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In this paper, we first point out a gap in Yang (2012). Next, strong convergence theorem for the common solution for a finite family of φ -strongly accretive operator equations in Banach spaces is established without any bounded assumption, and we also give an example to show universality of the result of this paper, which improve and extend some recent results. [ABSTRACT FROM AUTHOR]

Abstract: In this note we show that a multivalued bounded quasimonotone operator of type between a reflexive Banach space and its dual is pseudomonotone. Hence we deduce that a result of Su Ke and He Zhen [Su Ke, He Zhen, Solutions of nonlinear evolution inclusions, J. Math. Anal. Appl. 313 (2006) 84–97] follows from a theorem of Bian and Webb [W. Bian, J.R.L. Webb, Solutions of nonlinear evolution inclusions, Nonlinear Anal. 37 (1999) 915–932]. [Copyright &y& Elsevier]

This paper proposes and analyzes a mixed variational formulation for the Navier-Stokes equations with variable viscosity that depends nonlinearly on the velocity gradient. Differently from previous works in which augmented terms are added to the formulation, here we employ a technique that had been previously applied to the stationary Boussinesq problem and the Navier-Stokes equations with constant viscosity. Firstly, a modified pseudostress tensor is introduced involving the diffusive and convective terms and the pressure. Secondly, by using the incompressibility condition, the pressure is eliminated, and the gradient of velocity is incorporated as an auxiliary unknown to handle the aforementioned nonlinearity. As a consequence, a Banach spaces-based formulation is obtained, which can be written as a perturbed twofold saddle point operator equation. We address the continuous and discrete solvabilities of this problem by linearizing the perturbation and employing a fixed-point approach along with a particular case of a known abstract theory. Given an integer ℓ ≥ 0 , feasible choices of finite element subspaces include discontinuous piecewise polynomials of degree ≤ ℓ for each entry of the velocity gradient, Raviart-Thomas spaces of order ℓ for the pseudostress, and discontinuous piecewise polynomials of degree ≤ ℓ for the velocity as well. Finally, optimal a priori error estimates are derived, and several numerical results confirming in general the theoretical rates of convergence, and illustrating the good performance of the scheme, are reported. [ABSTRACT FROM AUTHOR]

For a quasinilpotent operator T on a Banach space X , Douglas and Yang defined k x = lim sup z → 0 ln ⁡ ‖ (z − T) − 1 x ‖ ln ⁡ ‖ (z − T) − 1 ‖ for each nonzero vector x ∈ X , and call Λ (T) = { k x : x ≠ 0 } the power set of T. They proved that the power set have a close link with T 's lattice of hyperinvariant subspaces. In this paper, we compute the power set of some weighted shifts on ℓ p for 1 ≤ p < ∞. The following results are obtained: (1) If T is an injective quasinilpotent forward unilateral weighted shift on ℓ p (N) , then Λ (T) = { 1 } when k e 0 = 1 , where { e n } n = 0 ∞ be the canonical basis for ℓ p (N) ; (2) There is a class of backward unilateral weighted shifts on ℓ p (N) whose power set is [ 0 , 1 ] ; (3) There exists a bilateral weighted shift on ℓ p (Z) with power set [ 1 2 , 1 ]. [ABSTRACT FROM AUTHOR]

The aim of this expository paper is to explain to graduate students and beginning researchers in the field of mathematics, statistics and engineering the fundamental concept of sparse machine learning in Banach spaces. In particular, we use binary classification as an example to explain the essence of learning in a reproducing kernel Hilbert space and sparse learning in a reproducing kernel Banach space (RKBS). We then utilize the Banach space ℓ 1 (N) to illustrate the basic concepts of the RKBS in an elementary yet rigorous fashion. This paper reviews existing results in the author's perspectives to reflect the state of the art of the field of sparse learning, and includes new theoretical observations on the RKBS. Several open problems critical to the theory of the RKBS are also discussed at the end of this paper. [ABSTRACT FROM AUTHOR]

