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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]

Abstract: An error is pointed out in the paper [J.R. Kang, Y.C. Kwun, J.Y. Park, Controllability of second order differential inclusion in Banach spaces, J. Math. Anal. Appl. 285 (2003) 537-550]. By an additional condition the considered system is controllable. [Copyright &y& Elsevier]

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]

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]

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]

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 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]

The article investigates the differential equations in Banach spaces with a right-hand side being more general than Lipschitz. It provides a detailed understanding on the proof of an existence theorem under the additional assumption that the right-hand side is uniformly continuous. It also discusses the problem that arises when trying to prove a result without the assumptions of the uniform convexity of X.

In this note, a small gap is corrected in the assumption of main theorem of T.M. Tuyen (Theorem 3.1, Regularization proximal point algorithm for common fixed points of nonexpansive mappings in Banach spaces, J. Optim. Theory Appl., 152:351-365, ). We give another assumption, which allows us to obtain the strong convergence of regularization algorithms. [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]

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]

We show that any Banach space X that has a sequence of unit vectors weakly converging to 0 has a closed and bounded convex set that is not remotal. This extends the main result of Sababheh and Khalil, (Numer. Funct. Anal. Optim. 2008; 29:1166-1170). [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]

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]

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]

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]

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]

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]

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]

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]

In this paper, we investigate the existence of solution and extremal solutions for an initial-value problem of perturbed functional fractional differential equations with Caputo-Hadamard derivative. Our analysis relies on the fixed point theorem of Burton and Kirk and the concept of upper and lower solutions combined with a fixed point theorem in ordered Banach space established by Dhage and Henderson. [ABSTRACT FROM AUTHOR]

In this paper, we propose an iterative algorithm, which is based on the Mann iterative method for solving simultaneously common fixed point problem with a finite family of demicontractive mappings and systems of variational inequalities involving an infinite family of strongly accretive operators. Under suitable assumptions, we prove the strong convergence of this algorithm in Banach spaces. Application to systems of constrained convex minimization problem is provided to support our main results. The results of this paper improve and extend results of [M. Eslamian, C. R. Math. Acad. Sci. Paris, 355 (2017), 1168–1177], and of many others. [ABSTRACT FROM AUTHOR]

Characterization of simultaneous similarity for commuting m- tuples of operators is an open problem even in finite-dimensional spaces; known as "A wild problem in linear algebra". In this paper we offer a criterion for simultaneous similarity of m-tuples of k-cyclic commuting operators on an arbitrary Banach space. Moreover, we obtain an additional equivalence condition in the case of finite dimensional Banach spaces, which extends the result found by Shekhtman [Math. Stat. 1 (2013), pp. 157–161] for pairs of cyclic commuting matrices. We also present two applications of our results, one in the case of general multiplication operators on Banach spaces of analytic function, and one for m-tuples of commuting square matrices. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a regularized iterative scheme for solving the split common fixed point problem of asymptotically demicontractive and demimetric mappings in real Banach spaces. We then prove that the iterative scheme converges strongly to a solution to the problem, which is also a solution to a variational inequality problem. A split feasibility problem is also considered with the aid of our scheme. [ABSTRACT FROM AUTHOR]

In this paper, we present a result on the existence of asymptotically stable solutions of infinite systems (IS) of quadratic Hammerstein integral equations (IEs). Our study will be conducted in the Banach space of functions, which are continuous and bounded on the half-real axis with values in the classical Banach sequence space consisting of real bounded sequences. The main tool used in our investigations is the technique associated with the measures of noncompactness (MNCs) and a fixed point theorem of Darbo type. The applicability of our result is illustrated by a suitable example at the end of the paper. [ABSTRACT FROM AUTHOR]

Conditions for the existence and uniqueness of mild solutions for a system of semilinear impulsive differential equations with infinite fractional Brownian movements and the Wiener process are established. Our approach is based on a novel application of Burton and Kirk's fixed point theorem in extended Banach spaces. This paper aims to extend current results to a differential-inclusions scenario. The motivation of this paper for impulsive neutral differential equations is to investigate the existence of solutions for impulsive neutral differential equations with fractional Brownian motion and a Wiener process (topics that have not been considered and are the main focus of this paper). [ABSTRACT FROM AUTHOR]

