On the semilinear integro-differential nonlocal Cauchy problem

Piotr Majcher; Magdalena Roszak

Displaying similar documents to “On the semilinear integro-differential nonlocal Cauchy problem”

A local Landau type inequality for semigroup orbits

Gerd Herzog, Peer Christian Kunstmann (2014)

Studia Mathematica

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Given a strongly continuous semigroup ${(S (t))}_{t \geq 0}$ on a Banach space X with generator A and an element f ∈ D(A²) satisfying $| | S (t) f | | \leq e^{- ω t} | | f | |$ and $| | S (t) A ² f | | \leq e^{- ω t} | | A ² f | |$ for all t ≥ 0 and some ω > 0, we derive a Landau type inequality for ||Af|| in terms of ||f|| and ||A²f||. This inequality improves on the usual Landau inequality that holds in the case ω = 0.

Bi-spaces global attractors in abstract parabolic equations

J. W. Cholewa, T. Dłotko (2003)

Banach Center Publications

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An abstract semilinear parabolic equation in a Banach space X is considered. Under general assumptions on nonlinearity this problem is shown to generate a bounded dissipative semigroup on $X^{α}$ . This semigroup possesses an $(X^{α} - Z)$ -global attractor that is closed, bounded, invariant in $X^{α}$ , and attracts bounded subsets of $X^{α}$ in a ’weaker’ topology of an auxiliary Banach space Z. The abstract approach is finally applied to the scalar parabolic equation in Rⁿ and to the partly dissipative system. ...

The distance between fixed points of some pairs of maps in Banach spaces and applications to differential systems

Cristinel Mortici (2006)

Czechoslovak Mathematical Journal

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Let $T$ be a $γ$ -contraction on a Banach space $Y$ and let $S$ be an almost $γ$ -contraction, i.e. sum of an $(ε, γ)$ -contraction with a continuous, bounded function which is less than $ε$ in norm. According to the contraction principle, there is a unique element $u$ in $Y$ for which $u = T u .$ If moreover there exists $v$ in $Y$ with $v = S v$ , then we will give estimates for $∥ u - v ∥ .$ Finally, we establish some inequalities related to the Cauchy problem.

Local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation in Besov spaces

Gang Wu, Jia Yuan (2007)

Applicationes Mathematicae

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We study local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation $\partial_{t} u - \partial ³_{t x x} u + 2 κ \partial_{x} u + \partial_{x} [g (u) / 2] = γ (2 \partial_{x} u \partial ²_{x x} u + u \partial ³_{x x x} u)$ for the initial data u₀(x) in the Besov space $B_{p, r}^{s} (ℝ)$ with max(3/2,1 + 1/p) < s ≤ m and (p,r) ∈ [1,∞]², where g:ℝ → ℝ is a given $C^{m}$ -function (m ≥ 4) with g(0)=g’(0)=0, and κ ≥ 0 and γ ∈ ℝ are fixed constants. Using estimates for the transport equation in the framework of Besov spaces, compactness arguments and Littlewood-Paley theory, we get a local well-posedness result.

Semigroups generated by certain pseudo-differential operators on the half-space $ℝ_{0 +}^{n + 1}$

Victoria Knopova (2004)

Colloquium Mathematicae

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The aim of the paper is two-fold. First, we investigate the ψ-Bessel potential spaces on $ℝ_{0 +}^{n + 1}$ and study some of their properties. Secondly, we consider the fractional powers of an operator of the form $- A_{\pm} = - ψ (D_{x^{'}}) \pm \partial / (\partial x_{n + 1})$ , $(x^{'}, x_{n + 1}) \in ℝ_{0 +}^{n + 1}$ , where $ψ (D_{x^{'}})$ is an operator with real continuous negative definite symbol ψ: ℝⁿ → ℝ. We define the domain of the operator $- {(- A_{\pm})}^{α}$ and prove that with this domain it generates an $L_{p}$ -sub-Markovian semigroup.

