New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the cond...New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the conditions 0 less than or equal to f(0)(+) < M-1, m(1) < f(infinity)(-)less than or equal to infinity or 0 less than or equal to f(infinity)(+)< M-1, m(1) < f (-)(0)less than or equal to infinity where f(0)(+) = lim(u -->0)f(u)/u, f(infinity)(-)= lim(u --> infinity)f(u)/u, f(0)(-)= lim(u -->0)f(u)/u, f(infinity)(+) = lim(u --> infinity)f(u)/u, g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.展开更多
In this paper the following result is obtained: Suppose f(x,u,v) is nonnegative, continuous in ( a, b)×R +×R +; f may be singular at x=a (and/or x=b ) and v=0; f is nondecreasing on u for each x,v,...In this paper the following result is obtained: Suppose f(x,u,v) is nonnegative, continuous in ( a, b)×R +×R +; f may be singular at x=a (and/or x=b ) and v=0; f is nondecreasing on u for each x,v, nonincreasing on v for each x,u; there exists a constant q∈(0,1) such that t qf(x,t -1 u,tu)f(x,u,u)λ qf(x,λ -1 u,λu),0<t<1<λ, u∈R +. Then a necessary and sufficient condition for the equation u″+f(x,u,u)=0 on the boundary condition αu(a)-βu′(a)=0, γ(b)+δu′(b)=0 to have C 1(I) nonzero solutions is that 0<∫ b af(x,e(x),e(x))dx<∞, where α,β,γ,δ are nonnegative real numbers, Δ=(b-a)αγ+αδ+βγ>0, e(x)=G(x,x), G(x,y) is Green's function of above mentioned boundary value problem (when f(x,u,v)≡0). Received September 9,1996. Revised March 31,1997. 1991 MR Subject Classification: 34B.展开更多
The convergence results of block iterative schemes from the EG （Explicit Group） family have been shown to be one of efficient iterative methods in solving any linear systems generated from approximation equations. A...The convergence results of block iterative schemes from the EG （Explicit Group） family have been shown to be one of efficient iterative methods in solving any linear systems generated from approximation equations. Apart from block iterative methods, the formulation of the MSOR （Modified Successive Over-Relaxation） method known as SOR method with red-black ordering strategy by using two accelerated parameters, ω and ω′, has also improved the convergence rate of the standard SOR method. Due to the effectiveness of these iterative methods, the primary goal of this paper is to examine the performance of the EG family without or with accelerated parameters in solving second order two-point nonlinear boundary value problems. In this work, the second order two-point nonlinear boundary value problems need to be discretized by using the second order central difference scheme in constructing a nonlinear finite difference approximation equation. Then this approximation equation leads to a nonlinear system. As well known that to linearize nonlinear systems, the Newton method has been proposed to transform the original system into the form of linear system. In addition to that, the basic formulation and implementation of 2 and 4-point EG iterative methods based on GS （Gauss-Seidel）, SOR and MSOR approaches, namely EGGS, EGSOR and EGMSOR respectively are also presented. Then, combinations between the EG family and Newton scheme are indicated as EGGS-Newton, EGSOR-Newton and EGMSOR-Newton methods respectively. For comparison purpose, several numerical experiments of three problems are conducted in examining the effectiveness of tested methods. Finally, it can be concluded that the 4-point EGMSOR-Newton method is more superior in accelerating the convergence rate compared with the tested methods.展开更多
This paper studies the positive solutions of the nonlinear second-order periodic boundary value problem u″（t） ＋ λ（t）u（t） = f（t,u（t））,a.e.t ∈ [0,2π],u（0） = u（2π）,u′（0） = u′（2π）,where f（t,u）...This paper studies the positive solutions of the nonlinear second-order periodic boundary value problem u″（t） ＋ λ（t）u（t） = f（t,u（t））,a.e.t ∈ [0,2π],u（0） = u（2π）,u′（0） = u′（2π）,where f（t,u） is a local Carath′eodory function.This shows that the problem is singular with respect to both the time variable t and space variable u.By applying the Leggett-Williams and Krasnosel＇skii fixed point theorems on cones,an existence theorem of triple positive solutions is established.In order to use these theorems,the exact a priori estimations for the bound of solution are given,and some proper height functions are introduced by the estimations.展开更多
The singular second-order m-point boundary value problem , is considered under some conditions concerning the first eigenvalue of the relevant linear operators, where (Lϕ)(x) = (p(x)ϕ′...The singular second-order m-point boundary value problem , is considered under some conditions concerning the first eigenvalue of the relevant linear operators, where (Lϕ)(x) = (p(x)ϕ′(x))′ + q(x)ϕ(x) and ξ<SUB> i </SUB>∈ (0, 1) with 0 【 ξ<SUB>1</SUB> 【 ξ<SUB>2</SUB> 【 · · · 【 ξ<SUB> m−2</SUB> 【 1, a <SUB>i </SUB>∈ [0, ∞). h(x) is allowed to be singular at x = 0 and x = 1. The existence of positive solutions is obtained by means of fixed point index theory. Similar conclusions hold for some other m-point boundary value conditions.展开更多
This paper investigates the existence of solutions of periodic boundary value problems for nonlinear second order functional differential equations. By establishing comparison results, criteria on the existence of max...This paper investigates the existence of solutions of periodic boundary value problems for nonlinear second order functional differential equations. By establishing comparison results, criteria on the existence of maximal and minimal solutions are obtained.展开更多
In this paper, we obtain the existence of positive solutions for singular second-order Neumann boundary value problem by using the fixed point indices, the result generalizes some present results.
