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This paper is dedicated to weighted composition semigroups on spaces of continuous functions and their subspaces. We consider semigroups induced by semiflows and semicocycles on Banach spaces F (Ω) of continuous functions on a Hausdorff space Ω such that the norm-topology is stronger than the compact-open topology like the Hardy spaces, the weighted Bergman spaces, the Dirichlet space, the Bloch type spaces, the space of bounded Dirichlet series and weighted spaces of continuous or holomorphic functions. It was shown by Gallardo-Gutiérrez, Siskakis and Yakubovich that there are no non-trivial norm-strongly continuous weighted composition semigroups on Banach spaces F (D) of holomorphic functions on the open unit disc D such that H ∞ ⊂ F (D) ⊂ B 1 where H ∞ is the Hardy space of bounded holomorphic functions on D and B 1 the Bloch space. However, we show that there are non-trivial weighted composition semigroups on such spaces which are strongly continuous w.r.t. the mixed topology between the norm-topology and the compact-open topology. We study such weighted composition semigroups in the general setting of Banach spaces of continuous functions and derive necessary and sufficient conditions on the spaces involved, the semiflows and semicocycles for strong continuity w.r.t. the mixed topology and as a byproduct for norm-strong continuity as well. Moreover, we give several characterisations of their generator and their space of norm-strong continuity. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we present a lemma that allows us to characterize a broad class of C0-semigroups on time scales, which can be applied to prove existence and uniqueness of solutions of systems of partial differential equations where the time domain is a time scale. The result obtained is applied to study the controllability of the heat equation on time scales. The lemma presented in this paper can be seen as a unification of the one proved by H. Leiva in [28], for semigroups in [0,∞). [ABSTRACT FROM AUTHOR]

This paper deals with a one-dimensional system in the linear isothermal theory of swelling porous elastic soils subject to fractional derivative-type boundary damping. We apply the semigroup theory. We prove well-posedness by the Lumer–Phillips theorem. We show the lack of exponential stability and strong stability is proved by using general criteria due to Arendt–Batty. Polynomial stability result is obtained by applying the Borichev–Tomilov theorem. [ABSTRACT FROM AUTHOR]

In the present paper, we address the asymptotic behavior of a fish population system structured in age and weight, while also incorporating spatial effects. Initially, we develop an abstract perturbation result concerning the essential spectral radius, employing the regular systems approach. Following that, we present the model in the form of a perturbed boundary problem, which involves unbounded operators on the boundary. Using time-invariant regular techniques, we construct the corresponding semigroup solution. Then, we designate an operator characteristic equation of the primary system via the radius of a bounded linear operator defined on the boundary space. Moreover, we provide a characterization of the uniform exponential stability and the asynchronous exponential growth property (AEG) by localizing the essential radius and proving the irreducibility of the perturbed semigroup. Finally, we precise the projection that emerged from the (AEG) property; this depends on the developed characteristic equation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of (ω , Q) -periodic mild solutions for the following problem x ′ (t) = A x (t) + f (t , x (t)) , t ∈ R , on a Banach space X. Here, A is a closed linear operator which generates an exponentially stable C 0 -semigroup and the nonlinearity f satisfies suitable properties. The approaches are based on the well-known Banach contraction principle. In addition, a sufficient criterion is established for the existence and uniqueness of (ω , Q) -periodic mild solutions to the Hopfield-type neural network model. [ABSTRACT FROM AUTHOR]

In this paper, we compare several Cesàro- and Kreiss-type boundedness conditions for a C 0 -semigroup on a Banach space and we show that those conditions are all equivalent for a positive semigroup on a Banach lattice. Furthermore, we give an estimate of the growth rate of a Kreiss bounded and eventually positive C 0 -semigroup (T t) t ≥ 0 on certain Banach lattices X. We prove that if X is an L p -space, 1 < p < + ∞ , then ‖ T t ‖ = O t / log (t) max (1 / p , 1 / p ′) , and if X is an (AL) or (AM) -space, then ‖ T t ‖ = O (t 1 - ϵ) for some ϵ ∈ (0 , 1) , improving previous estimates. [ABSTRACT FROM AUTHOR]