In this paper we introduce and analyze a Banach spaces-based approach yielding a fully-mixed finite element method for numerically solving the stationary chemotaxis-Navier-Stokes problem. This is a nonlinear coupled model representing the biological process given by the cell movement, driven by either an internal or an external chemical signal, within an incompressible fluid. In addition to the velocity and pressure of the fluid, the velocity gradient and the Bernouilli-type stress tensor are introduced as further unknowns, which allows to eliminate the pressure from the equations and compute it afterwards via a postprocessing formula. In turn, besides the cell density and the chemical signal concentration, the pseudostress associated with the former and the gradient of the latter are introduced as auxiliary unknowns as well. The resulting continuous formulation, posed in suitable Banach spaces, consists of a coupled system of three saddle point-type problems, each one of them perturbed with trilinear forms that depend on data and the unknowns of the other two. The well-posedness of it is analyzed by means of a fixed-point strategy, so that the classical Banach theorem, along with the Babuška-Brezzi theory in Banach spaces, allow to conclude, under a smallness assumption on the data, the existence of a unique solution. Adopting an analogue approach for the associated Galerkin scheme, and under suitable hypotheses on arbitrary finite element subspaces employed, we apply the Brouwer and Banach theorems to show existence and then uniqueness of the discrete solution. General a priori error estimates, including those for the postprocessed pressure, are also derived. Next, a specific set of finite element subspaces satisfying the required stability conditions, and yielding approximate local conservation of momentum, is introduced, which, given an integer k ≥ 0 , is defined in terms of Raviart-Thomas spaces of order k and piecewise polynomials of degree ≤ k only. The respective rates of convergence of the resulting Galerkin method are then provided. Finally, several numerical experiments confirming the latter and illustrating the good performance of the method, are reported. [ABSTRACT FROM AUTHOR]

In this paper, we propose the frozen form of nonstationary iterated Tikhonov regularization to solve the inverse problems and study its convergence analysis in the Banach space setting, where the iterates of proposed method are defined through minimization problems. These minimization problems involve uniformly convex penalty terms that may be non-smooth, which can be utilized in the reconstruction of several notable features (e.g., discontinuities and sparsity) of the sought solutions. We terminate our method through a discrepancy principle and show its regularizing nature. Moreover, we show the convergence of the iterates of the method with respect to Bregman distance as well as their strong convergence. In order to show the applicability of the proposed method, we show that it is applicable on two ill-posed inverse problems. Finally, by assuming that a stability estimate holds within a certain set, we derive the convergence rates of our method. • In this paper, the frozen form of nonstationary iterated Tikhonov regularization to solve the inverse problems is proposed. • The convergence analysis is studied in Banach spaces for both the noisy as well as non-noisy data. • The convergence rates of the method are derived through stability estimates. • Two concrete examples are discussed to showcase the validity of our results. [ABSTRACT FROM AUTHOR]

This paper focuses on the analysis of generalized quasivariational inequalities with non-self map. In Aussel et al., (2016), introduced the concept of the projected solution to study such problems. Subsequently, in the literature, this concept has attracted great attention and has been developed from different perspectives. The main contribution of this paper is to prove new existence results of the projected solution for generalized quasivariational inequality problems with non-self map in suitable infinite dimensional spaces. As an application, a quasiconvex quasioptimization problem is studied through a normal cone approach. [ABSTRACT FROM AUTHOR]

