Search

This paper deals with the approximate controllability for a class of fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space. We delete the compactness condition or Lipschitz condition for nonlocal term appearing in various literatures, and only need to suppose some weak growth condition on the nonlocal term. The discussion is based on the fixed point theorem, diagonal argument and approximation techniques. In the end, an example is presented to illustrate the abstract theory. [ABSTRACT FROM AUTHOR]

By using monotone iterative technique and operator semigroup theorem, we consider the existence of mild solutions for a class of nonlocal semi-linear evolution equation with not instantaneous impulses in ordered Banach spaces. Finally, an example is given to show the existence results. [ABSTRACT FROM AUTHOR]

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 D0q 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 C0-semigroup, f ∈ C(J×E×E;E), J = [0, a], a > 0 is a constant, T is a Volterra integral operator, 0 < t1 < t2 < ⋯ < tm < a, Ik ∈ C(E,E), k = 1, 2, ∈, m and x0 ∈ 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. [ABSTRACT FROM AUTHOR]

In this article, we apply the perturbation technique and monotone iterative method in the presence of the lower and the upper solutions to discuss the existence of the minimal and maximal mild solutions to the retarded evolution equations involving nonlocal and impulsive conditions in an ordered Banach space X U'(t) + Au(t) = f(t, u(t), ut), t ∈ [0; a], t ≠ tk, u(tk+) = u(tk-) + Ik(u(tk)), k = 1, 2, ..., m, u(s) = g(u)(s) + φ(s), s ≠ [-r, 0], where A : D(A) ⊂ X → X is a closed linear operator and -A generates a strongly continuous semigroup T(t) (t ≥ 0) on X, a, r > 0 are two constants, f : [0, a] x X x C0 → X is Carathéodory continuous, 0 < t1 < t2 < ... < tm < a are pre-fixed numbers, Ik ∈ C(X,X) for k = 1, 2,..., m, φ ∈ C0 is a priori given history, while the function g : Ca → C0 implicitly defines a complementary history, chosen by the system itself. Under suitable monotonicity conditions and noncompactness measure conditions, we obtain the existence of the minimal and maximal mild solutions, the existence of at least one mild solutions as well as the uniqueness of mild solution between the lower and the upper solutions. An example is given to illustrate the feasibility of our theoretical results. [ABSTRACT FROM AUTHOR]