Banach Space Research Articles

Recurrence formula for some higher order evolution equations

Riccati’s differential equation is formulated as abstract equation in finite or infinite dimensional Banach spaces. Since the Riccati’s differential equation with the Cole–Hopf transform shows a relation between the first order evolution equations and the second order evolution equations, its generalization suggests the existence of recurrence formula leading to a sequence of differential equations with different order. In conclusion, by means of the logarithmic representation of operators, a transform between the first order evolution equations and the higher order evolution equation is presented. Several classes of evolution equations with different orders are given, and some of them are shown as examples.

Integro-Differential Equations in Banach Spaces and Analytic Resolving Families of Operators

Integro-Differential Equations in Banach Spaces and Analytic Resolving Families of Operators

Krasnosel’skii iterative process for approximating fixed points of generalized Bianchini mappings in Banach space and applications to variational inequality and split feasibility problems

Krasnosel’skii iterative process for approximating fixed points of generalized Bianchini mappings in Banach space and applications to variational inequality and split feasibility problems

Fast hybrid method for solving variational inequalities beyond monotonicity in real Banach spaces

In this work, we introduce a new inertial hybrid algorithm that combines the subgradient extragradient, Halpern, and S-iteration methods. We provide an approximate solution to variational inequality and common fixed point problem of relatively non-expansive mappings in real Banach spaces, where our cost operator is Lipschitz continuous and quasimonotone, and our step size is self-adaptive, non-monotone, and non-reliant on any line-search principle. We provide two strong convergence results with (and without) recourse to the monotonicity property, and further conduct several relevant numerical experiments to reveal the efficiency of our method over others in the literature.

Approximating fixed point results for pseudo-contractive map and some remarks on demi-contractive map in the Banach space

Approximating fixed point results for pseudo-contractive map and some remarks on demi-contractive map in the Banach space

Hyers-Ulam stability of a-type additive functional equations in Banach space using direct method

Hyers-Ulam stability of a-type additive functional equations in Banach space using direct method

Long-term accuracy of numerical approximations of SPDEs with the stochastic Navier–Stokes equations as a paradigm

Abstract This work introduces a general framework for establishing the long time accuracy for approximations of Markovian dynamical systems on separable Banach spaces. Our results illuminate the role that a certain uniformity in Wasserstein contraction rates for the approximating dynamics bears on long time accuracy estimates. In particular, our approach yields weak consistency bounds on ${\mathbb{R}}^{+}$ while providing a means to sidestepping a commonly occurring situation where certain higher order moment bounds are unavailable for the approximating dynamics. Additionally, to facilitate the analytical core of our approach, we develop a refinement of certain ‘weak Harris theorems’. This extension expands the scope of applicability of such Wasserstein contraction estimates to a variety of interesting stochastic partial differential equation examples involving weaker dissipation or stronger nonlinearity than would be covered by the existing literature. As a guiding and paradigmatic example, we apply our formalism to the stochastic 2D Navier–Stokes equations and to a semi-implicit in time and spectral Galerkin in space numerical approximation of this system. In the case of a numerical approximation, we establish quantitative estimates on the approximation of invariant measures as well as prove weak consistency on ${\mathbb{R}}^{+}$. To develop these numerical analysis results, we provide a refinement of $L^{2}_{x}$ accuracy bounds in comparison to the existing literature, which are results of independent interest.

On obtaining convergence order of a fourth and sixth order method of Hueso et al. without using Taylor series expansion

Hueso et al. (2015) studied the fourth and sixth order methods to approximate a solution of a nonlinear equation in Rn, where the convergence analysis needs the involved operator to be five times differentiable and seven times differentiable for fourth-order and sixth-order methods, respectively. Also, they found no error estimate for those methods, as the Taylor series expansion played a leading role in proving the convergence. In this paper, we extended the method in the Banach space settings and relaxed the higher order derivative of the involved operator so that the methods can be used in a bigger class of problems which were not covered by the analysis in Hueso et al. (2015). Also, we obtained an error estimate without Taylor series expansion. This error estimate helps to get the number of iterations to achieve a given accuracy. Moreover, new sixth-order method is introduced by small modification and numerical examples were discussed for all these methods to validate our theoretical results and to study the dynamics.

Theory on Linear L-Fractional Differential Equations and a New Mittag–Leffler-Type Function

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard’s iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two.

Quantitative asymptotic regularity of the VAM iteration with error terms for $m$-accretive operators in Banach spaces

In this paper, we obtain, by using proof mining methods, quantitative results on the asymptotic regularity of the viscosity approximation method (VAM) with error terms for m -accretive operators in Banach spaces. For concrete instances of the parameter sequences, linear rates are computed by applying a lemma due to Sabach and Shtern.

A New Approach to Circular Inversion in l1-Normed Spaces

While there are well-known synthetic methods in the literature for finding the image of a point under circular inversion in l2-normed geometry (Euclidean geometry), there is no similar synthetic method in Minkowski geometry, also known as the geometry of finite-dimensional Banach spaces. In this study, we have succeeded in creating a synthetic construction of the circular inversion in l1-normed spaces, which is one of the most fundamental examples of Minkowski geometry. Moreover, this synthetic construction has been given using the Euclidean circle, independently of the l1-norm.

