The paper deals with the existence of three- solutions for the second- order differential equations with nonlinear boundary value conditions x″=f(t,x,x′) , t∈ [a,b], g1(x(a) ,x′(a) ) =0 , g2 (x(b) ,x′(...The paper deals with the existence of three- solutions for the second- order differential equations with nonlinear boundary value conditions x″=f(t,x,x′) , t∈ [a,b], g1(x(a) ,x′(a) ) =0 , g2 (x(b) ,x′(b) ) =0 , where f :[a,b]× R1× R1→ R1,gi:R1× R1→ R1(i=1 ,2 ) are continuous functions.The methods employed are the coincidence degree theory.As an application,the sufficient conditions under which there are arbitrary odd solutions for the BVP are obtained展开更多
This paper deals with the existence of positive solutions for the singular fourth order boundary value problem.A necessary and sufficient condition for the existence of C3 positive solution is given by means of the mo...This paper deals with the existence of positive solutions for the singular fourth order boundary value problem.A necessary and sufficient condition for the existence of C3 positive solution is given by means of the monotone iterative technique.Furthermore,the uniqueness of the C3 positive solution,and the iterative sequence of the C3 positive solution are also obtained.展开更多
By using cone theory and the MSnch fixed theorem combined with a monotone iterative technique, we investigate the existence of positive solutions for systems of second- order nonlinear singular differential equations ...By using cone theory and the MSnch fixed theorem combined with a monotone iterative technique, we investigate the existence of positive solutions for systems of second- order nonlinear singular differential equations with integral boundary conditions on infinite interval and establish the existence theorem of positive solutions and iterative sequence for approximating the positive solutions. The results in this paper improve some known results.展开更多
Making use of upper and lower solutions and analytical method, the author studies theexistence of positive solution for the singular equation x + f(t, z) = 0 satisfying nonlinear boundary conditions: x (0) = 0, h(x (1...Making use of upper and lower solutions and analytical method, the author studies theexistence of positive solution for the singular equation x + f(t, z) = 0 satisfying nonlinear boundary conditions: x (0) = 0, h(x (1), x’ (1)) = 0, g (z (0), x’(0)) = 0, and x (1) = 0,which extends the result of J. V. Baxley.展开更多
Let Ω be a bounded or unbounded domain in R~n. The initial-boundary value problem for the porous medium and plasma equation with singular terms is considered in this paper. Criteria for the appearance of quenching ph...Let Ω be a bounded or unbounded domain in R~n. The initial-boundary value problem for the porous medium and plasma equation with singular terms is considered in this paper. Criteria for the appearance of quenching phenomenon and the existence of global classical solution to the above problem are established. Also, the life span of the quenching solution is estimated or evaluated for some domains.展开更多
In the paper an inverse boundary value problem for a fourth order elliptic equation with an integral condition of the first kind is investigated. First, the given problem is reduced to an equivalent problem in a certa...In the paper an inverse boundary value problem for a fourth order elliptic equation with an integral condition of the first kind is investigated. First, the given problem is reduced to an equivalent problem in a certain sense. Then, using the Fourier method the equivalent problem is reduced to solving the system of integral equations. The existence and uniqueness of a solution to the system of integral equation is proved by the contraction mapping principle. This solution is also the unique solution to the equivalent problem. Finally, by equivalence, the theorem of existence and uniqueness of a classical solution to the given problem is proved.展开更多
