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In the framework of quasi‐regular strongly local Dirichlet form (E,D(E))$(\mathcal {E},D(\mathcal {E}))$ on L2(X;m)$L^2(X;\mathfrak {m})$ admitting minimal E$\mathcal {E}$‐dominant measure μ$\mu$, we construct a natural p$p$‐energy functional (Ep,D(Ep))$(\mathcal {E}^{\,p},D(\mathcal {E}^{\,p}))$ on Lp(X;m)$L^p(X;\mathfrak {m})$ and (1,p)$(1,p)$‐Sobolev space (H1,p(X),∥·∥H1,p)$(H^{1,p}(X),\Vert \cdot \Vert _{H^{1,p}})$ for p∈]1,+∞[$p\in]1,+\infty [$. In this paper, we establish the Clarkson‐type inequality for (H1,p(X),∥·∥H1,p)$(H^{1,p}(X),\Vert \cdot \Vert _{H^{1,p}})$. As a consequence, (H1,p(X),∥·∥H1,p)$(H^{1,p}(X),\Vert \cdot \Vert _{H^{1,p}})$ is a uniformly convex Banach space, hence it is reflexive. Based on the reflexivity of (H1,p(X),∥·∥H1,p)$(H^{1,p}(X),\Vert \cdot \Vert _{H^{1,p}})$, we prove that (generalized) normal contraction operates on (Ep,D(Ep))$(\mathcal {E}^{\,p},D(\mathcal {E}^{\,p}))$, which has been shown in the case of various concrete settings, but has not been proved for such a general framework. Moreover, we prove that (1,p)$(1,p)$‐capacity Cap1,p(A)<∞${\rm Cap}_{1,p}(A)<\infty$ for open set A$A$ admits an equilibrium potential eA∈D(Ep)$e_A\in D(\mathcal {E}^{\,p})$ with 0≤eA≤1$0\le e_A\le 1$m$\mathfrak {m}$‐a.e. and eA=1$e_A=1$m$\mathfrak {m}$‐a.e. on A$A$. [ABSTRACT FROM AUTHOR]

We obtain a complete description of the spectrum of the Fréchet algebra of symmetric analytic functions bounded on balls on the sequence space ℓ1$\ell _1$. This is achieved after proving that on the analogous algebra for ℓp$\ell _p$, 1≤p<∞$1\le p <\infty$, the radius function of any evaluation homomorphism δx,x∈ℓp$\delta _x, \nobreakspace x \in \ell _p$, coincides with the norm of x$x$. [ABSTRACT FROM AUTHOR]

The finite Hilbert transform T$T$, when acting in the classical Zygmund space LlogL$L\textnormal {log} L$ (over (−1,1)$(-1,1)$), was intensively studied in [8]. In this note, an integral representation of T$T$ is established via the L1(−1,1)$L^1(-1,1)$‐valued measure mL1:A↦T(χA)$m_{L^1}: A\mapsto T(\chi _A)$ for each Borel set A⊆(−1,1)$A\subseteq (-1,1)$. This integral representation, together with various non‐trivial properties of mL1$m_{L^1}$, allows the use of measure theoretic methods (not available in [8]) to establish new properties of T$T$. For instance, as an operator between Banach function spaces T$T$ is not order bounded, it is not completely continuous and neither is it weakly compact. An appropriate Parseval formula for T$T$ plays a crucial role. [ABSTRACT FROM AUTHOR]

We construct a compact Hausdorff space K$K$ such that the space P(K)$P(K)$ of Radon probability measures on K$K$ considered with the weak∗$\text{weak}^*$ topology (induced from the space of continuous functions C(K)$C(K)$) is countably tight that is a generalization of sequentiality (i.e., if a measure μ$\mu$ is in the closure of a set M$M$, there is a countable M′⊆M$M^{\prime }\subseteq M$ such that μ$\mu$ is in the closure of M′$M^{\prime }$) but K$K$ carries a Radon probability measure that has uncountable Maharam type (i.e., L1(μ)$L_1(\mu)$ is nonseparable). The construction uses (necessarily) an additional set‐theoretic assumption (the ◇$\diamondsuit$ principle) as it was already known, by a result of Fremlin, that it is consistent that such spaces do not exist. This should be compared with the result of Plebanek and Sobota who showed that countable tightness of P(K×K)$P(K\times K)$ implies that all Radon measures on K$K$ have countable type. So, our example shows that the tightness of P(K×K)$P(K\times K)$ and of P(K)×P(K)$P(K)\times P(K)$ can be different as well as P(K)$P(K)$ may have Corson property (C), while P(K×K)$P(K\times K)$ fails to have it answering a question of Pol. Our construction is also a relevant example in the general context of injective tensor products of Banach spaces complementing recent results of Avilés, Martínez‐Cervantes, Rodríguez, and Rueda Zoca. [ABSTRACT FROM AUTHOR]