Let B (X) be the algebra of all bounded linear operators on a complex or real Banach space X with dim ⁡ X ≥ 3. In this paper, we characterize the maps from B (X) into itself which preserve the ascent of product of operators or, they preserve the descent of product of operators. It turns out that both problems are connected with preservers of the rank-one nilpotency of the product. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyze a new mixed finite element method for a stationary magneto-hydrodynamic (MHD) model. The method is based on the utilization of a new dual-mixed formulation recently introduced for the Navier-Stokes problem, which is coupled with a classical primal formulation for the Maxwell equations. The latter implies that the velocity and a pseudostress tensor relating the velocity gradient with the convective term for the hydrodynamic equations, together with the magnetic field and a Lagrange multiplier related with the divergence-free property of the magnetic field, become the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree k for the aforementioned pseudostress tensor, discontinuous piecewise polynomial elements of degree k for the velocity, Nédélec elements of degree k for the magnetic field and Lagrange elements of degree k for the associated Lagrange multiplier. The analysis of the continuous and discrete problems are carried out by means of the Lax–Milgram lemma, the Banach–Nečas–Babuška and Banach fixed-point theorems, under a sufficiently small data assumption. In particular, the analysis of the discrete scheme requires a quasi-uniformity assumption on mesh. We also develop an a priori error analysis and show that the proposed finite element method is optimal convergent. Finally, some numerical results illustrating the good performance of the method are provided. [ABSTRACT FROM AUTHOR]

We assume that the generalized resolvent for a bounded linear operator pencil mapping one Banach space onto another has an isolated essential singularity at the origin and is analytic on some annular region of the complex plane centred at the origin. In such cases the resolvent operator can be represented on the annulus by a convergent Laurent series and the spectral set has two components—a bounded component inside the inner boundary of the annulus and an unbounded component outside the outer boundary. In this paper we prove that the complementary spectral separation projections on the domain space are uniquely determined by the respective generating subspaces for the associated infinite-length generalized Jordan chains of vectors and that the domain space is the direct sum of these two subspaces. We show that the images of the generating subspaces under the mapping defined by the pencil provide a corresponding direct sum decomposition for the range space and that this is simply the decomposition defined by the complementary spectral separation projections on the range space. If the domain space has a Schauder basis we show that the separated systems of fundamental equations are reduced to two semi-infinite systems of matrix equations which can be solved recursively to obtain a basic solution and thereby determine the Laurent series coefficients for the resolvent operator on the given annular region. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak invariance principle for stationary ortho-martingales with values in 2-smooth or cotype 2 Banach spaces. Then, with the help of a suitable maximal ortho-martingale approximation, we derive the weak invariance principle for stationary random fields in L p , 1 ≤ p ≤ 2 , under a condition in the spirit of Hannan. As an application, we get an asymptotic result for the L p -distances (1 ≤ p ≤ 2) between the common distribution function and the corresponding empirical distribution function of stationary random fields. [ABSTRACT FROM AUTHOR]

In the classical theory of Markov chains, one may study the mean time to reach some chosen state, and it is well-known that in the irreducible, finite case, such quantity can be calculated in terms of the fundamental matrix of the walk, as stated by the mean hitting time formula. In this work, we present an analogous construction for the setting of irreducible, positive, trace preserving maps. The reasoning on positive maps generalizes recent results given for quantum Markov chains, a class of completely positive maps acting on graphs, presented by S. Gudder. The tools employed in this work are based on a proper choice of block matrices of operators, inspired in part by recent work on Schur functions for closed operators on Banach spaces, due to F. A. Grünbaum and one of the authors. The problem at hand is motivated by questions on quantum information theory, most particularly the study of quantum walks, and provides a basic context on which statistical aspects of quantum evolutions on finite graphs can be expressed in terms of the fundamental matrix, which turns out to be an useful generalized inverse associated with the dynamics. As a consequence of the wide generality of the mean hitting time formula found in this paper, we have obtained extensions of the classical version, either by assuming only the knowledge of the probabilistic distribution for the initial state, or by enlarging the arrival state to a subset of states. [ABSTRACT FROM AUTHOR]

