Banach's Theorem Research Articles

On the Recovery of a Conformable Time-Dependent Inverse Coefficient Problem for Diffusion Equation of Periodic Constraints Type and Integral Over-Posed Data

In the utilized analysis, we consider the inverse coefficient problem of recovering the time-dependent diffusion coefficient along the solution of the conformable time-diffusion equation subject to periodic boundary conditions and an integral over-posed data. Along with this, the conformable time derivative with order 0 < η ≤ 1 is defined in the sense of a limit operator. The formal solution set for the considered inverse coefficient conformable problem is acquired via utilizing the Fourier expansion method. Under some conditions on the data and applicability of the Banach theorem, we insured the existence and uniqueness of the regular solution. Continuous dependence of the solutions set q t , u x , t in the given data is shown. Couples of illustrative examples in the form of data results and computational figures are also utilized. Future remarks, highlights, and work results are epitomized in the penultimate part. Finally, some latest used and focused references are given.

Some Fixed Point Theorems in Extended Cone \(b\)-Metric Spaces

In this paper, the notion of extended cone \(b\)-metric space is introduced, established the structure of the open ball and defined the notion of convergence of a sequence. Finally, restructured the Banach and Kannan contraction theorems without the normality condition in this new setting.

Convexity, Markov Operators, Approximation, and Related Optimization

The present review paper provides recent results on convexity and its applications to the constrained extension of linear operators, motivated by the existence of subgradients of continuous convex operators, the Markov moment problem and related Markov operators, approximation using the Krein–Milman theorem, related optimization, and polynomial approximation on unbounded subsets. In many cases, the Mazur–Orlicz theorem also leads to Markov operators as solutions. The common point of all these results is the Hahn–Banach theorem and its consequences, supplied by specific results in polynomial approximation. All these theorems or their proofs essentially involve the notion of convexity.

GLOBAL ATTRACTIVITY, ASYMPTOTIC STABILITY AND BLOW-UP POINTS FOR NONLINEAR FUNCTIONAL-INTEGRAL EQUATIONS’ SOLUTIONS AND APPLICATIONS IN BANACH SPACE BC(R+) WITH COMPUTATIONAL COMPLEXITY

Nonlinear functional-integral equations contain the unknown function nonlinearly, occurring extensively in theory developed to a certain extent toward different applied problems and solutions thereof. Nonlinear science serves to reveal the nonlinear descriptions of widely different systems, has had a fundamental impact on complex dynamics. Accordingly, this study provides the basic definitions and results regarding Banach spaces with the proof as well as applications. Besides, proving the existence and uniqueness of the solutions by the Banach theorem is carried out. The existence results of the equations are generally obtained based on fundamental methods by which the fixed-point theorems are frequently applied. Subsequently, the definition of the measure pertaining to noncompactness on Banach space along with properties is presented. Hence, the aim is to study nonlinear functional-integral equations in the Banach space [Formula: see text] using the measure of noncompactness. In addition, the proof of the existence of global attractors and asymptotic stabilities, having at least one solution which pertains to [Formula: see text] is looked into. As the last stage, the definitions of global attractivity and asymptotical stability for equations on Banach space [Formula: see text] are given while proving the existence of solutions and global attractivity. Besides these, the existence and asymptotical stability are proven. Furthermore, the proof of the existence of global attractors and asymptotic stabilities, having at least one solution which pertains to [Formula: see text] for an example nonlinear integral equation has been demonstrated by one example premediated and the existence of blow-up point has been examined in one and more than one points regarding the nonlinear integral equation. The example nonlinear integral equation as addressed in this study has been examined in terms of its blow-up point and asymptotic stability with computational complexity for the first time. By doing so, it has been demonstrated that the existence of blow-up point has been the case in one and more than one points and besides this, the examination of its asymptotic stability has shown that it is asymptotically stable at least in one point, yet has no blow-up point pertaining to [Formula: see text]. This has enabled thorough analysis of computational complexity of the related nonlinear functional-integral equations within the example both addressing dimension of input in Banach space and that of dataset with [Formula: see text] dimension. This applicable approach assures the accurate transformation of the complex problems towards optimized solutions through the generation of advanced mathematical models in nonlinear dynamic settings characterized by complexity.

Analysis of a semi-augmented mixed finite element method for double-diffusive natural convection in porous media

In this paper we study a stationary double-diffusive natural convection problem in porous media given by a Navier-Stokes/Brinkman type system, for describing the velocity and the pressure, coupled to a vector advection-diffusion equation relate to the heat and substance concentration, of a viscous fluid in a porous media with physical boundary conditions. The model problem is rewritten in terms of a first-order system, without the pressure, based on the introduction of the strain tensor and a nonlinear pseudo-stress tensor in the fluid equations. After a variational approach, the resulting weak model is then augmented using appropriate redundant penalization terms for the fluid equations along with a standard primal formulation for the heat and substance concentration. Then, it is rewritten as an equivalent fixed-point problem. Well-posedness results for both the continuous and the discrete schemes are stated, as well as the respective convergence result under certain regularity assumptions combined with the Lax-Milgram theorem, and the Banach and Brouwer fixed-point theorems. In particular, Raviart-Thomas elements of order k are used for approximating the pseudo-stress tensor, piecewise polynomials of degree ≤k and ≤k+1 are utilized for approximating the strain tensor and the velocity, respectively, and the heat and substance concentration are approximated by means of Lagrange finite elements of order ≤k+1. Optimal a priori error estimates are derived and confirmed through some numerical examples that illustrate the performance of the proposed semi-augmented mixed-primal scheme.

