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This paper introduces a novel category of nonlinear mappings and provides several theorems on their existence and convergence in Banach spaces, subject to various assumptions. Moreover, we obtain convergence theorems concerning iterates of α -Krasnosel'skiĭ mapping associated with the newly defined class of mappings. Further, we present that α -Krasnosel'skiĭ mapping associated with b -enriched quasinonexpansive mapping is asymptotically regular. Furthermore, some new convergence theorems concerning b -enriched quasinonexpansive mappings have been proved. [ABSTRACT FROM AUTHOR]

In this paper, we extend the results obtained by Ezearn on annihilated points for his higher-order nonexpansive mappings to the context of general higher-order nonexpansive mappings. Precisely in his thesis, Ezearn introduced the concept of annihilated points, which extends the notion of fixed points, and it is only meaningful in the context of higher-order nonexpansive mappings and gave some mild conditions when the annihilated points could exist in strictly convex Banach spaces. In the last direction, we also extend Ezearn's result on the approximate fixed point sequence for higher-order nonexpansive mappings to general higher-order nonexpansive mappings. [ABSTRACT FROM AUTHOR]

In this paper, we establish a generalization of the Galewski-Rădulescu nonsmooth global implicit function theorem to locally Lipschitz functions defined from infinite dimensional Banach spaces into Euclidean spaces. Moreover, we derive, under suitable conditions, a series of results on the existence, uniqueness, and possible continuity of global implicit functions that parametrize the set of zeros of locally Lipschitz functions. Our methods rely on a nonsmooth critical point theory based on a generalization of the Ekeland variational principle. [ABSTRACT FROM AUTHOR]

Let A and B be two closed linear relations acting between two Banach spaces X and Y , and let λ be a complex number. We study the stability of the nullity and deficiency of A when it is perturbed by λ B. In particular, we show the existence of a constant ρ > 0 for which both the nullity and deficiency of A remain stable under perturbation by λ B for all λ inside the disk λ < ρ. [ABSTRACT FROM AUTHOR]

In this paper, we study the split feasibility problem in Banach spaces. At first, we prove that a solution of this problem is a solution of the equivalent equation defined by using a metric projection, a generalized projection, and sunny generalized nonexpansive retraction, respectively. Then, using the hybrid method with these projections, we prove strong convergence theorems in mathematical programing in order to find a solution of the split feasibility problem in Banach spaces. [ABSTRACT FROM AUTHOR]

This paper deals with some existence of mild solutions for two classes of impulsive integrodifferential equations in Banach spaces. Our results are based on the fixed point theory and the concept of measure of noncompactness with the help of the resolvent operator. Two illustrative examples are given in the last section. [ABSTRACT FROM AUTHOR]

We prove Browder's convergence theorem for multivalued mappings in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm by using the notion of diametrically regular mappings. Our results are significant improvement on results of Jung (2007) and Panyanak and Suantai (2020). [ABSTRACT FROM AUTHOR]

In this paper, a cyclic algorithm for approximating a class of split variational inequality problem is introduced and studied in some Banach spaces. A strong convergence theorem is proved. Some applications of the theorem are presented. The results presented here improve, unify, and generalize certain recent results in the literature. [ABSTRACT FROM AUTHOR]

In the theory control systems, there are many various qualitative control problems that can be considered. In our previous work, we have analyzed the approximate controllability and observability of the nonautonomous Riesz-spectral systems including the nonautonomous Sturm-Liouville systems. As a continuation of the work, we are concerned with the analysis of stability, stabilizability, detectability, exact null controllability, and complete stabilizability of linear non-autonomous control systems in Banach spaces. The used analysis is a quasisemigroup approach. In this paper, the stability is identified by uniform exponential stability of the associated C0-quasisemigroup. The results show that, in the linear nonautonomous control systems, there are equivalences among internal stability, stabizability, detectability, and input-output stability. Moreover, in the systems, exact null controllability implies complete stabilizability. [ABSTRACT FROM AUTHOR]

Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X⁎. Let T:X⊇D(T)→2X⁎ be a maximal monotone operator and C:X⊇D(C)→X⁎ be bounded and continuous with D(T)⊆D(C). The paper provides new existence theorems concerning solvability of inclusion problems involving operators of the type T+C provided that C is compact or T is of compact resolvents under weak boundary condition. The Nagumo degree mapping and homotopy invariance results are employed. The paper presents existence results under the weakest coercivity condition on T+C. The operator C is neither required to be defined everywhere nor required to be pseudomonotone type. The results are applied to prove existence of solution for nonlinear variational inequality problems. [ABSTRACT FROM AUTHOR]

