Positive pseudo almost periodic solutions for some nonlinear infinite delay integral equations

Integration in some new concept of ergodic functions and application to some epidemiological models

“…Ding et al 8 studied the problem of existence of positive almost automorphic solutions for the neutral nonlinear delay integral equation $$$$ x(t)=\gamma x\left(t-\sigma \right)+\left(1-\gamma \right){\int}_{t-\sigma}^tf\left(s,x(s)\right) ds\kern0.5em \mathrm{for}\kern0.5em t\in \mathbb{R}, $$$$ where $$$ 0\le \delta <1,f\left(t,x\right)=\sum \limits_n^{i=1}{f}_i\left(t,x\right){g}_i\left(t,x\right),{f}_i\Big(t $$$,· $$$ \Big) $$$ is nondecreasing in $$$ {\mathbb{R}}^{+} $$$ and $$$ {g}_i\Big(t $$$,· $$$ \Big) $$$ is nonincreasing in $$$ {\mathbb{R}}^{+} $$$. Equation () is also studied in the almost periodic case by Ait Dads and Ezzinbi, 5 and by Ait Dads et al, 32 where $$$ f\Big(t $$$,· $$$ \Big) $$$ is nondecreasing in …”

“…To obtain our results, we use the contraction mapping principle associated with Hilbert's projective metric. Note that in the case when $$$ \alpha =0 $$$ and $$$ g=0 $$$, Equation () was considered using Hilbert's projective metric by Ait Dads et al 5 in the pseudo almost periodic case and by Cieutat and Ezzinbi 7 in the pseudo almost automorphic case.…”

“…Pseudo‐almost periodicity was introduced in the literature in the early nineties by Zhang, 1,2 as a natural generalization of the classical almost periodicity in the sense of Bohr. Since then, the existence of pseudo‐almost periodic solutions to differential equations, partial differential equations and functional differential equations has been of a great interest to several authors and hence generated various contributions: Ait Dads et al 3–5 obtained sufficient conditions for the existence of pseudo almost periodic solutions of some delay differential equations and other contributions upon pseudo‐almost periodic solutions to various differential equations have recently been made in previous works 6–10 . Diagana 11 introduced the concept of weighted pseudo almost periodic, which generalizes the one of the pseudo almost periodicity.…”