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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]

We briefly review some history and some of the main mathematical properties of the basic reproduction number, a concept encountered frequently in mathematical biology. We discuss some limitations on the practical use of basic reproduction numbers. One stems from the known fact that basic reproduction numbers are not unique. Another is related to the fact that basic reproduction numbers and the spectral radius of their associated linear operator could respond in opposite ways when controlled. [ABSTRACT FROM AUTHOR]

In this manuscript, we employ integrated resolvent operator to derive the variation of constants formula for the solution of the impulsive integro-differential system, in nonlocal domain with finite delay function. Using the Banach fixed point theorem and integrated resolvent operator, we explore the existence of mild solution of the aforementioned system. Additionally, we establish the Hyers–Ulam stability of the system. Finally, the main result is illustrated with the help of an example. [ABSTRACT FROM AUTHOR]

We prove a compactness result related to G-convergence for autonomous evolutionary equations in the sense of Picard. Compared to previous work related to applications, we do not require any boundedness or regularity of the underlying spatial domain; nor do we assume any periodicity or ergodicity assumption on the potentially oscillatory part. In terms of abstract evolutionary equations, we remove any compactness assumptions of the resolvent modulo kernel of the spatial operator. To achieve the results, we introduced a slightly more general class of material laws. As a by-product, we also provide a criterion for G-convergence for time-dependent equations solely in terms of static equations. [ABSTRACT FROM AUTHOR]

We show results on propagation of anisotropic Gabor wave front sets for solutions to a class of evolution equations of Schrödinger type. The Hamiltonian is assumed to have a real-valued principal symbol with the anisotropic homogeneity a (λ x , λ σ ξ) = λ 1 + σ a (x , ξ) for λ > 0 where σ > 0 is a rational anisotropy parameter. We prove that the propagator is continuous on anisotropic Shubin–Sobolev spaces. The main result says that the propagation of the anisotropic Gabor wave front set follows the Hamilton flow of the principal symbol. [ABSTRACT FROM AUTHOR]

We study a Euler–Bernoulli beam equation with localized discontinuous structural damping. As our main result, we prove that the associated $ C_0 $ C 0 -semigroup $ (S(t))_{t\geq 0} $ (S (t)) t ≥ 0 is of Gevrey class $ \delta \gt 24 $ δ > 24 for t> 0 , hence immediately differentiable. Moreover, we show that $ (S(t))_{t\geq 0} $ (S (t)) t ≥ 0 is exponentially stable. [ABSTRACT FROM AUTHOR]

We consider stochastic reaction-diffusion equations with colored noise on the space of real-valued and continuous functions on a compact subset of ℝd for d = 1 , 2 , 3. We prove Schauder-type estimates, which will depend on the color of the noise, for the stationary and evolution problems associated with the corresponding transition semigroup. [ABSTRACT FROM AUTHOR]

We investigate the existence of solutions for a class of fractional evolution equations with nonlocal initial conditions and superlinear growth nonlinear functions in Banach spaces. By using the compactness of semigroup generated by the linear operator, we neither assume any Lipschitz property of the nonlinear term nor the compactness of the nonlocal initial conditions. Moreover, the approximation technique coupled with the Hartmann-type inequality argument allows the treatment of nonlinear terms with superlinear growth. Then combining with the Leray-Schauder continuation principle, we prove the existence results. Finally, the results obtained are applied to fractional parabolic equations with continuous superlinearly growth nonlinearities and nonlocal initial conditions including periodic or antiperiodic conditions, multipoint conditions and integral-type conditions. [ABSTRACT FROM AUTHOR]

We investigate stochastic reaction-diffusion equations on finite metric graphs. On each edge in the graph a multiplicative cylindrical Gaussian noise driven reaction-diffusion equation is given. The vertex conditions are the standard continuity and generalized, non-local Neumann-Kirchhoff-type law in each vertex. The reaction term on each edge is assumed to be an odd degree polynomial, not necessarily of the same degree on each edge, with possibly stochastic coefficients and negative leading term. The model is a generalization of the problem in [14] where polynomials with much more restrictive assumptions are considered and no first order differential operator is involved. We utilize the semigroup approach from [15] to obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph. [ABSTRACT FROM AUTHOR]

This work deals with the existence and uniqueness of μ -pseudo almost periodic solutions to some transport processes along the edges of a finite network with inhomogeneous conditions in the vertices. For that, the strategy consists of seeing these systems as a particular case of the semilinear boundary evolution equations (S H B E) du dt = A m u (t) + f (t , u (t)) , t ∈ R , L u (t) = g (t , u (t)) , t ∈ R , where A : = A m | k e r L generates a C 0 -semigroup admitting an exponential dichotomy on a Banach space. Assuming that the forcing terms taking values in a state space and in a boundary space respectively are only μ -pseudo almost periodic in the sense of Stepanov, we show that (SHBE) has a unique μ -pseudo almost periodic solution which satisfies a variation of constant formula. Then we apply the previous result to obtain the existence and uniqueness of μ -pseudo almost periodic solution to our model of network. [ABSTRACT FROM AUTHOR]

We study an abstract linear viscoelastic equation in Hilbert space. It fits some convolution type of integro-differential equations which govern dynamics in viscoelastic or thermoviscoelastic materials. A new semigroup approach is proposed which is adapted from a semigroup approach designed originally to a second-order abstract linear integro-differential equation. As compared with the existing semigroup approach, the system contraction can be obtained readily. Sufficient conditions for the system exponential stability are also established. An illustrative example is given, and numerical simulations are provided. They are in good agreement with the theoretical analysis. [ABSTRACT FROM AUTHOR]

This article is dedicated to investigate the existence and uniqueness of solutions for abstract neutral differential equations with state-dependent time impulses. We employ the Banach contraction method and the Schauder’s fixed point approach to establish our results. In order to prove our findings, certain assumptions are made. Additionally, we provide an example to illustrate the application of our theoretical results. [ABSTRACT FROM AUTHOR]