Periodic Problems of Difference Equations and Ergodic Theory by B. A. Biletskyi, A. A. Boichuk, A. A. Pokutnyi

“…The necessary and sufficient conditions for solvability of the family of difference equations with periodic boundary condition were obtained using the notion of relative spectrum of linear bounded operator in the Banach space and the ergodic theorem. It is shown that when the condition of existence is satisfied, then such periodic solutions are built using the formula for the generalized inverse operator to the linear limited one.…”
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On the largest analytic set for cyclic operators by A. Bourhim

“…Furthermore, we show that two densely similar cyclic Banach-space operators possessing Bishop's property (β) have equal approximate point spectra.…”
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On an nth-order infinitesimal generator and time-dependent operator differential equation with a strongly almost periodic solution by Aribindi Satyanarayan Rao

“…In a Banach space, if u is a Stepanov almost periodic solution of a certain nth-order infinitesimal generator and time-dependent operator differential equation with a Stepanov almost periodic forcing function, then u,u′,…,u (n−2) are all strongly almost periodic and u (n−1) is weakly almost periodic.…”
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Obstacles to bounded recovery by Boris Shekhtman

“…Let X be a Banach space, V⊂X is its subspace and U⊂X*. Given x∈X, we are looking for v∈V such that u (v)=u (x) for all u∈U and ‖v‖ ≤M‖x‖. …”
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Fixed Points of Asymptotic Pointwise Nonexpansive Mappings in Modular Spaces by Xue Wang, Ying Chen, Rudong Chen

“…Kirk and Xu studied the existence of fixed points of asymptotic pointwise nonexpansive mappings in the Banach space. In this paper, we investigate these kinds of mappings in modular spaces. …”
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Schwarz-Pick Estimates for Holomorphic Mappings with Values in Homogeneous Ball by Jianfei Wang

“…Let BX be the unit ball in a complex Banach space X. Assume BX is homogeneous. The generalization of the Schwarz-Pick estimates of partial derivatives of arbitrary order is established for holomorphic mappings from the unit ball Bn to BX associated with the Carathéodory metric, which extend the corresponding Chen and Liu, Dai et al. results.…”
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Sensitivity of dynamical systems to parameters in a convex subset of a topological vector space by H.T. Banks, S. Dediu, H.K. Nguyen

“…We develop a theory for sensitivity with respect to parameters ina convex subset of a topological vector space of dynamical systemsin a Banach space. Specific motivating examples for probabilitymeasure dependent differential, partial differential and delaydifferential equations are given. …”
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An example for a one-parameter nonexpansive semigroup by Tomonari Suzuki

“…This example shows that there exists a one-parameter nonexpansive semigroup {T(t):t≥0} on a closed convex subset C of a Banach space E such that limt→∞‖(1/t)∫0tT(s)xds−x‖=0 for some x∈C, which is not a common fixed point of {T(t):t≥0}.…”
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Viscosity Methods of Asymptotically Pseudocontractive and Asymptotically Nonexpansive Mappings for Variational Inequalities by Xionghua Wu, Yeong-Cheng Liou, Zhitao Wu, Pei-Xia Yang

“…As an application, if 𝑇 be an asymptotically nonexpansive self-mapping of a nonempty bounded closed convex subset 𝐾 of a real Banach space with a uniformly Gateaux differentiable norm and which possesses uniform normal structure, we prove that the iterative process defined by 𝑧0∈𝐾,𝑧𝑛+1=(1−𝑡𝑛/𝑘𝑛)𝑓(𝑧𝑛)+(𝛼𝑡𝑛/𝑘𝑛)𝑇𝑛𝑧𝑛+(𝛽𝑡𝑛/𝑘𝑛)𝑧𝑛 converges strongly to the fixed point of 𝑇.…”
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Existence of the Mild Solutions for Delay Fractional Integrodifferential Equations with Almost Sectorial Operators by Fang Li