We consider applications of the g-Drazin inverse to some classes of abstract Cauchy problems, namely, the heat equation with operator coefficient and delay differential equations in Banach space. [ABSTRACT FROM AUTHOR]

We consider a class of nonlinear discrete-time Volterra equations in Banach spaces. Estimates for the norm of operator-valued functions and the resolvents of quasi-nilpotent operators are used to find sufficient conditions that all solutions of such equations are elements of an appropriate Banach space. These estimates give us explicit boundedness conditions. The boundedness of solutions to Volterra equations with infinite delay is also investigated. [ABSTRACT FROM AUTHOR]

Suppose C is a cone contained in real vector space V. When does V contain a hyperplane H that intersects each of the 0-rays in C∖{0} exactly once? We build on results found in Aliprantis, Tourky, and Klee Jr.’s work to give a partial answer to this question. We also present an example of a salient, closed Banach space cone C for which there does not exist a hyperplane that intersects each 0-ray in C∖{0} exactly once. [ABSTRACT FROM AUTHOR]

We obtain characterizations of compactness for resolvent families of operators and as applications we study the existence of mild solutions to nonlocal Cauchy problems for fractional derivatives in Banach spaces. We discuss here simultaneously the Caputo and Riemann-Liouville fractional derivatives in the cases 0<α<1 and 1<α<2. [ABSTRACT FROM AUTHOR]

Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X⁎. Let T:X⊇D(T)→2X⁎ and A:X⊇D(A)→2X⁎ be maximal monotone operators. The maximality of the sum of two maximal monotone operators has been an open problem for many years. In this paper, new maximality theorems are proved for T+A under weaker sufficient conditions. These theorems improved the well-known maximality results of Rockafellar who used condition D(T)∘∩D(A)≠∅ and Browder and Hess who used the quasiboundedness of T and condition 0∈D(T)∩D(A). In particular, the maximality of T+∂ϕ is proved provided that D(T)∘∩D(ϕ)≠∅, where ϕ:X→(-∞,∞] is a proper, convex, and lower semicontinuous function. Consequently, an existence theorem is proved addressing solvability of evolution type variational inequality problem for pseudomonotone perturbation of maximal monotone operator. [ABSTRACT FROM AUTHOR]

We study the local exponential stability of evolution difference systems with slowly varying coefficients and nonlinear perturbations. We establish the robustness of the exponential stability in infinite-dimensional Banach spaces, in the sense that the exponential stability for a given pseudolinear equation persists under sufficiently small perturbations. The main methodology is based on a combined use of new norm estimates for operator-valued functions with the “freezing” method. [ABSTRACT FROM AUTHOR]

We prove two results concerning the existence of solutions for functional differential inclusions that are governed by sweeping processes, with noncompact valued perturbations in Banach spaces. Indeed, we have two goals. The first is to give a technique that allows considering sweeping processes with noncompact valued perturbations and associated with a multivalued function depending on time. The second is to give a technique to overcome the arising problem from the nonlinearity of the normalized mappings, when we deal with sweeping processes with noncompact valued perturbations and associated with a multivalued function depending on time and state. [ABSTRACT FROM AUTHOR]

We are concerned with both regular and degenerate first-order identification problems related to systems of differential equations of weakly parabolic type in Banach spaces. Several applications to partial differential equations and systems will be given in a subsequent paper to show the fullness of our abstract results. [ABSTRACT FROM AUTHOR]

In this paper we established some vector-valued inequalities of Gronwall type in ordered Banach spaces. Our results can be applied to investigate systems of real-valued Gronwall-type inequalities. We also show that the classical Gronwall-Bellman-Bihari integral inequality can be generalized from composition operators to a variety of operators, which include integral operators, maximal operators, geometric mean operators, and geometric maximal operators. [ABSTRACT FROM AUTHOR]

The existence of uncountably many positive solutions and Mann iterative approximations for a nonlinear three-dimensional difference system are proved by using the Banach fixed point theorem. Four illustrative examples are also provided. [ABSTRACT FROM AUTHOR]

