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In the present paper we study the existence and stability problems of positive periodic solutions to a Gilpin–Ayala competitive model with periodic coefficients on time scales. Firstly, based on Schauder's fixed theorem, some sufficient conditions for the existence of positive periodic solution to the considered system are obtained. Furthermore, we establish asymptotic behavior by using the existence of periodic solutions. Since the considered system is based on an arbitrary time scale, our results are applicable to both discrete and continuous scenarios. We provide a specific example to verify the above results. [ABSTRACT FROM AUTHOR]

Define an infinite matrix D α = (d n , v α) by d n , v α = { v α σ (α) ⁢ (n) , v ∣ n , 0 , v ∤ n , where σ (α) ⁢ (n) is defined to be the sum of the 훼-th power of the positive divisors of n ∈ N , and construct the matrix domains ℓ p ⁢ (D α) ( 0 < p < ∞ ), c 0 ⁢ (D α) , c ⁢ (D α) and ℓ ∞ ⁢ (D α) defined by the matrix D α . We develop Schauder bases and determine 훼-, 훽- and 훾-duals of these new spaces. We characterize some matrix transformation from ℓ p ⁢ (D α) , c 0 ⁢ (D α) , c ⁢ (D α) and ℓ ∞ ⁢ (D α) to ℓ ∞ , 푐, c 0 and ℓ 1 . Furthermore, we determine some criteria for compactness of an operator (or matrix) from X ∈ { ℓ p ⁢ (D α) , c 0 ⁢ (D α) , c ⁢ (D α) , ℓ ∞ ⁢ (D α) } to ℓ ∞ , 푐, c 0 or ℓ 1 . [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of second-order impulsive delay differential systems. Firstly, we define cosine-type and sine-type delay matrix functions, which are used to derive the solutions of the impulsive delay differential systems. Secondly, based on the Schauder and Banach fixed-point theorems, we establish sufficient conditions that guarantee the existence and uniqueness of solutions to nonlinear impulsive delay differential systems. Finally, several examples are given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

We present some existence and localization results for periodic solutions of impulsive first‐order coupled nonlinear systems of two equations, without requiring periodicity for the nonlinearities. The arguments are based on Schauder's fixed point theorem together with the upper and lower solution methods, where the upper and lower solutions are not necessarily well ordered. In addition, results on equi‐regulated functions are required for the impulsive analysis. An application to a Wilson‐Cowan system of two strongly coupled neurons illustrates one of the main results. [ABSTRACT FROM AUTHOR]

We continue our study of the thresholding greedy algorithm when we restrict the vectors involved in our approximations so that they either are supported on intervals of N or have constant coefficients. We introduce and characterize what we call consecutive greedy bases and provide new characterizations of almost greedy and squeeze symmetric Schauder bases. Moreover, we investigate some cases involving greedy-like properties with constant 1 and study the related notion of Property (A, τ ). [ABSTRACT FROM AUTHOR]

The theory of fractal surfaces, in its basic setting, asserts the existence of a bivariate continuous function defined on a triangular domain. The extant literature on the construction of fractal surfaces over triangular domains use certain assumptions for the construction and deal primarily with the interpolation aspects. Working in the framework of fractal surfaces over triangular domains, this note has a two-fold target. Firstly, to revamp the existing constructions of fractal surfaces over triangular domains, and secondly to connect the idea of fractal surfaces further with the theory of approximation. To this end, in the same spirit of the so-called α -fractal functions on intervals and hyperrectangles, we study a salient subclass of fractal surfaces, which provides a parameterized family of bivariate fractal functions corresponding to a fixed continuous function defined on a triangular domain. Some elementary properties of the single-valued (linear and nonlinear) and multi-valued fractal operators associated with the α -fractal function formalism of the bivariate fractal functions on a triangular domain are recorded. A fractal approximation process, that is a sequence of single-valued fractal operators converging strongly to the identity operator on C (Δ , R) , the space of all real-valued continuous functions defined on a triangular domain Δ , is obtained. An approximation class of fractal functions, referred to as the fractal polynomials, is hinted at. The notion of α -fractal function and associated single-valued fractal operator in conjunction with appropriate stability results for Schauder bases provide Schauder bases consisting of self-referential functions for C (Δ , R) . [ABSTRACT FROM AUTHOR]