On semigroups with an infinitesimal operator

Jolanta Olko (2005)

Annales Polonici Mathematici

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Let $F^{t} : t \geq 0$ be an iteration semigroup of linear continuous set-valued functions. If the semigroup has an infinitesimal operator then it is a uniformly continuous semigroup majorized by an exponential semigroup. Moreover, for sufficiently small t every linear selection of $F^{t}$ is invertible and there exists an exponential semigroup $f^{t} : t \geq 0$ of linear continuous selections $f^{t}$ of $F^{t}$ .

Application of $(L)$ sets to some classes of operators

Kamal El Fahri, Nabil Machrafi, Jawad H&#039;michane, Aziz Elbour (2016)

Mathematica Bohemica

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The paper contains some applications of the notion of $Ł$ sets to several classes of operators on Banach lattices. In particular, we introduce and study the class of order $(L)$ -Dunford-Pettis operators, that is, operators from a Banach space into a Banach lattice whose adjoint maps order bounded subsets to an $(L)$ sets. As a sequence characterization of such operators, we see that an operator $T : X \to E$ from a Banach space into a Banach lattice is order $Ł$ -Dunford-Pettis, if and only if $| T (x_{n}) | \to 0$ for $σ (E, E^{'})$ for every...

Comparison of solutions and successive approximations in the theory of the equation $\partial^{2} z / \partial x \partial y = f (x, y, z, \partial z / \partial x, \partial z / \partial y)$

J. Kisyński, A. Pelczar

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CONTENTSIntroduction........................................................................................................................................................................................................... 5I. THE CAUCHY-DARBOUX PROBLEM IN FUNCTION CLASSES $C^{1}' * (Δ_{a, b}; E)$ AND $L_{1}^{1, *} (Δ_{a, b}; E)$ ......................... 71. Basic function classes ......................................................................................................................................................................................

On a question of Demailly-Peternell-Schneider

Meng Chen, Qi Zhang (2013)

Journal of the European Mathematical Society

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We give an affirmative answer to an open question posed by Demailly-Peternell-Schneider in 2001 and recently by Peternell. Let $f : X \to Y$ be a surjective morphism from a log canonical pair $(X, D)$ onto a $ℚ$ -Gorenstein variety $Y$ . If $- (K_{X} + D)$ is nef, we show that $- K_{Y}$ is pseudo-effective.

Order-bounded operators from vector-valued function spaces to Banach spaces

Marian Nowak (2005)

Banach Center Publications

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Let E be an ideal of L⁰ over a σ-finite measure space (Ω,Σ,μ). For a real Banach space $(X, | | \cdot | |_{X})$ let E(X) be a subspace of the space L⁰(X) of μ-equivalence classes of strongly Σ-measurable functions f: Ω → X and consisting of all those f ∈ L⁰(X) for which the scalar function ${| | f (\cdot) | |}_{X}$ belongs to E. Let E(X)˜ stand for the order dual of E(X). For u ∈ E⁺ let $D_{u} (= {f \in E (X) : | | f (\cdot) | |}_{X} \leq u)$ stand for the order interval in E(X). For a real Banach space $(Y, | | \cdot | |_{Y})$ a linear operator T: E(X) → Y is said to be order-bounded whenever for each u ∈...

On asymptotically symmetric Banach spaces

M. Junge, D. Kutzarova, E. Odell (2006)

Studia Mathematica

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A Banach space X is asymptotically symmetric (a.s.) if for some C < ∞, for all m ∈ ℕ, for all bounded sequences ${(x_{j}^{i})}_{j = 1}^{\infty} \subseteq X$ , 1 ≤ i ≤ m, for all permutations σ of 1,...,m and all ultrafilters ₁,...,ₘ on ℕ, $l i m_{n ₁, ₁} . . . l i m_{n ₘ, ₘ} | | \sum_{i = 1}^{m} x_{n_{i}}^{i} | | \leq C l i m_{n_{σ (1)},_{σ (1)}} . . . l i m_{n_{σ (m)},_{σ (m)}} | | \sum_{i = 1}^{m} x_{n_{i}}^{i} | |$ . We investigate a.s. Banach spaces and several natural variations. X is weakly a.s. (w.a.s.) if the defining condition holds when restricted to weakly convergent sequences ${(x_{j}^{i})}_{j = 1}^{\infty}$ . Moreover, X is w.n.a.s. if we restrict the condition further to normalized weakly null sequences. If X is a.s. then...

