具有脉冲条件混合型一阶脉冲积分微分方程解的存在性

在现实世界里脉冲现象是普遍存在的．例如最优控制、化学技术、医疗等方面都会有脉冲现象。事实上,有脉冲的变化过程意味着在某一时刻的状态将发生急速的变化,脉冲微分方程是一种来描述有脉冲影响的系统的数学方法。运用上,下解法和单调迭代技术论证具有脉冲条件混合型一阶脉冲积分微分方程解的存在性。     

Integral boundary value problems for first order integro-differential equations with impulsive integral conditions

This paper is concerned with the integral boundary value problems for first order integro-differential equations with impulsive integral conditions. Sufficient conditions are obtained for the existence of extremal solutions.     

Monotone and numerical-analytic methods for differential equations

The method of lower and upper solutions combined with monotone iterative techniques is used for ordinary differential equations with integral boundary conditions showing the existence of extremal solutions. Some existence results are also formulated using the numerical-analytic method. Sufficient conditions for existence are given.     

Optimal existence conditions for second order periodic solutions of delay differential equations with upper and lower solutions in the reverse order

In this paper we show that the monotone iterative technique provides two monotone sequences that converge uniformly to extremal (periodic) solutions of second order delay differential equations without assuming properties of monotonicity in the nonlinear part. Moreover, we obtain optimal existence conditions with upper and lower solutions in the reverse order. Our results are new even for ordinary differential equations.     

A new existence theory for single and multiple positive periodic solutions to volterra integro-differential equations with impulse effects

This paper presents a new existence theory for single and multiple positive periodic solutions to a kind of nonautonomous Volterra integro-differential equations with impulse effects. Existence is established by using Krasnoselskii fixed-point theorem in cones. This paper extends some previous results and reports some new results about impulsive differential equation     

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

This paper discusses anti-periodic boundary value problems of second order impulsive differential equations. By using the method of upper and lower solutions coupled with the monotone iterative technique, new existence results of coupled solutions and uniqueness of problems are obtained.     

Existence of optimal controls for SPDE with locally monotone coefficients

The aim of this paper is to investigate the existence of optimal controls for systems described by stochastic partial differential equations (SPDEs) with locally monotone coefficients controlled by external forces which are feedback controls. To attain our objective we adapt the argument of Lisei (2002) where the existence of optimal controls to the stochastic Navier–Stokes equation was studied. The results obtained in the present paper may be applied to demonstrate the existence of optimal controls to various types of controlled SPDEs such as: a stochastic nonlocal equation and stochastic semilinear equations which are locally monotone equations; we also apply the result to a monotone equation such as the stochastic reaction–diffusion equation and to a stochastic linear equation.     

Solutions to the Cauchy problem for differential equations in Banach spaces with fractional order

In this paper, we use the monotone iterative technique combined with cone theory to investigate the existence of solutions to the Cauchy problem for Caputo fractional differential equations in Banach spaces. New existence theorems are obtained for the case of a cone P being normal and fully regular respectively. Moreover, two examples are given to illustrate the abstract results.     

Nonnegative solutions of fractional functional differential equations

In this paper, we discuss the existence and monotone iterative method of nonnegative solutions for fractional functional differential equations. The main conclusion is that the nonnegative solutions can be derived from the monotone iterative method, which starts off with a nonnegative upper solution or the zero function under different conditions. Our approach of obtaining nonnegative solutions is feasible for computational purposes.     

On stability in nonlinear approximation of systems of impulsive differential equations

Theorems on existence of the Lyapunov functions for homogeneous systems of impulsive differential equations were proved. The results obtained were applied to studying stability in nonlinear approximation of the critical equilibria of the nonlinear systems of impulsive differential equations. An example of a second-order system was given.     

Hybrid fixed point theory for strictly monotone increasing multi-valued mappings with applications

In this paper, a general hybrid fixed point theorem for the strict monotone increasing multi-valued mappings in ordered Banach spaces is proved via measure of noncompactness and it is further applied to perturbed functional nonconvex differential inclusions for proving the existence results for the extremal solutions under mixed Lipschitz, compactness and strict monotonic conditions.     