In the present article, we provide a new nonlinear contraction for the Schauder type selections of multi-valued mappings in metric spaces which is a new spread of the Darbo theorem. Meanwhile, we apply the main results in coupled fixedpoint theory and functional integral equation. [ABSTRACT FROM AUTHOR]

Weaving frames have been introduced to deal with some problems in signal processing and wireless sensor networks. More recently, the notion of fractal operator and fractal convolutions have been linked with perturbation theory of Schauder bases and frames. However, the existing literature has established limited connections between the theory of fractals and frame expansions. In this paper we de ne weaving frames generated via fractal operators combined with fractal convolutions. The aim is to demonstrate how partial fractal convolutions are associated to Riesz bases, frames and the concept of weaving frames in a Hilbert space. The context deals with ones-sided convolutions i.e both left and right partial fractal convolution operators on Lebesgue space Lp (1 ⩽ p ⩽ ∞). Some applications using partial fractal convolutions with null function have been obtained for the perturbation theory of bases and weaving frames. [ABSTRACT FROM AUTHOR]

In this paper, we study the ideal strong lacunary ward compactness of a subset of a 2-normed space and the ideal strongly lacunary ward continuity of a function on . Here a subset of is said to be ideal strong lacunary ward compact if any sequence in has an ideal strong lacunary quasi-Cauchy subsequence. Additionally, a function on is said to be ideal strong lacunary ward continuous if it preserves ideal strong lacunary quasi-Cauchy sequences; an ideal is defined to be a hereditary and additive family of subsets of . We find that a subset of with a countable Hamel basis is totally bounded if and only if it is ideal strong lacunary ward compact. [ABSTRACT FROM AUTHOR]

The determination of an unknown time-dependent source term is investigated in a Kuramoto–Sivashinsky equation from given additional integral-type measurement. Based on Schauder's fixed point theorem, the existence and uniqueness of such inverse problem are obtained under certain assumptions on the input data. In order to calculate the unknown source term, a time-discrete system is established, and its solution shall be applied to approximate the unknown quantity. The existence, uniqueness and some estimates to the time-discrete system are derived, and the convergence rates are deduced rigorously for both exact and noisy observation, respectively. Finally, the theoretical convergence rate results are verified, and accurate and stable solutions to the inverse problem are computed numerically by two numerical experiments. [ABSTRACT FROM AUTHOR]

This paper consists of two parts. The first one deals with the generation of an iterative algorithm to obtain an approximate solution of a linear equation of the second kind in a Banach space. This generation is based on a perturbed version of the geometric series theorem which, in particular, allows us to find a family of unisolvent linear Fredholm integral equations of the second kind, even when the associated linear operator has norm greater than or equal to 1. When we consider Fredholm equations of this type and linear Volterra integral equations of second kind, the numerical schemes obtained when appropriate Schauder bases are also introduced in the spaces where the equations operate, enable us to approximate their respective solutions iteratively. The second part of this work focuses on the design of a numerical method for solving an inverse problem associated with a linear equation of the second kind in a Banach space, a method which we apply to problems of parameter estimation related to the two classes of integral equations mentioned above. [ABSTRACT FROM AUTHOR]

Schauder's fixed point theorem and Banach contraction principle are used to study a q-difference equation. We give sufficient conditions for the existence, uniqueness, and stability of the nondecreasing bounded continuous solutions. We also give the approximate sequences for the corresponding solutions. Finally, some examples are considered for our results. [ABSTRACT FROM AUTHOR]

