Banach Space Of Continuous Functions Research Articles

Positive solutions of nonlinear delayed differential equations with impulses

The paper is concerned with the long-term behavior of solutions to scalar nonlinear functional delayed differential equations y ̇ ( t ) = − f ( t , y t ) , t ≥ t 0 . It is assumed that f : [ t 0 , ∞ ) × C ↦ R is a continuous mapping satisfying a local Lipschitz condition with respect to the second argument and C ≔ C ( [ − r , 0 ] , R ) , r > 0 is the Banach space of continuous functions. The problem is solved of the existence of positive solutions if the equation is subjected to impulses y ( t s + ) = b s y ( t s ) , s = 1 , 2 , … , where t 0 ≤ t 1 < t 2 < ⋯ and b s > 0 , s = 1 , 2 , … . A criterion for the existence of positive solutions on [ t 0 − r , ∞ ) is proved and their upper estimates are given. Relations to previous results are discussed as well.

English

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation (P) $$\dot x\left( t \right) = \mathcal{L}\left( t \right)x\left( t \right) + f\left( {t,x\left( t \right)} \right),t \in \mathbb{R}$$ where {ℒ(t): t ∈ R} is a family of linear operators from a Banach space E into itself and f: R × E → E. By L(E) we denote the space of linear operators from E into itself. Furthermore, for a 0, we let C([−d, 0],E) be the Banach space of continuous functions from [−d, 0] into E and f d : [a, b] × C([−d, 0],E) → E. Let $$\hat {\mathcal{L}}:[a,b] \to L(E)$$ be a strongly measurable and Bochner integrable operator on [a, b] and for t ∈ [a, b] define τ t x(s) = x(t + s) for each s ∈ [−d, 0]. We prove that, under certain conditions, the differential equation with delay (Q) $$\dot x\left( t \right) = \hat {\mathcal{L}}\left( t \right)x\left( t \right) + {f^d}\left( {t,{\tau _t}x} \right),ift \in \left[ {a,b} \right],$$ has at least one weak solution and, under suitable assumptions, the differential equation (Q) has a solution. Next, under a generalization of the compactness assumptions, we show that the problem (Q) has a solution too.

On some geometric properties of Banach spaces of continuous functions on separable compact lines

We study properties of Banach spaces $C(L)$ of all continuous scalar (real or complex) functions on compact lines&nbsp;$L$. First we show that if $L$ is a separable compact line, then for every closed linear subspace $X$ of $C(L)$ with separable dual the

Functional Itô versus Banach space stochastic calculus and strict solutions of semilinear path-dependent equations

Functional Itô calculus was introduced in order to expand a functional [Formula: see text] depending on time [Formula: see text], past and present values of the process [Formula: see text]. Another possibility to expand [Formula: see text] consists in considering the path [Formula: see text] as an element of the Banach space of continuous functions on [Formula: see text] and to use Banach space stochastic calculus. The aim of this paper is threefold. (1) To reformulate functional Itô calculus, separating time and past, making use of the regularization procedures which match more naturally the notion of horizontal derivative which is one of the tools of that calculus. (2) To exploit this reformulation in order to discuss the (not obvious) relation between the functional and the Banach space approaches. (3) To study existence and uniqueness of smooth solutions to path-dependent partial differential equations which naturally arise in the study of functional Itô calculus. More precisely, we study a path-dependent equation of Kolmogorov type which is related to the window process of the solution to an Itô stochastic differential equation with path-dependent coefficients. We also study a semilinear version of that equation.

PFA and complemented subspaces of ℓ∞/c0

The Banach space $\ell_\infty/c_0$ is isomorphic to the linear space of continuous functions on $\mathbb N^*$ with the supremum norm, $C(\mathbb N^*)$. Similarly, the canonical representation of the $\ell_\infty$ sum of $\ell_\infty/c_0$ is the Banach space of continuous functions on the closure of any non-compact cozero subset of $\mathbb N^*$. It is important to determine if there is a continuous linear lifting of this Banach space to a complemented subset of $C(\mathbb N^*)$. We show that PFA implies there is no such lifting.

On isomorphisms of Banach spaces of continuous functions

We prove that if K and L are compact spaces and C(K) and C(L) are isomorphic as Banach spaces, then K has a π-base consisting of open sets U such that Ū is a continuous image of some compact subspace of L. This sheds new light on isomorphic classes of spaces of the form \(C({[0,1]^\kappa })\) and spaces C(K) where K is Corson compact.

Weak asymptotic methods for 3-D self-gravitating pressureless fluids. Application to the creation and evolution of solar systems from the fully nonlinear Euler-Poisson equations

We construct a family of classical continuous functions S(x, y, z, t, ϵ) which tend to satisfy asymptotically the system of selfgravitating pressureless fluids when ϵ → 0. This produces a weak asymptotic method in the sense of Danilov, Omel’yanov, and Shelkovich. The construction is based on a family of two ordinary differential equations (ODEs) (one for the continuity equation and one for the Euler equation) in classical Banach spaces of continuous functions. This construction applies to 3-D self-gravitating pressureless fluids even in presence of point and string concentrations of matter. The method is constructive which permits to check numerically from standard methods for ODEs that these functions tend to the known or admitted solutions when the latter exist. As a direct application, we present a simulation of formation and evolution of a planetary system from a rotating disk of dust: a theorem in this paper asserts that the observed results are a depiction of functions that satisfy the system with arbitrary precision.