Study of Nonlinear Fractional Order Delay Problem Under Mittag-Leffler Power Law

In the concerned article, the author's developed indispensable conditions for theresults of existence theory and Ulam type stability of the intended pantograph fractional differential equation. The desired results for existence of the solutions and stability analysis are investigated with a new class of fractional derivative introduced by Atangana-Baleanu-Caputo and so abbreviated as (ABC). The mentioned derivative contains Mittag-Leffler function as a nonsingular kernel instead of singular kernel. We use the tools of fixed point theory, i.e Banach's theorem and Krasnoselskii's type to develop the desired results. Furthermore, we investigate different types of stabilities results including generalized Ullam-Hyers and Rassias stabilities for the concerned problem. We also provide appropriate examples for illustrative purposes.

Classical vs. non-Archimedean analysis: an approach via algebraic genericity

In this paper, we show new results and improvements of the non-Archimedean counterpart of classical analysis in the theory of lineability. Besides analyzing the algebraic genericity of sets of functions having properties regarding continuity, discontinuity, Lipschitzianity, differentiability and analyticity, we also study the lineability of sets of sequences having properties concerning boundedness and convergence. In particular we show (among several other results) the algebraic genericity of: (i) functions that do not satisfy Liouville’s theorem, (ii) sequences that do not satisfy the classical theorem of Cèsaro, or (iii) functionals that do not satisfy the classical Hahn–Banach theorem.

On the Correlation between Banach Contraction Principle and Caristi’s Fixed Point Theorem in b-Metric Spaces

We solve a question posed by E. Karapinar, F. Khojasteh and Z.D. Mitrović in their paper “A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces”. We also characterize the completeness of b-metric spaces with the help of a variant of the contractivity condition introduced by the authors in the aforementioned article.

Algorithm of automatic digital cartographic generalisation with the use of contractive self-mapping

Abstract The research of modern cartography in the field of digital generalisation focuses on the development of such methods that would be fully automatic and give an unambiguously objective result. Devising them requires specific standards as well as unique and verifiable algorithms. In metric space, a proposal for such a method, based on contractive mapping, the Lipschitz and Cauchy conditions and the Banach theorem, using the Salishchev metric, was presented in the publication (Barańska et al., 2021). The method formulated there is dedicated to linear objects (polylines). The current work is a practical supplement to it. It presents the practical implementation of the algorithm for automatic and objective generalisation. The article describes an operational diagram of the subsequent stages of the proposed generalisation method. In the test example, a binary tree structure of an ordered polyline was created. It was simplified in two selected scales and its shape after generalisation was illustrated. The resulting polyline obtained by the fully automatic method was verified in terms of accuracy.

Comparison of the results of calculating sin1 ° using the al-Kashi method and modern technologies

The article discusses methods for calculating the value of sin1 ° using the al-Kashi method and interpreted object-oriented programming language Python. The calculation of the value by each method is presented and the comparison of the obtained results is presented. The convergence of the al-Kashi method using Banach's theorem has been established.

A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman–Forchheimer and double-diffusion equations

We propose and analyze a new mixed finite element method for the nonlinear problem given by the coupling of the steady Brinkman–Forchheimer and double-diffusion equations. Besides the velocity, temperature, and concentration, our approach introduces the velocity gradient, the pseudostress tensor, and a pair of vectors involving the temperature/concentration, its gradient and the velocity, as further unknowns. As a consequence, we obtain a fully mixed variational formulation presenting a Banach spaces framework in each set of equations. In this way, and differently from the techniques previously developed for this and related coupled problems, no augmentation procedure needs to be incorporated now into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Banach theorem, combined with classical results on nonlinear monotone operators and Babuška–Brezzi’s theory in Banach spaces, are applied to prove the unique solvability of the continuous and discrete systems. Appropriate finite element subspaces satisfying the required discrete inf-sup conditions are specified, and optimala priorierror estimates are derived. Several numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method.

Fractional dynamics and stability analysis of COVID-19 pandemic model under the harmonic mean type incidence rate

In this research, COVID-19 model is formulated by incorporating harmonic mean type incidence rate which is more realistic in average speed. Basic reproduction number, equilibrium points, and stability of the proposed model is established under certain conditions. Runge-Kutta fourth order approximation is used to solve the deterministic model. The model is then fractionalized by using Caputo-Fabrizio derivative and the existence and uniqueness of the solution are proved by using Banach and Leray-Schauder alternative type theorems. For the fractional numerical simulations, we use the Adam-Moulton scheme. Sensitivity analysis of the proposed deterministic model is studied to identify those parameters which are highly influential on basic reproduction number.

