Search Results (12)

The primary aim of this manuscript is to establish unique fixed point results for a class of $\Psi$-contraction operators in complete G-metric spaces. By combining and extending various fixed point theorems in the context of $\Psi$-contraction operators, we introduce a novel function, denoted as $\psi$, and explore its properties. Our work presents new theoretical results, supported by examples and applications, that enrich the study of G-metric spaces. These results not only generalize and unify a broad range of existing findings in the literature but also expand their use to boundary value problems, Fredholm-type integral equations, and nonlinear Caputo fractional differential equations. In doing so, we offer a more comprehensive understanding of fixed point theory in the G-metric space framework and broaden its scope in applied mathematics. We also offer a numerical spectral approach for solving fractional initial value problems, utilizing shifted Chebyshev polynomials to construct a semi-analytic solution that inherently satisfies the given homogeneous initial conditions. Full article

This article deals with a few outcomes ensuring the fixed points of a weak $(G,\psi )$-contraction map of metric spaces comprised with a reflexive and transitive digraph G. To validate our findings, we furnish several examples. The findings we obtain enable us to seek out the unique solution of a nonlinear integral equation. The outcomes presented herewith sharpen, subsume, unify, improve, enrich, and compile a number of existing theorems. Full article

One of the critical challenges facing the mining sector is related to the prevention and mitigation of catastrophic incidents associated with its tailing dams. As mining tailings are very heterogeneous and field characterization is expensive and complex, geotechnical properties of these materials are largely unknown. The seismic cone penetration test (SCPTu) provides a field approach to estimate a large array of geotechnical information, including the liquefaction potential of tailing dams. Yet, the exploration of strain softening behaviors in geomaterials under undrained loading, utilizing the state parameter ($\psi$) inferred from SCPTu tests initially applied to soft soils, has been often used for mining tailings. This study is concerned with the implementation of a tailing classification system which uses the ratio between the small strain shear modulus and the cone tip resistance (${\mathrm{G}}_0/{\mathrm{q}}_{\mathrm{t}}$). A series of laboratory tests was executed, and three different methodologies were adopted to assess the effects of (partial) drainage conditions based on 531.26 m of SCPTu measurements conducted at three different upstream iron ore tailing dams in Brazil. Furthermore, the ${\mathrm{G}}_0/{\mathrm{q}}_{\mathrm{t}}$ ratio is integrated with $\psi$ to assess the liquefaction tendencies of the investigated materials. The findings reveal the heterogeneous nature of the tailings, wherein indications of partial drainage are discernible across numerous records. Liquefaction analyses demonstrate that the tailings exhibit a contractive behavior in over 94% of the SCPTu data, confirming their susceptibility to flow liquefaction. Our findings are relevant for site characterization within iron ore tailing dams and other mining sites with similar geotechnical attributes. Full article

The state parameter allows the evaluation of the in situ state of soils, which can be particularly useful in tailings impoundments where flow liquefaction is the most common failure model. Positive state parameter values characterize a contractive response during shearing and, for non-plastic soils, can indicate flow liquefaction susceptibility. This paper presents a methodology to classify and estimate the state parameter (Ψ) for non-plastic silty soils based on seismic cone penetration measurements. The method expands on a previous methodology developed for sands that use the ratio of the small strain shear modulus and the cone tip resistance G₀/q_t for classification and Ψ assessment. For non-plastic silty soils, drainage conditions during cone penetration must be accounted for and are used to allow soil classification and correct the cone tip resistance. An empirical formulation is proposed to correct q_t for partial drainage measurements and predict Ψ for non-plastic silty soils. Mining tailings results of in situ and laboratory tests were used to validate the proposed methodology producing promising responses. The Ψ value estimated through the proposed methodology are in the range of those obtained from laboratory tests, indicating an adequate prediction of behavior for non-plastic silty soils. Full article

Analytical spherically symmetric static solution to the set of Einstein and Klein-Gordon equations in a synchronous reference frame is considered. In a synchronous reference frame, a static solution exists in the ultrarelativistic limit $p=-\epsilon /3$. Pressure p is negative when matter tends to contract. The solution pretends to describe a collapsed black hole. The balance at the boundary with dark matter ensures the static solution for a black hole. There is a spherical layer inside a black hole between two “gravitational” radii $r_g$ and $r_h>r_g$, where the solution exists, but it is not unique. In a synchronous reference frame, $\det g_{ik}$ and $g^{rr}$ do not change signs. The non-uniqueness of solutions with boundary conditions at $r=r_g$ and $r=r_h$ makes it possible to find the gravitational field both inside and outside a black hole. The synchronous reference frame allows one to find the remaining mass of the condensate. In the model “$\lambda {\left|\psi \right|}^4$”, total mass $M=3\left(c^2/2k\right)\hspace{0.17em}{r}_h$ is three times that of what a distant observer sees. This gravitational mass defect is spent for bosons to be in the bound ground state, and for the balance between elasticity and density of the condensate. Full article

