Nonlinear Impulsive Integro-differential Equations Research Articles

Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces

This paper investigates the existence of solutions for a class of second-order boundary-value problems with integral boundary conditions of nonlinear impulsive integro-differential equations in Banach spaces. The arguments are based upon the fixed point theorem of strict set contraction operators. Meanwhile, an example is worked out to demonstrate the main results.

A class of nonlinear integrodifferential impulsive periodic systems of mixed type and optimal controls on Banach space

A class of nonlinear integrodifferential impulsive periodic systems of mixed type on Banach space is considered. Existence of periodic PC-mild solutions is proved. Existence of periodic optimal pairs of systems governed by nonlinear impulsive integrodifferential equations of mixed type is also presented. An example is given for demonstration.

Existence of solutions for anti-periodic boundary value problems of nonlinear impulsive functional integro-differential equations of mixed type

We discuss the existence of minimal and maximal solutions for a class of first order nonlinear impulsive functional integro-differential equations of mixed type with anti-periodic boundary conditions. The main tool of study is the monotone iterative technique.

On the existence of solutions for nonlinear first-order implicit impulsive integro-differential equations

By using the Mönch fixed-point theorem, the purpose of this paper is to obtain the new existence theorems of solutions for a new class of initial value problems of nonlinear first-order implicit impulsive integro-differential equations in Banach spaces under some weaker conditions.

Second-order nonlinear impulsive integro-differential equations of mixed type with time-varying generating operators and optimal controls on Banach spaces

In this paper, a class of second-order nonlinear impulsive integro-differential equations of mixed type whose principle is time-varying generating operators with unbounded perturbation on Banach spaces is considered. Discussing the perturbation of time-varying operator matrix and constructing corresponding the evolution system generated by operator matrix, we introduce the reasonable mild solution of second-order nonlinear impulsive integro-differential equations of mixed type and prove the existence of mild solutions. The existence of optimal controls for a Lagrange problem of systems governed by the second-order nonlinear impulsive integro-equations of mixed type is also presented. An example is given for demonstration.

A unique positive solution for nth-order nonlinear impulsive singular integro-differential equations on unbounded domains in Banach spaces

In this paper, the fixed point theory combined with a monotone iterative technique is used to investigate the unique positive solution of a boundary problem for nth-order nonlinear impulsive singular integro-differential equations of mixed type on an infinite interval in a Banach space. The conditions for the existence of a unique positive solution are established. In addition, an explicit iterative sequence for approximating the solution of the boundary value problem is derived together with an error estimate. Furthermore, the conditions of the theorems can be easily verified.

New results for the periodic boundary value problem for impulsive integro-differential equations

This paper discusses the periodic boundary value problem for first-order nonlinear impulsive integro-differential equations. We prove some new comparison principles, and then establish new existence results for extremal solutions by using these principles and the monotone iterative technique. These results improve and extend all known relevant conclusions from the literature. Some examples are also given to illustrate the advantage of our results.

Existence of solutions of nth order impulsive integro-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.

Monotone method for a system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces

In this paper, by using a monotone iterative technique in the presence of lower and upper solutions, we discuss the existence of solutions for a new system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces. Under wide monotonicity conditions and the noncompactness measure conditions, we also obtain the existence of extremal solutions and a unique solution between lower and upper solutions.

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.

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.

Periodic boundary value problems for impulsive integro-differential equations of mixed type

By developing a 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 first order nonlinear impulsive integro-differential equations of mixed type.

New comparison results for impulsive integro-differential equations and applications

We prove some new maximum principles for ordinary integro-differential equations. This allows us to introduce a new definition of lower and upper solutions which leads to the development of the monotone iterative technique for a periodic boundary value problem related to a nonlinear first-order impulsive integro-differential equation.

On well-posedness of an initial value problem for nonlinear second-order impulsive integro-differential equations of Volterra type in Banach spaces

In this paper, we develop an existence and uniqueness theorem of the solution to an initial value problems for a class of second-order impulsive integro-differential equations of Volterra type in a real Banach space by using the generalized Banach fixed point theorem. An explicit iterative scheme for the solution and an error estimate of the approximation sequence for the initial value problem are also derived. Two examples are presented to demonstrate the application of our results.

Initial value problems for nonlinear second order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, through solving equations step by step, without any assumption of compactness-type conditions, we obtain unique solution of initial value problems of nonlinear second order impulsive integral-differential in Banach spaces. The results obtained generalize and improve the corresponding results of Guo and Chai in papers [D.J. Guo, Initial value problems for nonlinear second-order impulsive integro-differential equations in Banach spaces, J. Math. Anal. Appl. 200 (1996) 1–13; G.Q. Chai, Initial value problems for nonlinear second order impulsive integro-differential equations in Banach space, Acta Math. Sinica 20 (3) (2000) 351–359 (in Chinese)].

Extremal solutions for [formula omitted]th-order impulsive integro-differential equations on the half-line in Banach spaces

By establishing a comparison result and using the upper and lower solutions, the author obtains the existence of maximal and minimal solutions for a boundary value problem of n th-order nonlinear impulsive integro-differential equations on the half-line in Banach spaces.

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

In this paper, by using the cone theory and the monotone iterative technique, we investigate the existence of maximal and minimal solutions of the initial value problem for second order impulsive differential equations which nonlinearly involve the derivative x ′ in Banach space and give an example to illustrate our results.

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

In this paper, the author obtains the existence of multiple positive solutions for a boundary value problem of a class of nth-order nonlinear impulsive integro-differential equations on an infinite interval in a Banach space by means of the fixed point index theory of completely continuous operators.

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

In this paper, we develop a theorem for the existence of global solutions of an initial value problems for a class of second-order impulsive integro-differential equations of mixed type in a real Banach space. The results obtained are the generalizations and improvements of various recent results. As an application of the theorem, the existence of global solutions of two mixed boundary value problems for two classes of fourth-order impulsive differential equations are established.

Multiple positive solutions for [formula omitted]th-order impulsive integro-differential equations in Banach spaces

In this paper, the author obtains the existence of multiple positive solutions for a boundary value problem of n th-order nonlinear impulsive integro-differential equations on an infinite interval in a Banach space by means of fixed point index theory of completely continuous operators.