In this paper, we characterize bounded, compact, and order bounded sums of weighted composition-differentiation operators from Bergman-type spaces to weighted Banach spaces of analytic functions. [ABSTRACT FROM AUTHOR]

In this paper, we first obtain a refined version of the Bohr inequality of norm-type for holomorphic mappings with lacunary series on the polydisk in $ \mathbb {C}^n $ C n under some restricted conditions. Next, we determine the refined version of the Bohr inequality for holomorphic functions defined on a balanced domain G of a complex Banach space X and take values in the unit disk $ \mathbb {D} $ D . Furthermore, as a consequence of one of these results, we obtain a refined version of the Bohr-type inequality for harmonic functions $ f=h+\bar {g} $ f = h + g ¯ defined on a balanced domain $ G\subset X $ G ⊂ X. All the results are proved to be sharp. [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]

We investigate the following fractional p-Laplacian convex-concave problem: ( P λ ) (− Δ) p s u = λ | u | q − 2 u + | u | p s * − 2 u in Ω , u = 0 in R n \ Ω , where Ω is a bounded C1,1 domain in R n , s ∈ (0, 1), p > q > 1, n > sp, λ > 0, and p s * = n p n − s p is the critical Sobolev exponent. Our analysis extends classical works (A. Ambrosetti, H. Brezis, and G. Cerami, "Combined effects of concave and convex nonlinearities in some elliptic problems," J. Funct. Anal., vol. 122, no. 2, pp. 519–543, 1994, B. Barrios, E. Colorado, R. Servadei, and F. Soria, "A critical fractional equation with concave-convex power nonlinearities," Ann. Inst. Henri Poincare Anal. Non Lineaire, vol. 32, no. 4, pp. 875–900, 2015, J. García Azorero, J. Manfredi, and I. Peral Alonso, "Sobolev versus Hölder local minimizer and global multiplicity for some quasilinear elliptic equations," Commun. Contemp. Math., vol. 2, no. 3, pp. 385–404, 2000) to fractional p-Laplacian. Owing to the nonlinear and nonlocal properties of (− Δ) p s , we need to overcome many difficulties and apply notably different approaches, due to the lack of Picone identity, the stability theory, and the strong comparison principle. We show first a dichotomy result: a positive W 0 s , p (Ω) solution of (Pλ) exists if and only if λ ∈ (0, Λ] with an extremal value Λ ∈ (0, ∞). The W 0 s , p (Ω) regularity for the extremal solution seems to be unknown regardless of whether s = 1 or s ∈ (0, 1). When p ≥ 2, p − 1 < q < p and n > s p (q + 1) q + 1 − p , we get two positive solutions for (Pλ) with small λ > 0. Here the mountain pass structure is more involved than the classical situations due to the lack of explicit minimizers for the Sobolev embedding, we should proceed carefully and simultaneously the construction of mountain pass geometry and the estimate for mountain pass level. Finally, we show another new result for (Pλ) and all p > q > 1: without sign constraint, (Pλ) possesses infinitely many solutions when λ > 0 is small enough. Here we use the Z 2 -genus theory, based on a space decomposition for reflexible and separable Banach spaces, which has its own interest. [ABSTRACT FROM AUTHOR]

Consider L^0, the F-space of all equivalence classes of measurable functions on a finite measure space equipped with the topology of convergence in measure. Inspired by Nikishin's classical result on the factorization of sublinear continuous operators from a Banach space to L^0, we prove a theorem that characterizes those maps from any quasi-metric space into L^0 that factor strongly through Marcinkiewicz weighted spaces. We show applications to sublinear operators on a certain class of quasi-Banach spaces with generalized Rademacher type generated by Orlicz sequence spaces. [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]

