Integro-differential Equations Of Mixed Type Research Articles

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.

Nonlinear boundary value problem for second order impulsive integro-differential equations of mixed type in Banach space

Using the cone theory and lower and upper solutions, we investigate the existence of extremal solutions of nonlinear boundary value problem for second order impulsive integro-differential equations, which involve the derivative x ′ and deviating argument x ( β ( t ) ) in Banach space.

Positive solutions for [formula omitted]th-order nonlinear impulsive singular integro-differential equations on infinite intervals in Banach spaces

In this paper, the cone theory and Mönch fixed point theorem combined with a monotone iterative technique are used to investigate the positive solutions of a class of boundary problems for n th-order nonlinear impulsive singular integro-differential equations of mixed type on an infinite interval in Banach spaces. The conditions for the existence of a positive solution are established. In addition, an explicit iterative approximation of the solution for the boundary value problem is derived.

New existence theorem for impulsive integro-differential equations of mixed type

In this paper, we investigate the nonlinear boundary value problem for first order impulsive integro-differential equations of mixed type. New existence theorem of extremal solutions is established by new comparison principles and the monotone iterative technique. The result improves and extends the known relevant conclusion in the literature. Example is also provided to illustrate the advantage of our result.

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.

Fixed point theorems for a class of nonlinear operators in Banach spaces and applications

In this paper, we study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using the cone theory and Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions. The results in this paper are applied to a class of abstract semilinear evolution equations with noncompact semigroup in Banach spaces and the initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. The results obtained here improve and generalize many known results.

Nonlinear Integrodifferential Equations of Mixed Type in Banach Spaces

We prove two existence theorems for the integrodifferential equation of mixed type:x'(t)=f(t,x(t),∫0tk1(t,s)g(s,x(s))ds,∫0ak2(t,s)h(s,x(s))ds),x(0)=x0, where in the first part of this paperf, g, h, xare functions with values in a Banach spaceEand integrals are taken in the sense of Henstock-Kurzweil (HK). In the second partf, g, h, xare weakly-weakly sequentially continuous functions and integrals are taken in the sense of Henstock-Kurzweil-Pettis (HKP) integral. Additionally, the functionsf, g, h, xsatisfy some conditions expressed in terms of the measure of noncompactness or the measure of weak noncompactness.

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.

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

In this paper, the global solutions of initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces are investigated. The existence and uniqueness of solutions and their iterative approximation are obtained by using Mönch fixed point theorem and new comparison results. The results presented here essentially improve, generalize and unify many well-known results.

Nonlinear boundary value problem for first order impulsive integro-differential equations of mixed type

This paper discusses nonlinear boundary value problem for first order impulsive integro-differential equations of mixed type. The lower and upper solutions and monotone iterative techniques are used to achieve the existence results. Two examples are provided to illustrate the efficiency of the obtained results.

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

In this paper, the minimal and maximal solutions of initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces are investigated by using new comparison results and the partial order method. The results presented here essentially improve, generalize and unify many well-known 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)].

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.

Extremal solutions of periodic boundary value problems for first order integro-differential equations of mixed type

This paper investigates the maximal and minimal solutions of periodic boundary value problems for first order nonlinear integro-differential equations of mixed type by establishing a comparison result and using the monotone iterative technique.

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

This paper investigates the existence of minimal and maximal solutions of periodic boundary value problem for first order impulsive integro-differential equations of mixed type by establishing a comparison result and using the method of upper and lower solutions and the monotone iterative technique.

Existence, Uniqueness and Data Dependence for the Solutions of some Integro-Differential Equations of Mixed Type in Banach Space

Existence, Uniqueness and Data Dependence for the Solutions of some Integro-Differential Equations of Mixed Type in Banach Space

Existence theorems of global solutions of initial value problems for nonlinear integrodifferential equations of mixed type in banach spaces and applications

This paper investigates the existence of global solutions for initial value problems of the nonlinear integrodifferential equations in Banach spaces. We obtain the existence of global solutions by Schauder's fixed-point theorem. Our results are the generalizations and improvements of recent results. As applications of our results, the existence of global solutions of two classes of third-order mixed boundary value problem is obtained, and an example of infinite system for nonlinear integrodifferential equations is worked out to illustrate our main results.

A unique solution of initial value problems for first order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, we investigate initial value problem for first order impulsive integro-differential equation of mixed type in a Banach space. Without any compactness-type assumption, we obtain a unique solution of the equation, an explicit iterative approximation of the solution and an error estimate of the approximation sequence. Our result improves and extends many recent results.