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In a recently published theorem on the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings, Tang et al. (J. Inequal. Appl. 2015:305, 2015) studied a uniformly convex and 2-uniformly smooth real Banach space with the Opial property and best smoothness constant κ satisfying the condition 0 < κ < 1 2 , as a real Banach space more general than Hilbert spaces. A well-known example of a uniformly convex and 2-uniformly smooth real Banach space with the Opial property is E = l p , 2 ≤ p < ∞ . It is shown in this paper that, if κ is the best smoothness constant of E and satisfies the condition 0 < κ ≤ 1 2 , then E is necessarily l 2 , a real Hilbert space. Furthermore, some important remarks concerning the proof of this theorem are presented. [ABSTRACT FROM AUTHOR]

Very recently I. Beg et al. [I. Beg, A. Bartwal, S. Rawat and R.C. Dimri, Best proximity points in noncommutative Banach spaces, Comp. Appl. Math., 41, 41 (2022)] introduced the concepts of k-ordered proximal contraction of the first kind and k-ordered proximal contraction of the second kind for non-self mappings and investigated the existence of best proximity points for these two classes of mappings. In the current paper, we show that their existence results are straightforward consequences of the same conclusions in fixed point theory. [ABSTRACT FROM AUTHOR]

The present paper is in two folds. In the first fold, we prove the Lipschitz continuity of the proximal mapping associated with a general strongly H-monotone mapping and compute an estimate of its Lipschitz constant under some mild assumptions imposed on the mapping H involved in the proximal mapping. We provide two examples to show that a maximal monotone mapping need not be a general H-monotone for a single-valued mapping H from a Banach space to its dual space. A class of multi-valued nonlinear variational inclusion problems is considered, and by using the notion of proximal mapping and Nadler's technique, an iterative algorithm with mixed errors is suggested to compute its solutions. Under some appropriate hypotheses imposed on the mappings and parameters involved in the multi-valued nonlinear variational inclusion problem, the strong convergence of the sequences generated by the proposed algorithm to a solution of the aforesaid problem is verified. The second fold of this paper investigates and analyzes the notion of Cn-monotone mappings defined and studied in [S.Z. Nazemi, A new class of monotone mappings and a new class of variational inclusions in Banach spaces, J. Optim. Theory Appl. 155(3)(2012) 785-795]. Several comments related to the results and algorithm appeared in the above mentioned paper are given. [ABSTRACT FROM AUTHOR]

Let ℓ be a Banach sequence space with a monotone norm in which the canonical system (en) is an unconditional basis. We show that if there exists a continuous linear unbounded operator between ℓ -Köthe spaces, then there exists a continuous unbounded quasidiagonal operator between them. Using this result, we study the corresponding Köthe matrices when every continuous linear operator between ℓ -Köthe spaces is bounded. As an application, we observe that the existence of an unbounded operator between ℓ -Köthe spaces, under a splitting condition, causes the existence of a common basic subspace. [ABSTRACT FROM AUTHOR]

This short and concise note provides a detailed exposition of the approach and results established by (Lin et al, 2015, pp.271-279). We show that the obtained results are not particularly surprising and new. Namely, using an old result due to K. Deimling it is indicated that tvs-cone metric spaces over solid cones are actually cone metric spaces over normal solid cones. Hence, there are only cone metric spaces over normal solid cones or over normal non-solid cones. One question still unanswered is whether an ordered topological vector space with a non-normal non-solid cone exists. [ABSTRACT FROM AUTHOR]

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]

We obtain results on the existence and approximation of fixed points of enriched contractions in quasi-Banach spaces and thus extend the previous results for enriched contractions defined on Banach spaces [Berinde, V.; Păcurar, M. Approximating fixed points of enriched contractions in Banach spaces. J. Fixed Point Theory Appl. 22 (2020), no. 2, Paper No. 38, 10 pp.]. The theoretical results are illustrated by means of an appropriate example of enriched contraction on a quasi-Banach space which is not a Banach space and thus show that our new results are effective generalizations of the previous ones in literature. [ABSTRACT FROM AUTHOR]