The focus of our investigation was on determining the existence of solutions for fractional differential equations (FDEs) of order 1 < γ ≤ 2 involving the boundary conditions κ0ϕ(0) + η0ϕ(v) = μ0, and κ1ϕ′(0) + η1ϕ′ (v) = μ1, for κi, ηi, μi ∊ R+. The existence results were based on the Schauder fixed point theorem and the nonlinear alternative of the Leray-Schauder type. Examples were provided to illustrate the results. [ABSTRACT FROM AUTHOR]

The main purpose of this investigation is to revisit solvability process of the Cauchy problems of Caputo fractional two-term initial value problems. To this aim, the Green function technique has chosen to make a bridge between the operator and the fixed point theories. The appeared Green functions in this paper are constructed by the Fox-Wright functions. Our solvability tools include the existence and uniqueness criteria as novel refinements of the Banach contraction principal and Schauder fixed point theorem. This investigation will be finalized by presenting some numerical applications that illustrate proposed solvability criteria. [ABSTRACT FROM AUTHOR]

In this paper, keeping in view the idea of difference sequence space E(Δ) of Kızmaz [20], we availed an opportunity to introduce new kind of summing squence spaces E(∇), E ∊ {l∞,c,c0} by exploring the sum of two consecutive terms. In addition to this we computed the continuous as well Köthe-Toeplitz duals of these spaces. Like E(Δ) (the difference sequence spaces of Kızmaz) new sequence spaces E(∇) turned out to be much wider than E. [ABSTRACT FROM AUTHOR]

Let X, Y be Banach spaces and S : X →Y be an invertible Lipschitz map. Let T :X →Y be a map and there exist 1, 2 = [0,1) such that Then we prove that T is an invertible Lipschitz map. This is non-linear version of 26 years old Casazza-Kalton-Christensen-van Eijndhoven perturbation. It also a non-linear version of 29 years old Soderlind-Campanato perturbation and 3 years old Barbagallo-Ernst-Thera perturbation. We give applications to the theory of metric frames. The notion of Lipschitz atomic decomposition for Banach spaces is also introduced. [ABSTRACT FROM AUTHOR]

In this paper, we prove an existence and uniqueness result for a class of Dirichlet boundary value problems whose model is − Δ p u = β | ∇ u | q + c | u | p − 2 u + f in   Ω , u = 0 on   ∂ Ω , where Ω is an open bounded subset of R N , N ≥ 2 , 1 < p < N , Δ p u is the so-called p-Laplace operator, and p − 1 < q < p . We assume that β is a positive constant, c and f are measurable functions belonging to suitable Lorentz spaces. Our approach is based on Schauder fixed point theorem. [ABSTRACT FROM AUTHOR]

The concept of adjusted sublevel set for a quasiconvex function was introduced by Aussel and Hadjisavvas who proved the local existence of a norm-to-weak ∗ upper semicontinuous base-valued submap of the normal operator associated with the adjusted sublevel set. When the space is finite-dimensional, a globally defined upper semicontinuous base-valued submap is obtained by taking the intersection of the unit sphere, which is compact, with the normal operator, which is closed. Unfortunately, this technique does not work in the infinite-dimensional case. We propose a partition of unity technique to overcome this problem in Banach spaces. An application is given to a quasiconvex quasioptimization problem through the use of a new existence result for generalized quasivariational inequalities which is based on the Schauder fixed point theorem. [ABSTRACT FROM AUTHOR]

We provide criteria for the existence of solutions of nonlinear discrete-time boundary value problems on infinite-time intervals. The problems are formulated as nonlinear operator equations on sequence spaces and the tools of nonlinear functional analysis are employed throughout the paper. [ABSTRACT FROM AUTHOR]

