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To describe the dynamics of a size-structured population and its unstructured resource, we formulate bookkeeping equations in two different ways. The first, called the PDE formulation, is rather standard. It employs a first order partial differential equation, with a non-local boundary condition, for the size-density of the consumer, coupled to an ordinary differential equation for the resource concentration. The second is called the DELAY formulation and employs a renewal equation for the population level birth rate of the consumer, coupled to a delay differential equation for the (history of the) resource concentration. With each of the two formulations we associate a constructively defined semigroup of nonlinear solution operators. The two semigroups are intertwined by a non-invertible operator. In this paper we delineate in what sense the two semigroups are equivalent. In particular, we (i) identify conditions on both the model ingredients and the choice of state space that guarantee that the intertwining operator is surjective, (ii) focus on large time behaviour and (iii) consider full orbits, i.e., orbits defined for time running from $-\infty $ to $+\infty $. Conceptually, the PDE formulation is by far the most natural one. It has, however, the technical drawback that the solution operators are not differentiable, precluding rigorous linearisation. For the delay formulation, one can (under certain conditions concerning the model ingredients) prove the differentiability of the solution operators and establish the Principle of Linearised Stability. Next the equivalence of the two formulations yields a rather indirect proof of this principle for the PDE formulation., Comment: To appear in Mathematical Models and Methods in the Applied Sciences

Let Bj (t) (j = 1,..., m) and B(t, τ) (t ≥ 0, 0 ≤ τ ≤ 1) be bounded variable operators in a Banach space. We consider the equation u ′ ( t ) = ∑ k = 1 m B k ( t ) u ( t - h k ( t ) ) + ∫ 0 1 B ( t , τ ) u ( t - h 0 ( τ ) ) d τ ( t ≥ 0 ) , u'\left( t \right) = \sum\limits_{k = 1}^m {{B_k}\left( t \right)u\left( {t - {h_k}\left( t \right)} \right)} + \int\limits_0^1 {B\left( {t,\tau } \right)u\left( {t - {h_0}\left( \tau \right)} \right)d\tau \,\,\,\,\left( {t \ge 0} \right),} where hk (t) (t ≥ 0; k = 1, ..., m) and h 0(τ) are continuous nonnegative bounded functions. Explicit delay-dependent exponential stability conditions for that equation are established. Applications to integro-differential equations with delay are also discussed

In the paper, we discuss the existence and uniqueness of mild solutions of a class of fractional functional differential equations in Hilbert space separable using the Banach fixed point theorem technique. In this sense, Faedo-Galerkin approximation to the solution is studied and demonstrated some convergence results.

The objective of this article is to discuss the existence and uniqueness of mild solutions for a class of non-autonomous semilinear differential equations with nonlocal condition via monotone iterative method with upper and lower solutions in an ordered complete norm space X, using evolution system and measure of noncompactness.

We consider the well-posedness of the second-order degenerate differential equations with infinite delay (P2): (Mu′)′(t)+(Lu)′(t)=Au(t)+ ∫ −∞ta(t−s)Bu(s)ds+f(t), (0≤t≤2π) with periodic boundary conditions u(0)=u(2π), (Mu′)(0)=(Mu′)(2π), in Lebesgue–Bochner spaces Lp(𝕋;X) and periodic Besov spaces Bp,qs(𝕋;X), where A, B, L and M are closed linear operators in a Banach space X satisfying D(A)∩D(B)⊂D(M)∩D(L), D(A)∩D(B)≠{0} and a∈L1(ℝ+). We completely characterize the well-posedness of (P2) in the above function spaces by using known operator-valued Fourier multiplier theorems. We also give concrete examples to support our abstract results.

We study the existence and regularity of solutions for neutral integro-differential equations with nonlocal condition by applying the theory of resolvent operators established recently in the literature. Since the nonlinear terms of the systems involve spacial derivatives, we make full use of the theory of fractional power, the [math] -norm, and Schauder’s fixed point theorem to discuss the problems. An example is given to illustrate the applications of the obtained results.

This paper is devoted to study the existence and regularity of mild solutions in some interpolation spaces for a class of functional partial differential equations with nonlocal initial conditions. The linear part is assumed to be a sectorial operator in Banach space X. The fractional power theory and α-norm are used to discuss the problem so that the obtained results can be applied to equations with terms involving spatial derivatives. Moreover, we present an example to illustrate the application of main results.

In this paper, we are devoted to study the existence of mild solutions for delay evolution equations with nonlocal conditions. By using tools involving the Kuratowski measure of noncompactness and fixed point theory, we establish some existence results of mild solutions without the assumption of compactness on the associated semigroup. Our results improve and generalize some related conclusions on this issue. Moreover, we present an example to illustrate the application of the main results.

