Existence of solutions for a damped nonlinear impulsive problem with Dirichlet boundary conditions

In this paper, we consider the existence of solutions for second‐order nonlinear damped impulsive differential equations with Dirichlet boundary condition. By critical point theory, we obtain some existence theorems of solutions for the nonlinear problem. We extend and improve some recent results. Copyright © 2013 John Wiley & Sons, Ltd.     

一类含非线性边界条件的高阶微分方程解的存在性(英文)

将上下解方法和Leray-Shauder度应用到一类含有非线性边界条件的n阶微分方程,得到了至少存在一个解的结果,并且改进和推广了文献中的某些结果.     

Existence of solutions for nonlinear boundary value problems of impulsive differential equations

With the help of the coincidence degree continuation theorem, we generalize to the case of impulsive systems of the second order some existence results obtained by Gaines and Mawhin in the ordinary case. In particular, for the impulsive functions the treatment does not assume any monotonicity conditions, which are necessary in earlier papers treated by S.Hu and V.Lakshmikantham, L.H.Erbe and X.Liu with other methods.     

Existence of positive solutions for singular impulsive differential equations with integral boundary conditions

In this paper, we study the existence of positive solutions for singular impulsive differential equations with integral boundary conditions where the nonlinearity f(t,u,v) may be singular at u = 0 and v = 0. The proof is based on the theory of Leray–Schauder degree, together with a truncation technique. Some recent results in the literature are generalized and improved. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions

This paper describes the method of quasilinearization for first-order nonlinear impulsive functional differential equations with anti-periodic boundary conditions. A monotone iterative technique coupled with lower and upper solutions is employed to obtain sequences of approximate solutions converging monotonically and quadratically to the unique solution of the problem at hand.     

Existence of positive solutions for singular ordinary differential equations with nonlinear boundary conditions

We prove the existence of nonnegative solutions of the problem , , for a physically motivated class of nonlinearity . The results, which are established using a ``forbidden value' argument, are new even in the case of linear .

     

Unique existence results and numerical solutions for fourth-order impulsive differential equations with nonlinear boundary conditions

The work is concerned with three kinds of fourth-order impulsive differential equations with nonlinear boundary conditions. We at first focused on studying the existence and uniqueness of positive solutions for these kinds of problems. By converting the problem to an equivalent integral equation, then applying the new class of fixed point theorems for the sum operator on cone, we obtain the sufficient conditions which not only guarantee the existence of a unique positive solution, but also be applied to construct two iterative sequences for approximating it. Further, we present the numerical methods for solving the fourth-order differential equations. At last, some examples are given with numerical verifications to illustrate the main results.     

Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions

The existence, uniqueness and continuous dependence of a mild solution of an impulsive neutral functional differential evolution nonlocal Cauchy problem in general Banach spaces are studied, by using the fixed point technique and semigroup of operators.     

Multiplicity of solutions for a class of impulsive differential equations with Dirichlet boundary conditions via variant fountain theorems

In this paper, we study the existence of infinitely many classical solutions for a class of second-order impulsive differential equations. By using two new fountain theorems, we deal with two cases: that when the nonlinearity is superlinear and that when it is asymptotically linear at infinity. Some recent results are extended and improved.     

Existence result of solutions to differential equations of variable-order with nonlinear boundary value conditions

In this work, a differential equation of variable-order with nonlinear boundary value conditions is discussed. By some analysis techniques and Arzela–Ascoli theorem, existence result of solution is obtained.     

Existence and uniqueness of solutions for nonlinear impulsive partial differential equations with delay

In this paper, a system of nonlinear impulsive partial differential equations with delay is investigated by the method of upper–lower solutions. The existence-uniqueness result of this system and the comparison principle of the corresponding equations are obtained. An application is given to some model problems in ecology.     

Existence of solutions for fractional differential equations with multi-point boundary conditions

We discuss the existence of solutions for a nonlinear multi-point boundary value problem of integro-differential equations of fractional order q ∈ (1, 2]. Our analysis relies on the contraction mapping principle and the Krasnoselskii’s fixed point theorem. Example is provided to illustrate the theory.     

Positive solutions for second order impulsive differential equations with integral boundary conditions

This paper is concerned with a class of second order impulsive differential equations with integral boundary conditions. Under different combinations of superlineary and sublinearity of nonlinear term and the impulses, various existence, multiplicity, and nonexistence results for positive solutions are derived in terms of the parameter lies in some intervals. The results obtained herein generalize and improve some known results.     

Existence and boundary behavior for singular nonlinear differential equations with arbitrary boundary conditions

Existence theory is developed for the equation ?(u)=F(u), where ? is a formally self-adjoint singular second-order differential expression and F is nonlinear. The problem is treated in a Hilbert space and we do not require the operators induced by ? to have completely continuous resolvents. Nonlinear boundary conditions are allowed. Also, F is assumed to be weakly continuous and monotone at one point. Boundary behavior of functions associated with the domains of definitions of the operators associated with ? in the singular case is investigated. A special class of self-adjoint operators associated with ? is obtained.     

Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations

In this paper, we investigate existence and uniqueness of solutions for nonlinear impulsive fractional differential equations. By utilizing the well-known fixed point theorems, we obtain some sufficient conditions for the uniqueness and existence of the solutions. At the end of the paper, an example is given to illustrate our main results.     

Existence of mild solutions for semilinear differential equations with nonlocal and impulsive conditions

This paper is concerned with the existence of mild solutions for impulsive semilinear differential equations with nonlocal conditions. Using the technique of measures of noncompactness in Banach and Fréchet spaces of piecewise continuous functions, existence results are obtained both on bounded and unbounded intervals, when the impulsive functions and the nonlocal item are not compact in the space of piecewise continuous functions but they are continuous and Lipschitzian with respect to some measure of noncompactness, and the linear part generates only a strongly continuous evolution system.     

一类二阶微分议程边值问题三解的存在性

§ 1　 IntroductionWe are interested in the existence ofthree-solutions ofthe following second-order dif-ferential equations with nonlinear boundary value conditionsx″=f( t,x,x′) ,　　 t∈ [a,b] ,( 1 .1 )g1 ( x( a) ,x′( a) ) =0 ,　　 g2 ( x( b) ,x′( b) ) =0 ,( 1 .2 )where f:[a,b]×R1 ×R1 →R1 ,gi:R1 ×R1 →R1 ( i=1 ,2 ) are continuous functions.The study ofthe existence of three-solutions ofboundary value prolems forsecond or-der differential equations was initiated by Amann[1 ] .In[1 …     

Existence and compactness of solutions to impulsive differential equations with nonlocal conditions

The new type of nonlinear integral inequalities of Volterra–Fredholm type for discontinuous functions is investigated. Then, by using these inequalities and Schaefer fixed‐point theorem, we present new existence results for impulsive semilinear differential equations with nonlocal conditions. Moreover, the compactness of solution sets can be shown in some certain conditions. Copyright © 2015 John Wiley & Sons, Ltd.