In this paper, we show the weak and strong well-posedness of density dependent stochastic differential equations driven by α -stable processes with α ∈ (1 , 2). The existence part is based on Euler's approximation as Hao et al. (2021), while, the uniqueness is based on the Schauder estimates in Besov spaces for nonlocal Fokker–Planck equations. For the existence, we only assume the drift being continuous in the density variable. For the weak uniqueness, the drift is assumed to be Lipschitz in the density variable, while for the strong uniqueness, we also need to assume the drift being β 0 -order Hölder continuous in the spatial variable, where β 0 ∈ (1 − α / 2 , 1). [ABSTRACT FROM AUTHOR]

In this study, we investigate the existence of mild solutions for a class of Hilfer fractional stochastic evolution equations with order μ ∈ (1 , 2) and type ν ∈ [ 0 , 1 ] . We prove the existence of mild solutions of Hilfer fractional stochastic evolution equations when the semigroup is compact as well as noncompact. Our approach is based on the Schauder fixed point theorem, the Ascoli–Arzelà theorem and the Kuratowski measure of noncompactness. An example is also provided, to demonstrate the efficacy of this method. [ABSTRACT FROM AUTHOR]

We present some existence and localization results for periodic solutions of first-order coupled nonlinear systems of two equations, without requiring periodicity for the nonlinearities. The arguments are based on Schauder's fixed point theorem together with not necessarily well-ordered upper and lower solutions. A real-case scenario shows the applicability of our results to some population dynamics models, describing the interaction between a criminal and a non-criminal population with a law enforcement component. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of positive mild solutions and controllability for fractional differential evolution equations of order [Math Processing Error] γ ∈ (1 , 2) with nonlocal conditions in Banach spaces. Our approach is based on Schauder's fixed point theorem, Krasnoselskii's fixed point theorem, and the Arzelà-Ascoli theorem. Finally, we include an example to verify our theoretical results. [ABSTRACT FROM AUTHOR]

Previous studies have shown that fractional derivative operators have become an integral part of modeling natural and physical phenomena. During the progress and evolution of these operators, it has become clear to researchers that each of these operators has special capacities for investigating phenomena in engineering sciences, physics, biological mathematics, etc. Fixed point theory and its famous contractions have always served as useful tools in these studies. In this regard, in this work, we considered the Hilfer-type fractional operator to study the proposed integrodifferential equation. We have used the capabilities of measure theory and fixed point techniques to provide the required space to guarantee the existence of the solution. The Schauder and Arzela-Ascoli theorems play a fundamental role in the existence of solutions. Finally, we provided two examples with some graphical and numerical simulation to make our results more objective. [ABSTRACT FROM AUTHOR]

In the 1987 publication [5] we showed that for second order semilinear elliptic problems a solution exists between a subsolution and a supersolution without using monotone iteration and under minimal assumptions. The alternative is based on a Schauder fixed point argument. Filling some missing details of that paper we noticed that even nonlocal operators are allowed on the right hand side as long as these are quasimonotone. The setting uses 'weak'-solutions which are continuous on the closure of the domain and solve the equation in distributional sense. As such they differ from the standard weak solutions. We conclude with some examples concerning the setting. [ABSTRACT FROM AUTHOR]

This paper deals with the analysis of the internal control of a free-boundary problem for the 1D heat equation with local and nonlocal nonlinearities. We prove a local null controllability result with distributed controls, locally suported in space. The proof is based on Schauder's fixed point theorem combined with some appropriate specific estimates. [ABSTRACT FROM AUTHOR]

Consider the space of weighted binomial matrices in the Nakano sequence space of soft functions. We have offered some geometric and topological structures of the multiplication operator acting on this space and its associated operator ideal. The existence of a fixed point of the Kannan contraction operator in this prequasioperator ideal is confirmed. Finally, we discuss many applications of solutions to nonlinear stochastic dynamical matrix systems and illustrative examples of our findings. [ABSTRACT FROM AUTHOR]