In this paper, by utilizing a fixed point theorem on cone in Banach space, the author disscusses the periodic boundary value problem of second order differential systems. Some results on the existence of positive solu...In this paper, by utilizing a fixed point theorem on cone in Banach space, the author disscusses the periodic boundary value problem of second order differential systems. Some results on the existence of positive solutions are derived.展开更多
In this paper, we establish two existence theorems of twin positive solutions for a class of nonlinear second-order three-point boundary value problems, and concentrate on the case when nonlinear term does not satisfy...In this paper, we establish two existence theorems of twin positive solutions for a class of nonlinear second-order three-point boundary value problems, and concentrate on the case when nonlinear term does not satisfy usual conditions.展开更多
We investigate the existence, uniqueness and asymptotic estimates of solutions for a class of second order quasi-linear singularly perturbed boundary value problem on the infinite interval when its reduced problem has...We investigate the existence, uniqueness and asymptotic estimates of solutions for a class of second order quasi-linear singularly perturbed boundary value problem on the infinite interval when its reduced problem has weakly discontinuous solutions. In this paper, a class of quasi-linear turning point problem on the finite interval is discussed.展开更多
In this paper, we study a singular third-order three-point boundary value problem. By using a fixed-point theorem of cone expansion-compression type,we establish results on the existence of at least one, at least two,...In this paper, we study a singular third-order three-point boundary value problem. By using a fixed-point theorem of cone expansion-compression type,we establish results on the existence of at least one, at least two, and n positive solutions to the boundary value problem. Finally we give an example.展开更多
This article shows the existence and asymptotic estimates of solutions of singularly perturbed boundary value problems for a class of third order nonlinear differential equations εx'' = f(t,x,x',ε), x(0)...This article shows the existence and asymptotic estimates of solutions of singularly perturbed boundary value problems for a class of third order nonlinear differential equations εx'' = f(t,x,x',ε), x(0) = A, x'(0) = x'(1), x'(0) = x'(1).展开更多
文摘New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the conditions 0 less than or equal to f(0)(+) < M-1, m(1) < f(infinity)(-)less than or equal to infinity or 0 less than or equal to f(infinity)(+)< M-1, m(1) < f (-)(0)less than or equal to infinity where f(0)(+) = lim(u -->0)f(u)/u, f(infinity)(-)= lim(u --> infinity)f(u)/u, f(0)(-)= lim(u -->0)f(u)/u, f(infinity)(+) = lim(u --> infinity)f(u)/u, g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.
文摘In this paper the following result is obtained: Suppose f(x,u,v) is nonnegative, continuous in ( a, b)×R +×R +; f may be singular at x=a (and/or x=b ) and v=0; f is nondecreasing on u for each x,v, nonincreasing on v for each x,u; there exists a constant q∈(0,1) such that t qf(x,t -1 u,tu)f(x,u,u)λ qf(x,λ -1 u,λu),0<t<1<λ, u∈R +. Then a necessary and sufficient condition for the equation u″+f(x,u,u)=0 on the boundary condition αu(a)-βu′(a)=0, γ(b)+δu′(b)=0 to have C 1(I) nonzero solutions is that 0<∫ b af(x,e(x),e(x))dx<∞, where α,β,γ,δ are nonnegative real numbers, Δ=(b-a)αγ+αδ+βγ>0, e(x)=G(x,x), G(x,y) is Green's function of above mentioned boundary value problem (when f(x,u,v)≡0). Received September 9,1996. Revised March 31,1997. 1991 MR Subject Classification: 34B.