Starting from a characterization of polynomial dichotomy by means of admissibility, recently proved in [Dragicevic, D.; Sasu, A. L.; Sasu, B. Admissibility and polynomial dichotomy of discrete nonautonomous systems. Carpath. J. Math. 38 (2022), 737-762.], the aim of this paper is to explore the roughness of polynomial dichotomy in the presence of perturbations and to obtain a new robustness criterion. We show that the polynomial dichotomy is robust when subjected to linear additive perturbations which are bounded by a well-chosen sequence. We emphasize that the new bounds imposed to the perturbation family improve and extend the previous approaches. Furthermore, we mention that the main result applies to discrete nonautonomous systems in Banach spaces with the only requirement that their propagators exhibit a polynomial growth. [ABSTRACT FROM AUTHOR]

In this paper, we develop some new multiple fixed point theorems for the sum of two operators T + S where I - T is Lipschitz invertible and S is a k-set contraction on translate of a cone in a Banach space. New existence criteria for multiple positive solutions of a singular generalized Sturm-Liouville multipoint boundary value problem are established. The article ends with an illustrative example. [ABSTRACT FROM AUTHOR]

In this paper, we study the infinite-dimensional endpoint state-constrained optimal control problem for fractional evolution equations. The state equation is modeled by the -valued left Caputo fractional evolution equation with the analytic semigroup, where is a Banach space. The objective functional is formulated by the Bolza form, expressed in terms of the left Riemann-Liouville (RL) fractional integral running and initial/terminal costs. The endpoint state constraint is described by initial and terminal state values within convex subsets of. Under this setting, we prove the Pontryagin maximum principle. Unlike the existing literature, we do not assume the strict convexity of , the dual space of. This assumption is particularly important, as it guarantees the differentiability of the distance function of the endpoint state constraint. In the proof, we relax this assumption via a separation argument and constructing a family of spike variations for the Ekeland variational principle. Subsequently, we prove the maximum principle, including nontriviality, adjoint equation, transversality, and Hamiltonian maximization conditions, by establishing variational and duality analysis under the finite codimensionality of initial- and end-point variational sets. Our variational and duality analysis requires new representation results on left Caputo and right RL linear fractional evolution equations in terms of (left and right RL) fractional state transition operators. Indeed, due to the inherent complex nature of the problem of this paper, our maximum principle and its proof technique are new in the optimal control context. As an illustrative example, we consider the state-constrained fractional diffusion PDE control problem, for which the optimality condition is derived by the maximum principle of this paper. [ABSTRACT FROM AUTHOR]

The aim of the paper is to develop some representation formulae for strongly continuous operator semigroups on Banach spaces, in terms of limits of integrated means with respect to some given family of probability Borel measures and other parameters. The cases where these limits hold true pointwise or uniformly on compact subintervals are discussed separately. In order to face them different methods have been required: the former case has been studied by using purely functional-analytic methods, the latter one by involving methods arising from Approximation Theory. The paper also contains some estimates of the rate of convergence in terms of the rectified modulus of continuity and the second modulus of continuity. In a final section some illustrative examples and applications are provided. [ABSTRACT FROM AUTHOR]

In this paper, the existence of optimal solutions for problems governed by differential equations involving feedback controls is established for when the problem must account for a Volterra-type distributed delay and is subject to the action of impulsive external forces. The problem is reformulated within the class of impulsive semilinear integro-differential inclusions in Banach spaces and is studied by using topological methods and multivalued analysis. The paper concludes with an application to a population dynamics model. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide new results on the Wolf and Schechter essential pseudospectra of bounded linear operators on a Banach space. More precisely, we characterize the Wolf and Schechter essential pseudospectra by using the notion of Fredholm perturbation. Also, we state the condition under which the Wolf (respectively, Schechter) essential pseudospectrum of two different bounded linear operators coincides. Furthermore, we give some characterizations of the Wolf and Schechter essential pseudospectra of 3 × 3 upper triangular block operator matrices. [ABSTRACT FROM AUTHOR]

In the paper, we introduce a new class of noncyclic φ-contractions as a generalization of the class of noncyclic contractions which was first introduced in the paper [R. Esp'ınola, M. Gabeleh, On the structure of minimal sets of relatively nonexpansive mappings, Numerical Functional Analysis and Optimization 34 (8), 845-860, 2013] and study the existence, uniqueness and convergence of a fixed point for such class of noncyclic mapping in the framework of uniformly convex Banach spaces. We obtain existence results of the best proximity points for cyclic φ-contractions as a consequence of our main theorems. [ABSTRACT FROM AUTHOR]