In this paper, we define LGc-fuzzy Euclidean topological space with countable basis, which L denotes a complete distributive lattice and we show that each LGc-fuzzy open covering of this space can be refined to an LGc-fuzzy open covering that is locally finite. We introduce C∞ LG-fuzzy manifold (X, Tc), with countable basis of LG-fuzzy open sets which X is an L-fuzzy subset of a crisp set M and T : LXM → L, is an L-gradation of openness on X. We prove that for any LG-fuzzy topological manifold (X,T), there exists an LG-fuzzy exhaustion. We prove LG-Urysohn lemma and also existence of LG-partitions of unity on every LG-fuzzy topological manifold. [ABSTRACT FROM AUTHOR]

Schauder's fixed point theorem and its applications to delay differential equations (DDEs) cannot be over emphasized. In this work, interesting results regarding the sufficient conditions for nonoscillation of more generalized forms of DDEs are established and provides improvements on the results obtained in the past. However, applying the theorem and the characteristic equation of DDE with constant coefficients helps to determine those conditions. Lastly, to ascertain a claim from an existing literature for a solution that is nonoscillatory and has such larger solutions, comparison theorem is used and the results are found to be true. [ABSTRACT FROM AUTHOR]

In this paper, we present some results on existence and uniqueness for a class of boundary value problems for iterative fractional differential equations with generalized Caputo fractional derivative. For our proofs, we employ some suitable fixed point theorems. Finally, we provide an illustration for more clarity. [ABSTRACT FROM AUTHOR]

Making use of the Boyd-Wong fixed point theorem, we establish a new existence and uniqueness result and an approximation process of the fixed point for the product of two nonlinear operators in Banach algebras. This provides an adequate tool for deriving the existence and uniqueness of solutions of two interesting type of nonlinear functional equations in Banach algebras, as well as for developing an approximation method of their solutions. In addition, to illustrate the applicability of our results we give some numerical examples. [ABSTRACT FROM AUTHOR]

Fractal functions are constructed through iterated function systems on suitable Banach spaces. In this article, first, we introduce a new class of fractal approximants with variable scaling parameters, namely, Bernstein–Kantorovich α -fractal functions. The proposed class of functions is non-differentiable and has good properties of convergence. The construction of bases consisting of self-referential functions in different Banach spaces plays an important role in the studies of non-linearity. In this paper, we show the existence of fractal bases consisting of a new class of self-referential functions. In particular, we establish the existence of frames and fractal bases for the proposed class of functions in the space L p (I) , where I is a compact interval in R . In the end, we study some special properties with the proposed self-referential functions in the Hilbert space L 2 (I) . [ABSTRACT FROM AUTHOR]

Let X be a Banach space with a normalized (Schauder) basis (b k) k and let { T (t) } t ≥ 0 be a C 0 -semigroup on X with generator A : D (A) ⊂ X → X . The main contribution of this paper is a new estimate for the semigroup orbits of initial data in D (A 2) that have with respect to (b k) k "absolutely summable graph norm." This result is noteworthy because (1) there is no requirement that z ↦ R (z , A) x has a bounded analytic extension to C + [e.g. Van Neerven (Semigroup Forum 53(2):155–161, 1996)] and (2) the only structural condition on X is the existence of (b k) k [e.g. Wrobel (Indiana Univ. Math. J. 38(1):101–114, 1989)]. Two directions for additional research are in some detail discussed as well at the end of this paper. [ABSTRACT FROM AUTHOR]

In [16], T. Oikhberg introduced and studied variants of the greedy and weak greedy algorithms for sequences with gaps. In this paper, we extend some of the notions that appear naturally in connection with these algorithms to the context of sequences with gaps. In particular, we will consider sequences of natural numbers for which the inequality nk+1 ≤ Cnk or nk+1 ≤ C+nk holds for a positive constant C and all k, and find conditions under which the extended notions are equivalent their regular counterparts. In this context, we study an extension of democratic bases, proving that if n = (nk)k=1∞ is an increasing sequence of natural numbers such that either (nk+1 -nk=1∞ is bounded and (ek)k=1∞ is a Markushevich basis or (nk+1/nk)k=1∞ is bounded and (ek)k=1∞ is a Schauder basis, then n-democracy is equivalent to democracy. Additionally, we give examples proving that these results are optimal, and we obtain similar results for some of the other properties that appear naturally in the study of the greedy algorithm. [ABSTRACT FROM AUTHOR]