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, 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]

We consider periodic difference equations in infinite-dimensional Banach spaces possessing compact asymptotically stable set $ {\mathcal A} $ A . It is established that such $ {\mathcal A} $ A persists under spatial discretizations by means of projection methods as nearby closed and bounded uniformly asymptotically stable sets, which moreover converge to $ {\mathcal A} $ A in the Hausdorff metric for increasingly more accurate schemes. The proof is based on a Lyapunov function guaranteed by an ambient converse theorem and a pullback construction. As application serves integrodifference equations on spaces of continuous and p-integrable functions over a compact habitat. [ABSTRACT FROM AUTHOR]

We consider a C 0 -semigroup on a Banach space such that the resolvent is uniformly bounded on the right half-plane. In this paper we provide a condition on the resolvent which is sufficient and necessary for the uniform exponential stability of such a semigroup. As a consequence, we give an alternative proof of Gearhart's theorem (Trans. Amer. Math. Soc. 236, 385–394 (1978)). The approach lies on a complex inversion formula and tempered distributions. [ABSTRACT FROM AUTHOR]

This study uses a classic fixed point theorem of cone type in a Banach space to identify the eigenvalue intervals of parameters for which an iterative system of a Hadamard fractional boundary value problem has at least one positive solution. To the best of our knowledge, no attempt has been made to obtain such results for Hadamard-type problems in the literature. We provided an example to illustrate the feasibility of our findings in order to show how effective they are. [ABSTRACT FROM AUTHOR]

This paper is focussed on some properties of paramonotone operators on Banach spaces and their application to certain feasibility problems for convex sets in a Hilbert space and convex systems in the Euclidean space. In particular, it shows that operators that are simultaneously paramonotone and bimonotone are constant on their domains, and this fact is applied to tackle two particular situations. The first one, closely related to simultaneous projections, deals with a finite amount of convex sets with an empty intersection and tackles the problem of finding the smallest perturbations (in the sense of translations) of these sets to reach a nonempty intersection. The second is focussed on the distance to feasibility; specifically, given an inconsistent convex inequality system, our goal is to compute/estimate the smallest right-hand side perturbations that reach feasibility. We advance that this work derives lower and upper estimates of such a distance, which become the exact value when confined to linear systems. [ABSTRACT FROM AUTHOR]

In this article, we introduce the structure of a new extended cone b-metric-like space over a Banach algebra. In this generalized space, we define the notion of generalized Reich-type mappings and prove a fixed point result. We also prove a fixed point result using α ∗ − ψ multivalued contraction and provide some of its consequences. Finally, we furnish with applications to establish the validity of our results. [ABSTRACT FROM AUTHOR]

This manuscript introduces a novel three-variable cubic functional equation and derives its general solution. Employing both the direct and fixed-point methods, we investigate the orthogonal stability of this equation within the frameworks of quasi- β -Banach spaces and multi-Banach spaces. Additionally, the study explores the stability of the equation in various extended Banach spaces and provides a specific example illustrating the absence of stability in certain cases. [ABSTRACT FROM AUTHOR]

Inequalities are obtained which connect the probability tails and moments of functions of the nth partial sums of independent random variables taking values in a separable Banach space and those for the accompanying infinitely divisible laws. Some applications to empirical processes are studied. [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]

Since it is unclear how to define a domain that is strictly pseudoconvex in the infinite-dimensional setting, we develop a general theory with Banach spaces in mind. We first focus on the finite dimensional case and eliminate the need for two degrees of differentiability of the boundary of a domain, since differentiable functions are difficult to find in infinite dimensions. We introduce ℓ -strict pseudoconvexity for ℓ ≥ 1 , 1-strict pseudoconvexity at the boundary, ℓ -uniform pseudoconvexity for ℓ ≥ 0 , and finally strong pseudoconvexity. Defining ℓ -strict pseudoconvexity and ℓ -uniform pseudoconvexity for ℓ < 2 depends on extending a notion of strict plurisubharmonicity to cases lacking C 2 -smoothness, first studying it in the sense of distribution and then considering it in infinite dimensions. Examples of strictly plurisubharmonic functions as well as strongly pseudoconvex domains related to important classical Banach spaces are presented. Finally, some related solutions to the inhomogeneous Cauchy–Riemann equations for ∂ ¯ -closed (0, 1)-forms in infinite-dimensional domains are shown. [ABSTRACT FROM AUTHOR]