This paper investigates the boundedness and convergence properties of two general iterative processes which involve sequences of self-mappings on either complete metric or Banach spaces. The sequences of self-mappings considered in the first iterative scheme are constructed by linear combinations of a set of self-mappings, each of them being a weighted version of a certain primary self- mapping on the same space. The sequences of self-mappings of the second iterative scheme are powers of an iteration-dependent scaled version of the primary self-mapping. Some applications are also given to the important problem of global stability of a class of extended nonlinear polytopic-type parameterizations of certain dynamic systems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce and analyze hybrid viscosity methods for a general system of variational inequalities (GSVI) with hierarchical fixed point problem constraint in the setting of real uniformly convex and 2-uniformly smooth Banach spaces. Here, the hybrid viscosity methods are based on Korpelevich's extragradient method, viscosity approximation method, and hybrid steepest-descent method. We propose and consider hybrid implicit and explicit viscosity iterative algorithms for solving the GSVI with hierarchical fixed point problem constraint not only for a nonexpansive mapping but also for a countable family of nonexpansive mappings in X, respectively. We derive some strong convergence theorems under appropriate conditions. Our results extend, improve, supplement, and develop the recent results announced by many authors. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present two new forward-backward splitting schemes with relaxations and errors for finding a common element of the set of solutions to the variational inclusion problem with two accretive operators and the set of fixed points of strict pseudocontractions in infinite-dimensional Banach spaces. Under mild conditions, some weak and strong convergence theorems for approximating these common elements are proved. The methods in the paper are novel and different from those in the early and recent literature. Further, we consider the problem of finding a common element of the set of solutions of a mathematical model related to equilibrium problems and the set of fixed points of a strict pseudocontractions. [ABSTRACT FROM AUTHOR]

We obtain inequalities of Holder and Minkowski type with weights generalizing both the case of weights with alternating signs and the classical case of nonnegative weights. [ABSTRACT FROM AUTHOR]

We show a necessary and sufficient condition for the existence of metric projection on a class of half-space ... in Banach space. Two representations of metric projections ... and ... are given, respectively, where ... stands for dual half-space of ... in dual space X*. By these representations, a series of continuity results of the metric projections ... and ... are given. We also provide the characterization that a metric projection is a linear bounded operator. [ABSTRACT FROM AUTHOR]

We introduce a new generalized system of nonlinear variational inequality problems (GSNVIP) by using the generalized projection method. Moreover, we introduce an iterative scheme for finding a solution to this problem. Moreover, some existence and strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces under suitable conditions. The results presented in the paper improve and extend some recent results. [ABSTRACT FROM AUTHOR]

Ideas and techniques from standard and nonstandard theories of measure spaces and Banach spaces are brought together to give a new approach to the study of the extension of vector measures. Applications of our results lead to simple new proofs for theorems of classical measure theory. The novelty lies in the use of the principle of extension by continuity (for which we give a nonstandard proof) to obtain in an unified way some notable theorems which have been obtained by Fox, Brooks, Ohba, Diestel, and others. The methods of proof are quite different from those used by previous authors, and most of them are realized by means of nonstandard analysis. [ABSTRACT FROM AUTHOR]

Let H(D) denote the space of all holomorphic functions on the unit disk D of C, u ∈ H(D) and let n be a positive integer, φ a holomorphic self-map of D, and μ a weight. In this paper, we investigate the boundedness and compactness of a weighted differentiation composition operator Dφ,un f(z) = u(z) f(n) (φ(z)), f ∈ H(D), from the logarithmic Bloch spaces to the Zygmund-type spaces. [ABSTRACT FROM AUTHOR]

We first investigate the construction of a fusion frame system in a finite-dimensional Hilbert space Fn when its fusion frame operator matrix is given and provides a corresponding algorithm. The matrix representations of its local frame operators and inverse frame operators are naturally obtained. We then study the related properties of the constructed fusion frame systems. Finally, we implement the construction of fusion frame systems which behave optimally for erasures in some special sense in signal transmission. [ABSTRACT FROM AUTHOR]

This paper is concerned with some properties of a periodic two-component Camassa-Holm system. By constructing two sequences of solutions of the two-component Camassa-Holm system, we prove that the solution map of the Cauchy problem of the two-component Camassa-Holm system is not uniformly continuous in Hs (핊), s > 5/2. [ABSTRACT FROM AUTHOR]