We establish that the summability of the series ∑ ε n is the necessary and sufficient criterion ensuring that every (1 + ε n) -bounded Markushevich basis in a separable Hilbert space is a Riesz basis. Further we show that if n ε n → ∞ , then in ℓ 2 there exists a (1 + ε n) -bounded Markushevich basis that under any permutation is non-equivalent to a Schauder basis. We extend this result to any separable Banach space. Finally we provide examples of Auerbach bases in 1-symmetric separable Banach spaces whose no permutations are equivalent to any Schauder basis or (depending on the space) any unconditional Schauder basis. [ABSTRACT FROM AUTHOR]

Existence of global finite-time bounded entropy solutions to a parabolic–parabolic system proposed in Bellomo et al. (2010) is established in bounded domains under no-flux boundary conditions for nonnegative bounded initial data. This modification of the classical Keller–Segel model features degenerate diffusion and chemotaxis that are both subject to flux-saturation. The approach is based on Schauder's fixed point theorem and calculus of functions of bounded variation. [ABSTRACT FROM AUTHOR]

We provide explicit sequence space representations for the test function and distribution spaces occurring in the Valdivia-Vogt structure tables by making use of Wilson bases generated by compactly supported smooth windows. Furthermore, we show that these kind of bases are common unconditional Schauder bases for all separable spaces occurring in these tables. Our work implies that the Valdivia-Vogt structure tables for test functions and distributions may be interpreted as one large commutative diagram. [ABSTRACT FROM AUTHOR]

In this manuscript we introduce a quadratic integral equation of the Urysohn type of fractional variable order. The existence and uniqueness of solutions of the proposed fractional model are studied by transforming it into an integral equation of fractional constant order. The obtained new results are based on the Schauder's fixed-point theorem and the Banach contraction principle with the help of piece-wise constant functions. Although the used methods are very powerful, they are not applied to the quadratic integral equation of the Urysohn type of fractional variable order. With this research we extend the applicability of these techniques to the introduced the Urysohn type model of fractional variable order. The applicability of the new results are demonstrated by providing Ulam–Hyers stability criteria and an example. Moreover, the presented results lead to future progress and expansion of the theory of fractional-order models, as well as of the concept of entropy in the framework of fractional calculus. Further, an example is constructed to demonstrate the reasonableness and effectiveness of the observed results. [ABSTRACT FROM AUTHOR]

We show that if A is a spectrally bounded algebra, then all functions operate spectrally on A if and only if SpAx is finite for every x ∈ A. We also prove that if A is a commutative Q-l.m.c.a, then all functions operate spectrally on A if and only if A/RadA is algebraic. Furthermore, if A is a semi-simple commutative Q-l.m.c.a. which is a Baire space, all functions operate spectrally on A if and only if it is isomorphic to Cn for some n ∈ N. A structure result concerning semi-simple commutative complete m-convex algebras of countable dimension is also given. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of integrable solutions for initial value problems for fractional order implicit differential equations with Hadamard fractional derivative. Our results are based on Schauder's fixed point theorem and the Banach contraction principle fixed point theorem. [ABSTRACT FROM AUTHOR]