We analyze the lumpability of linear systems on Banach spaces, namely, the possibility of projecting the dynamics by a linear reduction opera-tor onto a smaller state space in which a self-contained dynamical description exists. We obtain conditions for lumpability of dynamics defined by unbounded operators using the theory of strongly continuous semigroups. We also derive results from the dual space point of view using sun dual theory. Furthermore, we connect the theory of lumping to several results from operator factoriza-tion. We indicate several applications to particular systems, including delay differential equations. © 2017, American Institute of Mathematical Sciences. All rights reserved.

The purpose of this paper is to study a three species competition model with diffusion. It is well known that there exists a family of traveling wave solutions connecting two equilibria $(0,1,1)$ and $(1,0,0)$. In this paper, we first establish the exact asymptotic behavior of the traveling wave profiles at $\pm \infty$. Then, by constructing a pair of explicit upper and lower solutions via the combination of traveling wave solutions, we derive the existence of some new entire solutions which behave as two traveling fronts moving towards each other from both sides of $x$-axis. Such entire solution provides another invasion way of the stronger species to the weak ones.

The well-posedness and the flow invariance are studied for a semilinear functional differential equation governed by a family of non-densely defined operators in a general Banach space. The notion of mild solutions is introduced through a new type of variation of constants formula and the well-posedness is established under a semilinear stability condition with respect to a metric-like functional and a subtangential condition. The abstract result is applied to a size-structured model with birth delay.

In this paper, we consider a class of nonlocal dispersal equation with nonlocal time-delayed reaction. We prove that all noncritical wavefronts are globally exponentially stable by the weighted energy method and comparison principle. However, for the critical wavefronts, we prove that they are globally asymptotically stable.

In this paper we prove the global existence and attractivity of mild solutions for neutral semilinear evolution equations with state-dependent delay in a Banach space.

This paper is concerned with the existence of mild solutions, strong solutions and strict solutions for a class of neutral integrodifferential equations with nonlocal conditions in Banach space. Since the nonlinear terms of the systems involve spacial derivatives, the theory of fractional power and $\alpha$-norm is used to discuss the problem. In the end an example is provided to illustrate the applications of the obtained results.

The aim of this work is to investigate the asymptotic behavior of solutions near hyperbolic equilibria for nonlinear partial neutral functional differential equations. We suppose that the linear part $A$ satisfies the Hille-Yosida condition on a Banach space and is not necessarily densely defined; the delayed part is assumed to be Lipschitz. We show the existence of stable and unstable manifolds near hyperbolic equilibria when the neutral operator is stable and the semigroup generated by the part of $A$ in $\overline{D(A)}$ is compact. Local stable and unstable manifolds are also obtained when the undelayed part is a C$^{1}$ function in a neighborhood of the equilibria.

The paper will investigate the local existence, uniqueness of solution to integrodifferential equations with infinite delay. We assume that the linear part is not necessarily densely defined and satisfies a Hille-Yosida condition. Moreover, the continuity of solutions with respect to initial conditions is also studied.

We study the controllability problem for a system governed by a degenerate semilinear functional differential inclusion in a Banach space with infinite delay. Notice that we are not assuming that the generalized semigroup generated by the linear part of inclusion is compact. Instead we suppose that the multivalued nonlinearity satisfies the regularity condition expressed in terms of the Hausdorff measure of noncompactness. It allows to obtain the general controllability principle in the terms of the topological degree theory for condensing multivalued operators. Two realizations of this principle are considered.

In this paper, we shall establish sufficient conditions for the existence of integral solutions and extremal integral solutions for some non-densely defined impulsive semilinear functional differential equations in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators. The question of controllability of these equations are also considered.

In this work, we discuss the existence, regularity and stability of solutions for some partial functional differential equations with infinite delay. We assume that the linear part generates an analytic semigroup on a Banach space $X$ and the nonlinear part is a Lipschitz continuous function with respect to the fractional power norm of the linear part.

In this paper, we obtain two new existence theorems for mild solutions and classical solutions to nonautonomous functional equations with infinite delay in Banach spaces.

The homogenization of $3 \times 3$ system of differential equations related to the Coriolis and Lorentz forces are studied. It generates memory effects. The memory (or nonlocal) kernel is described by the Volterra integral equation. When the coefficient is independent of time, the memory kernel can be characterized explicitly in terms of Young’s measure. The kinetic formulation of the homogenized equation is also obtained.

In this work, we consider a class of nonlinear partial neutral functional di¤erential equations with a nondensely defined Hille-Yosida operator. We first prove the local existence, uniqueness and regularity of solutions. Second, we study the global existence and stability. In the end, we extend in the autonomous case, results of Hale ([21], [23]) concerning dissipativeness and existence of a global attractor to our situation.