Variational methods to the second-order impulsive differential equation with Dirichlet boundary value problem

In this paper, we study the existence of solutions for a class of second-order impulsive differential equation. By using the critical point theorem of Y. Jabri and an even functional theorem, we give some new criteria to guarantee that the impulsive differential equation has at least one solution, infinitely many solutions under the assumption that a nonlinear term satisfies sublinear, superlinear, asymptotically linear, respectively. Some recent results are extended and conditions of assumptions are simplified. Finally, some examples are presented to illustrate our main results.     

Existence of Optimal Mild Solutions and Controllability of Fractional Impulsive Stochastic Partial Integro‐Differential Equations with Infinite Delay

In this paper, we investigate a new class of fractional impulsive stochastic partial integro‐differential equations with infinite delay in Hilbert spaces. By using the stochastic analysis theory, fractional calculus, analytic α‐resolvent operator and the fixed point technique combined with fractional powers of closed operators, we firstly give the existence of of mild solutions and optimal mild solutions for the these equations. Next, the controllability of the controlled fractional impulsive stochastic partial integro‐differential systems with not instantaneous impulses is presented. Finally, examples are also given to illustrate our results.     

Existence results for impulsive neutral stochastic functional integro-differential inclusions with nonlocal initial conditions

In this paper, we prove the existence of mild solutions for a class of impulsive neutral stochastic functional integro-differential inclusions with nonlocal initial conditions and resolvent operators. Sufficient conditions for the existence are derived with the help of the fixed point theorem for multi-valued operators due to Dhage and the fractional power of operators. An example is provided to illustrate the theory.     

Nonhomogeneous boundary value problem for first-order impulsive differential equations with delay

The authors employ the method of upper and lower solution coupled with the monotone iterative technique to obtain results of existence and uniqueness for a nonhomogeneous boundary value problem of impulsive differential equations with delay.     

Control problems for semi-linear retarded integro-differential equations by the Fredholm theory

ABSTRACT

In this paper, we deal with the approximate controllability for semi-linear retarded functional integro-differential equations by using the Fredholm theory in Hilbert spaces. We no longer require the compactness of structural operators to obtain the approximate controllability for the nonlinear differential system, but instead we use the theory of interpolation spaces and the regularity of solutions of semi-linear given equations with unbounded principal operators. Finally, based on the properties of general degree theory in infinite dimensional spaces, we investigate the relation between the reachable set of trajectories of the semi-linear retarded functional integro-differential system and that of its corresponding linear system excluded by the nonlinear term.     

Invariant imbedding and inverse problems with applications to radiative transfer

In this paper, we developed two methods to solve the inverse problem of a nonlinear integro-differential equation. Both methods are based on the principle of invariant imbedding. The first method involves two auxiliary integro-differential equations. The inverse problem is solved by a sequence of approximation solutions of linear equations. The second method involves algebraic equations of scattering matrices under the so-called star-product. An application to a radiative transfer problem such as correction of atmospheric effects on remote sensing is discussed.     

Solution of impulsive boundary value problems on infinite intervals

In this paper, the upper and lower solution method and monotone iterative technique are employed to study the boundary value problem for a class of first-order impulsive ordinary differential equations with Carathéodory condition on infinite intervals. Several existence theorems are obtained.     

A note on the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations with non-Lipschitz coefficients

In this note, we study the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations under some Carathéodory-type conditions on the coefficients by means of the successive approximation. In particular, we generalize and improve the results that appeared in Govindan [T.E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77 (2005) 139–154] and Bao and Hou [J. Bao, Z. Hou, Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients, Comput. Math. Appl. 59 (2010) 207–214].     

Antiperiodic boundary value problem for first-order impulsive ordinary differential equations

This paper discusses the antiperiodic boundary value problem for first-order impulsive ordinary differential equations. We establish several existence results by using the Leray-Schauder alternative, the lower and upper solution method and the monotone iterative technique.