In this paper, we present a new analytical scheme to generate atomic solutions of partial differential equations. The theory of tensor product in Banach spaces coupled with some features of atomics operators are utilized to attain our results. Some demonstrative examples are given for completeness. [ABSTRACT FROM AUTHOR]

Let X be a complex Banach space, and denote by \mathcal {B}(X) the algebra of all bounded linear operators on X. Let C,D\in \mathcal {B} \left (X\right) be fixed operators. In this paper, we characterize linear, continuous and bijective maps \varphi and \psi on \mathcal {B}\left (X\right) for which there exist invertible operators T_0, W_0 \in \mathcal { B}(X) such that \varphi (T_0), \psi (W_0) \in \mathcal {B}(X) are both invertible, having the property that \varphi \left (A\right) \psi \left (B\right) =D in \mathcal {B}(X) whenever AB=C in \mathcal {B}(X). As a corollary, we deduce the form of linear, bijective and continuous maps \varphi on \mathcal {B}(X) having the property that \varphi \left (A\right) \varphi \left (B\right) =D in \mathcal {B}(X) whenever AB=C. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the existence, uniqueness, and stability properties for a class of fractional weighted Cauchy-type problem in the variable exponent Lebesgue space Lp(.). The obtained results are set up by employing generalized intervals and piece-wise constant functions so that the Lp(.) is transformed into the classical Lebesgue spaces. Moreover, the usual Banach Contraction Principle is utilized, and the Ulam-Hyers (UH) stability is studied. At the final stage, we provide an example to support the accuracy of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in studying Bishop–Phelps–Bollobás type properties related to the denseness of the operators which attain their numerical radius. We prove that every Banach space with a micro-transitive norm and the second numerical index strictly positive satisfies the Bishop–Phelps–Bollobás point property, and we see that the one-dimensional space is the only one with both the numerical index 1 and the Bishop–Phelps–Bollobás point property. We also consider two weaker properties L $ {}_{p,p} $ p , p -nu and L $ {}_{o,o} $ o , o -nu, the local versions of Bishop–Phelps–Bollobás point and operator properties respectively, where the η which appears in their definition does not depend just on $ \varepsilon > 0 $ ε > 0 but also on a state $ (x, x^*) $ (x , x ∗) or on a numerical radius one operator T. We address the relation between the L $ {}_{p,p} $ p , p -nu and the strong subdifferentiability of the norm of the space X. We show that finite dimensional spaces and $ c_0 $ c 0 are examples of Banach spaces satisfying the L $ {}_{p,p} $ p , p -nu, and we exhibit an example of a Banach space with a strongly subdifferentiable norm failing it. We finish the paper by showing that finite dimensional spaces satisfy the L $ {}_{o,o} $ o , o -nu and that, if X has a strictly positive numerical index and has the approximation property, this property is equivalent to finite dimensionality. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative scheme to approximate the fixed point of generalized α-nonexpansive mappings. we first prove that the proposed iteration process is faster than all of Picard, Mann, Ishikawa, Noor, Agarwal, Abbas and Thakur processes for contractive mappings. We also obtain some weak and strong convergence theorems for generalized α-nonexpansive mappings. Using the example presented in [R. Pant and R. Shukla, Approximating fixed point of generalized α-nonexpansive mappings in Banach spaces, J. Numer. Funct. Anal. Optim. 38(2017) 248-266.], we compare the convergence behavior of the new iterative process with other iterative processes. [ABSTRACT FROM AUTHOR]

In this paper we correct an error made in our paper [Blouhi, T.; Caraballo, T.; Ouahab, A. Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion. Stoch. Anal. Appl. 34 (2016), no. 5, 792-834]. In fact, in this corrigendum we present the correct hypotheses and results, and highlight that the results can be proved using the same method used in the original work. The main feature is that we used a result which has been proved only when the diffusion term does not depend on the unknown. [ABSTRACT FROM PUBLISHER]