Let H be a complex Hilbert space; denote by Alg N and Cp(H) the atomic nest algebra associated with the atomic nest N on H and the space of Schatten-p class operators on, H respectively. Let Cp (H) ∩ Alg N be the space of Schatten-p class operators in Alg N. When 1 ≤ p < + ∞ and p ≠ 2, we give a complete characterization of nonlinear surjective isometries on Cp (H) ∩ Alg N. If p = 2, we also prove that a nonlinear surjective isometry on C2 (H) ∩ Alg N is the translation of an orthogonality preserving map. [ABSTRACT FROM AUTHOR]

In the framework of a real Banach space with uniformly Gateaux differentiable norm, some new viscosity iterative sequences {xn} are introduced for an infinite family of asymptotically nonexpansive mappings {Ti}i=1∞ in this paper. Under some appropriate conditions, we prove that the iterative sequences {xn} converge strongly to a common fixed point of the mappings {Ti}i=1∞, which is also a solution of a variational inequality. Our results extend and improve some recent results of other authors. [ABSTRACT FROM AUTHOR]

Let ... be the familiar class of normalized univalent functions in the unit disk. Fekete and Szegö proved the well-known result maxf∈... |a3 - λa2²| = 1 + 2e-2λ/(1-λ) for λ ∈ [0, 1]. We investigate the corresponding problem for the class of starlike mappings defined on the unit ball in a complex Banach space or on the unit polydisk in Cn, which satisfies a certain condition. [ABSTRACT FROM AUTHOR]

An explicit projection algorithm with viscosity technique is constructed for finding the fixed points of the pseudocontractive mapping in Hilbert spaces. Strong convergence theorem is demonstrated. Consequently, as an application, we can approximate to the minimum-norm fixed point of the pseudocontractive mapping. [ABSTRACT FROM AUTHOR]

We consider error bound issue for conic inequalities in Hilbert spaces. In terms of proximal subdifferentials of vector-valued functions, we provide sufficient conditions for the existence of a local error bound for a conic inequality. In the Hilbert space case, our result improves and extends some existing results on local error bounds. [ABSTRACT FROM AUTHOR]

The existence of piecewise almost periodic solutions for impulsive neutral functional differential equations in Banach space is investigated. Our results are based on Krasnoselskii's fixed-point theorem combined with an exponentially stable strongly continuous operator semigroup. An example is given to illustrate the theory. [ABSTRACT FROM AUTHOR]

We prove that if T is an m-isometry on a Hilbert space and Q an n-nilpotent operator commuting with T, then T + Q is a (2n + m - 2)-isometry. Moreover, we show that a similar result for (m, q)-isometries on Banach spaces is not true. [ABSTRACT FROM AUTHOR]

We are concerned with determining values of λ, for which there exist positive solutions of the nonlinear elliptic problem -Δu = λa(x)f(u) in Ω, ∂u/∂n + b(x)g(u) = 0 on ∂Ω. The proof of our main results is based upon unilateral global bifurcation theorem of López-Gómez. [ABSTRACT FROM AUTHOR]

We first prove characterizations of p-uniform convexity and q-uniform smoothness. We next give a formulation on absolute normalized norms on ℂ². Using these, we present some examples of Banach spaces. One of them is a uniformly convex Banach space which is not p-uniformly convex. [ABSTRACT FROM AUTHOR]

Let K be a nonempty closed convex subset of a real Banach space E, let S : K → K be nonexpansive, and let T : K → K be Lipschitz strongly pseudocontractive mappings such that p ∈ F(S) ∩ F(T) = {x ∈ K : Sx = Tx = x} and ∥x - Sy∥ ≤ ∥Sx - Sy∥ and ∥x - Ty∥ ≤ ∥Tx - Ty∥ for all x, y, ∈ K. Let {βn} be a sequence in [0, 1] satisfying (i) ∑n = 1∞ βn = ∞; (ii) limn → ∞βn = 0. For arbitrary x0 ∈ K, let {xn} be a sequence iteratively defined by xn = syn, yn = (1 - βn)xn-1 + βnTxn, n ≥ 1. Then the sequence {xn} converges strongly to a common fixed point p of S and T. [ABSTRACT FROM AUTHOR]

We construct an implicit sequence suitable for the approximation of solutions of K-positive definite operator equations in real Banach spaces. Furthermore, implicit error estimate is obtained and the convergence is shown to be faster in comparsion to the explicit error estimate obtained by Osilike and Udomene (2001). [ABSTRACT FROM AUTHOR]

