Search Results (84)

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

In this paper, we extend the study of contravariant Einstein-like metrics to Poisson doubly warped product manifolds (PDWPMs). We derive the necessary and sufficient conditions under which the base and fiber manifolds of a PDWPM inherit Einstein-like structures from the total space. As applications, we construct Einstein-like Poisson doubly warped product structures belonging to classes $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{P}$ in various spacetime models, including generalizations of Reissner–Nordström, standard static, and Robertson–Walker spacetimes. Full article

The aims of this paper are (a) to introduce the concept of the 0-complete m-metric spaces, (b) to obtain the results for $m^w$-Caristi mapping using Kirk’s approach, (c) to investigate the problem of non-cooperative equilibrium (abbreviated as NCE) in two- and three-person games in the structure of game theory and find the solution by employing coupled and tripled fixed-point results within the framework of 0-complete m-metric spaces (m-metric spaces, respectively), and (d) to establish some coupled fixed-point results which extend the scope of metric fixed point theory. We provide some examples to support the concepts and results presented in this paper. As an application of our results in this paper, we obtain the existence of a solution for a nonlinear integral equation. Full article

In this paper, we extend the concept of b-metric spaces to the vectorial case, where the distance is vector-valued, and the constant in the triangle inequality axiom is replaced by a matrix. For such spaces, we establish results analogous to those in the b-metric setting: fixed-point theorems, stability results, and a variant of Ekeland’s variational principle. As a consequence, we also derive a variant of Caristi’s fixed-point theorem. 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 a connected-image set for a mapping, which extends the notion of edge-preserving properties with respect to mapping. We also present novel definitions of connected-image contractions, with a focus on fixed-point theorems involving auxiliary functions in b-metric spaces. The relationships between these mathematical concepts are explored, along with their applications to solving differential and integral equations. In particular, we discuss existence results for solving integral equations and second-order ordinary differential equations with inhomogeneous Dirichlet boundary conditions, as well as theorems related to contractions of the integral type. Full article

In this paper, we introduce several new concepts: generalized neutrosophic rectangular b-metric-like spaces (GNRBMLSs), generalized intuitionistic rectangular b-metric-like spaces (GIRBMLSs), and generalized fuzzy rectangular b-metric-like spaces (GFRBMLSs). These innovative spaces can expand various topological spaces, including neutrosophic rectangular extended b-metric-like spaces, intuitionistic fuzzy rectangular extended b-metric-like spaces, and fuzzy rectangular extended b-metric-like spaces. Moreover, we establish Banach’s fixed point theorem and Ćirić’s quasi-contraction theorem with respect to these spaces, and we explore an application regarding the existence and uniqueness of solutions for fuzzy fractional delay integro-differential equations, as derived from our main results. 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

Background: The invasion of glioblastoma cells beyond the visible tumor margin depicted by conventional neuroimaging is believed to mediate recurrence and predict poor survival. Radiomic biomarkers that are associated with the direction and extent of tumor infiltration are, however, non-existent. Methods: Patients from a single center with newly diagnosed glioblastoma (n = 7) underwent preoperative Q-space magnetic resonance imaging (QSI; 3T, 64 gradient directions, b = 1000 s/mm²) between 2018 and 2019. Tumors were manually segmented, and patterns of inter-voxel coherence spatially intersecting each segmentation were generated to represent tumor-associated tractography. One patient additionally underwent regional biopsy of diffusion tract- versus non-tract-associated tissue during tumor resection for RNA sequencing. Imaging data from this cohort were compared with a historical cohort of n = 66 glioblastoma patients who underwent similar QSI scans. Associations of tractography-derived metrics with survival were assessed using t-tests, linear regression, and Kaplan–Meier statistics. Patient-derived glioblastoma xenograft (PDX) mice generated with the sub-hippocampal injection of human-derived glioblastoma stem cells (GSCs) were scanned under high-field conditions (QSI, 7T, 512 gradient directions), and tumor-associated tractography was compared with the 3D microscopic reconstruction of immunostained GSCs. Results: In the principal enrollment cohort of patients with glioblastoma, all cases displayed tractography patterns with tumor-intersecting tract bundles extending into brain parenchyma, a phenotype which was reproduced in PDX mice as well as in a larger comparison cohort of glioblastoma patients (n = 66), when applying similar methods. Reconstructed spatial patterns of GSCs in PDX mice closely mirrored tumor-associated tractography. On a Kaplan–Meier survival analysis of n = 66 patients, the calculated intra-tumoral mean diffusivity predicted the overall survival (p = 0.037), as did tractography-associated features including mean tract length (p = 0.039) and mean projecting tract length (p = 0.022). The RNA sequencing of human tissue samples (n = 13 tumor samples from a single patient) revealed the overexpression of transcripts which regulate cell motility in tract-associated samples. Conclusions: QSI discriminates tumor-specific patterns of inter-voxel coherence believed to represent white matter pathways which may be susceptible to glioblastoma invasion. These findings may lay the groundwork for future work on therapeutic targeting, patient stratification, and prognosis in glioblastoma. Full article