The Massera Theorem for almost periodic solutions of linear periodic ordinary differential equations of the form $(*)x^{\prime}=A(t)x+f(t)$ , where $f$ is almost periodic, is stated and proved. Furthermore, it is extended to abstract functional differential equations $(**)x^{\prime}=Ax+F(t)x_{t}+f(t)$ , where $A$ is the generator of a compact semigroup, $F$ is periodic and $f$ is almost periodic. The main techniques used in the proofs involve a new variation of constants formula in the phase space and a decomposition theorem for almost periodic solutions.

We prove the equivalence of the well-posedness of a partial differential equation with nonautonomous past and an associated abstract Cauchy problem. This equivalence is then used to derive sufficient conditions for exponential stability of the solutions.

For asymptotically almost periodic functional differential equations with infinite delay in a Banach space, some stability properties of a bounded solution are deduced from stabilities in a certain limiting equation which is obtained by employing the Bohr topology.

Evolution equations are considered as operator equations involving a sum of the time-derivative operator $ \partial _{0} $ regarded as a normal operator in a suitable Hilbert space setting and another fairly arbitrary spatial operator $ A$ acting in a Hilbert space $ H $. As an extension of the linear theory the case of a class of nonlinear constitutive laws with memory effects is considered.

We consider the semilinear non-autonomous evolution equation $\frac{d}{dt}u(t)=Au(t)+G(t,u(t))$, $t\geq s\geq 0,$ where $(A,D(A))$ is a Hille-Yosida operator on a Banach space $X$ and $G$ is a continuous function on $\mathbb R_+\times \overline{D(A)}$ with values in the extrapolated Favard class corresponding to $A$. In our main results we present principles of linearized stability and instability for a solution of such an equation. Our approach is based on the theory of extrapolation spaces. We apply the results to non-autonomous semilinear retarded differential equations.

Results on (a) the existence and (b) asymptotic stability of mild and of strong solutions to the nonlinear partial functional differential equation with delay $ (FDE) \; \, \dot{u} (t) + B u(t) \ni F(u_t), \; t \geq 0 , \; u_0 = \varphi \in E, $ are presented. The `partial differential expression' $B$ will be a, generally multivalued, accretive operator, and the history-responsive operator $F$ will be allowed to be (defined and) Lipschitz continuous on `thin' subsets of the initial-history space $E $ of functions from an interval $I \subset (-\infty,0] $ to the state Banach space $X.\,$ As one of the main results, it is shown that the well-established solution theory on strong, mild and integral solutions to the undelayed counterpart to (FDE) of the nonlinear initial-value problem $ (CP) \; \, \dot{u} (t) + B u(t) \ni f(t), \; t \geq 0 , \; u(0) = u_0 \in X, $ can fully be extended to the more general initial-history problem (FDE). The results are based on the relation of the solutions to (FDE) to those of an associated nonlinear Cauchy problem in the initial-history space $E. $ Applications to models from population dynamics and biology are presented.

The differential equation $d(Mu(t))/dt=-Lu(t)+L_1u(t-1), $ $t\geq0, $ $u(t)=\varphi(t),$ $ -1\leq t\leq0$, for a given strongly continuous $X$-valued function $\varphi$ on $[-1,0]$ is studied, where $M, L, L_1$ are closed linear operators from the complex Banach space $X$ into itself, and $L$ is invertible. Though already in the finite dimensional case in general existence of continuous solutions on $[-1,\infty)$ may fail or it is possible to have continuous solutions only on a finite interval, we indicate classes of operators for which existence results analogous to the ones for regular equations $M=I$ hold. In particular, solutions are given explicitly by a recovery formula.

The differentiability with respect to the delay of the solution of the retarded differential equation (1.1) below will be shown by using elementary arguments. The investigation is extended to the infinite-dimensional case. An application to heat conduction is also given.

This paper is devoted to two problems for the nonautonomous integrodifferential equation in a Banach space $X$ of hyperbolic type $$ \text{$u'(t) = A(t)u(t)+ \int_0^t B(t,s)u(s) ds + f(t)$ for $t\in[0,T]$, and $u(0) = u_0$.} $$ One is the problem of existence and uniqueness of classical solutions without assuming that the common domain of $A(t)$ is dense in $X$, and the other is the regularity problem in the case where the common domain of $A(t)$ is dense in $X$. The regularity result will play a role in developing an abstract theory which can be applied to the second-order integrodifferential equations with the third kind boundary conditions.

On demontre deux resultats d'existence locale concernant des equations differentielles fonctionnelles fortement non lineaires

A general class of linear and nonautonomous delay differential equations with initial data in a separable Hilbert space is treated. The classic questions of existence, uniqueness, and regularity of solutions are addressed. Moreover, the semigroup approach typically adopted in the autonomous case for continuous initial functions is extended, and thus the existence of an equivalent abstract ordinary formulation is shown to hold. Finally, the existence of infinitely many Lyapunov exponents for the associated evolution is proven and their meaning is discussed.