Celebrating 100 years of the Banach contraction principle, we prove some fixed point theorems having all ingredients of the principle, but dealing with common fixed points of a contractive semigroup of nonlinear mappings acting in a modulated topological vector space. This research follows the ideas of the author's recent papers ['On modulated topological vector spaces and applications', Bull. Aust. Math. Soc. 101 (2020), 325–332, and 'Normal structure in modulated topological vector spaces', Comment. Math. 60 (2020), 1–11]. Modulated topological vector spaces generalise, among others, Banach spaces and modular function spaces. The interest in modulars reflects the fact that the notions of 'norm like' but 'noneuclidean' (and not even necessarily convex) constructs to measure a level of proximity between complex objects are frequently used in science and technology. To prove our fixed point results in this setting, we introduce a new concept of Opial sets using analogies with the norm-weak and modular versions of the Opial property. As an example, the results of this work can be applied to spaces like $L^p$ for $p> 0 $ , variable Lebesgue spaces $L^{p(\cdot)}$ where $1 \leq p(t) , Orlicz and Musielak–Orlicz spaces. [ABSTRACT FROM AUTHOR]

Consider an infinite-dimentional Banach space denoted as X, and designate ∂(X) as the algebra of all bounded linear operators on X. Moreover, let s (A) denote the spectrum of ..., and ...(A)) indicate the boundary of s (A). A map ... is termed a ρ-spectrum if ... (A) for all A ... In this paper, we characterize all surjective maps f1 and f2 on ρ(X) satisfying ... = (AB) for all A,B(X). [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new Bregman extragradient method with a different line-search process for solving variational inequality problems in reflexive Banach spaces. Precisely, we prove that the sequence generated by our proposed iterative algorithm converges strongly to an element of the solution sets of variational inequality problems. Moreover, some numerical examples are given to show the effectiveness of the proposed algorithm. The results obtained in this paper extend and improve many recent ones in the literature. [ABSTRACT FROM AUTHOR]

This paper investigates analytic Besov functions of n variables which act on the generators of n commuting C0-semigroups on a Banach space. The theory for n = 1 has already been published, and the present paper uses a different approach to that case as well as extending to the cases when n ≥ 2. It also clarifies some spectral mapping properties and provides some operator norm estimates. [ABSTRACT FROM AUTHOR]

In this paper, the semi-local convergence of the Cordero's sixth-order iterative method in Banach space was proved by the method of recursion relation. In the process of proving, the auxiliary sequence and three increasing scalar functions can be derived using Lipschitz conditions on the first-order derivatives. By using the properties of auxiliary sequence and scalar function, it was proved that the iterative sequence obtained by the iterative method was a Cauchy sequence, then the convergence radius was obtained and its uniqueness was proven. Compared with Cordero's process of proving convergence, this paper does not need to ensure that G (s) is continuously differentiable in higher order, and only the first-order Fréchet derivative was used to prove semi-local convergence. Finally, the numerical results showed that the recursion relationship is reasonable. [ABSTRACT FROM AUTHOR]

In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order p of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order p + 3 (for p ≥ 3 ) and have extended it to a generalized (m + 2) -step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to p + 3 m , m ∈ N . We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations. In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order (for ) and have extended it to a generalized -step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to , . We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations. [ABSTRACT FROM AUTHOR]

Let n be an integer greater than 1 . In this paper, we obtained the stability of the multivariable Cauchy-Jensen functional equation n f (x 1 + ⋯ + x n , y 1 + ⋯ + y n n) = ∑ 1 ≤ i , j ≤ n f (x i , y j) in Banach spaces, quasi-Banach spaces, and normed 2 -Banach spaces. Let be an integer greater than . In this paper, we obtained the stability of the multivariable Cauchy-Jensen functional equation in Banach spaces, quasi-Banach spaces, and normed -Banach spaces. [ABSTRACT FROM AUTHOR]

Let Q be a bounded domain of RN (Nx1) with boundary of class C2. In the present paper we shall study a variational problem relating the weighted Hardy inequalities with sharp missing terms established in [8]. As weights we treat non-doubling functions of the distance to the boundary . [ABSTRACT FROM AUTHOR]

In this study, we extended the applicability of a derivative-free algorithm to encompass the solution of operators that may be either differentiable or non-differentiable. Conditions weaker than the ones in earlier studies are employed for the convergence analysis. The earlier results considered assumptions up to the existence of the ninth order derivative of the main operator, even though there are no derivatives in the algorithm, and the Taylor series on the finite Euclidian space restricts the applicability of the algorithm. Moreover, the previous results could not be used for non-differentiable equations, although the algorithm could converge. The new local result used only conditions on the divided difference in the algorithm to show the convergence. Moreover, the more challenging semi-local convergence that had not previously been studied was considered using majorizing sequences. The paper included results on the upper bounds of the error estimates and domains where there was only one solution for the equation. The methodology of this paper is applicable to other algorithms using inverses and in the setting of a Banach space. Numerical examples further validate our approach. [ABSTRACT FROM AUTHOR]

