Monotone iterative technique for fractional partial differential equations with impulses

摘要

In this article, we use a monotone iterative technique based on the presence of lower and upper solutions to discuss the existence of mild solutions for the initial value problem of the impulsive time fractional order partial differential equation of Volterra type in an ordered Banach space E where D_0~q is the Caputo fractional derivative of order q, 0 < q < 1, A : D(A) ? E → E is a closed linear operator and -A is a generator of equicontinuous C_0 -semigroup, f ∈ C(J×E×E, E), J = [0, a], a > 0 is a constant, T is a Volterra integral operator, 0 < t_1 < t_2 < · · · < t_m < a, I_k ∈ C(E, E), k = 1, 2, · · · , m and x_0 ∈ E. Under wide monotone conditions and the noncompactness measure condition of nonlinearity f, we obtain the existence of extremal mild solutions and unique mild solution between lower and upper solutions. The results obtained generalize the recent conclusions on this topic. An example is also given to illustrate that our results are valuable.