Operators on two Banach spaces of continuous functions on locally compact spaces of ordinals

Denote by $[0,\omega_1)$ the set of countable ordinals, equipped with the order topology, let $L_0$ be the disjoint union of the compact ordinal intervals $[0,\alpha]$ for $\alpha$ countable, and consider the Banach spaces $C_0[0,\omega_1)$ and $C_0(L_0)$ consisting of all scalar-valued, continuous functions which are defined on the locally compact Hausdorff spaces $[0,\omega_1)$ and $L_0$, respectively, and which vanish eventually. Our main result states that a bounded operator $T$ between any pair of these two Banach spaces fixes a copy of $C_0(L_0)$ if and only if the identity operator on $C_0(L_0)$ factors through $T$, if and only if the Szlenk index of $T$ is uncountable. This implies that the set $\mathscr{S}_{C_0(L_0)}(C_0(L_0))$ of $C_0(L_0)$-strictly singular operators on $C_0(L_0)$ is the unique maximal ideal of the Banach algebra $\mathscr{B}(C_0(L_0))$ of all bounded operators on $C_0(L_0)$, and that $\mathscr{S}_{C_0(L_0)}(C_0[0,\omega_1))$ is the second-largest proper ideal of $\mathscr{B}(C_0[0,\omega_1))$. Moreover, it follows that the Banach space $C_0(L_0)$ is primary and complementably homogeneous.

Evolution Systems for Differential Equations with Variable Time Lags

We consider nonautonomous retarded functional differential equations under hypotheses which are designed for the application to equations with variable time lags, which may be unbounded, and construct an evolution system of solution operators which are continuously differentiable. These operators are defined on manifolds of continuously differentiable functions. The results apply to pantograph equations and to nonlinear Volterra integro-differential equations, for example. For linear equations we also provide a simpler evolution system with solution operators on a Banach space of continuous functions.

Existence and multiplicity of positive solutions for one-dimensional prescribed mean curvature equations

Abstract In this work, we establish the existence and multiplicity results of positive solutions for one-dimensional prescribed mean curvature equations. Our approach is based on fixed point index theory for completely continuous operators which leave invariant a suitable cone in a Banach space of continuous functions. MSC:34B10, 34B18.

Numerical approaches for systems of Volterra–Fredholm integral equations

In this work we introduce two numerical methods for solving systems of Volterra–Fredholm integral equations.In the nonlinear case we suggest a fixed point method, where the iterations are perturbed in a suitable way according to a Schauder basis in the Banach space of continuous functions C[a,b]2.In the linear case we propose a collocation method based on a particular class of approximating functions. In both methods, convergence analysis and/or low computational cost are analysed, taking into account the properties of the basis under consideration. Numerical results confirm the theoretical study.

A stopping rule for an iterative algorithm in systems of integral equations

In this paper, an important inequality for \(T\), a contraction integral operator, is obtained. From a practical programming point of view, this inequality allows us to express our iterative algorithm with a “for loop” rather than a “while loop”. The main tool used in our research is the fixed point theorem in the Banach space of continuous functions, \(X:=C([a , b],\mathbb{R }^k)\).

A Banach space in which every injective operator is surjective

We construct an infinite-dimensional Banach space of continuous functions C(K) such that every one-to-one operator T:C(K)→ C(K) is onto.

On Bounded Weak and Strong Solutions of Non Linear Differential Equations with and without Delay in Banach Spaces

Assume that $E$ is a Banach space, $B_{r}=\{x\in E:\Vert x\Vert \le r\}$ and $C([-d,0],B_{r})$ is the Banach space of continuous functions from $[-d,0]$ into $B_{r}$. Consider $f:\mathbf{R}^+\times E\to E$; $f^{d}:[0,T]\times C([-d,0],B_{r})\to E$; for each $t\in [0,T]$ the mapping $\theta_{t}\in C([-d,0],B_{r})$ is defined by $\theta_{t}x(s)= x(t+s)$, $s\in [-d,0]$ and let $A(t)$ be a linear operator from $E$ into itself. In this paper we give existence theorems for bounded weak and strong solutions of the nonlinear differential equation 26767 \dot{x}(t)=A(t)x+f(t,x),\qquad t\in \mathbf{R}^+, 26767 and we prove that, with certain conditions, the differential equation with delay 26767 \dot{x}(t)=L(t)x(t)+f^{d}(t,\theta_{t}x),\qquad \text{if}\quad t\in [0,T] \qquad\qquad(\mathrm{Q}) 26767 has at least one weak solution where $L(t)$ is a linear operator from $E$ into $E$. Moreover, under suitable assumptions, the problem $(\mathrm{Q})$ has a solution. Furthermore under a generalization of the compactness assumptions, we show that $(\mathrm{Q})$ has a solution too.