This paper explores the existence and stability of implicit neutral Caputo fractional qdifference equations within four distinct classes, incorporating various delay types such as finite, infinite, and state-dependent delays. To establish the existence of solutions, we utilize the fixed point theorem of Krasnoselskii in Banach spaces. The concluding section provides illustrative examples that highlight the obtained results. [ABSTRACT FROM AUTHOR]

This paper first shows that for any p ∈ (1, 2) there exists a continuously differentiable function f on lp (and Lp) such that the proximal subdifferential of f is empty everywhere, and hence it is not suitable to develop theory on proximal subdifferential in the classical Banach spaces lp and LP with p ∈ (1, 2). On the other hand, this paper establishes variational analysis based on the proximal subdifferential in the framework of smooth Banach spaces of power type 2, which conclude all Hilbert spaces and all the classical spaces lp and Lp with p ∈ (2, +∞). In particular, in such a smooth space, we provide the proximal subdifferential rules for sum functions, product functions, composite functions and supremum functions, which extend the basic results on the proximal subdifferential established in the framework of Hilbert spaces. Some of our main results are new even in the Hilbert space case. As applications, we provide KKT-like conditions for nonsmooth optimization problems in terms of proximal subdifferential. [ABSTRACT FROM AUTHOR]

Let γ ∈ ℝ ∖ { 0 } and X (ℝ n) be a ball Banach function space satisfying some extra mild assumptions. Assume that Ω = ℝ n or Ω ⊂ ℝ n is an (, ∞) -domain for some ∈ (0 , 1 ]. In this paper, the authors prove that a function f belongs to the homogeneous ball Banach Sobolev space Ẇ 1 , X (Ω) if and only if f ∈ L loc 1 (Ω) and sup λ ∈ (0 , ∞) λ ∫ { y ∈ Ω :   | f (⋅) − f (y) | > λ | ⋅ − y | 1 + γ p } ⋅ − y γ − n d y 1 p X (Ω) < ∞ , where p ∈ [ 1 , ∞) is related to X (ℝ n). This result is of wide generality and can be applied to various specific Sobolev-type function spaces, including Morrey [Bourgain–Morrey-type, weighted (or mixed-norm or variable) Lebesgue, local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, which is new even in all these special cases; in particular, it coincides with the well-known result of H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung when X (Ω) : = L q (ℝ n) with 1 < p = q < ∞ , while it is still new even when X (Ω) : = L q (ℝ n) with 1 ≤ p < q < ∞. The novelty of this paper exists in that, to establish the characterization of Ẇ 1 , X (Ω) , the authors provide a machinery via using the generalized Brezis–Seeger–Van Schaftingen–Yung formula on X (ℝ n) , the extension theorem on Ẇ 1 , X (Ω) , the Bourgain–Brezis–Mironescu-type characterization of the inhomogeneous ball Banach Sobolev space W 1 , X (Ω) , and the method of extrapolation to overcome those difficulties caused by that X (ℝ n) might be neither the rotation invariance nor the translation invariance and that the norm of X (ℝ n) has no explicit expression. [ABSTRACT FROM AUTHOR]

In this paper, we explore the spectral properties of unbounded generalized Fredholm operators acting on a non-reflexive Banach space X. The results are formulated in terms of some topological conditions made on X or on its dual X * . In addition, we introduce the concept of the so-called g-g-Riesz linear operators as an extension of Riesz operators. The obtained results are used to discuss the incidence of the behavior of generalized essential spectra. Furthermore, a relation between the generalized essential spectrum and the left (resp. the right) essential spectrum by means of g-Riesz perturbation is provided. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and minimal wave speed of traveling wavefronts and the existence of periodic waves for a modified Leslie–Gower model with diffusion and chemotaxis. The existence of traveling wavefronts is proved by applying the perturbation method. Our approach is based on an abstract formulation of the wave profile as a solution of an operational equation in a certain Banach space, coupled with the Fredholm theory and the Banach contraction mapping principle. Moreover, we study the minimal wave speed of traveling wavefronts by using the standard stability analysis, and investigate the existence condition of periodic waves when traveling wavefronts disappears. Some numerical simulations are presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, the behavior of derivative-free techniques to approximate solutions of nonlinear Hammerstein-type integral equations in Banach space C ([ α , β ]) as alternatives against the well-known Newton's method is examined. In particular, from the well-known Kurchatov method, we construct a uniparametric family of derivative-free iterative schemes in order to achieve an approximation of a solution of a Hammerstein-type integral equation with a non-differentiable Nemyskii operator. We compare the uniparametric family built and the Kurchatov method, obtaining improvements in the precision and also in the accessibility through studying its dynamic behavior. [ABSTRACT FROM AUTHOR]