文摘The convergence results of block iterative schemes from the EG （Explicit Group） family have been shown to be one of efficient iterative methods in solving any linear systems generated from approximation equations. Apart from block iterative methods, the formulation of the MSOR （Modified Successive Over-Relaxation） method known as SOR method with red-black ordering strategy by using two accelerated parameters, ω and ω′, has also improved the convergence rate of the standard SOR method. Due to the effectiveness of these iterative methods, the primary goal of this paper is to examine the performance of the EG family without or with accelerated parameters in solving second order two-point nonlinear boundary value problems. In this work, the second order two-point nonlinear boundary value problems need to be discretized by using the second order central difference scheme in constructing a nonlinear finite difference approximation equation. Then this approximation equation leads to a nonlinear system. As well known that to linearize nonlinear systems, the Newton method has been proposed to transform the original system into the form of linear system. In addition to that, the basic formulation and implementation of 2 and 4-point EG iterative methods based on GS （Gauss-Seidel）, SOR and MSOR approaches, namely EGGS, EGSOR and EGMSOR respectively are also presented. Then, combinations between the EG family and Newton scheme are indicated as EGGS-Newton, EGSOR-Newton and EGMSOR-Newton methods respectively. For comparison purpose, several numerical experiments of three problems are conducted in examining the effectiveness of tested methods. Finally, it can be concluded that the 4-point EGMSOR-Newton method is more superior in accelerating the convergence rate compared with the tested methods.
基金Supported by National Natural Science Foundation of China(Grant No.11071109)
文摘This paper studies the positive solutions of the nonlinear second-order periodic boundary value problem u″（t） ＋ λ（t）u（t） = f（t,u（t））,a.e.t ∈ [0,2π],u（0） = u（2π）,u′（0） = u′（2π）,where f（t,u） is a local Carath′eodory function.This shows that the problem is singular with respect to both the time variable t and space variable u.By applying the Leggett-Williams and Krasnosel＇skii fixed point theorems on cones,an existence theorem of triple positive solutions is established.In order to use these theorems,the exact a priori estimations for the bound of solution are given,and some proper height functions are introduced by the estimations.
文摘The singular second-order m-point boundary value problem , is considered under some conditions concerning the first eigenvalue of the relevant linear operators, where (Lϕ)(x) = (p(x)ϕ′(x))′ + q(x)ϕ(x) and ξ<SUB> i </SUB>∈ (0, 1) with 0 【 ξ<SUB>1</SUB> 【 ξ<SUB>2</SUB> 【 · · · 【 ξ<SUB> m−2</SUB> 【 1, a <SUB>i </SUB>∈ [0, ∞). h(x) is allowed to be singular at x = 0 and x = 1. The existence of positive solutions is obtained by means of fixed point index theory. Similar conclusions hold for some other m-point boundary value conditions.
基金Supported by NNSF-China (No.10071043)the YNSF of Shandong Province (No.Y2000A06)
文摘This paper investigates the existence of solutions of periodic boundary value problems for nonlinear second order functional differential equations. By establishing comparison results, criteria on the existence of maximal and minimal solutions are obtained.
文摘In this paper, we obtain the existence of positive solutions for singular second-order Neumann boundary value problem by using the fixed point indices, the result generalizes some present results.
基金Supported by the Natural Science Foundation of Guangdong Province(No. 032469).
文摘In this paper, by utilizing a fixed point theorem on cone in Banach space, the author disscusses the periodic boundary value problem of second order differential systems. Some results on the existence of positive solutions are derived.
文摘In this paper, we establish two existence theorems of twin positive solutions for a class of nonlinear second-order three-point boundary value problems, and concentrate on the case when nonlinear term does not satisfy usual conditions.
文摘We investigate the existence, uniqueness and asymptotic estimates of solutions for a class of second order quasi-linear singularly perturbed boundary value problem on the infinite interval when its reduced problem has weakly discontinuous solutions. In this paper, a class of quasi-linear turning point problem on the finite interval is discussed.
基金the National Natural Science Foundation of China(11261053)The Natural Science Foundation of Gansu Province(1308RJZA125)
文摘In this paper, we study a singular third-order three-point boundary value problem. By using a fixed-point theorem of cone expansion-compression type,we establish results on the existence of at least one, at least two, and n positive solutions to the boundary value problem. Finally we give an example.
文摘This article shows the existence and asymptotic estimates of solutions of singularly perturbed boundary value problems for a class of third order nonlinear differential equations εx'' = f(t,x,x',ε), x(0) = A, x'(0) = x'(1), x'(0) = x'(1).