Operator theoretic properties of semigroups in terms of their generators

S. Blunck, L. Weis (2001)

Studia Mathematica

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Let $(T_{t})$ be a C₀ semigroup with generator A on a Banach space X and let be an operator ideal, e.g. the class of compact, Hilbert-Schmidt or trace class operators. We show that the resolvent R(λ,A) of A belongs to if and only if the integrated semigroup $S_{t} : = \int_{0}^{t} T_{s} d s$ belongs to . For analytic semigroups, $S_{t} \in$ implies $T_{t} \in$ , and in this case we give precise estimates for the growth of the -norm of $T_{t}$ (e.g. the trace of $T_{t}$ ) in terms of the resolvent growth and the imbedding D(A) ↪ X.

On the K-theory of the $C^{*}$ -algebra generated by the left regular representation of an Ore semigroup

Joachim Cuntz, Siegfried Echterhoff, Xin Li (2015)

Journal of the European Mathematical Society

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We compute the $K$ -theory of $C^{*}$ -algebras generated by the left regular representation of left Ore semigroups satisfying certain regularity conditions. Our result describes the $K$ -theory of these semigroup $C^{*}$ -algebras in terms of the $K$ -theory for the reduced group $C^{*}$ -algebras of certain groups which are typically easier to handle. Then we apply our result to specific semigroups from algebraic number theory.

Weakly null sequences with upper estimates

Daniel Freeman (2008)

Studia Mathematica

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We prove that if $(v_{i})$ is a seminormalized basic sequence and X is a Banach space such that every normalized weakly null sequence in X has a subsequence that is dominated by $(v_{i})$ , then there exists a uniform constant C ≥ 1 such that every normalized weakly null sequence in X has a subsequence that is C-dominated by $(v_{i})$ . This extends a result of Knaust and Odell, who proved this for the cases in which $(v_{i})$ is the standard basis for $ℓ_{p}$ or c₀.

Spaces of multipliers and their preduals for the order multiplication on [0, 1]

Savita Bhatnagar, H. L. Vasudeva (2002)

Colloquium Mathematicae

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Let I = [0, 1] be the compact topological semigroup with max multiplication and usual topology. C(I), $L^{p} (I)$ , 1 ≤ p ≤ ∞, are the associated Banach algebras. The aim of the paper is to characterise $H o m_{C (I)} (L^{r} (I), L^{p} (I))$ and their preduals.

Weak precompactness and property (V*) in spaces of compact operators

Ioana Ghenciu (2015)

Colloquium Mathematicae

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We give sufficient conditions for subsets of compact operators to be weakly precompact. Let $L_{w *} (E *, F)$ (resp. $K_{w *} (E *, F)$ ) denote the set of all w* - w continuous (resp. w* - w continuous compact) operators from E* to F. We prove that if H is a subset of $K_{w *} (E *, F)$ such that H(x*) is relatively weakly compact for each x* ∈ E* and H*(y*) is weakly precompact for each y* ∈ F*, then H is weakly precompact. We also prove the following results: If E has property (wV*) and F has property (V*), then $K_{w *} (E *, F)$ has property (wV*). Suppose...