This paper explores a 2D non-linear fractional integral equation of the Riemann–Liouville type. The authors establish the existence of at least one solution for this 2D integral equation in the Banach space of all real continuous functions on [ 0 , c ] × [ 0 , d ] , under some weaker conditions. The main tools used in our considerations are the concept of a measure of noncompactness and Petryshyn's fixed point theorem. Illustrative examples highlight the broad applicability of our findings across a diverse spectrum of integral equations. [ABSTRACT FROM AUTHOR]

We extend some of main results of Abasov et al., 2016, Fotiy et al., (2020), Mykhaylyuk et al., (2015), Pliev and Fang, (2017), Pliev and Popov, (2014) to the setting of orthogonally additive operators on lattice-normed spaces. We introduce a new class of C -complete lattice-normed spaces which strictly includes a class of Banach–Kantorovich spaces. The first main result of the paper asserts that every laterally-to-norm continuous C -compact orthogonally additive operator T : X → Y from an atomless C -complete lattice-normed space X to a Banach space Y is narrow. As a non expecting consequence of the first main result we obtain necessary and sufficient conditions for a nonlinear superposition operator T N : E (X) → E (X) to be C -compact. We also show that the sum of two orthogonally additive operators G and T , where G is narrow and T is laterally-to-norm continuous and C -compact, is a narrow operator. In the last part of the article we investigate dominated orthogonally additive operators. In particular we show the narrowness of dominated operators taking values in a Banach sequence space Y or in a separable symmetric operator space C E. [ABSTRACT FROM AUTHOR]

In this paper we investigate an evolution problem (SDQHVI) which constitutes of the second order differential inclusion driven by a quasi-variational–hemivariational inequality (QHVI) with perturbation operator in Banach spaces. Unlike the existing literature on differential inclusions, on the one hand, the second order differential operator is not assumed to be compact; on the other hand, the solvability of the inequality is proved under non-coercive condition. More precisely, first, it is shown that the solution set of the QHVI with perturbation operator is nonempty, bounded, closed and convex under the KKM theorem and the Minty formulation. Then, an associated multivalued map with the solution set of the QHVI with perturbation operator is introduced, and we prove that it is upper semicontinuous and measurable. Finally, by utilizing Kakutani–Fan–Glicksberg fixed point theorem, a weak topology technique and properties of strong continuous cosine operators, we prove the existence of mild solutions for SDQHVI. [ABSTRACT FROM AUTHOR]

This study explores the Sugeno exponential function, which is the solution to a first order differential equation with respect to nonadditive measures, specifically distorted Lebesgue measures. We define k -distorted semigroup property of the Sugeno exponential function, introduce a new addition operation on a set of distortion functions, and discuss some related results. Furthermore, m -Bernoulli inequality, a more general inequality than the well-known Bernoulli inequality on the real line, is established for the Sugeno exponential function. Additionally, the above concept is extended to a system of differential equations with respect to the distorted Lebesgue measure which gives rise to the study of a matrix m -exponential function. Finally, we present an appropriate m -distorted logarithm function and describe its behavior when applied to various functions, such as the sum, product, quotient, etc., while maintaining basic algebraic structures. The results are illustrated throughout the paper with a variety of examples. • The study analyzes the Sugeno exponential function in a distorted Lebesgue measure's differential equation. • It introduces k -distorted semigroup, broadens k -Bernoulli inequality, explores matrix m -exponential in differential equations. • It introduces k -distorted logarithm analyses in sums/products preserves algebraic structures. • Numerous examples illustrate the findings. [ABSTRACT FROM AUTHOR]

In this paper we study the Controllability and the generalized Ulam-Hyers stability of Riemann-Liouville fractional order linear differential equations with variable coefficients in Banach spaces. We obtained the desired results by means of fixed point principle and fractional calculus. Furthermore, we used the Gronwall's inequality to investigate above type of stability for Riemann-Liouville fractional order linear differential equation with time dependent coefficients under some specific assumptions. An example is also provided at the end to illustrate the theoretical work. [ABSTRACT FROM AUTHOR]

