Search Results (41)

Since the monotonicity of the best approximant is crucial to establish partial ordering methods, in this paper, we, respectively, characterize the best approximants in Banach function spaces and Lorentz spaces ${\mathrm{\Gamma }}_{p,w}$, in which we especially focus on the monotonicity characterizations. We first study monotonicity characterizations of the metric projection operator onto sublattices in general Banach function spaces by the property $H_g$. The sufficient and necessary conditions for monotonicity of the metric projection onto cones and sublattices are then, respectively, established in ${\mathrm{\Gamma }}_{p,w}$. The Lorentz spaces ${\mathrm{\Gamma }}_{p,w}$ are also shown to be reflexive under the condition $R{B}_p$, which is the basis for the existence of the best approximant. As applications, by establishing the partial ordering methods based on the obtained monotonicity characterizations, the solvability and approximation theorems for best proximity points are deduced without imposing any contractive and compact conditions in ${\mathrm{\Gamma }}_{p,w}$. Our results extend and improve many previous results in the field of the approximation and partial ordering theory. Full article

This study introduces a new metric structure called the Graphical Fuzzy Cone Metric Space (GFCMS) and explores its essential properties in detail. We examine its topological aspects in detail and introduce the notion of Hausdorff distance within this setting—an advancement not previously explored in any graphical structure. Furthermore, a fixed-point result is proven within the framework of GFCMS, accompanied by examples that demonstrate the applicability of the theoretical results. As a significant application, we construct fractals within GFCMS, marking the first instance of fractal generation in a graphical structure. This pioneering work opens new avenues for research in graph theory, fuzzy metric spaces, topology, and fractal geometry, with promising implications for diverse scientific and computational domains. Full article

In this monograph, motivated by the work of Aphane, Gaba, and Xu, we explore fixed-point theory within the framework of $C^{\star }$-algebra-valued bipolar b-metric spaces, characterized by a non-solid positive cone. We define and analyze $(F_{\mathcal{H}}-G_{\mathcal{H}})$-contractions, utilizing positive monotone functions to extend classical contraction principles. Key contributions include the existence and uniqueness of fixed points for mappings satisfying generalized contraction conditions. The interplay between the non-solidness of the cone, the $C^{\star }$-algebra structure, and the completeness of the space is central to our results. We apply our results to find uniqueness of solutions to Fredholm integral equations and differential equations, and we extend the Ulam–Hyers stability problem to non-solid cones. This work advances the theory of metric spaces over Banach algebras, providing foundational insights with applications in operator theory and quantum mechanics. Full article

This paper introduces a new concept of random Sehgal contraction in the setting of random cone metric spaces. We explore the modern advancements of traditional fixed-point theorems in a random setting, elaborating on the Sehgal–Guseman fixed-point theorem within the realm of random cone metric spaces. A significant aspect of our research is the interplay between symmetry and randomness; while symmetry provides a framework for understanding structural properties, randomness introduces complexity, which can lead to unexpected behaviors. Our research provides a deeper understanding of the classical results and incorporates a detailed example to illustrate our findings. In addition, major random fixed-point results are also established, which could be applied to nonlinear random fractional differential equations (FDEs) and integral equations as well as to random boundary value problems (BVPs) related to homogeneous random transverse bars. Full article

This paper introduces the idea of a cone m-hemi metric space, which extends the idea of an m-hemi metric space. By presenting non-trivial examples, we demonstrate the superiority of cone m-hemi metric spaces over m-hemi metric spaces. Further, we extend the Banach contraction principle and Krasnoselskii, Meir–Keeler, Boyd–Wong, and some other fixed-point results in the setting of complete and compact cone m-hemi metric spaces. Furthermore, we provide several non-trivial examples and applications to the Fredholm integral equation and dynamic market equilibrium to demonstrate the validity of the main results. Full article

The objective of this research article is to introduce Kikkawa and Suzuki-type contractions in the setting of topological vector space-valued cone metric space with a solid cone and establish some new fixed point results for multi-valued mappings. The problem of finding fixed points for multi-valued mappings satisfying locally contractive conditions on a closed ball is also addressed. Our findings generalize a number of well-established results in the literature. To highlight the uniqueness of our key finding, we present an example. As a demonstration of the applicability of our principal theorem, we prove a result in homotopy theory. Full article

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 ${\mathsf{\Phi }}_s$ functions in the b-metric space to the extended b-metric space and the cone b-metric space, we introduce the class of ${\mathsf{\Phi }}_M$ 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. Full article

Motivated by two definitions of distance, “pre-symmetric w-distance” and “w-cone distance”, we define the concept of a pre-symmetric w-cone distance in a TVS-CMS and introduce its properties and examples. Also, we discuss the TVS-cone version of the recent results obtained by Romaguera and Tirado. Meanwhile, using Minkowski functionals, we show the equivalency between some consequences concerning a pre-symmetric w-distance in a usual metric space and a pre-symmetric w-cone distance in a TVS-CMS. Then, some types of various w-cone-contractions and the relations among them are investigated. Finally, as an application, a characterization of the completeness of TVS-cone metric regarding pre-symmetric concept is performed, which differentiates our results from former characterizations. Full article