The primary goal of this article is devoted to the study of fractal bases and fractal frames for L 2 (I × J) , the collection of all square integrable functions on the rectangle I × J . The fractal function when recognized as an internal binary operation paved way for the construction of right and left partial fractal convolution operators on L 2 (I) , for any real compact interval I. The aforementioned operators defined on one dimensional space have been applied to obtain operators on the space L 2 (I × J) by the identification of L 2 (I × J) with the tensor product space L 2 (I) ⊗ L 2 (J) . In this paper, we establish properties of this bounded linear operator which eventually helps to prove that L 2 (I × J) admits Bessel sequences, Riesz bases and frames consisting of products of fractal (self-referential) functions in a nice way. [ABSTRACT FROM AUTHOR]

Let X be a Banach space with a Schauder basis (x m) m = 0 ∞ , and I be the identity operator on X. It is known, at least in essence, that if (T m) m = 0 ∞ is a sequence of bounded linear operators on X such that ∑ m = 0 ∞ ∥ I − T m ∥ < ∞ , then (T m (x m)) m = 0 ∞ is also a basis. The first part of this work acts as an expository note to formally record the aforementioned stability result. In the second part, we apply this stability result to construct a Schauder basis consisting of bivariate fractal functions for the space of continuous functions defined on a rectangle. To this end, fractal perturbations of the elements in the classical bivariate Faber–Schauder system are formulated using a sequence of bounded linear fractal operators close to the identity operator in accordance with the stability result mentioned above. This illustration, although emphasized only for the bivariate case, can easily be extended to higher dimensions. Further, the perturbation technique used here acts as a companion for a few researches on fractal bases in the univariate setting. [ABSTRACT FROM AUTHOR]

In the present study, we have constructed new Banach sequence spaces ℓ p L , c 0 L , c L , and ℓ ∞ L , where L = l v , k is a regular matrix defined by l v , k = l k / l v + 2 − v + 2 , 0 ≤ k ≤ v , 0 , k > v , for all v , k = 0 , 1 , 2 , ⋯ , where l = l k is a sequence of Leonardo numbers. We study their topological and inclusion relations and construct Schauder bases of the sequence spaces ℓ p L , c 0 L , and c L. Besides, α -, β - and γ -duals of the aforementioned spaces are computed. We state and prove results of the characterization of the matrix classes between the sequence spaces ℓ p L , c 0 L , c L , and ℓ ∞ L to any one of the spaces ℓ 1 , c 0 , c , and ℓ ∞. Finally, under a definite functional ρ and a weighted sequence of positive reals r , we introduce new sequence spaces c 0 L , r ρ and ℓ p L , r ρ . We present some geometric and topological properties of these spaces, as well as the eigenvalue distribution of ideal mappings generated by these spaces and s -numbers. [ABSTRACT FROM AUTHOR]

First of all, in this paper we obtain a perturbed version of the geometric series theorem, which allows us to present an iterative numerical method to approximate the fixed point of a contractive affine operator. This result requires some approximations that we obtain using the projections associated with certain Schauder bases. Next, an algorithm is designed to approximate the solution of Fredholm's linear integral equation, and we illustrate the behavior of the method with some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, under some sufficient conditions on the nonlinearity, we established the existence of nonnegative solutions for a hybrid nonlinear differential equation with iterative terms. Our approach is based on the application of Schauder's fixed point theorem for proving the results. Finally, an example is given to illustrate our results. [ABSTRACT FROM AUTHOR]

We provide topological interpretations of the recently introduced notions of amenability and congeniality of bases of innite dimensional algebras. In order not to restrict our attention only to the countable dimension case, the uniformity of the topologies involved is analyzed and therefore the pertinent ideas about uniform topological spaces are surveyed. A basis B over an innite dimensional F-algebra A is called amenable if FB, the direct product indexed by B of copies of the eld F, can be made into an A-module in a natural way. (Mutual) congeniality is a relation that serves to identify cases when different amenable bases yield isomorphic A-modules. (Not necessarily mutual) congeniality between amenable bases yields an epimorphism of the modules they induce. We prove that this epimorphism is one-to-one only if the congeniality is mutual, thus establishing a precise distinction between the two notions.