The paper focuses on establishing sufficient conditions for the existence of the solutions in some functional q-integral equations, particularly in Banach spaces. In this method, the technique of measures of noncompactness and Petryshyn's fixed point theorem in Banach space is used. We provide some examples of equations, which confirm that our result is applicable to a wide class of integral equations. [ABSTRACT FROM AUTHOR]

Let Ω be a nonempty closed convex subset of a real Hilbert space H . Let ℑ be a nonspreading mapping from Ω into itself. Define two sequences { ψ n } n = 1 ∞ and { ϕ n } n = 1 ∞ as follows: { ψ n + 1 = π n ψ n + (1 − π n) ℑ ψ n , ϕ n = 1 n ∑ n t = 1 ψ t , for n ∈ N , where 0 ≤ π n ≤ 1 , and π n → 0 . In 2010, Kurokawa and Takahashi established weak and strong convergence theorems of the sequences developed from the above Baillion-type iteration method (Nonlinear Anal. 73:1562–1568, 2010). In this paper, we prove weak and strong convergence theorems for a new class of (η , β) -enriched strictly pseudononspreading ((η , β) -ESPN) maps, more general than that studied by Kurokawa and W. Takahashi in the setup of real Hilbert spaces. Further, by means of a robust auxiliary map incorporated in our theorems, the strong convergence of the sequence generated by Halpern-type iterative algorithm is proved thereby resolving in the affirmative the open problem raised by Kurokawa and Takahashi in their concluding remark for the case in which the map ℑ is averaged. Some nontrivial examples are given, and the results obtained extend, improve, and generalize several well-known results in the current literature. [ABSTRACT FROM AUTHOR]

In this paper, we study split null point problem in reexive Banach spaces. Using the Bregman technique together with a modified inertial Halpern method, we approximate a solution of split null point problem. Also, we establish a strong convergence result for approximating the solution of the aforementioned problems. It is worth mentioning that the iterative algorithm employ in this study is design in such a way that it does not require prior knowledge of operator norm. We display some numerical examples to illustrate the performance of the proposed iterative method. The result discuss in this paper extends and complements many related results in literature. [ABSTRACT FROM AUTHOR]

In this paper, for a bounded linear operator defined on a complex Banach space of infinite dimension, we consider the set of isolated points in its approximate point spectrum, which are eigenvalues of finite multiplicity; this set can be equal to the spectrum of the operator but without its upper semi-Fredholm spectrum, and this relation or equality defines in the literature a new spectral property called the property (ωπ) and is a generalization of the classical a-Weyl theorem. We establish some characterizations and consequences about the property (ωπ), some with topological aspects. Furthermore, we study this property through the Riesz functional calculus. Part of the spectral structure of a linear operator verifying property (ωπ) is described, obtaining some associated properties. [ABSTRACT FROM AUTHOR]

