### A fixed point theorem of Leggett-Williams type with applications to single- and multivalued equations.

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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.

Existence results of nonnegative solutions of asymptotically linear, nonlinear integral equations are studied.

We study positive linear Volterra integro-differential equations in Banach lattices. A characterization of positive equations is given. Furthermore, an explicit spectral criterion for uniformly asymptotic stability of positive equations is presented. Finally, we deal with problems of robust stability of positive systems under structured perturbations. Some explicit stability bounds with respect to these perturbations are given.

Sufficient conditions which guarantee that certain linear integro-differential equation cannot have a positive solution are established.

An existence theorem is proved for the scalar convolution type integral equation $x\left(t\right)={\int}_{-\infty}^{\infty}h(t-s)f(s,x\left(s\right))ds$.