In recent years, a vast amount of work has been done on initial value problems for important nonlinear evolution equations like the nonlinear Schrödinger equation (NLS) and the Korteweg-de Vries equation (KdV...In recent years, a vast amount of work has been done on initial value problems for important nonlinear evolution equations like the nonlinear Schrödinger equation (NLS) and the Korteweg-de Vries equation (KdV). No comparable attention has been given to mixed initial-boundary value problems for these equations, i.e. forced nonlinear systems. But in many cases of physical interest, the mathematical model leads precisely to the forced problems. For example, the launching of solitary waves in a shallow water channel, the excitation of ion-acoustic solitons in a double plasma machine, etc. In this article, we present the PDE (Partial Differential Equation) method to study the following iut = uxx - g|u|pu, g ∈ R, p > 3, x?∈ Ω = [0,L], 0 ≤?t?u (x,0) = u0 (x) ∈?H2 (Ω) and Robin inhomogeneous boundary condition ux (0,t) + αu (0,t) = R1(t), t ≥ 0 and ux (L,t) + αu (L,t) = R2 (t), t ≥ 0 (here?α?is a real number). The equation is posed in a semi-infinite strip on a finite domain Ω. Such problems are called forced problems and have many applications in other fields like physics and chemistry. The main tool of PDE method is semi-group theory. We are able to prove local existence and uniqueness theorem for the nonlinear Schrödinger equation under initial condition and Robin inhomogeneous boundary condition.展开更多
In this paper nonlinear boundary value problems for discontinuous delayed differen- tial equations are considered. Some existence and boundedness results of solutions are obtained via the method of upper and lower sol...In this paper nonlinear boundary value problems for discontinuous delayed differen- tial equations are considered. Some existence and boundedness results of solutions are obtained via the method of upper and lower solutions, which may be discontinuous. Our analysis can be applied to those phenomena from physics and control theory which have been successfully described by delayed differential equations with discontinuous functions, such as electric, pneu- matic, and hydraulic networks. It is also an important step to precede the design of control signals when finite-time or practical stability control are concerned.展开更多
In this paper, two kinds of initial boundary value problems for Kuramoto_Sivashinsky equation are considered. Some prior estimates are derived by Galerkin methods. The existence, uniqueness and regularities of the gen...In this paper, two kinds of initial boundary value problems for Kuramoto_Sivashinsky equation are considered. Some prior estimates are derived by Galerkin methods. The existence, uniqueness and regularities of the generalized global solutions and the classical global solutions for the equation are proved. Morever, the asymptotic behavior of these solutions are considered under some conditions.展开更多
This paper is concerned with the following n-th ordinary differential equation:{u~(n)(t)=f(t,u(t),u~(1)(t),···,u~(n-1) (t)),for t∈(0,1),u~(i) (0)=0,0 ≤i≤n3,au~(n-2)(0)du~(n-1)(0)=0,cu~(n-2)(1)...This paper is concerned with the following n-th ordinary differential equation:{u~(n)(t)=f(t,u(t),u~(1)(t),···,u~(n-1) (t)),for t∈(0,1),u~(i) (0)=0,0 ≤i≤n3,au~(n-2)(0)du~(n-1)(0)=0,cu~(n-2)(1)+du~(n-1)(1)=0,where a,c ∈ R,,≥,such that a~2 + b~2 >0 and c~2+d~2>0,n ≥ 2,f:[0,1] × R → R is a continuous function.Assume that f satisfies one-sided Nagumo condition,the existence theorems of solutions of the boundary value problem for the n-th-order nonlinear differential equations above are established by using Leray-Schauder degree theory,lower and upper solutions,a priori estimate technique.展开更多
In this paper we are concerned with the following nonlinear degenerate parabolic systems u_t=△x(gradψ(u))+D_xb(u)+f(x.t.u)with Dirichlet boundary conditions,where u,gradψ(u),b and f are vector valued functions and ...In this paper we are concerned with the following nonlinear degenerate parabolic systems u_t=△x(gradψ(u))+D_xb(u)+f(x.t.u)with Dirichlet boundary conditions,where u,gradψ(u),b and f are vector valued functions and xUnder some structure conditions on the terms of the systems,we have established theresults on existence and uniquence of global solutions of the systems.展开更多