A novel fixed point iteration process applied in solving the Caputo type fractional differential equations in Banach spaces

We introduce the modified Picard–Ishikawa hybrid iterative scheme and establish some strong convergence results for the class of asymptotically generalized [Formula: see text]-pseudocontractive mappings in the intermediate sense in Banach spaces and approximate the fixed point of this class of mappings via the newly introduced iteration scheme. We construct some numerical examples to support our results. Furthermore, we apply the Picard–Ishikawa hybrid iteration scheme in solving the nonlinear Caputo type fractional differential equations. Our results generalize, extend and unify several existing results in literature.

Multistep Iterative Methods for Solving Equations in Banach Space

The novelty of this article lies in the fact that we extend the use of a multistep method for developing a sequence whose limit solves a Banach space-valued equation. We suggest the error estimates, local convergence, and semi-local convergence, a radius of convergence, the uniqueness of the required solution that can be computed under ω-continuity, and conditions on the first derivative, which is on the method. But, earlier studies used high-order derivatives, even though those derivatives do not appear in the body structure of the proposed method. In addition to this, they did not propose computable estimates and semi-local convergence. We checked the applicability of our study to three real-life problems for semi-local convergence and two problems chosen for local convergence. Based on the obtained results, we conclude that our approach improves its applicability and makes it suitable for challenges in applied science.

Projective tensor products of approximation spaces associated with positive operators

In this paper the projective tensor products of approximation spaces associated with positive operators in Banach spaces are characterized. We show that the tensor products of approximation spaces can be considered as the interpolation spaces generated by $K$-method of real interpolation. The inequalities that provide a sharp estimates of best approximations by analytic vectors of positive operators on projective tensor products are established. Application to spectral approximations of the regular elliptic operators on projective tensor products of Lebesgue spaces is shown.

Some results on ultrametric 2-normed spaces

In this paper, we study the ultrametric 2-normed spaces and the ultrametric 2-Banach spaces. In particular, we establish some results on Cauchy sequences in ultrametric 2-normed spaces. Also, we introduce and study the notion of bounded linear 2-functionals on ultrametric 2-Banach spaces and we give some of its properties. On the other hand, the new norm on the ultrametric 2-normed space is constructed. The concepts of closed operators between ultrametric 2-normed spaces and $b$-linear functionals in ultrametric 2-normed spaces are introduced. Finally, a necessary and sufficient condition for a linear operator to be closed in terms of its graph is proved and some results on bounded $b$-linear functionals in ultrametric 2-normed spaces are given.

New Extension of Darbo’s Fixed Point Theorem and Its Application to a System of Weighted-Fractional-Type Integral Equations

In this article, we introduce several new extensions of Darbo’s fixed point theorem with newly constructed contraction functions associated with the measure of noncompactness. In applying our new extensions, we prove the existence of solutions to a system of weighted fractional types of integral equations in Banach space BC(R+). At the end, we establish an example to show the applicability of our discovery.

A Novel Fixed-Point Iteration Approach for Solving Troesch’s Problem

This paper introduces a novel F fixed-point iteration method that leverages Green’s function for solving the nonlinear Troesch problem in Banach spaces, which are symmetric spaces. The Troesch problem, characterized by its challenging boundary conditions and nonlinear nature, is significant in various physical and engineering applications. The proposed method integrates fixed-point theory with Green’s function techniques to develop an iteration process that ensures convergence, stability, and accuracy. The numerical experiments demonstrate the method’s efficiency and robustness, highlighting its potential for broader applications in solving nonlinear differential equations in Banach spaces.

On fuzzy fractional differential inclusion driven by variational–hemivariational inequality in Banach spaces

The aim of this paper is to examine an evolution problem (FFDIVHVI) involving a fuzzy fractional differential inclusion and a variational–hemivariational inequality (VHVI) in Banach spaces. First, we show a uniqueness and existence theorem for VHVI under the theory of monotone operators and the surjectivity theorem. Then, by utilizing fixed point theorem for multivalued contraction mapping and fuzzy set theory, we establish the existence result for FFDIVHVI. In addition, it is proven that the collection of all mild trajectories of FFDIVHVI exhibits compactness. Finally, we illustrate the applicability of the abstract theory by a nonlinear quasistatic thermoelastic frictional contact problem for which we provide existence results.

Approximate norm‐additive maps on Banach spaces

AbstractLet be two Banach spaces, be an ‐norm‐additive map for some , that is, In this paper, we prove that if is surjective with , then there exists a linear surjective isometry such that The estimate is sharp. We also approximate standard surjective ‐norm‐additive maps between the positive cones of continuous function spaces by linear isometries within a sharp approximation error .

Uniqueness of algebra norm on quotients of the algebra of bounded operators on a Banach space

We show that the quotient algebra B(X)/I has a unique algebra norm for every closed ideal I of the Banach algebra B(X) of bounded operators on X, where X denotes any of the following Banach spaces:•(⨁n∈Nℓ2n)c0 or its dual space (⨁n∈Nℓ2n)ℓ1,•(⨁n∈Nℓ2n)c0⊕c0(Γ) or its dual space (⨁n∈Nℓ2n)ℓ1⊕ℓ1(Γ) for an uncountable cardinal number Γ,•C0(KA), the Banach space of continuous functions vanishing at infinity on the locally compact Mrówka space KA induced by an uncountable, almost disjoint family A of infinite subsets of N, constructed such that C0(KA) admits “few operators”. Equivalently, this result states that every homomorphism from B(X) into a Banach algebra is continuous and has closed range. The key step in our proof is to show that the identity operator on a suitably chosen Banach space factors through every operator in B(X)∖I with control over the norms of the operators used in the factorization. These quantitative factorization results may be of independent interest.