“…This paper is concerned with the existence of mild solutions for the fractional integrodifferential equations with finite delay and almost sectorial operators in a separable Banach space X. We obtain existence theorem for mild solutions to the above-mentioned equations, by means of measure of noncompactness and the resolvent operators associated with almost sectorial operators. …”
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Q-Type Spaces of Harmonic Mappings by Alaa Kamal

“…In this paper, we attempt to investigate a new harmonic mapping class denoted by the Banach space QHp in the unit disk. Several characteristics of the class Qp are examined. …”
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A Nonlocal Cauchy Problem for Fractional Integrodifferential Equations by Fang Li, Jin Liang, Tzon-Tzer Lu, Huan Zhu

“…This paper is concerned with a nonlocal Cauchy problem for fractional integrodifferential equations in a separable Banach space X. We establish an existence theorem for mild solutions to the nonlocal Cauchy problem, by virtue of measure of noncompactness and the fixed point theorem for condensing maps. …”
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Exact Null Controllability for Fractional Nonlocal Integrodifferential Equations via Implicit Evolution System by Amar Debbouche, Dumitru Baleanu

“…We introduce a new concept called implicit evolution system to establish the existence results of mild and strong solutions of a class of fractional nonlocal nonlinear integrodifferential system, then we prove the exact null controllability result of a class of fractional evolution nonlocal integrodifferential control system in Banach space. As an application that illustrates the abstract results, two examples are provided.…”
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Existence Results for an Implicit Coupled System Involving $\xi$-Caputo and $p$-Laplacian Operators by Walid Benhadda, Abderrazak Kassidi, Ali El Mfadel, Elomari M'hamed

“…This paper aims to establish the existence and uniqueness of a solution to a coupled system of $\xi$-Caputo fractional differential equations involving the $p$-Laplacian operator in an arbitrary Banach space. The results of the existence derived  using the tools of topological degree theory for condensing maps and the Banach contraction principle. …”
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Some Intersections of the Weighted -Spaces by F. Abtahi, H. G. Amini, H. A. Lotfi, A. Rejali

“…Also, some sufficient conditions for that space to be a Banach space are provided. Furthermore, for an arbitrary subset of and a positive submultiplicative weight function on , Banach subspace of is introduced. …”
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Fixed Point of Strong Duality Pseudocontractive Mappings and Applications by Baowei Liu

“…Let E be a smooth Banach space with the dual , an operator is said to be α-strong duality pseudocontractive if , for all , where α is a nonnegative constant. …”
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Optimal Controls for a Class of Impulsive Katugampola Fractional Differential Equations with Nonlocal Conditions by Xingru Chen, Haibo Gu, Yu Sun

“…In this paper, we investigate a class of impulsive Katugampola fractional differential equations with nonlocal conditions in a Banach space. First, by using a fixed-point theorem, we obtain the existence results for a class of impulsive Katugampola fractional differential equations. …”
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The Closed Graph Theorem and the Space of Henstock-Kurzweil Integrable Functions with the Alexiewicz Norm by Luis Ángel Gutiérrez Méndez, Juan Alberto Escamilla Reyna, Maria Guadalupe Raggi Cárdenas, Juan Francisco Estrada García

“…Based on this fact we show that there exists a norm on ℋ𝒦([a,b]) under which it is a Banach space. Therefore if we equip ℋ𝒦([a,b]) with the Alexiewicz topology then ℋ𝒦([a,b]) is not K-Suslin, neither infra-(u) nor a webbed space.…”
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An existence result for a new coupled system of differential inclusions involving with Hadamard fractional orders by Lili Zhang, Chengbo Zhai

“…By applying a fixed point theorem for three operators containing $ x\in{AxBx+Cx} $ in Banach algebras, we get an existence result for the discussed system via multi-valued maps in a Banach space. An example is provided to support the validation of the theoretical result achieved.…”
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The approximation property of some vector valued Sobolev-Slobodeckij spaces by Carlos Bosch, Salvador Pérez-Esteva, Joaquín Motos

“…We prove that Wm,p(ℜn,E) has the approximation property provided E has it, furthermore if E is a Banach space with the strict approximation property then Wm,p(ℜn,E) has this property.…”
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