... dt the Caputo derivative of order α ∈ (0, 1), A : ℓ2 → ℓ2 is defined as follows (Av)i = vi+ 1 − (2 + λ )vi + vi 1 , v ∈ ℓ2 , ρ : R+ → [0, h] is a continuous function, λ > We give the following assumptions ... following problem (I) m ∑ ∂u (t, x) − ∆x u(t, x) + λu(t, x) = f (x, u(t, x)) + bi (x )vi (t), x ∈ Ω, t > 0, ∂t i=1 [∫ ] ∫ vi (t) ∈ k1,i (y)u(t − h, y)dy, k2,i (y)u(t − h, y)dy , ≤ i ≤ m, O O u(t, x) = ... domain in Rn m ∑ ∂u (t, x) − ∆x u(t, x) + λu(t, x) = f (x, u(t, x)) + bi (x )vi (t), x ∈ Rn , t > 0, ∂t i=1 ] [∫ ∫ vi (t) ∈ k1,i (y)u(t − h, y)dy, k2,i (y)u(t − h, y)dy , ≤ i ≤ m, O O u(s, x)...