Dissipativity and stability for a nonlinear differential equation with distributed order symmetrized fractional derivative

We study the solvability, dissipativity and stability for the equation d2dt2u(t)+b∫01±ETαu(t)ϕ(α)dα+F(u(t))=0,t∈[0,T],T>0, where ∫01±ETαu(t)ϕ(α)dα is the distributed order symmetrized Caputo fractional derivative of uu, ϕ(α),α∈(0,1)ϕ(α),α∈(0,1), is a positive integrable function or a distribution of the form ∑i=0ncαiδ(α−αi), 0≤α0<α1<⋯<αn≤10≤α0<α1<⋯<αn≤1, cαi≥0cαi≥0, i=0,1,…,ni=0,1,…,n, and F(u)F(u), u∈Ru∈R, is a locally Lipschitz continuous function on RR.