In this paper we provide sufficient conditions for perturbed saddle-point formulations in Banach spaces and their associated Galerkin schemes to be well-posed. Our approach, which extends a similar procedure employed with Hilbert spaces, proceeds in two slightly different ways depending on whether the kernel of the adjoint operator induced by one of the bilinear forms is trivial or not. If the latter holds, we make use of an equivalence result between a couple of inf-sup conditions involved, which, differently from the Hilbertian case, turns out to hold with different constants. While this fact causes no inconvenient in the continuous analysis, it does become a delicate issue at the discrete level, and hence the corresponding inf-sup conditions need to be assumed separately with respective constants independent of the finite element subspaces employed. In turn, if that kernel is trivial, then we employ a suitable characterization of a closed range injective adjoint operator, so that only one of the aforementioned inf-sup conditions is then required for the analysis. The applicability of the continuous solvability is illustrated with a mixed formulation for the decoupled Nernst-Planck equation. [ABSTRACT FROM AUTHOR]

In this paper we develop an a posteriori error analysis of a new momentum conservative mixed finite element method recently introduced for the steady-state Navier–Stokes problem in two and three dimensions. More precisely, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces, such as local estimates, and suitable Helmholtz decompositions, we derive a reliable and efficient residual-based a posteriori error estimator for the corresponding mixed finite element scheme on arbitrary (convex or non-convex) polygonal and polyhedral regions. On the other hand, inverse inequalities, the localization technique based on bubble functions, among other tools, are employed to prove the efficiency of the proposed a posteriori error indicator. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported. [ABSTRACT FROM AUTHOR]

This paper establishes a theory of calculus for fuzzy-number-valued functions whose range is embedded in a specific Banach space of fuzzy numbers. These spaces are generated by strongly linearly independent (SLI) fuzzy numbers whose operations are induced by an isomorphism with R n. We use the notion of Fréchet differentiability and Riemann integrability for these functions and present a fundamental theorem of calculus. Lastly, we develop a theory of fuzzy differential equations and provide an existence and uniqueness theorem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of backward heat equations with time-dependent coefficient in the Banach space L p (R) , (1 < p < ∞). For this problem, we first prove the stability estimates of Hölder type. After that the Tikhonov-type regularization is applied to solve the problem. A priori and a posteriori parameter choice rules are investigated, which yield error estimates of Hölder type. Numerical implementations are presented to show the validity of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we consider hemivariational–variational inequalities driven by uniformly monotone or d -monotone operators in Banach spaces. We establish related minimization principles leading to the existence and uniqueness of solutions to the inequality considered as well as we suggest the Ritz type numerical approximations. The theoretical results obtained are next applied to some problems inspired by models from contact mechanics. [ABSTRACT FROM AUTHOR]

In this paper we extend recent results obtained for the Navier–Stokes equations to propose and analyze a new fully mixed virtual element method (mixed-VEM) for the stationary two-dimensional Boussinesq equations appearing in non-isothermal flow phenomena. The model consists of a Navier–Stokes type system, modeling the velocity and the pressure of the fluid, coupled to an advection-diffusion equation for the temperature. The variational formulation is based on the introduction of the additional unknowns given by a modified pseudostress tensor, which depends on the pressure, and the diffusive and convective terms of the fluid, and the pseudoheat vector, which involves the temperature, its gradient, and the velocity. As a consequence of the former, the pressure is eliminated from the system, but computed afterwards via a post-processing formula. In turn, for the Galerkin approximation we follow the approach employed in a previous work introducing for the first time an L p spaces-based mixed-VEM for the Navier–Stokes equations, and couple it with a similar mixed-VEM for the convection–diffusion equation modeling the temperature. The solvability analysis of the resulting coupled discrete scheme is carried out by using appropriate fixed-point arguments, along with the discrete versions of the Babuška–Brezzi theory and the Banach-Nečas-Babuška theorem, both in subspaces of Banach spaces. The first Strang lemma is applied to derive the a priori error estimates for the virtual element solution as well as for the fully computable approximation of the pseudostress tensor, the pseudoheat vector, and the post-processed pressure. Finally, several numerical results, illustrating the performance of the mixed-VEM scheme and confirming the rates of convergence predicted by the theory, are reported. [ABSTRACT FROM AUTHOR]

