General Banach Spaces Research Articles

Some common fixed point theorems of p- contraction and (α, β) composition contraction in Generalized Banach space

In this work, we introduce Some common fixed point theorems of contraction and (α, ) composition contraction in Generalized Banach space. The provided theorem is a generalization and extension of many well-known theorems.

Projections in Lipschitz‐free spaces induced by group actions

AbstractWe show that given a compact group G acting continuously on a metric space by bi‐Lipschitz bijections with uniformly bounded norms, the Lipschitz‐free space over the space of orbits (endowed with Hausdorff distance) is complemented in the Lipschitz‐free space over . We also investigate the more general case when G is amenable, locally compact or SIN and its action has bounded orbits. Then, we get that the space of Lipschitz functions is complemented in . Moreover, if the Lipschitz‐free space over , , is complemented in its bidual, several sufficient conditions on when is complemented in are given. Some applications are discussed. The paper contains preliminaries on projections induced by actions of amenable groups on general Banach spaces.

Fixed point theorems for non-self mappings on strictly star-shaped sets in generalized Banach spaces

Fixed point theorems for non-self mappings on strictly star-shaped sets in generalized Banach spaces

Integral Functionals on Nonseparable Banach Spaces With Applications

In this paper, we study integral functionals defined on spaces of functions with values on general (non-separable) Banach spaces. We introduce a new class of integrands and multifunctions for which we obtain measurable selection results. Then, we provide an interchange formula between integration and infimum, which enables us to get explicit formulas for the conjugate and Clarke subdifferential of integral functionals. Applications to expected functionals from stochastic programming, optimality conditions for a calculus of variation problem and sweeping processes are given.

On symplectic Banach spaces

We extend and generalize the result of Kalton and Swanson (Z_2 is a symplectic Banach space with no Lagrangian subspace) by showing that all higher order Rochgberg spaces {mathfrak {R}}^{(n)} are symplectic Banach spaces with no Lagrangian subspaces. The nontrivial symplectic structure on Rochberg spaces of even order is the one induced by the natural duality; while the nontrivial symplectic structure on Rochberg spaces of odd order requires perturbation with a complex structure. We will also study symplectic structures on general Banach spaces and, motivated by the unexpected appearance of complex structures, we introduce and study almost symplectic structures.

Iterative schemes for numerical reckoning of fixed points of new nonexpansive mappings with an application

<abstract><p>The goal of this manuscript is to introduce a new class of generalized nonexpansive operators, called $ (\alpha, \beta, \gamma) $-nonexpansive mappings. Furthermore, some related properties of these mappings are investigated in a general Banach space. Moreover, the proposed operators utilized in the $ K $-iterative technique estimate the fixed point and examine its behavior. Also, two examples are provided to support our main results. The numerical results clearly show that the $ K $-iterative approach converges more quickly when used with this new class of operators. Ultimately, we used the $ K $-type iterative method to solve a variational inequality problem on a Hilbert space.</p></abstract>

Interpretation on nonlocal neutral functional differential equations with delay

<abstract><p>This work deals with the existence and continuous dependence of an integral solution for neutral integro-differential equations with a nonlocal condition. This result is established by using an integrated resolvent operator under conditions of Lipschitz continuity and uniqueness via the Banach fixed point technique. We also study the existence of a strict solution on reflexive and general Banach spaces. In the last section, an example is provided related to this theory.</p></abstract>

On $ \mathcal{A B C} $ coupled Langevin fractional differential equations constrained by Perov's fixed point in generalized Banach spaces

<abstract><p>Nonlinear differential equations are widely used in everyday scientific and engineering dynamics. Problems involving differential equations of fractional order with initial and phase changes are often employed. Using a novel norm that is comfortable for fractional and non-singular differential equations containing Atangana-Baleanu-Caputo fractional derivatives, we examined a new class of initial values issues in this study. The Perov fixed point theorems that are utilized in generalized Banach spaces form the foundation for the new findings. Examples of the numerical analysis are provided in order to safeguard and effectively present the key findings.</p></abstract>

On the approximate boundary controllability of some partial functional integrodifferential equations with finite delay in Banach spaces

<p style='text-indent:20px;'>This work concerns the study of approximate boundary controllability for some nonlinear partial functional integrodifferential equations with finite delay arising in the modeling of materials with memory, in the framework of general Banach spaces. We give sufficient conditions that ensure the approximate controllability of the system by supposing that its linear undelayed part is approximately controllable, admits a resolvent operator in the sense of Grimmer, and by making use of the Banach fixed-point Theorem and the continuity of the resolvent operator in the uniform norm-topology. As a result, we obtain a generalization of several important results in the literature, without assuming the compactness of the resolvent operator and the uniform boundedness of the nonlinear term. An example of applications is given for illustration.</p>

Lorentz spaces and embeddings induced by almost greedy bases in superreflexive Banach spaces

The aim of this paper is to show that almost greedy bases induce tighter embeddings in superreflexive Banach spaces than in general Banach spaces. More specifically, we show that an almost greedy basis in a superreflexive Banach space $$\mathbb{X}$$ induces embeddings that allow squeezing $$\mathbb{X}$$ between two superreflexive Lorentz sequence spaces that are close to each other in the sense that they have the same fundamental function.