Background: In this study, we present a quantitative method to evaluate the motion artifact correction (MAC) technique through the morphological analysis of blood vessels in the images before and after MAC. Methods: Cone-beam computed tomography (CBCT) scans of 37 patients who underwent transcatheter chemoembolization were obtained, and images were reconstructed with and without the MAC technique. First, two interventional radiologists selected the blood vessels corrected by MAC. We devised a motion-corrected index (MCI) metric that analyzed the morphology of blood vessels in 3D space using information on the centerline of blood vessels, and the blood vessels selected by the interventional radiologists were quantitatively evaluated using MCI. In addition, these blood vessels were qualitatively evaluated by two interventional radiologists. To validate the effectiveness of the devised MCI, we compared the MCI values in a blood vessel corrected by MAC and one non-corrected by MAC. Results: The visual evaluation revealed that motion correction was found in the images of 23 of 37 patients (62.2%), and a performance evaluation of MAC was performed with 54 blood vessels in 23 patients. The visual grading analysis score was 1.56 ± 0.57 (radiologist 1) and 1.56 ± 0.63 (radiologist 2), and the proposed MCI was 0.67 ± 0.11, indicating that the vascular morphology was well corrected by the MAC. Conclusions: We verified that our proposed method is useful for evaluating the MAC technique of CBCT, and the MAC technique can correct the blood vessels distorted by the patient’s movement and respiration. Full article

The purpose of this paper is to generalize the concept of classical fuzzy set to vector-valued fuzzy set which can attend values not only in the real interval $[0, 1]$, but in an ordered interval of a Banach algebra as well. This notion allows us to introduce the concept of vector-valued fuzzy metric space which generalizes, extends and unifies the notion of classical fuzzy metric space and complex-valued fuzzy metric space and permits us to consider the fuzzy sets and metrics in a larger domain. Some topological properties of such spaces are discussed and some fixed point results in this new setting are proved. Multifarious examples are presented which clarify and justify our claims and results. Full article

We geometrically derive the explicit form of the unitary representation of the Poincaré group for vector-valued wave functions and use it to apply speed-of-light boosts to a simple polarization basis to end up with a Hawton–Baylis photon position operator with commuting components. We give explicit formulas for other photon boost eigenmodes. We investigate the underlying affine connections on the light cone in momentum space and find that while the Pryce connection is metric semi-symmetric, the flat Hawton–Baylis connection is not semi-symmetric. Finally, we discuss the localizability of photon states on closed loops and show that photon states on the circle, both unnormalized improper states and finite-norm wave packet smeared-over washer-like regions are strictly localized not only with respect to Hawton–Baylis operators with commuting components but also with respect to the noncommutative Jauch–Piron–Amrein POV measure. Full article

The equation $G(x,x)=\tilde{y},$ where $G:X\times X\to Y,$ and $X,Y$ are vector metric spaces (meaning that the values of a distance between the points in these spaces belong to some cones $E_{+},M_{+}$ of a Banach space E and a linear space M, respectively), is considered. This operator equation is compared with a “model” equation, namely, $g(t,t)=0,$ where a continuous map $g:E_{+}\times E_{+}\to M_{+}$ is orderly covering in the first argument and antitone in the second one. The idea to study equations comparing them with “simpler” ones goes back to the Kantorovich fixed-point theorem for an operator acting in a Banach space. In this paper, the conditions under which the solvability of the “model” equation guarantees the existence of solutions to the operator equation are obtained. The statement proved extends the recent results about fixed points and coincidence points to more general equations in more general vector metric spaces. The results obtained for the operator equation are then applied to the study of the solvability, as well as to finding solution estimates, of the Cauchy problem for a functional differential equation. Full article

When we consider a non-definite pseudo-Riemannian manifold, we obtain lightlike tangent vectors that constitute the null tangent bundle, whose fibers are lightlike cones in the corresponding tangent spaces. In this paper, we define and study a class of “g-natural” metrics on the tangent bundle of a pseudo-Riemannian manifold and we investigate the geometry of the null tangent bundle as a lightlike hypersurface equipped with an induced g-natural metric. Full article

In the present paper, we discuss a nonlocal modification of the Kerr metric. Our starting point is the Kerr–Schild form of the Kerr metric $g_{\mu \nu }={\eta }_{\mu \nu }+\Phi l_{\mu }{l}_{\mu }$. Using Newman’s approach, we identify a shear free null congruence $\mathit{l}$ with the generators of the null cone with apex at a point p in the complex space. The Kerr metric is obtained if the potential $\Phi$ is chosen to be a solution of the flat Laplace equation for a point source at the apex p. To construct the nonlocal modification of the Kerr metric, we modify the Laplace operator $▵$ by its nonlocal version $exp(-{\ell }^2▵)▵$. We found the potential $\Phi$ in such an infinite derivative (nonlocal) model and used it to construct the sought-for nonlocal modification of the Kerr metric. The properties of the rotating black holes in this model are discussed. In particular, we derived and numerically solved the equation for a shift of the position of the event horizon due to nonlocality. AlbertaThy 5–23. Full article

In this manuscript, the nonlinear Burgers equations are studied via a fractal fractional (FF) Caputo operator. The results of coupled fixed point theorems in cone metric space are used to discuss the uniqueness of solution to the proposed coupled equations. The solution of the proposed equation is computed via Natural transform associated with the Adomian decomposition method (NADM). The acquired results are graphically presented for some values of fractional order and fractal dimensions. The accuracy and consistency of the applied method is verified through error analysis. Full article