Let (H , 〈 , 〉) be a complex Hilbert space and B (H) the space of bounded linear operators in H. Any other equivalent inner product in H is of the form 〈 f , g 〉 A = 〈 A f , g 〉 (f , g ∈ H) for some positive invertible operator A ∈ B (H). In this paper we study the bundle M which consist of the unit sphere { f ∈ H : 〈 f , f 〉 A = 1 } over each (equivalent) inner product 〈 , 〉 A , which due to the observation above can be defined M = { (A , f) ∈ B (H) × H : A is positive and invertible and 〈 A f , f 〉 = 1 }. We prove that M is a complemented submanifold of the Banach space B (H) × H and a homogeneous space of the Banach-Lie group G (H) ⊂ B (H) of invertible operators. We introduce a reductive structure in M , and study properties of the geodesics of the linear connection induced by this reductive structure. We consider certain submanifolds of M , for instance, the one obtained when the positive elements A describing the inner products lie in a prescribed C⁎-algebra A ⊂ B (H). [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the stability of some B-essential spectra for closed linear operators in a Banach space under finite rank perturbations and to provide a characterization of specific B-essential spectra associated with a 3 × 3 unbounded operator matrix defined over its maximal domain. As an application, grounded in this novel investigation, we present the diffusion problem of radiative transfer equations in a channel. [ABSTRACT FROM AUTHOR]

This paper studies the variation of spectra of closed subspaces (i.e., linear relations) in Banach spaces under some perturbations. A concept of commutativity for subspaces in Banach spaces is introduced and its properties are studied. It is shown that boundedness of the inverse of a closed subspace is preserved under relatively bounded and small gap perturbations and perturbation about spectral condition, separately. By using these results, upper semi-continuity of spectra of closed subspaces is obtained and an error estimate of spectra of closed subspaces is given under bounded perturbation. In the special case that the space is a Hilbert space, invariance of self-adjointness of subspaces under relatively bounded and small gap perturbations and the variation of spectrum of a self-adjoint subspace under bounded and Hermitian perturbation are discussed. The results obtained in the present paper generalize the corresponding results for closed operators to closed subspaces and some of which weaken certain assumptions of the related existing results. [ABSTRACT FROM AUTHOR]

C ⁎ -algebras, group algebras, and the algebra A (X) of approximable operators on a Banach space X having the bounded approximation property are known to be zero product determined. In this paper we give a quantitative estimate of this property by showing that, for the Banach algebra A , there exists a constant α with the property that for every continuous bilinear functional φ : A × A → C there exists a continuous linear functional ξ on A such that sup ‖ a ‖ = ‖ b ‖ = 1 ⁡ | φ (a , b) − ξ (a b) | ≤ α sup ‖ a ‖ = ‖ b ‖ = 1 , a b = 0 ⁡ | φ (a , b) | in each of the following cases: (i) A is a C ⁎ -algebra, in which case α = 8 ; (ii) A = L 1 (G) for a locally compact group G , in which case α = 60 27 1 + sin ⁡ π 10 1 − 2 sin ⁡ π 10 ; (iii) A = A (X) for a Banach space X having property (A) (which is a rather strong approximation property for X), in which case α = 60 27 1 + sin ⁡ π 10 1 − 2 sin ⁡ π 10 C 2 , where C is a constant associated with the property (A) that we require for X. [ABSTRACT FROM AUTHOR]

