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

We study the solvability, dissipativity and stability for the equation d 2 d t 2 u ( t ) + b ∫ 0 1 ± E T α u ( t ) ϕ ( α ) d α + F ( u ( t ) ) = 0 , t ∈ [ 0 , T ] , T > 0 , where ∫ 0 1 ± E T α u ( t ) ϕ ( α ) d α is the distributed order symmetrized Caputo fractional derivative of u , ϕ ( α ) , α ∈ ( 0 , 1 ) , is a positive integrable function or a distribution of the form ∑ i = 0 n c α i δ ( α − α i ) , 0 ≤ α 0 < α 1 < ⋯ < α n ≤ 1 , c α i ≥ 0 , i = 0 , 1 , … , n , and F ( u ) , u ∈ R , is a locally Lipschitz continuous function on R .