On the Cauchy problem for convolution equations

(2013)

Colloquium Mathematicae

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We consider one-parameter (C₀)-semigroups of operators in the space $^{'} (ℝ ⁿ; ℂ^{m})$ with infinitesimal generator of the form ${(G *) |}_{^{'} (ℝ ⁿ; ℂ^{m})}$ where G is an $M_{m \times m}$ -valued rapidly decreasing distribution on ℝⁿ. It is proved that the Petrovskiĭ condition for forward evolution ensures not only the existence and uniqueness of the above semigroup but also its nice behaviour after restriction to whichever of the function spaces $(ℝ ⁿ; ℂ^{m})$ , $_{L^{p}} (ℝ ⁿ; ℂ^{m})$ , p ∈ [1,∞], $(_{a}) (ℝ ⁿ; ℂ^{m})$ , a ∈ ]0,∞[, or the spaces $_{L^{q}}^{'} (ℝ ⁿ; ℂ^{m})$ , q ∈ ]1,∞], of bounded distributions.

The algebra of the subspace semigroup of $M ₂ (_{q})$

Jan Okniński (2002)

Colloquium Mathematicae

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The semigroup $S = S (M ₂ (_{q}))$ of subspaces of the algebra $M ₂ (_{q})$ of 2 × 2 matrices over a finite field $_{q}$ is studied. The ideal structure of S, the regular -classes of S and the structure of the complex semigroup algebra ℂ[S] are described.

On the opial type criterion for the well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations

Malkhaz Ashordia (2016)

Mathematica Bohemica

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The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense $d x (t) = d A_{0} (t) \cdot x (t) + d f_{0} (t)$ , $x (t_{0}) = c_{0}$ $(t \in I)$ with a unique solution $x_{0}$ is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems $d x (t) = d A_{k} (t) \cdot x (t) + d f_{k} (t)$ , $x (t_{k}) = c_{k}$ $(k = 1, 2, \dots)$ to have a unique solution $x_{k}$ for any sufficiently large $k$ such that $x_{k} (t) \to x_{0} (t)$ uniformly on $I$ . Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems....

The Lindelöf property in Banach spaces

B. Cascales, I. Namioka, J. Orihuela (2003)

Studia Mathematica

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A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space $M^{D}$ the following four conditions are equivalent: (i) K is fragmented by $d_{D}$ , where, for each S ⊂ D, $d_{S} (x, y) = s u p ϱ (x (t), y (t)) : t \in S$ . (ii) For each countable subset...

Property $(𝐰𝐋)$ and the reciprocal Dunford-Pettis property in projective tensor products

Ioana Ghenciu (2015)

Commentationes Mathematicae Universitatis Carolinae

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A Banach space $X$ has the reciprocal Dunford-Pettis property ( $R D P P$ ) if every completely continuous operator $T$ from $X$ to any Banach space $Y$ is weakly compact. A Banach space $X$ has the $R D P P$ (resp. property $(w L)$ ) if every $L$ -subset of $X^{*}$ is relatively weakly compact (resp. weakly precompact). We prove that the projective tensor product $X \otimes_{π} Y$ has property $(w L)$ when $X$ has the $R D P P$ , $Y$ has property $(w L)$ , and $L (X, Y^{*}) = K (X, Y^{*})$ .

Inverses of generators of nonanalytic semigroups

Ralph deLaubenfels (2009)

Studia Mathematica

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Suppose A is an injective linear operator on a Banach space that generates a uniformly bounded strongly continuous semigroup ${e^{t A}}_{t \geq 0}$ . It is shown that $A^{- 1}$ generates an $O (1 + τ) A {(1 - A)}^{- 1}$ -regularized semigroup. Several equivalences for $A^{- 1}$ generating a strongly continuous semigroup are given. These are used to generate sufficient conditions on the growth of ${e^{t A}}_{t \geq 0}$ , on subspaces, for $A^{- 1}$ generating a strongly continuous semigroup, and to show that the inverse of -d/dx on the closure of its image in L¹([0,∞)) does not generate...

Dieudonné operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

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A bounded linear operator between Banach spaces is called a Dieudonné operator ( = weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let (Ω,Σ,μ) be a finite measure space, and let X and Y be Banach spaces. We study Dieudonné operators T: L¹(X) → Y. Let $i_{\infty} : L^{\infty} (X) \to L ¹ (X)$ stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then $T \circ i_{\infty} : L^{\infty} (X) \to Y$ is a weakly compact operator. Moreover, we obtain that...