This paper studies the regularity problem for block uniformly elliptic operators in divergence form with complex bounded measurable coefficients. We consider the case where the boundary data belongs to Lebesgue spaces with weights in the Muckenhoupt classes. Our results generalize those of S. Mayboroda (and those of P. Auscher and S. Stahlhut employing the first order method) who considered the unweighted case. To obtain our main results we use the weighted Hardy space theory associated with elliptic operators recently developed by the last two named authors. One of the novel contributions of this paper is the use of an "inhomogeneous" vertical square function which is shown to be controlled by the gradient of the function to which is applied in weighted Lebesgue spaces. [ABSTRACT FROM AUTHOR]

In this paper we present the abstract approach to Feynman's operational calculus in the most general setting, in which the time-ordering measures are allowed to have arbitrarily supported discrete parts. Two approaches to the operational calculus are presented in detail and examples of each method are presented. In particular, for the second method we present an evolution equation satisfied by the operational calculus and examples (Feynman–Kac formulas with Lebesgue–Stieltjes measures) are considered. Furthermore, a basic stability (or continuity) result is presented. [ABSTRACT FROM AUTHOR]

We study linear time-dependent Maxwell equations on a cuboid consisting of two homogeneous subcuboids. At the interface, we allow for nonzero surface charge density and surface current. This model is a first step towards a detailed mathematical analysis of the interaction of single-layer materials with electromagnetic fields. The main results of this paper provide several wellposedness and regularity statements for the solutions of the Maxwell system. To prove the statements, we employ extension arguments using interpolation theory, as well as semigroup theory and regularity theory for elliptic transmission problems. [ABSTRACT FROM AUTHOR]

Due to the seminal works of Hochbruck and Ostermann (Appl Numer Math 53(2–4):323–339, 2005, Acta Numer 19:209–286, 2010) exponential splittings are well established numerical methods utilizing operator semigroup theory for the treatment of semilinear evolution equations whose principal linear part involves a sectorial operator with angle greater than π 2 (meaning essentially the holomorphy of the underlying semigroup). The present paper contributes to this subject by relaxing the sectoriality condition, but in turn requiring that the semigroup operators act consistently on an interpolation couple (or on a scale of Banach spaces). Our conditions (on the semigroup and on the semilinearity) are inspired by the approach of Kato (Math Z 187(4):471–480, 1984) to the local solvability of the Navier–Stokes equation, where the L p - L r -smoothing of the Stokes semigroup was fundamental. The present abstract operator theoretic result is applicable for this latter problem (as was already the result of Hochbruck and Ostermann), or more generally in the setting of Hochbruck and Ostermann (2005), but also allows the consideration of examples, such as non-analytic Ornstein–Uhlenbeck semigroups or the Navier–Stokes flow around rotating bodies. [ABSTRACT FROM AUTHOR]

In this short paper, we study a well-posed logarithmic counterpart of an ill-posed Cauchy problem associated with an abstract evolution equation of third order in time. [ABSTRACT FROM AUTHOR]

