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A general class of semilinear fractional evolution equations of mixed type with nonlocal conditions on infinite dimensional Banach spaces is concerned. Under more general conditions, the existence of mild solutions and positive mild solutions is obtained by utilizing a new estimation technique of the measure of noncompactness and a new fixed point theorem with respect to convexpower condensing operator. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and uniqueness of mild solutions for a second order evolution equation initial value problem in a Banach space, which can model an elastic system with structural damping. The discussion is based on the operator semigroups theory and fixed point theorem. In addition, an example is presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

The existence of positive solutions for Neumann boundary value problem of second-order impulsive differential equations -u''(t) + Mu(t) = f(t, u(t), t ∊ J, t≠ tk, -?u'|t=tk = Ik(u(tk)), k = 1, 2, . . ., m, u'(0) = u'(1) = ?, in an ordered Banach space E was discussed by employing the fixed point index theory of condensing mapping, where M > 0 is a constant, J = [0, 1], f ∊ C(J×K,K), Ik ∊ C(K,K), k = 1, 2, . . ., m, andK is the cone of positive elements in E. Moreover, an application is given to illustrate the main result. [ABSTRACT FROM AUTHOR]

We study the periodic boundary value problem for semilinear fractional differential equations in an ordered Banach space. The method of upper and lower solutions is then extended. The results on the existence of minimal and maximal mild solutions are obtained by using the characteristics of positive operators semigroup and the monotone iterative scheme. The results are illustrated by means of a fractional parabolic partial differential equations. [ABSTRACT FROM AUTHOR]

By means of the fixed point theory of strict set contraction operators, we establish a new existence theorem on multiple positive solutions to a singular boundary value problem for second-order impulsive differential equations with periodic boundary conditions in a Banach space. Moreover, an application is given to illustrate the main result. [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]

In this article, we study the existence of piecewise-continuous mild solutions for the initial value problems for a class of semilinear evolution equations. These equations have non-instantaneous impulses in Banach spaces and the corresponding solution semigroup is noncompact. We assume that the nonlinear term satisfies certain local growth condition and a noncompactness measure condition. Also we assume the non-instantaneous impulsive functions satisfy some Lipschitz conditions. An example is given to illustrate our results. [ABSTRACT FROM AUTHOR]