P.k. Jain And K Ahmad Metric Spaces Pdf Free 'LINK' Download

In this manuscript, the existence theorem for a unique solution to a coupled system of impulsive fractional differential equations in complex-valued fuzzy metric spaces is studied and the fuzzy version of some fixed point results by using the definition and properties of a complex-valued fuzzy metric space is presented. Ultimately, some appropriate examples are constructed to illustrate our theoretical results.

In addition to the FMS, there are still several extensions of metric space terms in metric FP theory. Azam et al. [16] established a new approach by replacing the set of real numbers with the set of complex numbers endowed with an ordered structure. In this regard, they introduced the notion of complex-valued metric spaces (CVMSs). In recent years, Shukla et al. [17] extended the setting of FMSs to complex-valued fuzzy metric spaces (CVFMSs). They also obtained FP theorems of contractive mappings on CVFMSs.

P.k. Jain And K Ahmad Metric Spaces Pdf Free Download

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Motivated by the above discussion, in this manuscript, some coupled fixed point consequences in the framework of complex-valued fuzzy metric spaces are proved. Thereafter, the theoretical results are involved to find the existence and uniqueness of a coupled system of nonlinear fractional differential equations with impulses in the form of

Due to the many applications in which the fixed point method is involved, this is an important pillar and a good tool in nonlinear analysis. This technique mainly participates in the complex analysis in terms of studying the existence and uniqueness of the FP within the various spaces that include complex numbers such as complex-valued metric space, complex-valued b-metric space, etc. In the setting of the fuzzy set framework, this formalism is devoted to obtaining the FP for single and multivalued mappings via suitable conditions. Nonlinear analysis and fixed point theory play a prominent role in many fields of mathematics, and by the technique of FP, we can solve several existing problems in mathematics. Blood flow systems, aerodynamics, the nonlinear oscillation of earthquake, the fluid-dynamic traffic model, and control theory both can be studied mathematically by fractional calculus. Because of that, this discipline has turned around many researchers and readers especially when we deal with these problems in a fixed point fashion. One of the important branches of fractional calculus is impulsive fractional differential equations since the pulse effect is significant in many processes and phenomena. For example, in biological systems such as heartbeats, blood flows, mechanical systems with impact, population dynamical systems, and so on. By successful applications of our derived results, we have studied the existence and uniqueness solutions for a coupled system of impulsive fractional differential equations via coupled FP techniques in the setting of CVFMSs. In addition, some theoretical results and nontrivial examples have been also presented.

From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. The spectral theorem provides a canonical form for symmetric, unitary and more generally normal operators on finite dimensional inner product spaces. A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces.[13]

Minkowski addressed the problems in relativity theory by the use of the geometry, now known as Minkowski geometry. He generalized Riemannian-metric problems into Lorentz-Minkowski spaces using a pseudo-Riemannian metric. In Minkowski space-, vectors are divided into timelike, lightlike, and spacelike by the Lorentz-Minkowski metric. These vectors have a causal nature, which can make seemingly straightforward problems complicated, particularly those involving null-vectors, such as pseudo-null-curves, null-curves, marginally trapped surfaces, and B-scrolls. One issue is the inability to accurately measure angles related to lightlike-vectors, which limits some studies. On the non-degenerate surfaces, timelike surfaces, and spacelike surfaces in Minkowski space-, numerous investigations have been done. For example, Treibergs [43] has investigated spacelike hypersurfaces of constant mean-curvature in the Minkowski space-. For timelike-surfaces with a defined Gauss-map, Aledo et al. [44] get a Lelievvre-type representation. In the Minkowski space-, Abdel-Baky and Abd-Ellah [45] investigate both (spacelike and timelike) governed W-surfaces. Brander et al. [46] used the non-compact real form SU to construct spacelike constant mean-curvature surfaces in the Minkowski space-. Lin [47] studied the impacts of curvature restrictions on timelike-surfaces in the Minkowski space- that are convex in the same way as are the surfaces in the Euclidean space-. Kossowski [48] explored zero mean-curvature surface constraints in the Euclidean space-. In his work, Georgiev [49] found sufficient conditions for Bzier surfaces to be spacelike. Ugail et al. [50] analyzed Bzier surfaces in the three-dimensional Minkowski space-, considering both timelike and spacelike cases, and sought to determine the surfaces that are extremals of the Dirichlet functional. Kuak Samanc and Celik [51] presented a geometric viewpoint of Bzier surfaces in Minkowski space and determined the shape operator of both timelike and spacelike Bzier surfaces in . In this work, we find the fundamental coefficients, Gauss-curvature, mean-curvature, and shape-operator of the timelike and spacelike shifted-knots Bzier surface. The results are then used to illustrate the scheme for the associated shape-operator of the timelike and spacelike bi-quadratic and bi-cubic SKBS in the Minkowski space-. 17dc91bb1f