Periodic boundary value problems for second-order impulsive integro-differential equations

By developing a new comparison result and using the monotone iterative technique, we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for second-order nonlinear impulsive integro-differential equations of mixed type.     

Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces

This paper investigates the multiplicity of positive solutions for a class of nonlinear boundary-value problems of fourth-order differential equations with integral boundary conditions in ordered Banach spaces. The arguments are based upon a specially constructed cone and the fixed point index theory in cone for strict set contraction operator. The nonexistence of positive solution is also studied. Meanwhile, an example is worked out to demonstrate the main results.     

The existence of solutions of infinite boundary value problems for first-order impulsive differential systems in Banach spaces

In this paper, the existence of solutions of infinite boundary value problems for first-order impulsive differential systems is obtained by means of the Schauder fixed point theorem in a Banach space.     

Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations

This paper is devoted to study the existence of multiple positive solutions for the second-order multi-point boundary value problem with impulse effects. The arguments are based upon fixed-point theorems in a cone. An example is worked out to demonstrate the main results.     

Boundary value problems for first order functional differential equations with impulsive integral conditions

In this paper, by using the method of lower and upper solutions coupled with the monotone iterative technique, we give conditions for existence of extreme solutions for first order functional differential equations with a new impulsive integral condition. Some comparison results are also formulated.     

Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces

This paper investigates the existence and multiplicity of positive solutions for a class of nonlinear boundary-value problem of second-order differential equations with integral boundary conditions in ordered Banach spaces. The arguments are based upon a specially constructed cone and the fixed point theory in a cone for strict set contraction operators. The nonexistence of a positive solution is also studied.     

Second-order boundary value problems with integral boundary conditions

In this paper we investigate the existence of positive solutions of nonlocal second-order boundary value problems with integral boundary conditions.     

Existence and nonexistence of positive solutions for a class of nth-order three-point boundary value problems in Banach spaces

By using the fixed-point principle in cone and the fixed-point index theory for strict-set-contraction operator, this paper investigates the existence, nonexistence, and multiplicity of positive solutions for a class of nonlinear three-point boundary value problems of nth-order differential equations in ordered Banach spaces. In addition, an example is included to demonstrate the main results.     

Existence and nonexistence of solutions of second-order nonlinear boundary value problems

We study the nonlinear boundary value problem consisting of the equation −y^″+q(t)y=w(t)f(y) on [a,b]$-y^{″}+q\left(t\right)y=w\left(t\right)f\left(y\right)\text{ on }[a,b]$ and a general separated homogeneous linear boundary condition. By comparing this problem with a corresponding linear Sturm–Liouville problem we obtain conditions for the existence and nonexistence of solutions of this problem. More specifically, let _λn,n=0,1,2,…${\lambda }_n,n=0,1,2,\dots$, be the n$n$-th eigenvalues of the corresponding linear Sturm–Liouville problem. Then under certain assumptions, the boundary value problem has a solution with exactly n$n$ zeros in (a,b)$(a,b)$ if _λn${\lambda }_n$ is in the interior of the range of f(y)/y,y∈(0,∞)$f\left(y\right)/y,y\in (0,\infty )$; and does not have any solution with exactly n$n$ zeros in (a,b)$(a,b)$ if _λn${\lambda }_n$ is outside of the range of f(y)/y,y∈(0,∞)$f\left(y\right)/y,y\in (0,\infty )$. These conditions become necessary and sufficient when f(y)/y$f\left(y\right)/y$ is monotone. The existences of multiple and even an infinite number of solutions are derived as consequences. We also discuss the changes of the number and the types of nontrivial solutions as the interval [a,b]$[a,b]$ shrinks, as q$q$ increases in a given direction, and as the boundary condition changes.     

Existence of solutions for the integral boundary value problems of fractional order impulsive differential equations

In this paper, we study a class of integral boundary value problem for fractional order impulsive differential equations, where both the nonlinearity and the impulsive terms contain the fractional order derivatives. By using fixed‐point theorems, the existence results of solution for the boundary value problem are established. Finally, some examples are presented to illustrate the existence results. Copyright © 2015 John Wiley & Sons, Ltd.     

Periodic boundary value problems of second-order impulsive differential equations

This paper discusses periodic boundary value problems of second-order impulsive differential equations. By using Schaeffer’s fixed-point theorem and monotone iterative technique, some existence results are obtained.     

Positive solutions for 2p-order and 2q-order systems of singular boundary value problems with integral boundary conditions

In this paper, we study the existence, multiplicity and nonexistence of positive solutions for 2p-order and 2q-order systems of singular boundary value problems with integral boundary conditions. The results are based upon the fixed-point theorem of cone expansion and compression type due to Krasnosel’skill. Moreover, it generalizes and includes some known results.     

Existence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order

This paper studies the existence and uniqueness of solutions of second-order three-point boundary value problems with lower and upper solutions in the reversed order, obtains the sufficient conditions for the existence and uniqueness of solutions by use of the monotone iterative method, and gives the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution.     

Global solutions of nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, by using the generalization of Darbo’s fixed point theorem, we establish the existence of global solutions of an initial value problem for a class of second-order impulsive integro-differential equations of mixed type in a real Banach space. Our results generalize and improve on the results of Guo et al. [F. Guo, L.S. Liu, Y.H. Wu, P. Siew, Global solutions of initial value problems for nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces, Nonlinear Anal. 61 (2005) 1363–1382] in the sense that the conditions for existence of global solution in our theorem is simpler and less strict. To demonstrate the application of the theorem, we give the global solutions of two mixed boundary value problems for two classes of fourth order impulsive integro-differential equations.     

Second order singular boundary value problems with integral boundary conditions

We study the second order singular boundary value problem