In this paper, we expanded from the convex case to the nonconvex case in the setting of reflexive smooth Banach spaces, the concept of the f-generalized projection πfS: X* ∞ S initially introduced for convex sets and convex functions in [19, 20]. Indeed, we defined the (f, λ)-generalized projection operator πSf: X* ∞ S from X* onto a nonempty closed set S. We proved many properties of πf,λS for any closed (not necessarily convex) set S and for any lower semicontinuous function f. Our principal results broaden the scope of numerous theorems established in [19, 20] from the convex setting to the nonconvex setting. An application of our main results to solutions of nonconvex variational problems is stated at the end of the paper. [ABSTRACT FROM AUTHOR]

Let D be a strongly double triangle subspace lattice on a nonzero complex reflexive Banach space. In this paper, we characterize the linear maps δ, τ: AlgD → AlgD satisfying δ(A)B+ Aτ(B) = 0 for any A, B ∈ AlgD with AB = 0. This result can be used to characterize linear maps derivable (centralized) at zero point and local centralizers on AlgD, respectively. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form D^{\alpha }_Cu(t)=Au(t)+f(t), u(0)=x, 0<\alpha \le 1, (*) where D^{\alpha }_Cu(t) is the derivative of the function u in the Caputo's sense, A is a linear operator in a Banach space \mathbb {X} that may be unbounded and f satisfies the property that \lim _{t\to \infty } (f(t+1)-f(t))=0 which we will call asymptotic 1-periodicity. By using the spectral theory of functions on the half line we derive analogs of Katznelson-Tzafriri and Massera Theorems. Namely, we give sufficient conditions in terms of spectral properties of the operator A for all asymptotic mild solutions of Eq. (*) to be asymptotic 1-periodic, or there exists an asymptotic mild solution that is asymptotic 1-periodic. [ABSTRACT FROM AUTHOR]

In this paper, we present new existence results for the quasivariational inequality problem (QV I) in reflexive Banach spaces using the fixed point method with quasi-monotonicity and local upper sign-continuity assumptions. These results improve upon previous ones which only required a weaker monotonicity condition and did not impose compactness on the involved set. [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 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]

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]

Green function of a 2n-th order differential equation with normal coefficients on the half-axis is studied. We first consider the Green function of our equation with “frozen” coefficients. Using Levi’s method, we obtain a Fredholm-type integral equation for the Green function of our problem, whose kernel is a Green function of a problem with constant coefficients. We prove an existence and uniqueness theorem for this integral equation in some Banach spaces of operator-valued functions. The main result of this paper is a theorem stating that the solution of the obtained integral equation is a Green function of our problem. [ABSTRACT FROM AUTHOR]

This paper introduces a novel F fixed-point iteration method that leverages Green's function for solving the nonlinear Troesch problem in Banach spaces, which are symmetric spaces. The Troesch problem, characterized by its challenging boundary conditions and nonlinear nature, is significant in various physical and engineering applications. The proposed method integrates fixed-point theory with Green's function techniques to develop an iteration process that ensures convergence, stability, and accuracy. The numerical experiments demonstrate the method's efficiency and robustness, highlighting its potential for broader applications in solving nonlinear differential equations in Banach spaces. [ABSTRACT FROM AUTHOR]

Over the past two decades, significant advancements have been made in understanding the stability according to Hyers–Ulam involving different functional equations (FEs). This study investigates the generalized stability of norm-based (norm-additive) FEs within the framework of arbitrary (noncommutative) groups and p-uniformly convex spaces. Specifically, we analyze two key functional equations, ∥ η (g h) ∥ = ∥ η (g) + η (h) ∥ and ∥ η (g h − 1) ∥ = ∥ η (g) − η (h) ∥ for every g , h ∈ G , where (G , ·) denotes an arbitrary group and B is considered to be a p-uniformly convex space. The surjectivity of the function η : G → B is a critical assumption in our analysis. Drawing upon the foundational works of L. Cheng and M. Sarfraz, this paper applies the large perturbation method tailored for p-uniformly convex spaces, where p ≥ 1 . This study extends previous research by offering a deeper exploration of the conditions under which these functional equations demonstrate Hyers–Ulam stability. In this study, the additive functional equation demonstrates a fundamental form of symmetry, where the order of operands does not affect the results. This symmetry under permutation of arguments is crucial for the analysis of stability. In the context of norm-additive FEs, this stability criterion investigates how small changes in the inputs of a functional equation affect the outputs, especially when the function is expected to follow an additive form. [ABSTRACT FROM AUTHOR]