Some Common Fixed Point Theorems of Rational Contractions Condition in Generalized Banach Space

In this work, we introduce some common fixed points theorems type rational contractions on generalized Banach space. The provided theorem is a generalization and an extension of many well-known theorems; we present a few examples to illustrate the validity of the results obtained in the paper.

"Solutions of Split Equality Hammerstein Type Equation Problems in Reflexive Real Banach Spaces"

"The purpose of this study is to introduce an inertial algorithm for approximating a solution of the split equality Hammerstein type equation problem in general reflexive real Banach spaces. Strong convergence results are established under the assumption that the associated mappings are monotone and uniformly continuous. The results in this paper generalize and improve many of the existing results in the literature in the sense that the underlying mappings are relaxed from Lipschitz continuous to uniformly continuous and the spaces under consideration are extended from Hilbert spaces to reflexive real Banach spaces with a more general problem which includes the Hammerstein type equation problems."

Fixed point theorems in generalized Banach spaces under G-weak topology features

Fixed point theorems in generalized Banach spaces under G-weak topology features

Norm-attainable operators on involutive stereotype tubes with algebraically connected component of the identity

This work is an in-depth study of the class of norm-attainable operators in a general Banach space setting. We give characterizations of norm-attainable operators on involutive stereotype tubes with algebraically connected component of the identity. In particular, we prove reflexivity, boundedness and compactness properties when the set of these operators contains unit balls with involution for the tubes when they are of stereotype category

A strong law of large numbers for non identically distributed Pettis integrable random sets in a general Banach space

A multivalued version of Cesàro mean convergence theorem is established. As a consequence, various convergence results of strong law of large numbers for pairwise independent Pettis integrable random sets in a Banach space without any additional geometric condition on the space are provided.

Note on the Fekete–Szegö Problem for Spirallike Mappings in Banach Spaces

In this note we present a remark on the paper “On the coefficient inequalities for a class of holomorphic mappings associated with spirallike mappings in several complex variables” by Y. Lai and Q. Xu [10] published recently in the journal Results in Mathematics. We show that one of the theorems in [10] concerning the finite-dimensional space \({{\mathbb {C}}}^n\) is a direct consequence of another one, so it does not need an independent proof. Moreover, we prove that a sharp norm estimate on the Fekete–Szegö functional over spirallike mappings in a general Banach space can be deduced from a result in [10].

Coupled system of $\psi$--Caputo fractional differential equations without and with delay in generalized Banach spaces

The main objective of this research manuscript is to establish various existence and uniqueness results as well as the Ulam--Hyers stability of solutions to a Coupled system of $\psi$--Caputo fractional differential equations without and with delay in generalized Banach spaces. Existence and uniqueness results are obtained by applying Krasnoselskii's type fixed point theorem, Schauder's fixed point theorem in generalized Banach spaces, and Perov's fixed point theorem combined with the Bielecki norm. While Urs's approach is used to analyze the Ulam--Hyers stability of solutions for the proposed problem. Finally, Some examples are given to illustrate the obtained results.

Generalized Hopf Bifurcation of Semilinear FDEs in General Banach Spaces

In this paper, we investigate the generalized Hopf bifurcation for semilinear functional differential equations in general Banach spaces by applying the Lyapunov–Schmidt reduction process. We not only obtain explicit expressions about the parameters of the original system, but also use these expressions to determine whether there are several periodic solutions when the parameters are changed. Moreover, an example is given to illustrate the theoretical results.

ISOMETRIES ON SOME GENERAL FAMILY FUNCTION SPACES AMONG COMPOSITION OPERATORS

In this paper, we discuss the isometries of composition operators on the holomorphic general family function spaces F(p, q, s). First, we classify the isometric composition operators acting on a general Banach spaces. For 1 < p < 2, we display that an isometry of Cφ is caused only by a rotation of the disk. We scrutinize the previous work on the case for p ≥ 2. Also, we characterize many of the foregoing results about all α-Besov-type spaces F(p, αp − 2, s), α > 0. We exhibit that in every classes F(p, αp − 2, s) except for the Dirichlet space D = F(2, 0, 0), rotations are the only that produce isometries.

Existence Results of Global Solutions for a Coupled Implicit Riemann-Liouville Fractional Integral Equation via the Vector Kuratowski Measure of Noncompactness

The main goal of this study is to demonstrate an existence result of a coupled implicit Riemann-Liouville fractional integral equation. First, we prove a new fixed point theorem in spaces with an extended norm structure. That theorem generalized Darbo’s theorem associated with the vector Kuratowski measure of noncompactness. Second, we employ our obtained fixed point theorem to investigate the existence of solutions to the coupled implicit fractional integral equation on the generalized Banach space C([0,1],R)×C([0,1],R).