In this paper, we studied the AA-iterative algorithm for finding fixed points of the class of nonlinear generalized (α, β)-nonexpansive mappings. First, we proved weak convergence and then proved several strong convergence results of the scheme in a ground setting of uniformly convex Banach spaces. We gave a few numerical examples of generalized (α, β)-nonexpansive mappings to illustrate the major outcomes. One example was constructed over a subset of a real line while the other one was on the two dimensional space with a taxicab norm. We considered both these examples in our numerical computations to show that our iterative algorithm was more effective in the rate of convergence corresponding to other fixed point algorithms of the literature. Some 2D and 3D graphs were obtained that supported graphically our results and claims. As applications of our major results, we solved a class of fractional differential equations, 2D Voltera differential equation, and a convex minimization problem. Our findings improved and extended the corresponding results of the current literature [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the existence of solutions of set-valued quasiequilibrium problems in real Banach spaces. We introduce a concept of Brezis pseudomonotonicity for set-valued mappings. By constructing appropriate constraint conditions for the weak coercive function and the objective mapping, we obtain a solution-existence result for usual set-valued equilibrium problems. By a sequence of penalized set-valued equilibrium problems, we obtain a solution-existence result for set-valued quasi-equilibrium problems. These results do not require the assumption of compactness of the constraint set, and improve the corresponding ones in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we offer Ostrowski-type inequalities that extend the findings that have been proven for functions of one variable with values in Banach spaces, conducted in a remarkable study by Dragomir, to functions of two variables containing values in the product Banach spaces. Our findings are also an extension of several previous findings that have been established for functions of two variable functions. Prior studies on Ostrowski-type inequalities incriminated functions that have values in Banach spaces or Hilbert spaces. This study is unique and significant in the field of mathematical inequalities, and specifically in the study of Ostrowski-type inequalities, because they have been established for functions having values in a product of two Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the delayed measure differential equations with nonlocal conditions via measure of noncompactness in ordered Banach spaces. Combining (β , γ k) -resolvent family, regulated functions and fixed point theorem with respect to convex-power condensing operator and measure of noncompactness, we investigate the existence of positive mild solutions for the mentioned system under the situation that the nonlinear function satisfies measure conditions and order conditions. In addition, we provide an example to verify the rationality of our conclusion. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to analyze the solvability of a class of stochastic functional integral equations by utilizing the measure of non-compactness with Petryshyn’s fixed point theorem in a Banach space. The results obtained in this paper cover numerous existing results concluded under some weaker conditions by many authors. An example is given to support our main theorem. [ABSTRACT FROM AUTHOR]

The computational study of fixed-point problems in distance spaces is an active and important research area. The purpose of this paper is to construct a new iterative scheme in the setting of Banach space for approximating solutions of fixed-point problems. We first prove the strong convergence of the scheme for a general class of contractions under some appropriate assumptions on the domain and a parameter involved in our scheme. We then study the qualitative aspects of our scheme, such as the stability and order of convergence for the scheme. Some nonlinear problems are then considered and solved numerically by our new iterative scheme. The numerical simulations and graphical visualizations prove the high accuracy and stability of the new fixed-point scheme. Eventually, we solve a 2D nonlinear Volterra Integral Equation (VIE) via the application of our main outcome. Our results improve many related results in fixed-point iteration theory. [ABSTRACT FROM AUTHOR]

In 2021, Z. Fu, Y. Chen and B. Han introduced an inexact Newton regularization (REGINN-IT) using an idea involving the non-stationary iterated Tikhonov regularization scheme for solving nonlinear ill-posed operator equations. In this paper, we suggest a simplified version of the REGINN-IT scheme by using the Bregman distance, duality mapping and a suitable parameter choice strategy to produce an approximate solution. The method is comprised of inner and outer iteration steps. The outer iterates are stopped by a Morozov-type stopping rule, while the inner iterate is executed by making use of the non-stationary iterated Tikhonov scheme. We have studied convergence of the proposed method under some standard assumptions and utilizing tools from convex analysis. The novelty of the method is that it requires computation of the Fréchet derivative only at an initial guess of an exact solution and hence can be identified as more efficient compared to the method given by Z. Fu, Y. Chen and B. Han. Further, in the last section of the paper, we discuss test examples to inspect the proficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem of Keller–Segel–Navier–Stokes system with fractional diffusion. Making use of Fourier localization technique and Littlewood-Paley theory, we establish the global well-posedness of mild solution for small initial data in mixed time-space Besov spaces. Furthermore, we obtain the well-posedness of mild solution in time-weighted Besov spaces by Banach fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we established two results concerning non-surjective (ε, s)-isometries of uniformly convex Banach spaces, which extended some known results of Dolinar and Jung. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the split equality Minty variational inequality problem in reflexive real Banach spaces. Then we construct a single projection inertial algorithm for solving the introduced problem. We establish a strong convergence result with the assumption that the mappings under consideration are Lipschitz continuous and quasimonotone. We give some specific applications of the main result and finally provide a numerical example to demonstrate the workability of our method. [ABSTRACT FROM AUTHOR]