We consider the boundary value problem for the second order quasilinear differential equationwhere f is allowed to change sign, φ(v) = \v\p-2v, p > 1. Using a new fixed point theorem in double cones, we show the e...We consider the boundary value problem for the second order quasilinear differential equationwhere f is allowed to change sign, φ(v) = \v\p-2v, p > 1. Using a new fixed point theorem in double cones, we show the existence of at least two positive solutions of the boundary value problem.展开更多
In this paper, out main purpose is to establish the existence of nonnegative solu-tions for a class quasilinear ordinary differential equation by modifying the method ofAnuradha et al. [4]. The main results in present...In this paper, out main purpose is to establish the existence of nonnegative solu-tions for a class quasilinear ordinary differential equation by modifying the method ofAnuradha et al. [4]. The main results in present paper are new and extend the resultsof the [4].展开更多
In this paper, we investigate the existence of positive solutions for singular fourthorder integral boundary-value problem with p-Laplacian operator by using the upper and lower solution method and fixed point theorem...In this paper, we investigate the existence of positive solutions for singular fourthorder integral boundary-value problem with p-Laplacian operator by using the upper and lower solution method and fixed point theorem. Nonlinear term may be singular at t= 0 and/or t - 1 and x =0.展开更多
运用迭代技巧考虑了带积分边界条件的三阶边值问题:{u'''(t)+f(t,u(t),u′(t))=0,t∈[0,1],u(0)=0,u′(0)=integral(g(t)u′(t)dt,u″(1)=0) from 1 to 0不仅获得了其单调正解的存在性,而且迭代列的初值是简单的零函数或一...运用迭代技巧考虑了带积分边界条件的三阶边值问题:{u'''(t)+f(t,u(t),u′(t))=0,t∈[0,1],u(0)=0,u′(0)=integral(g(t)u′(t)dt,u″(1)=0) from 1 to 0不仅获得了其单调正解的存在性,而且迭代列的初值是简单的零函数或一次函数.展开更多
文摘The paper deals with the existence of three- solutions for the second- order differential equations with nonlinear boundary value conditions x″=f(t,x,x′) , t∈ [a,b], g1(x(a) ,x′(a) ) =0 , g2 (x(b) ,x′(b) ) =0 , where f :[a,b]× R1× R1→ R1,gi:R1× R1→ R1(i=1 ,2 ) are continuous functions.The methods employed are the coincidence degree theory.As an application,the sufficient conditions under which there are arbitrary odd solutions for the BVP are obtained
文摘This paper deals with the existence of positive solutions for the singular fourth order boundary value problem.A necessary and sufficient condition for the existence of C3 positive solution is given by means of the monotone iterative technique.Furthermore,the uniqueness of the C3 positive solution,and the iterative sequence of the C3 positive solution are also obtained.
基金SuppoSed by the NSF of Anhui Provincial Education Depaxtment(KJ2012A265,KJ2012B187)
文摘By using cone theory and the MSnch fixed theorem combined with a monotone iterative technique, we investigate the existence of positive solutions for systems of second- order nonlinear singular differential equations with integral boundary conditions on infinite interval and establish the existence theorem of positive solutions and iterative sequence for approximating the positive solutions. The results in this paper improve some known results.
文摘Making use of upper and lower solutions and analytical method, the author studies theexistence of positive solution for the singular equation x + f(t, z) = 0 satisfying nonlinear boundary conditions: x (0) = 0, h(x (1), x’ (1)) = 0, g (z (0), x’(0)) = 0, and x (1) = 0,which extends the result of J. V. Baxley.
文摘Let Ω be a bounded or unbounded domain in R~n. The initial-boundary value problem for the porous medium and plasma equation with singular terms is considered in this paper. Criteria for the appearance of quenching phenomenon and the existence of global classical solution to the above problem are established. Also, the life span of the quenching solution is estimated or evaluated for some domains.
文摘In the paper an inverse boundary value problem for a fourth order elliptic equation with an integral condition of the first kind is investigated. First, the given problem is reduced to an equivalent problem in a certain sense. Then, using the Fourier method the equivalent problem is reduced to solving the system of integral equations. The existence and uniqueness of a solution to the system of integral equation is proved by the contraction mapping principle. This solution is also the unique solution to the equivalent problem. Finally, by equivalence, the theorem of existence and uniqueness of a classical solution to the given problem is proved.
