Initial value problems for nonlinear impulsive integro-differential equations in Banach space

In this paper, the fixed point theorem is applied to investigate the existence of solutions of Sturm-Liouville boundary value problems for nonlinear second order impulsive differential equations in Banach spaces.

In this article, we are concerned with the existence of extremal solutions to the initial value problem of impulsive differential equations in ordered Banach spaces. The existence and uniqueness theorem for the solution of the associated linear impulsive differential equation is established. With the aid of this theorem, the existence of minimal and maximal solutions for the initial value problem of nonlinear impulsive differential equations is obtained under the situation that the nonlinear term and impulsive functions are not monotone increasing by using perturbation methods and monotone iterative technique. The results obtained in this paper improve and extend some related results in abstract differential equations. An example is also given to illustrate the feasibility of our abstract results.

The initial value problems of first order impulsive differential equations in Banach spaces

Existence and uniqueness of solutions for initial value problems of second order ordinary differential equations in Banach spaces

In this paper, we prove the existence of solutions of initial value problems for nth order nonlinear impulsive integro-differential equations of mixed type on an infinite interval with an infinite number of impulsive times in Banach spaces. Our results are obtained by introducing a suitable measure of noncompactness.

Study of singular boundary value problems for second order impulsive differential equations

Multiple positive solutions of a boundary value problem for nth-order impulsive integro-differential equations in Banach spaces

Anti-periodic boundary value problems of second order impulsive differential equations

Existence theorems of solutions for two-point boundary value problem of second order ordinary differential equations in Banach spaces

The analytic and numerical stability of stiff impulsive differential equations in Banach space

This paper defines a concept of a semi-k-set-contraction operator, and establishes a degree theory for it. As its application, we discuss the existence for the solution of two-point boundary value problems for nonlinear second order integro-differential equations in Banach spaces.

In this paper, we discuss the existence of weak solutions for a nonlinear boundary value problem of fractional q-difference equations in Banach space. Our analysis relies on the Mönch's fixed-point theorem combined with the technique of measures of weak noncompactness. MSC:26A33, 34B15.

In this work, we study Bishop–Phelps cones (briefly, BP cones) given by an equation in Banach spaces. Due to the special form, these cones enjoy interesting properties. We show that nontrivial BP cones have given by an equation form a ‘large family’ in some sense in any Banach space and they can be used to characterize the strict convexity of the space. We obtain conditions for a convex cone to be included in or to be itself a BP cone given by an equation. Furthermore, we give an affirmative answer to the open question of whether these cones may possess a nonempty interior in infinite-dimensional spaces and introduce Lorentz cones in some classical Banach spaces of sequences as illustrations. We also obtain explicit representations for all BP cones given by an equation in some classical Banach spaces. Finally, we present some short applications in optimal control and approximation.

The solutions of initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces

Existence and regularity of solutions for some partial functional integrodifferential equations in Banach spaces