Representation of uniform boundedness principle and Hahn–Banach theorem in linear n-normed space

The concept of b-linear functional and its different types of continuity in linear n-normed space are presented and some of their properties are being established. We derive the Uniform Boundedness Principle and Hahn-Banach extension Theorem with the help of bounded b-linear functionals in the case of linear n-normed spaces and discuss some examples and applications. Finally, we present the concept of weak*convergence for the sequence of bounded b-linear functionals in linear n-normed space.

On a class of boundary value problems under ABC fractional derivative

In this work, we establish some necessary results about existence theory to a class of boundary value problems (BVPs) of hybrid fractional differential equations (HFDEs) in the frame of Atangana–Baleanu–Caputo (ABC) fractional derivative. Making use of Krasnoselskii and Banach theorems, we obtain the required conditions. Some appropriate results of Hyers–Ulam (H–U) stability corresponding to the considered problem are also established. Also a pertinent example is given to demonstrate the results.

An Lpspaces-based formulation yielding a new fully mixed finite element method for the coupled Darcy and heat equations

AbstractIn this work we present and analyse a new fully mixed finite element method for the nonlinear problem given by the coupling of the Darcy and heat equations. Besides the velocity, pressure and temperature variables of the fluid, our approach is based on the introduction of the pseudoheat flux as a further unknown. As a consequence of it, and due to the convective term involving the velocity and the temperature, we arrive at saddle point-type schemes in Banach spaces for both equations. In particular, and as suggested by the solvability of a related Neumann problem to be employed in the analysis, we need to make convenient choices of the Lebesgue and ${\textrm {H}}(div)$-type spaces to which the unknowns and test functions belong. The resulting coupled formulation is then written equivalently as a fixed-point operator, so that the classical Banach theorem, combined with the corresponding Babuška–Brezzi theory, the Banach–Nečas–Babuška theorem, suitable operators mapping Lebesgue spaces into themselves, regularity assumptions and the aforementioned Neumann problem, are employed to establish the unique solvability of the continuous formulation. Under standard hypotheses satisfied by generic finite element subspaces, the associated Galerkin scheme is analysed similarly and the Brouwer theorem yields existence of a solution. The respective a priori error analysis is also derived. Then, Raviart–Thomas elements of order $k\ge 0$ for the pseudoheat and the velocity and discontinuous piecewise polynomials of degree $\le k$ for the pressure and the temperature are shown to satisfy those hypotheses in the two-dimensional case. Several numerical examples illustrating the performance and convergence of the method are reported, including an application into the equivalent problem of miscible displacement in porous media.

On the pillars of Functional Analysis

Many authors consider that the main pillars of Functional Analysis are the Hahn–Banach Theorem, the Uniform Boundedness Principle and the Open Mapping Principle. The first one is derived from Zorn’s Lemma, while the latter two usually are obtained from Baire’s Category Theorem. In this paper we show that these three pillars should be either just two or at least eight, since the Uniform Boundedness Principle, the Open Mapping Principle and another five theorems are equivalent, as we show in a very elemental way. Since one can give an almost trivial proof of the Uniform Boundedness Principle that does not require the Baire’s theorem, we conclude that this is also the case for the other equivalent theorems that, in this way, are simultaneously proved in a simple, brief and concise way that sheds light on their nature.

Order Versions of the Hahn–Banach Theorem and Envelopes. II. Applications to Function Theory

Order Versions of the Hahn–Banach Theorem and Envelopes. II. Applications to Function Theory

Convex geometry of Markovian Lindblad dynamics and witnessing non-Markovianity

We develop a theory of linear witnesses for detecting non-Markovianity, based on the geometric structure of the set of Choi states for all Markovian evolutions having Lindblad-type generators. We show that the set of all such Markovian Choi states form a convex and compact set under the small time interval approximation. Invoking geometric Hahn–Banach theorem, we construct linear witnesses to separate a given non-Markovian Choi state from the set of Markovian Choi states. We present examples of such witnesses for dephasing channel and Pauli channel in case of qubits. Furthermore, we have devised another method of detection NM of various qubit channels, by using projective measurements in the Bell basis. This can be done without the knowledge of what specific kind of channel we are dealing with. This gives us a huge operational advantage for NM detection. We further investigate the geometric structure of the Markovian Choi states to find that they do not form a polytope. This presents a platform to consider nonlinear improvement of non-Markovianity witnesses.

A Perov Version of Fuzzy Metric Spaces and Common Fixed Points for Compatible Mappings

In this paper, we define and study the Perov fuzzy metric space and the topology induced by this space. We prove Banach contraction theorems. Moreover, we devised new results for Kramosil and Michálek fuzzy metric spaces. In the process, some results about multidimensional common fixed points as coupled/tripled common fixed point results are derived from our main results.

Quadruple fixed point result in m-metric space

The Banach Contraction theorem is the basis of the theory of fixed points. Many scholars have been working in different directions on the Banach Theorem. Further theorems on coupled, tripled and quadruple fixed points are established. A quadruple fixed point result was developed in this paper for partial order M-metric space using a monotone property. This theorem is the extension and alteration of literary outcomes. An illustration to verify the outcome has been given.Mathematics Subject Classification: 46T99, 54H25, 47H10, 54E50