The purpose of this paper is to present some new contraction mappings via control functions. In addition, some fixed point results for $\left(\mathrm{\Theta },\alpha ,\theta ,\mathrm{\Psi }\right)$ contraction, rational $\left(\mathrm{\Theta },\alpha ,\theta ,\mathrm{\Psi }\right)$ contraction and almost $\left(\mathrm{\Theta },\alpha ,\theta ,\mathrm{\Psi }\right)$ contraction mappings are obtained. Moreover, under contraction mappings of types (I), (II), and (III) of ${\left(\mathrm{\Theta },\theta ,\mathrm{\Psi }\right)}_{{\upsilon }_0}$, several fixed circle solutions are provided in the setting of a G-Metric space. Our results extend, unify, and generalize many previously published papers in this direction. In addition, some examples to show the reliability of our results are presented. Finally, a supporting application that discusses the possibility of a solution to a nonlinear integral equation is incorporated. Full article

The present article is devoted to prove some fixed point results for $(\psi ,\varphi )$-contractions in the framework of metric space equipped with a locally finitely $\mathcal{G}$-transitive relation. The results proved in this article improve and weaken some existing fixed point results available in the literature. Finally, an example is provided for attesting to the credibility of my results. Full article

The existence of a parametric fractional integral equation and its numerical solution is a big challenge in the field of applied mathematics. For this purpose, we generalize a special type of fixed-point theorems. The intention of this work is to prove fixed-point theorems for the class of $\beta -$G, $\psi -$G contractible operators of Darbo type and demonstrate the usability of obtaining results for solvability of fractional integral equations satisfying some local conditions in Banach space. In this process, some recent results have been generalized. As an application, we establish a set of conditions for the existence of a class of fractional integrals taking the parametric Riemann–Liouville formula. Moreover, we introduce numerical solutions of the class by using the set of fixed points. Full article

This paper is designed to display some results which generalize the recent results that cannot be established from the corresponding results in other spaces and do not satisfy the remarks of Jleli et al. (Fixed Point Theor Appl. 210, 2012) and Samet et al. (Int. J. Anal. Article ID 917158, 2013). We obtain unique fixed-point for mapping satisfying $\beta$-$\check{\psi }$ contraction only on a closed $G_d$ ball in complete dislocated $G_d$-metric space. An example is also discussed to shed light on the main result. Full article

We have introduced the new notions of R-weakly graph preserving and R-weakly $\alpha$ -admissible pair of multivalued mappings which includes the class of graph preserving mappings, weak graph preserving mappings as well as $\alpha$ -admissible mappings of type S, ${\alpha }_{*}$ -admissible mappings of type S and ${\alpha }_{*}$ - orbital admissible mappings of type S respectively. Some generalized contraction and rational contraction classes are also introduced for a pair of multivalued mappings and common fixed point theorems are proved in a b-metric space endowed with a graph. We have also applied our results to obtain common fixed point theorems for R-weakly $\alpha$ -admissible pair of multivalued mappings in a b-metric space which are the proper extension and generalization of many known results. Proper examples are provided in support of our results. Our main results and its consequences improve, generalize and extend many known fixed point results existing in literature. Full article

In this paper, we introduce the $(C,{\Psi }^{*},G)$ class of contraction mappings using C-class functions and some improved control functions for a pair of set valued mappings as well as a pair of single-valued mappings, and prove common fixed point theorems for such mappings in a metric space endowed with a graph. Our results unify and generalize many important fixed point results existing in literature. As an application of our main result, we have derived fixed point theorems for a pair of $\alpha$ -admissible set valued mappings in a metric space. Full article

Inspired by a metrical-fixed point theorem from Choudhury et al. (Nonlinear Anal. 2011, 74, 2116–2126), we prove some order-theoretic results which generalize several core results of the existing literature, especially the two main results of Harjani and Sadarangani (Nonlinear Anal. 2009, 71, 3403–3410 and 2010, 72, 1188–1197). We demonstrate the realized improvement obtained in our results by using a suitable example. As an application, we also prove a result for mappings satisfying integral type ${(\mathsf{\psi },\mathsf{\phi })}_g$ -generalized weakly contractive conditions. Full article