevant literature. The numerous implications that this technique has across the literature improve and integrate our findings. Applications of some of the results obtained are also incorporated.

Some fixed point results on ultrametric spaces endowed with a graph

Abstract The present article deals with new fixed point theorems by means of G G -strongly contractive maps. The research findings are demonstrated in a spherically complete ultrametric space with a graph for single-valued mappings. The special cases of the results that extend the current ones are offered, along with some examples that illustrate our results. Besides, an application utilized in dynamic programming that endorses the acquired observations is also provided.

Iterative learning control for differential inclusion systems with random fading channels by varying average technique.

The aim of this paper is to study iterative learning control for differential inclusion systems with random fading channels between the plant and the controller. In reality, the phenomenon of fading will inevitably occur in network transmission, which will greatly affect the tracking ability of output trajectory. This study discusses the impact of fading channel on tracking performance at the input and output sides, respectively. First, a set-valued mapping in a differential inclusion system with uncertainty is converted into a single-valued mapping by means of a Steiner-type selector. Then, to offset the effect of the fading channel and improve the tracking ability, a variable local average operator is constructed. The convergence of the learning control algorithm designed by the average operator is proved. The results show that the parameters in the varying local average operator can be adjusted to trade-off between the learning rate and the fading offset rate. Finally, the theoretical results are verified by numerical simulation of the switched reluctance motors.

Common Fixed Points of Generalized α-Nonexpansive Multivalued Mappings via Modified S-Type Iteration

In this paper, we introduce a class of generalized α-nonexpansive multivalued mapping and study some of its important properties. In particular, this class is a multivalued version of the single-valued nonexpansive mapping, called generalized α-nonexpansive mapping proposed by Pant and Shukla [11]. A modified S-type iteration scheme is proposed to approximate the common fixed point of two multivalued mappings. Our algorithm provides a multivalued extension of the method given by Khan et al. [6]. Strong and weak convergence of the iterative process are also proved under suitable assumptions.

On $ \theta $-hyperbolic sine distance functions and existence results in complete metric spaces

<p>In this paper, we first introduced the notion of $ \theta $-hyperbolic sine distance functions on a metric space and studied their properties. We investigated the existence and uniqueness of fixed points for some classes of single-valued mappings defined on a complete metric space and satisfying contractions involving the $ \theta $-hyperbolic sine distance function.</p>

Quantized iterative learning control for impulsive differential inclusion systems with data dropouts

This paper studies the quantized iterative learning control with encoding–decoding mechanism of a class of impulsive differential inclusion systems with random data dropouts. First, the set-valued mappings in the differential inclusion systems are transformed into single-valued mappings by using the Steiner-type selector. Then, a learning algorithm based on the intermittent update principle is designed to address the data asynchronism problem caused by two-sided data dropouts. If the data are successfully transmitted at the actuator and measurement sides, then the control input is effectively updated. Furthermore, a suitable scaling sequence is introduced to ensure the system output to achieve zero-error tracking performance for a desired trajectory. An upper bound of the quantization level is determined such that the quantization error is always bounded. The results show that the quantization method reduces the burden of network communication at the cost of increasing the amount of computation, and the learning algorithm does not require the data dropouts to satisfy a certain probability distribution. Finally, the effectiveness of the learning algorithm is verified by numerical simulations of the switched reluctance motor system.

Existence and Uniqueness Results of Fractional Differential Inclusions and Equations in Sobolev Fractional Spaces

In this work, by using the iterative method, we discuss the existence and uniqueness of solutions for multiterm fractional boundary value problems. Next, we examine some existence and uniqueness returns for semilinear fractional differential inclusions and equations for multiterm problems by using some notions and properties on set-valued maps and give some examples to explain our main results. We explore and use in this paper the fundamental properties of set-valued maps, which are needed for the study of differential inclusions. It began only in the mid-1900s, when mathematicians realized that their uses go far beyond a mere generalization of single-valued maps.

An algorithmic approach to solving split common fixed point problems for families of demicontractive operators in Hilbert spaces

In this article, we propose a new iterative algorithm for solving split common fixed point problems for finite families of single-valued demicontractive mappings in the setting of real Hilbert spaces. In order to solve this problem, we introduce a Halpern method with with an Armijo-line search for approximating the solution of the aforementioned problems. We establish a strong convergence result of the proposed method and display some numerical examples to illustrate the performance of our main result. Our result complements and generalizes some related results in literature.

Single-Valued Demicontractive Mappings: Half a Century of Developments and Future Prospects

Demicontractive operators form an important class of nonexpansive type mappings whose study led researchers to the creation of some beautiful results in the framework of metric fixed-point theory. This article aims to provide an overview of the most relevant results on the approximation of fixed points of single-valued demicontractive mappings in Hilbert spaces. Subsequently, we exhibit the role of additional properties of demicontractive operators, as well as the main features of the employed iterative algorithms to ensure weak convergence or strong convergence. We also include commentaries on the use of demicontractive mappings to solve some important nonlinear problems with the aim of providing a comprehensive starting point to readers who are attempting to apply demicontractive mappings to concrete applications. We conclude with some brief statements on our view on relevant and promising directions of research on demicontractive mappings in nonlinear settings (metric spaces) and some application challenges.

Once again on topological spaces containing a dense completely metrizable subset

We characterize the existence of a dense completely metrizable subset in a regular topological space X via the existence of residually defined continuous single-valued selections for certain set-valued mappings with values in X. We show that the property “X contains a dense completely metrizable subset” is preserved by continuous and demi-open single-valued mappings. We show also that the existence of residually defined continuous selections implies the Closed Graph and the Open Mapping theorems for completely metrizable topological vector spaces.

Quantitative Stability of Optimization Problems with Stochastic Constraints

In this paper, we consider optimization problems with stochastic constraints. We derive quantitative stability results for the optimal value function, the optimal solution set and the feasible solution set of optimization models in which the underlying stochastic constraints involve the mathematical expectation of random single-valued and set-valued mappings, respectively. New primal sufficient conditions are developed for the uniform error bound property of the stochastic constraint system for the single-valued case.