In this paper we present new results on the preservation of polynomial stability of damped wave equations under addition of perturbing terms. We in particular introduce sufficient conditions for the stability of perturbed two-dimensional wave equations on rectangular domains, a one-dimensional weakly damped Webster's equation, and a wave equation with an acoustic boundary condition. In the case of Webster's equation, we use our results to compute explicit numerical bounds that guarantee the polynomial stability of the perturbed equation. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. First, we use a classical method to establish Gaussian bounds of the fundamental matrix of a generalized parabolic Lamé system with only bounded and measurable coefficients. Second, we derive a maximal L 1 regularity result for the abstract Cauchy problem associated with a composite operator. In a concrete example, we also obtain maximal L 1 regularity for the Lamé system, from which it follows that the Lipschitz seminorm of the solutions to the Lamé system is globally L 1 -in-time integrable. As an application, we use a Lagrangian approach to prove a global-in-time well-posedness result for a viscous pressureless flow in a perturbation framework, but with possibly discontinuous densities. The method established in this paper might be a useful tool for studying many issues arising from viscous fluids with truly variable densities. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss the existence of mild solutions and positive mild solutions for a general class of semilinear fractional retarded evolution equations subjected to mixed nonlocal plus local initial conditions on infinite dimensional Banach spaces. Under the situation that the nonlinear term and nonlocal function satisfy some appropriate growth conditions and a noncompactness measure condition, we obtained the existence of mild solutions and positive mild solutions by utilizing a generalized Darbo's fixed point theorem and a new estimation technique of the measure of noncompactness. In addition, the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example about the retarded parabolic partial differential equation involving a general mixed nonlocal plus local initial conditions is also given to illustrate the feasibility of our abstract results. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with the VIP of fractional fuzzy evolution equations in the space of triangular fuzzy numbers. The continuous dependence of two kinds of fuzzy mild solutions on initial values and orders for the studied problem is obtained. The results obtained in this paper improve and extend some related conclusions on this topic. [ABSTRACT FROM AUTHOR]

In this paper, we prove that it is always possible to define a realization of the Laplacian Δ κ , θ on L 2 (Ω) subject to nonlocal Robin boundary conditions with general jump measures on arbitrary open subsets of R N . This is made possible by using a capacity approach to define an admissible pair of measures (κ , θ) that allows the associated form E κ , θ to be closable. The nonlocal Robin Laplacian Δ κ , θ generates a sub-Markovian C 0 -semigroup on L 2 (Ω) which is not dominated by the Neumann Laplacian semigroup unless the jump measure θ vanishes. [ABSTRACT FROM AUTHOR]

In this paper we investigate the following fractional order in time integrodifferential problem D t α u (t) + A u (t) = f (t , u (t)) + ∫ − ∞ t k (t − s) g (s , u (s)) d s , t ∈ R. Here, D t α is the Caputo derivative. We obtain results on the existence and uniqueness of (ω , c) -periodic mild solutions assuming that −A generates an analytic semigroup on a Banach space X and f, g, and k satisfy suitable conditions. Finally, an interesting example that fits our framework is given. [ABSTRACT FROM AUTHOR]

In this paper, we consider an abstract third-order differential equation and deduce some results on the maximal regularity of its strict solution. We assume that the inhomogeneity appearing in the right-hand term of this equation belongs to some anistropic Hölder spaces. We illustrate our results by a BVP involving a 3D Laplacian posed in a cusp domain of ℝ4. [ABSTRACT FROM AUTHOR]

In this paper, we give some new results on the maximal regularity of an elliptic differential equation set in three habitats with skewness boundary conditions at the interfaces in UMD spaces. This work completes naturally the results obtained in our paper (Labbas et al. in Mediterr J Math 15(3):128, 2018) studied in H ölder spaces. Our techniques use essentially the semigroup theory and the celebrated Dore–Venni theorem (Math Z 196:124–136, 1987). [ABSTRACT FROM AUTHOR]

This paper discusses the abstract time-space fractional evolution equation with the Caputo derivative of order α ∈ (0 , 1) and fractional power operator - A β , β ∈ (0 , 1) , where - A generates a C 0 -semigroup on a Banach space. The compactness and exponential stability of the semigroup which is generated by fractional power operator - A β are investigated. With the aid of the properties of the semigroup, the existence and global asymptotic behavior of S-asymptotically periodic solutions are obtained by some fixed point theorems and related inequalities. An example to the time-space fractional diffusion equation with fractional Laplacian will be shown. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new properties of (ω , c) -periodic and asymptotically (ω , c) -periodic functions, then we apply them to study the existence and uniqueness of mild solutions of these types to the following semilinear fractional differential equations: (1) { c D t α u (t) = A u (t) + c D t α − 1 f (t , u (t)) , 1 < α < 2 , t ∈ R , u (0) = 0 and (2) { c D t α u (t) = A u (t) + c D t α − 1 f (t , u (t − h)) , 1 < α < 2 , t , h ∈ R + , u (0) = 0 where c D t α (⋅) (1 < α < 2) stands for the Caputo derivative and A is a linear densely defined operator of sectorial type on a complex Banach space X and the function f (t , x) is (ω , c) -periodic or asymptotically (ω , c) -periodic with respect to the first variable. Our results are obtained using the Leray–Schauder alternative theorem, the Banach fixed point principle and the Schauder theorem. Then we illustrate our main results with an application to fractional diffusion-wave equations. [ABSTRACT FROM AUTHOR]