We introduce and study a new general system of nonlinear variational inclusions involving generalized m-accretive mappings in Banach space. By using the resolvent operator technique associated with generalized m-accretive mappings due to Huang and Fang, we prove the existence theorem of the solution for this variational inclusion system in uniformly smooth Banach space, and discuss convergence and stability of a class of new perturbed iterative algorithms for solving the inclusion system in Banach spaces. Our results presented in this paper may be viewed as an refinement and improvement of the previously known results. [ABSTRACT FROM AUTHOR]

We present a new iterative scheme with errors to solve the problems of finding common zeros of finite m-accretive mappings in a real Banach space. Strong convergence theorems are established, which extend the corresponding works given by some authors. Moreover, the relationship between zeros of m-accretive mappings and one kind of nonlinear elliptic systems is investigated, from which we can see that some restrictions imposed on the iterative scheme are valid and the solution of one kind of nonlinear elliptic systems can be approximated by a suitably defined iterative sequence. [ABSTRACT FROM AUTHOR]

We show that a local minimizer of Φ in the C¹ topology must be a local minimizer in the H¹ ∩ LP topology, under suitable assumptions for the functional Φ = (1/2)∫Ω|∇u|² + (1/p)∫Ω/u/P - ∫Ω/u/P-∫Ω F(x,u) with supercritical exponent p > 2* = 2n/(n-2). This result can be used to establish a solution to the corresponding equation admitting sub- and supersolution. Hence, we extend the conclusion proved by Brezis and Nirenberg (1993), the subcritical and critical case. [ABSTRACT FROM AUTHOR]

The problem of packing spheres in Orlicz sequence space lΦ,pequipped with the p-Amemiya norm is studied, and a geometric characteristic about the reflexivity of lΦ,p is obtained, which contains the relevant work about lp (p > 1) and classical Orlicz spaces lΦdiscussed by Rankin, Burlak, and Cleaver. Moreover the packing constant as well as Kottman constant in this kind of spaces is calculated. [ABSTRACT FROM AUTHOR]

We introduce the dominated farthest points problem in Banach lattices. We prove that for two equivalent norms such that X becomes an STM and LLUM space the dominated farthest points problem has the same solution. We give some conditions such that under these conditions the Fréchet differentiability of the farthest point map is equivalent to the continuity of metric antiprojection in the dominated farthest points problem. Also we prove that these conditions are equivalent to strong solvability of the dominated farthest points problem. We prove these results in STM, reflexive STM, and UM spaces. Moreover, we give some applications of the stated results in Musielak-Orlicz spaces Lø(μ) and Eø(μ) over nonatomic measure spaces in terms of the function ø. We will prove that the Fréchet differentiability of the farthest point map and the conditions ø ∈ Δ2 and ø > 0 in reflexive Musielak-Orlicz function spaces are equivalent. [ABSTRACT FROM AUTHOR]

The intent of this paper is to introduce the notion of compatible mappings for n-tupled coincidence points due to (Imdad et al. (2013)). Related examples are also given to support our main results. Our results are the generalizations of the results of (Gnana Bhaskar and Lakshmikantham (2006), Lakshmikantham and Ćirić (2009), Choudhury and Kundu (2010), and Choudhary et al. (2013)). [ABSTRACT FROM AUTHOR]

By using the classic metric projection method, we obtain sufficient conditions for Hölder continuity of the nonunique solution mapping for a parametric generalized variational inequality with respect to data perturbation. The result is different from the recent ones in the literature and has a strong geometric flavor. [ABSTRACT FROM AUTHOR]

For two complex Banach spaces X and Y, in this paper, we study the generalized spectrum ℳb (X,Y) of all nonzero algebra homomorphisms from ℋb(X), the algebra of all bounded type entire functions on X, ℋb(Y). We endow ℳb(X,Y) with a structure of Riemann domain over ℒ (X*,Y*) whenever X is symmetrically regular. The size of the fibers is also studied. Following the philosophy of (Aron et al., 1991), this is a step to study the set ℳb,∞ (X, By) of all nonzero algebra homomorphisms from ℋb(X) into ℋ∞(By) of bounded holomorphic functions on the open unit ball of and ℳ∞(BX, BY) of all nonzero algebra homomorphisms from ℋ∞(BY). [ABSTRACT FROM AUTHOR]

The analytic space can be embedded into a Bloch-type space. We establish a distance formula from Bloch-type functions F (p, q, s) to, which generalizes the distance formula from Bloch functions to BMOA by Peter Jones, and to F(p, p-2, s) by Zhao. [ABSTRACT FROM AUTHOR]