We first present a new form of a local K-convoluted C-group on a Banach space X, and then deduce some basic properties of a nondegenerate local K-convoluted C-group on X and some generation theorems of local K-convoluted C-groups, which can be applied to obtain some equivalence relations between the generation of a nondegenerate local K-convoluted C-group on X with subgenerator A and the unique existence of solutions of the abstract Cauchy problem ACP (A,f,x). [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]

Here we research the multivariate quantitative approximation of Banach space valued continuous multivariate functions on a box or RN, N Є N; by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We investigate also the case of approximation by iterated multilayer neural network operators of the last four types. These approximations are achieved by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a q-deformed and λ-parametrized A-generalized logistic function, which is a sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network are with one or multi hidden layers. [ABSTRACT FROM AUTHOR]

This research is concerned with the existence and uniqueness of solutions for a coupled system of Ψ –Riemann–Liouville fractional differential equations. To achieve this objective, we establish a set of necessary conditions by formulating the problem as an integral equation and utilizing well-known fixed-point theorems. By employing these mathematical tools, we demonstrate the existence and uniqueness of solutions for the proposed system. Additionally, to illustrate the practical implications of our findings, we provide several examples that showcase the main results obtained in this study. [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]

There is a long-standing problem, posed by A.T.-M. Lau [ Fixed point theory and its applications , Academic Press, New York-London, 1976, pp. 121–129], whether left amenability is sufficient to ensure the existence of a common fixed point for every jointly weak^{\ast } continuous nonexpansive semigroup action on a nonempty weak^{\ast } compact convex set in a dual Banach space. In this note we discuss the current status of this problem and give a partial solution in the case of weak^{\ast } compact convex sets with the Radon–Nikodým property. [ABSTRACT FROM AUTHOR]

Let \Sigma be a \sigma-algebra of subsets of a set \Omega and X be a Banach space, and let B(\Sigma,X) stand for the Banach space of all X-valued totally \Sigma-measurable functions on \Omega, equipped with the sup-norm. We study nuclear operators T:B(\Sigma,X)\rightarrow Y between Banach spaces B(\Sigma,X) and Y in terms of their representing measures m:\Sigma \rightarrow \mathcal {N}(X,Y), where \mathcal {N}(X,Y) stands for the Banach space of all nuclear operators U:X\rightarrow Y, equipped with the nuclear norm. We establish the relationship between nuclearity of a bounded linear operator T:B(\Sigma,X)\rightarrow Y and nuclearity of its conjugate operator. [ABSTRACT FROM AUTHOR]

Using elementary probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, we prove that for every infinite compact spaces K and L the product K\times L admits a sequence \langle \mu _n\colon n\in \mathbb {N}\rangle of normalized signed measures with finite supports which converges to 0 with respect to the weak* topology of the dual Banach space C(K\times L)^*. Our approach is completely constructive—the measures \mu _n are defined by an explicit simple formula. We also show that this result generalizes the classical theorem of Cembranos [Proc. Amer. Math. Soc. 91 (1984), pp. 556–558] and Freniche [Math. Ann. 267 (1984), pp. 479–486] which states that for every infinite compact spaces K and L the Banach space C(K\times L) contains a complemented copy of the space c_0. [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 work, the notion of function of bounded variation in 2-normed spaces was established (Definition 4.2), the set of functions of bounded (2, k)-variation was endowed with a norm (Theorem 4.5), and it was proved that such a set is a Banach space (Theorem 4.6). In addition, the fundamental properties of the functions of bounded (2, k)-variation in the formalism of 2-Hilbert and 2- normed spaces were studied (see Theorems 4.1, 4.3, 4.4). Also, it was shown how to endow a 2-normed space with a function of bounded (2, k)-variation from a classical Hilbert space (Proposition 4.1). A series of examples and counterexamples are presented that enrich the results obtained in this work (4.1 and 4.2) [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]