Almost disjoint families of countable sets and separable complementation properties

We study the separable complementation property (SCP) and its natural variations in Banach spaces of continuous functions over compacta KA induced by almost disjoint families A of countable subsets of uncountable sets. For these spaces, we prove among other things that C(KA) has the controlled variant of the separable complementation property if and only if C(KA) is Lindelöf in the weak topology if and only if KA is monolithic. We give an example of A for which C(KA) has the SCP while KA is not monolithic and an example of a space C(KA) with controlled and continuous SCP which has neither a projectional skeleton nor a projectional resolution of the identity. Finally, we describe the structure of almost disjoint families of cardinality ω1 which induce monolithic spaces of the form KA: they can be obtained from countably many ladder systems and pairwise disjoint families by applying simple operations.

On positive embeddings of C(K) spaces

We investigate isomorphic embeddings $T: C(K)\to C(L)$ between Banach spaces of continuous functions. We show that if such an embedding $T$ is a positive operator then $K$ is an image of $L$ under a upper semicontinuous set-function having finite values. Moreover we show that $K$ has a $\pi$-base of sets which closures a continuous images of compact subspaces of $L$. Our results imply in particular that if $C(K)$ can be positively embedded into $C(L)$ then some topological properties of $L$, such as countable tightness of Frechetness, pass to the space $K$. We show that some arbitrary isomorphic embeddings $C(K)\to C(L)$ can be, in a sense, reduced to positive embeddings.

On existence solutions and solution sets of differential equations and differential inclusions with delay in Banach spaces

Let C E ([− d , 0]) (resp. C B (0, T ) ([− d , 0]) be the Banach space of continuous functions from [− d , 0] into a Banach space E (resp. into B (0, T )), where B (0, T ) = { x ∈ E : ∥ x ∥ ⩽ T } and let C ∈ C E ( [ - d , 0 ] ) . In this paper we prove an existence theorem for the differential equation with delay ( P ) x ˙ ( t ) = f d ( t , θ ¯ t x ) , t ∈ [ 0 , T ] , x = C , on [ - d , 0 ] , where θ ¯ t : C B ( 0 , T ) ( [ - d , t ] ) → C E ( [ - d , 0 ] ) is such that θ ¯ t x ( s ) = x ( t + s ) for all s ∈ [− d , 0] and for all x ∈ C B (0, T ) ([− d , t ]) while f d is a function from [0, T ] × C B (0, T ) ([− d , 0]) into E . By using ( R E , N , p ) – measure of noncompactness and under a generalization of the compactness assumptions, we prove an existence theorem and give some topological properties of solution sets of the problem ( Q ) x ˙ ( t ) ∈ A ( t ) x ( t ) + F d ( t , θ t x ) , t ∈ [ 0 , T ] , x = C , on [ - d , 0 ] , where F d : [0, T ] × C E ([− d , 0]) → P fc ( E ), P fc ( E ) is the set of all nonempty closed convex subsets of E while θ t : C E ([− d , t ]) → C E ([− d , 0]) defined by θ t x ( s ) = x ( t + s ) ∀ x ∈ C E ([− d , t ]), ∀ s ∈ [− d , 0] and { A ( t ) : 0 ⩽ t ⩽ b } is a family of densely defined closed linear operators generating a continuous evolution operator S ( t , s ) .

ERM learning algorithm for multi-class classification

The multi-class classification problem is considered by an empirical risk minimization (ERM) approach. The hypothesis space for the learning algorithm is taken to be a ball of a Banach space of continuous functions. When the regression function lies in some interpolation space, satisfactory learning rates for the excess misclassification error are provided in terms of covering numbers of the unit ball of the Banach space. A comparison theorem is proved and is used to bound the excess misclassification error by means of the excess generalization error.

PLB-spaces of holomorphic functions with logarithmic growth conditions

Countable projective limits of countable inductive limits, called PLB-spaces, of weighted Banach spaces of continuous functions have recently been investigated by Agethen, Bierstedt and Bonet. In a previous article, the author extended their investigation to the case of holomorphic functions and characterized when spaces over the unit disc w.r.t. weights whose decay, roughly speaking, is neither faster nor slower than that of a polynomial are ultrabornological or barrelled. In this note, we prove a similar characterization for the case of weights which tend to zero logarithmically.

Strictly convex norms, $G_{𝛿}$-diagonals and non-Gruenhage spaces

We present an example in ZFC of a locally compact, scattered Hausdorff non-Gruenhage space $D$ having a $\G_delta$-diagonal. This answers a question posed by Orihuela, Troyanski and the author in a study of strictly convex norms on Banach spaces. In addition, we show that the Banach space of continuous functions $C_0(D)$ admits a $C^\infty$-smooth bump function.