文摘In recent years, a vast amount of work has been done on initial value problems for important nonlinear evolution equations like the nonlinear Schrödinger equation (NLS) and the Korteweg-de Vries equation (KdV). No comparable attention has been given to mixed initial-boundary value problems for these equations, i.e. forced nonlinear systems. But in many cases of physical interest, the mathematical model leads precisely to the forced problems. For example, the launching of solitary waves in a shallow water channel, the excitation of ion-acoustic solitons in a double plasma machine, etc. In this article, we present the PDE (Partial Differential Equation) method to study the following iut = uxx - g|u|pu, g ∈ R, p > 3, x?∈ Ω = [0,L], 0 ≤?t?u (x,0) = u0 (x) ∈?H2 (Ω) and Robin inhomogeneous boundary condition ux (0,t) + αu (0,t) = R1(t), t ≥ 0 and ux (L,t) + αu (L,t) = R2 (t), t ≥ 0 (here?α?is a real number). The equation is posed in a semi-infinite strip on a finite domain Ω. Such problems are called forced problems and have many applications in other fields like physics and chemistry. The main tool of PDE method is semi-group theory. We are able to prove local existence and uniqueness theorem for the nonlinear Schrödinger equation under initial condition and Robin inhomogeneous boundary condition.
基金Supported by the National Natural Science Foundation of China(Grants No.70703016 and No.10001024)Research Grant of the Business School of Nanjing University
文摘In this paper nonlinear boundary value problems for discontinuous delayed differen- tial equations are considered. Some existence and boundedness results of solutions are obtained via the method of upper and lower solutions, which may be discontinuous. Our analysis can be applied to those phenomena from physics and control theory which have been successfully described by delayed differential equations with discontinuous functions, such as electric, pneu- matic, and hydraulic networks. It is also an important step to precede the design of control signals when finite-time or practical stability control are concerned.
文摘In this paper, two kinds of initial boundary value problems for Kuramoto_Sivashinsky equation are considered. Some prior estimates are derived by Galerkin methods. The existence, uniqueness and regularities of the generalized global solutions and the classical global solutions for the equation are proved. Morever, the asymptotic behavior of these solutions are considered under some conditions.
文摘This paper is concerned with the following n-th ordinary differential equation:{u~(n)(t)=f(t,u(t),u~(1)(t),···,u~(n-1) (t)),for t∈(0,1),u~(i) (0)=0,0 ≤i≤n3,au~(n-2)(0)du~(n-1)(0)=0,cu~(n-2)(1)+du~(n-1)(1)=0,where a,c ∈ R,,≥,such that a~2 + b~2 >0 and c~2+d~2>0,n ≥ 2,f:[0,1] × R → R is a continuous function.Assume that f satisfies one-sided Nagumo condition,the existence theorems of solutions of the boundary value problem for the n-th-order nonlinear differential equations above are established by using Leray-Schauder degree theory,lower and upper solutions,a priori estimate technique.
基金The project supported by the Natural Science Foundation of FuJian Province of China
文摘In this paper we are concerned with the following nonlinear degenerate parabolic systems u_t=△x(gradψ(u))+D_xb(u)+f(x.t.u)with Dirichlet boundary conditions,where u,gradψ(u),b and f are vector valued functions and xUnder some structure conditions on the terms of the systems,we have established theresults on existence and uniquence of global solutions of the systems.
基金The project is supported by the National Natural Science Foundation of China(19871005)the Scientific Research Foundation of the Education Department of Hebei Province(2001111).
文摘We consider the boundary value problem for the second order quasilinear differential equationwhere f is allowed to change sign, φ(v) = \v\p-2v, p > 1. Using a new fixed point theorem in double cones, we show the existence of at least two positive solutions of the boundary value problem.
基金Supported by Natural Science Foundations of the Committee on Science and Technology of HenanProvince(984050400).
文摘In this paper, out main purpose is to establish the existence of nonnegative solu-tions for a class quasilinear ordinary differential equation by modifying the method ofAnuradha et al. [4]. The main results in present paper are new and extend the resultsof the [4].
基金Supported by the National Natural Science Foundation of China (Grant No. 10971179)the Natural Science Foundation of Liaocheng University (Grant No. 31805)
文摘In this paper, we investigate the existence of positive solutions for singular fourthorder integral boundary-value problem with p-Laplacian operator by using the upper and lower solution method and fixed point theorem. Nonlinear term may be singular at t= 0 and/or t - 1 and x =0.
文摘运用迭代技巧考虑了带积分边界条件的三阶边值问题:{u'''(t)+f(t,u(t),u′(t))=0,t∈[0,1],u(0)=0,u′(0)=integral(g(t)u′(t)dt,u″(1)=0) from 1 to 0不仅获得了其单调正解的存在性,而且迭代列的初值是简单的零函数或一次函数.