We research extremal S-asymptotically ω-periodic mild solutions to elastic systems by using upper and lower solutions. In our first result, it is assumed that the semigroup is compact. In addition, in our second result, the semigroup does not need to be compact and not assuming lower and upper solutions. Our conclusion is drawn using measures of noncompactness. Finally, the existence of mild solutions for damped equations is discussed as an application. [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]

This article introduces a multistep method for developing sequences that solve Banach space-valued equations. It provides error estimates, a radius of convergence, and uniqueness results. Our approach improves the applicability of the recommended method and addresses challenges in applied science. The theoretical advancements are supported by comprehensive computational results, demonstrating the practical applicability and robustness of the earlier method. We ensure more reliable and precise solutions to Banach space-valued equations by providing computable error estimates and a clear radius of convergence for the considered method. We conclude that our work significantly improves the practical utility of multistep methods, offering a rigorous and computable approach to solving complex equations in Banach spaces, with strong theoretical and computational results. [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]

We focus on the measures of Stepanov-like pseudoasymptotically Bloch τ-periodicity and its applications. First, we define a new notion of measures for Stepanov-like pseudoasymptotically Bloch periodic functions and discuss some of its fundamental properties. Then the obtained results are used to investigate the existence and uniqueness of the measure Stepanov-like pseudoasymptotically Bloch periodic mild solutions to semilinear delay differential equations in Banach spaces. Finally, an application is presented to illustrate the efficiency of the obtained results. [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]

We introduce the notion of a nonlinear smooth space generalizing both \operatorname {CAT}(0) spaces as well as smooth Banach spaces. We show that this notion allows for a unified treatment of several results in functional analysis. Namely, we substantiate the usefulness of this setting by establishing a nonlinear generalization of an important result due to Reich in Banach spaces. On par with the linear context, this nonlinear version entails a convergence proof of several other methods. Here, we establish the convergence of a general form of the Halpern-type schema for resolvent-like families of functions. We furthermore prove the convergence of the viscosity generalization of Halpern's iteration (even for families of maps) generalizing a result due to Chang. This work is set in the context of the 'proof mining' program, and the results are complemented with quantitative information like rates of convergence and of metastability (in the sense of T. Tao). [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]

The purpose of our paper is to present a few convergence results for the Krasnoselskii–Mann iteration and also the modified Krasnoselskii–Mann iteration used for approximating the fixed point of k-strictly pseudo contractive map in the Banach space, which can be derived from the corresponding convergence theorems in the class of non-expansive maps. Also, using the enrichment techniques for contractive type map T, i.e., the averaged operator T λ (u) = (1 - λ) u + λ T u , where λ ∈ (0 , 1 ] , we identified that pseudo-contractive and demi-contractive maps are an unsaturated class of mappings in the Banach space. Illustrative examples and numerical calculations are given to support the obtained results, which are in fact the generalizations of approximations results in the Banach space from the Hilbert space. [ABSTRACT FROM AUTHOR]

We introduce the space of holomorphic growth spaces with values in a Banach lattice. We provide norm and essential norm estimates of the embedding operator, and we completely characterize the bounded and compact embeddings of the growth spaces using vector-valued Carleson measures. As an application, we prove a characterization of weighted composition operators. [ABSTRACT FROM AUTHOR]

We study a class of equations in Banach spaces with a Riemann–Liouville-type integrodifferential operator with an operator-valued convolution kernel. The properties of k-resolving operators of such equations are studied and the class Am,K,χ of linear closed operators is defined so that the belonging to this class is necessary and, in the case of commutation of the operator with the convolution kernel, sufficient for the existence of analytic in the sector k-resolving families of operators of the equation under study. Under certain additional conditions on the convolution kernel, we prove theorems on the unique solvability of the nonhomogeneous linear equation of the class under consideration if the nonhomogeneity is continuous in the norm of the graph of the operator from the equation or Hölder continuous. We obtain a theorem on sufficient conditions on an additive perturbation of an operator of the class Am,K,χ in order that the perturbed operator also belong to such a class. Abstract results are used in the study of initial-boundary value problems for a system of partial differential equations with several fractional Riemann–Liouville derivatives of different orders with respect to time and for an equation with a fractional Prabhakar derivative with respect to time. [ABSTRACT FROM AUTHOR]