The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points, for both ordinary and stochastic differential equations while using either ergodic trajectories or i.i.d. samples. We illustrate these bounds by means of an example with the Ornstein–Uhlenbeck process. Moreover, we extend our analysis to (stochastic) nonlinear control-affine systems. We prove error estimates for a previously proposed approach that exploits the linearity of the Koopman generator to obtain a bilinear surrogate control system and, thus, circumvents the curse of dimensionality since the system is not autonomized by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the bilinear approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled. [ABSTRACT FROM AUTHOR]

In this paper, we study the solvability of Cauchy problem for a class of non-autonomous fractional evolution equation with Caputo's fractional derivative of order α ∈ (1 , 2) , which can be applied to model the time dependent coefficients fractional differential systems. We first introduce an operator family and analyze its properties, by the iterative method, we construct a solution to an operator-valued Volterra equation, which is the most critical ingredient to prove solvability of the problem. Finally, based on the solution operators we establish the existence and uniqueness of classical solutions. [ABSTRACT FROM AUTHOR]

In this paper, we solve the Cauchy problem for a hyperbolic system of first-order PDEs defined on a certain Banach space X. The system has a special semilinear structure because, on the one hand, the evolution law can be expressed as the sum of a linear unbounded operator and a nonlinear Lipschitz function but, on the other hand, the nonlinear perturbation takes values not in X but on a larger space Y which is related to X. In order to deal with this situation we use the theory of dual semigroups. Stability results around steady states are also given when the nonlinear perturbation is Fréchet differentiable. These results are based on two propositions: one relating the local dynamics of the nonlinear semiflow with the linearised semigroup around the equilibrium, and a second relating the dynamical properties of the linearised semigroup with the spectral values of its generator. The later is proven by showing that the Spectral Mapping Theorem always applies to the semigroups one obtains when the semiflow is linearised. Some epidemiological applications involving gut bacteria are commented [ABSTRACT FROM AUTHOR]

In this paper, based on the monotone iterative technique and the operator semigroup theory, a class of nonlocal problem of structural damped elastic systems with delay is studied in the case of noncompact semigroups in ordered Banach space. First, on the premise of the existence of upper and lower mild solutions, the existence and uniqueness of mild solutions for the elastic system are obtained. Then, an existence theorem of positive mild solutions for elastic system is obtained without assuming lower and upper mild solutions. Finally, as the application of abstract results, the existence and uniqueness of mild solutions and positive mild solutions for two classes of nonlocal damped beam vibration equations with delay are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a discrete-time fractional model of abstract form involving the Riemann-Liouville-like difference operator. On account of the C0-semigroups generated by a closed linear operator A and based on a distinguished class of sequences of operators, we show the existence of stable solutions for the nonlinear Cauchy problem by means of fixed point technique and the compact method. Moreover, we also establish the Ulam-Hyers-Rassias stability of the proposed problem. Two examples are presented to explain the main results. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence of mild solutions for a class of neutral partial integrodifferential equations with lack of compactness. The results are obtained using noncompact resolvent operators and a new fixed point theorem of Monch-Krasnosel’skii type. Our results are applied to a large variety of partial differential equations in which memory effects are considered. An example is provided at the end of the paper to illustrate the main results of this work. [ABSTRACT FROM AUTHOR]

The well established mixed monotone iterative technique is utilized explicitly to study the existence of the extremal solutions of a class of impulsive system with Hilfer fractional order in this paper. The procedure of finding mild L-quasi solutions of such impulsive evolution equation with non-compact semigroups involves measure of non-compactness and Sadovskii's fixed point theorem as well. An example is provided to illustrate the main results. [ABSTRACT FROM AUTHOR]