The notions of modular b-metric and modular b-metric space were introduced by Ege and Alaca as natural generalizations of the well-known and featured concepts of modular metric and modular metric space presented and discussed by Chistyakov. In particular, they stated generalized forms of Banach’s contraction principle for this new class of spaces thus initiating the study of the fixed point theory for these structures, where other authors have also made extensive contributions. In this paper we endow the modular b-metrics with a metrizable topology that supplies a firm endorsement of the idea of convergence proposed by Ege and Alaca in their article. Moreover, for a large class of modular b-metric spaces, we formulate this topology in terms of an explicitly defined b-metric, which extends both an important metrization theorem due to Chistyakov as well as the so-called topology of metric convergence. This approach allows us to characterize the completeness for this class of modular b-metric spaces that may be viewed as an offsetting of the celebrated Caristi–Kirk theorem to our context. We also include some examples that endorse our results. Full article

This study investigated the potential of petal colorimetric data to classify vase life stages in cut lisianthus flowers (Eustoma grandiflorum). We analyzed the changes in the petal color space over time, focusing on the b* value as an indicator of senescence due to increasing yellowing caused by copigmentation. A comparative analysis was conducted between two cultivation methods: soil (S) and hydroponic (H) cultivation. The objective was to evaluate the performance of machine learning models trained to classify vase life stages based on petal color data. Automated machine learning models exhibited better performance in H-cultivated cut flowers, effectively distinguishing days within the vase life stages from Days 1 to 14 for H cultivation. Cut flowers cultivated under S conditions showed less variation in the color space from Days 1 to 9, maintaining a relatively uniform color range. This made it more difficult to distinguish the vase life stages compared to H cultivation. These findings demonstrate that petal color metrics can serve as reliable indicators of cut flower senescence and potentially facilitate nondestructive methods for classifying vase life stages. This technology holds promise for wider applications in the floriculture industry, improving quality control, and extending the vase life of various cut-flower crops. Full article

This paper explores fixed points for both contractive and non-contractive mappings in traditional b-metric spaces, preordered b-metric spaces, and random b-metric spaces. Our findings provide insights into the behavior of mappings under various constraints and extend our approach to include coincidence and common fixed-point theorems in these spaces. We present new examples and graphical representations for the first time, offering novel results and enhancing several related findings in the literature, while broadening the scope of earlier works of Ran and Reurings, Nieto and Rodríguez-López, Górnicki, and others. Full article