In this paper we introduce and analyze, up to our knowledge for the first time, Banach spaces-based mixed variational formulations for nearly incompressible linear elasticity and Stokes models. Our interest in this subject is motivated by the respective need that arises from the solvability studies of nonlinear coupled problems in continuum mechanics that involve these equations. We consider pseudostress-based approaches in both cases and apply a suitable integration by parts formula for ad-hoc Sobolev spaces to derive the corresponding continuous schemes. We utilize known and new preliminary results, along with the Babuška-Brezzi theory in Banach spaces, to establish the well-posedness of the formulations for a particular range of the indexes of the Lebesgue spaces involved. Among the aforementioned new results from us, we highlight the construction of a particular operator mapping a tensor Lebesgue space into itself, and the generalization of a classical estimate in L 2 for deviatoric tensors, which plays a key role in the Hilbertian analysis of linear elasticity, to arbitrary Lebesgue spaces. No discrete analysis is performed in this work. [ABSTRACT FROM AUTHOR]

The objective of the present paper is to provide a detailed account of a local min-orthogonal method for finding multiple coexisting solutions to cooperative p -Laplacian systems. A local L -⊥ selection mapping on a product space of two Banach spaces is introduced in order to develop a local characterization of the coexisting solutions. Using this local characterization we propose a numerical algorithm for finding multiple coexisting solutions to the system and provide a subsequence convergence result of the algorithm. We also discuss the discretization of the problem using the finite element method and the associated convergence results of the numerical solutions. Finally, we provide the numerical results to demonstrate the efficiency of the proposed method and also to confirm its related theoretical results. [ABSTRACT FROM AUTHOR]

Let K (X , Y) be the space of compact operators. For a proximinal subspace Z ⊂ Y , this paper deals with the question, when does every Y -valued compact operator admit a Z -valued compact best approximation? For any reflexive Banach space X and for a L 1 -predual space Y , if Z ⊂ Y is a strongly proximinal subspace of finite codimension, we show that K (X , Z) is a proximinal subspace of K (X , Y) under an additional condition on the position of K (X , Z). When Y is a c 0 -direct sum of finite dimensional spaces we achieve a strong transitivity result by showing that for any proximinal subspace of finite codimension Z ⊂ Y , every Y -valued bounded operator admits a best Z -valued compact approximation. [ABSTRACT FROM AUTHOR]

In this paper we study a class of quasi-variational-hemivariational inequalities in reflexive Banach spaces. The inequalities contain a convex potential, a locally Lipschitz superpotential, and a implicit obstacle set of constraints. Solution existence and compactness of the solution set to the inequality problem are established based on the Kakutani–Ky Fan fixed point theorem. The applicability of the results is illustrated by the steady-state Oseen model of a generalized Newtonian incompressible fluid with mixed boundary conditions. The latter involve a unilateral boundary condition, the Navier slip condition, a nonmonotone version of the nonlinear Navier-Fujita slip condition, and a generalization of the threshold slip and leak condition of frictional type. [ABSTRACT FROM AUTHOR]

The goal of this paper is to introduce and study a new class of fractional differential hemivariational inequality formulated by an evolutionary hemivariational inequality and a fractional differential equation in Banach spaces. By employing the Rothe method and the surjectivity result, we derive the existence of unique solution for such a problem under some mild conditions. Moreover, we use the fully discrete scheme to approximate the fractional differential hemivariational inequality and provide an error estimate for the approximation. Finally, the main results are applied to obtain the unique solvability as well as the numerical analysis for a viscoelastic frictional contact problem with adhesion. [ABSTRACT FROM AUTHOR]

The theory of τ -summing and σ -nuclear linear operators on Banach spaces was developed by Pietsch [20, Chapter 23] . Extending the linear case to the range p > 1 and generalizing all cases to the multilinear setting, in this paper we introduce the concept of σ ( p ) -nuclear linear and multilinear operators. In order to develop the duality theory for the spaces of such operators, we introduce the concept of quasi- τ ( p ) -summing linear/multilinear operators and prove Pietsch-type domination theorems for such operators. The main result of the paper shows that, under usual conditions, linear functionals on the space of σ ( p ) -nuclear n -linear operators are represented, via the Borel transform, by quasi- τ ( p ) -summing n -linear operators. As far as we know, this result is new even in the linear case n = 1 . [ABSTRACT FROM AUTHOR]