We show that for the case of uniformly convex Banach spaces, the conditions of Brøndsted's fixed point theorem can be relaxed. [ABSTRACT FROM AUTHOR]

Let $\mathcal {F}$ be a hereditary collection of finite subsets of $\mathbb {N}$. In this paper, we introduce and characterize $\mathcal {F}$ -(almost) greedy bases. Given such a family $\mathcal {F}$ , a basis $(e_n)_n$ for a Banach space X is called $\mathcal {F}$ -greedy if there is a constant $C\geqslant 1$ such that for each $x\in X$ , $m \in \mathbb {N}$ , and $G_m(x)$ , we have $$ \begin{align*} \|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}. \end{align*} $$ Here, $G_m(x)$ is a greedy sum of x of order m , and $\mathbb {K}$ is the scalar field. From the definition, any $\mathcal {F}$ -greedy basis is quasi-greedy, and so the notion of being $\mathcal {F}$ -greedy lies between being greedy and being quasi-greedy. We characterize $\mathcal {F}$ -greedy bases as being $\mathcal {F}$ -unconditional, $\mathcal {F}$ -disjoint democratic, and quasi-greedy, thus generalizing the well-known characterization of greedy bases by Konyagin and Temlyakov. We also prove a similar characterization for $\mathcal {F}$ -almost greedy bases. Furthermore, we provide several examples of bases that are nontrivially $\mathcal {F}$ -greedy. For a countable ordinal $\alpha $ , we consider the case $\mathcal {F}=\mathcal {S}_{\alpha }$ , where $\mathcal {S}_{\alpha }$ is the Schreier family of order $\alpha $. We show that for each $\alpha $ , there is a basis that is $\mathcal {S}_{\alpha }$ -greedy but is not $\mathcal {S}_{\alpha +1}$ -greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals $\alpha , $$ \begin{align*} \mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_{\alpha}\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_{\beta}\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}. \end{align*} $$ [ABSTRACT FROM AUTHOR]

In this work, we introduce a new definition for the fractional differential operator that generalizes several well-known fractional differential operators. In fact, we introduce the notion of the p -proportional ω -weighted κ -Hilfer derivative includes an exponential function, D a , λ σ , ρ , p , κ , ω , and then we consider a non-instantaneous impulse differential inclusion containing D a , λ σ , ρ , p , κ , ω with order σ ∈ (1 , 2) and of kind ρ ∈ [ 0 , 1 ] in Banach spaces. We deduce the relevant relationship between any solution to the studied problem and the integral equation that corresponds to it, and then, by using an appropriate fixed-point theorem for multi-valued functions, we give two results for the existence of these solutions. In the first result, we show the compactness of the solution set. Next, we introduce the concept of the (p , ω , κ) -generalized Ulam-Hyeres stability of solutions, and, using the properties of the multi-valued weakly Picard operator, we present a result regarding the (p , ω , κ) -generalized Ulam-Rassias stability of the objective problem. Since many fractional differential operators are particular cases of the operator D a , λ σ , ρ , p , κ , ω , our work generalizes a number of recent findings. In addition, there are no past works on this kind of fractional differential inclusion, so this work is original and enjoyable. In the last section, we present examples to support our findings. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of finding the order 0<α<1$$ 0<\alpha <1 $$ in the fractional discrete system: C∇αun=Aun,n≥1,u0=x0,$$ \left\{\begin{array}{cll}{\kern0.1em }_C{\nabla}^{\alpha }{u}^n& =& A{u}^n,\kern0.30em n\ge 1,\\ {}{u}^0& =& {x}_0,\end{array}\right. $$where A$$ A $$ is a closed linear operator defined in a Banach space X,x0∈X$$ X,{x}_0\in X $$, and C∇αun$$ {\kern0.1em }_C{\nabla}^{\alpha }{u}^n $$ is the discrete Caputo fractional derivative of a given vector‐valued sequence (un)n∈ℕ0$$ {\left({u}^n\right)}_{n\in {\mathrm{\mathbb{N}}}_0} $$. [ABSTRACT FROM AUTHOR]