Background: Diffusion-weighted imaging (DWI), a pivotal component of multiparametric magnetic resonance imaging (mpMRI), plays a pivotal role in the detection, diagnosis, and evaluation of gastric cancer. Despite its potential, DWI is often marred by substantial anatomical distortions and sensitivity artifacts, which can hinder its practical utility. Presently, enhancing DWI’s image quality necessitates reliance on cutting-edge hardware and extended scanning durations. The development of a rapid technique that optimally balances shortened acquisition time with improved image quality would have substantial clinical relevance. Objectives: This study aims to construct and evaluate the unsupervised learning framework called attention dual contrast vision transformer cyclegan (ADCVCGAN) for enhancing image quality and reducing scanning time in gastric DWI. Methods: The ADCVCGAN framework, proposed in this study, employs high b-value DWI (b = 1200 ${\mathrm{s}/\mathrm{mm}}^2$) as a reference for generating synthetic b-value DWI (s-DWI) from acquired lower b-value DWI (a-DWI, b = 800 ${\mathrm{s}/\mathrm{mm}}^2$). Specifically, ADCVCGAN incorporates an attention mechanism CBAM module into the CycleGAN generator to enhance feature extraction from the input a-DWI in both the channel and spatial dimensions. Subsequently, a vision transformer module, based on the U-net framework, is introduced to refine detailed features, aiming to produce s-DWI with image quality comparable to that of b-DWI. Finally, images from the source domain are added as negative samples to the discriminator, encouraging the discriminator to steer the generator towards synthesizing images distant from the source domain in the latent space, with the goal of generating more realistic s-DWI. The image quality of the s-DWI is quantitatively assessed using metrics such as the peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), feature similarity index (FSIM), mean squared error (MSE), weighted peak signal-to-noise ratio (WPSNR), and weighted mean squared error (WMSE). Subjective evaluations of different DWI images were conducted using the Wilcoxon signed-rank test. The reproducibility and consistency of b-ADC and s-ADC, calculated from b-DWI and s-DWI, respectively, were assessed using the intraclass correlation coefficient (ICC). A statistical significance level of p < 0.05 was considered. Results: The s-DWI generated by the unsupervised learning framework ADCVCGAN scored significantly higher than a-DWI in quantitative metrics such as PSNR, SSIM, FSIM, MSE, WPSNR, and WMSE, with statistical significance (p < 0.001). This performance is comparable to the optimal level achieved by the latest synthetic algorithms. Subjective scores for lesion visibility, image anatomical details, image distortion, and overall image quality were significantly higher for s-DWI and b-DWI compared to a-DWI (p < 0.001). At the same time, there was no significant difference between the scores of s-DWI and b-DWI (p > 0.05). The consistency of b-ADC and s-ADC readings was comparable among different readers (ICC: b-ADC 0.87–0.90; s-ADC 0.88–0.89, respectively). The repeatability of b-ADC and s-ADC readings by the same reader was also comparable (Reader1 ICC: b-ADC 0.85–0.86, s-ADC 0.85–0.93; Reader2 ICC: b-ADC 0.86–0.87, s-ADC 0.89–0.92, respectively). Conclusions: ADCVCGAN shows excellent promise in generating gastric cancer DWI images. It effectively reduces scanning time, improves image quality, and ensures the authenticity of s-DWI images and their s-ADC values, thus providing a basis for assisting clinical decision making. Full article

In this study, we significantly extend the concept of modular metric-like spaces to introduce the notion of b-metric-like spaces. Furthermore, by incorporating a binary relation $\mathfrak{R}$, we develop the framework of $\mathfrak{R}$-modular b-metric-like spaces. We establish a groundbreaking fixed point theorem for certain extensions of Geraghty-type contraction mappings, incorporating both $𝒵$ simulation function and $E$-type contraction within this innovative structure. Moreover, we present several novel outcomes that stem from our newly defined notations. Afterwards, we introduce an unprecedented concept, the graphical modular b-metric-like space, which is derived from the binary relation $\mathfrak{R}$. Finally, we examine the existence of solutions for a class of functional equations that are pivotal in dynamic programming and in solving initial value problems related to the electric current in an RLC parallel circuit. Full article

This paper aims to give an extended class of contractive mappings combining types of Suzuki contractions α-admissible mapping and Wardowski F-contractions in b-metric-like spaces. Our results cover and generalize many of the recent advanced results on the existence and uniqueness of fixed points and fulfill the Suzuki-type nonlinear hybrid contractions on various generalized metrics. Full article