This paper is devoted to discuss Hilfer fractional evolution equations through its controllability and stability in a Banach space. We achieve our claims and conclusions by first demonstrating the nature and uniqueness of the suggested set of equations with their mild solutions. The outcomes of existence and uniqueness are obtained with the aid of the propagation family theory, non-compactness calculation methods and the fixed point technique. An example is also provided for the description of our main results. [ABSTRACT FROM AUTHOR]

In this paper, the notion of generalized split feasibility problem (GSFP) is studied in p -uniformly convex Banach spaces. Some special cases of the GSFP are highlighted. A self-adaptive step-size iterative algorithm which converges strongly to solution of the GSFP is proved. The implementation of the method is demonstrated with two numerical examples. Our method does not require prior information of operator norms. Our results extend, improve and enrich recently announced related results in the literature. [ABSTRACT FROM AUTHOR]

In this paper we prove existence and uniqueness of a mild solution to the Young equation d y (t) = A y (t) d t + σ (y (t)) d x (t) , t ∈ [ 0 , T ] , y (0) = ψ. Here, A is an unbounded operator which generates a semigroup of bounded linear operators (S (t)) t ≥ 0 on a Banach space X , x is a real-valued η -Hölder continuous. Our aim is to reduce, in comparison to Gubinelli et al. (2006) and Addona et al. (2022) (see also Deya et al. (2012) and Gubinelli and Tindel, (2010)), the regularity requirement on the initial datum ψ eventually dropping it. The main tool is the definition of a sewing map for a new class of increments which allows the construction of a Young convolution integral in a general interval [ a , b ] ⊂ R when the X α -norm of the function under the integral sign blows up approaching a and X α is an intermediate space between X and D (A). [ABSTRACT FROM AUTHOR]

This paper focuses on the analysis of generalized quasi-variational inequality problems with non-self constraint map. To study such problems, in Aussel et al. (2016) the authors introduced the concept of the projected solution and proved its existence in finite-dimensional spaces. The main contribution of this paper is to prove the existence of a projected solution for generalized quasi-variational inequality problems with non-self constraint map on real Banach spaces. Then, following the multistage stochastic variational approach introduced in Rockafellar and Wets (2017), we introduce the concept of the projected solution in a multistage stochastic setting, and we prove the existence of such a solution. We apply this theoretical result in studying an electricity market with renewable power sources. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new iterative method that is the combination of the proximal point algorithm, viscosity approximation method and alternating resolvent method for finding the common zeros of two accretive operators in Banach spaces. And we will prove the strong convergence theorems for the iterative algorithms and give the example of the main theorems. The results of this paper are improvements and extensions of the corresponding ones announced by many others. [ABSTRACT FROM AUTHOR]

Industrial manipulators have good repeatability but poor accuracy. To improve the accuracy of an industrial manipulator, using an external large-scale measurement system to provide the accurate positioning information is another good selection in addition to the error mapping and other calibration methods. In this paper, we present the model explaining how the accuracy of a manipulator is traced back to an external measurement system. The model analyzes the traceability problem in two parts: the position and the attitude. In both parts, movement sequences of the manipulator are constructed. The accuracy of the manipulator can be traced back to the external measurement system as long as the sequences of movements are convergent. However, the sequences of movements are not always convergent unless we provide a reasonable condition. In this paper, we find a conservative estimate of the condition by the Banach fixed-point theorem. Then, we set up an experiment to verify the accuracy traceability model. The results indicate that the proposed model exists, and is very effective in improving the positioning accuracy of a manipulator compared to conventional methods. [ABSTRACT FROM AUTHOR]