Existence of periodic solutions of functional differential equations with parameters such as Nicholson’s blowflies model call for the investigation of integral equations with parameters defined over spaces with periodic structures. In this paper, we study one such equation $\varphi \left(x\right)=\lambda {\int }_{[x,x+\omega ]\cap \Omega }K(x,y)h\left(y\right)f(y,\varphi (y-\tau \left(y\right)))dy$, x ∈ Ω, by means of the proper value theory of operators in Banach spaces with cones. Existence, uniqueness and continuous dependence of proper solutions are established.

We show that as soon as $c_0$ embeds complementably into the space of all weakly compact operators from $X$ to $Y$, then it must live either in $X^{*}$ or in $Y$.

Let Π₂ be the operator ideal of all absolutely 2-summing operators and let $I_m$ be the identity map of the m-dimensional linear space. We first establish upper estimates for some mixing norms of $I_m$. Employing these estimates, we study the embedding operators between Besov function spaces as mixing operators. The result obtained is applied to give sufficient conditions under which certain kinds of integral operators, acting on a Besov function space, belong to Π₂; in this context, we also...

In the paper the conditions for the existence of a $2\pi$-periodic solution in $t$ of the system $u_{tt}-u_{rr}-(2/r)u_r=ϵf(t,r,u,u_t,u_r)$, $\left|u(t,0)\right|<+\infty ,u(t,\pi )=0$ are investigated provided that $f$ is sufficiently smooth and $2\pi$-periodic in $t$.

We prove that for symbols in the modulation spaces ${ℳ}^{p,q}$, p ≥ q, the associated multilinear pseudodifferential operators are bounded on products of appropriate modulation spaces. In particular, the symbols we study here are defined without any reference to smoothness, but rather in terms of their time-frequency behavior.

One investigates the existence of an $\omega$-periodic solution of the problem $u_t=u_{xx}+cu+g(t,x)+ϵf(t,x,u,u_x,ϵ),u(t,0)=h_0\left(t\right)+ϵ{\chi }_0(t,u(t,0),u(t,\pi )),u(t,\pi )=h_1\left(t\right)+ϵ{\chi }_1(t,u(t,0),u(t,\pi ))$, provided the functions $g,f,h_0,h_1,{\chi }_0,{\chi }_1$ are sufficiently smooth and $\omega$-periodic in $t$. If $c\ne k^2$, $k$ natural, such a solution always exists for sufficiently small $ϵ>0$. On the other hand, if $c=l^2$, $l$ natural, some additional conditions have to be satisfied.

Let E be a Banach function space over a finite and atomless measure space (Ω,Σ,μ) and let $(X,|\left|·\right|{|}_X)$ and $(Y,|\left|·\right|{|}_Y)$ be real Banach spaces. A linear operator T acting from the Köthe-Bochner space E(X) to Y is said to be absolutely continuous if $\left|\right|T\left(1_{Aₙ}f\right){\left|\right|}_Y\to 0$ whenever μ(Aₙ) → 0, (Aₙ) ⊂ Σ. In this paper we examine absolutely continuous operators from E(X) to Y. Moreover, we establish relationships between different classes of linear operators from E(X) to Y.

Consider boundary value problems for a functional differential equation $$\left\{\begin{array}{cc}{x}^{\left(n\right)}\left(t\right)=\left(T^{+}x\right)\left(t\right)-\left(T^{-}x\right)\left(t\right)+f\left(t\right),\hfill & t\in [a,b],\hfill \\ lx=c,\hfill \end{array}\right.$$ where $T^{+},T^{-}:𝐂[a,b]\to 𝐋[a,b]$ are positive linear operators; $l:{\mathrm{𝐀𝐂}}^{n-1}[a,b]\to {ℝ}^n$ is a linear bounded vector-functional, $f\in 𝐋[a,b]$, $c\in {ℝ}^n$, $n\ge 2$. Let the solvability set be the set of all points $({𝒯}^{+},{𝒯}^{-})\in {ℝ}_2^{+}$ such that for all operators $T^{+}$, $T^{-}$ with $\parallel T^{\pm }{\parallel }_{𝐂\to 𝐋}={𝒯}^{\pm }$ the problems have a unique solution for every $f$ and $c$. A method of finding the solvability sets are proposed. Some new properties of these sets are obtained in various cases. We continue the investigations of the solvability sets started...