In this work, we are concerned with the existence and multiplicity of positive solutions for singular boundary value problems on the half-line. Two problems from epi- demiology and combustion theory set on the positiv...In this work, we are concerned with the existence and multiplicity of positive solutions for singular boundary value problems on the half-line. Two problems from epi- demiology and combustion theory set on the positive half-line are investigated upper and lower solution techniques combined with fixed point index on cones in priate Banach spaces. The results complement recent ones in the literature. We use appropriate Banach spaces. The results complement recent ones in the literature.展开更多
We show an example of a bounded potential on the half-line obtained as the image of an Inverse Transformation Operator of the Bessel singular potential of the Reduced Radial Schrödinger Equation, and show us ...We show an example of a bounded potential on the half-line obtained as the image of an Inverse Transformation Operator of the Bessel singular potential of the Reduced Radial Schrödinger Equation, and show us the Estimates of the A(α) amplitude.展开更多
This paper investigates existence of positive solutions of singular sub-linear boundary value problems on a half-line. Necessary and sufficient conditions for the existence of positive continuous solutions or smooth s...This paper investigates existence of positive solutions of singular sub-linear boundary value problems on a half-line. Necessary and sufficient conditions for the existence of positive continuous solutions or smooth solutions on [0, ∞] are given by constructing new lower and upper solutions.展开更多
We consider solving integral equations of the second kind defined on the half-line [0, infinity) by the preconditioned conjugate gradient method. Convergence is known to be slow due to the non-compactness of the assoc...We consider solving integral equations of the second kind defined on the half-line [0, infinity) by the preconditioned conjugate gradient method. Convergence is known to be slow due to the non-compactness of the associated integral operator. In this paper, we construct two different circulant integral operators to be used as preconditioners for the method to speed up its convergence rate. We prove that if the given integral operator is close to a convolution-type integral operator, then the preconditioned systems will have spectrum clustered around 1 and hence the preconditioned conjugate gradient method will converge superlinearly. Numerical examples are given to illustrate the fast convergence.展开更多
In this paper, we are concerned with the existence of a second-order differential equation with m-point boundary conditions on the half line. The existence results are proved by the Leray-Scha¨uder point theorem ...In this paper, we are concerned with the existence of a second-order differential equation with m-point boundary conditions on the half line. The existence results are proved by the Leray-Scha¨uder point theorem in a special Banach space. Green function plays an important role in the proof and its discussion is very interesting. The results obtained generalize those in the previous references.展开更多
面向“碳达峰、碳中和”目标,发展半波长输电技术可为构建以新能源为主体的新型电力系统提供能源保障。半波长技术因其输电距离远、经济性好等独特优势,在“西电东送”能源战略中有较好的应用前景。针对输电线路自然长度不足工频半波长...面向“碳达峰、碳中和”目标,发展半波长输电技术可为构建以新能源为主体的新型电力系统提供能源保障。半波长技术因其输电距离远、经济性好等独特优势,在“西电东送”能源战略中有较好的应用前景。针对输电线路自然长度不足工频半波长现象以及固定调谐的不足,文章提出一种变阻抗型可控调谐方法,基于可控串补(thyristor controlled series compensator,TCSC)阻抗控制策略构造可微调的调谐电感,基于电容三级投切策略构造可分级的调谐电容。对可控调谐可行性、故障过电压特征以及串补过压抑制进行仿真分析。结果表明,可控调谐能够消除电气长度误差,使线路保持半波长特性,线路补偿效果与可控调谐长度相关。展开更多
运用Mawhin重合度理论,讨论一类半直线上三阶多点边值问题(q(t)x″(t))′=f(t,x(t),x′(t),x″(t)),a.e.t∈[0,+∞);■(η■在dim Ker L=2共振情形下的可解性,获得了该边值问题至少存在一个解的充分条件.这里f:[0,1]×R^(3)→R满足L...运用Mawhin重合度理论,讨论一类半直线上三阶多点边值问题(q(t)x″(t))′=f(t,x(t),x′(t),x″(t)),a.e.t∈[0,+∞);■(η■在dim Ker L=2共振情形下的可解性,获得了该边值问题至少存在一个解的充分条件.这里f:[0,1]×R^(3)→R满足L^(1)[0,+∞)-Carathéodory条件,αi,βj∈R(1≤i≤m,1≤j≤n),0<ξ_(1)<ξ_(2)<…<ξ_(m)<+∞,0<η_(1)<η_(2)<…<η_(n)<+∞(m,n∈Z+),q(t)>0,q(t)∈C[0,+∞)∩C^(2)(0,+∞),1/q(t)∈L^(1)[0,+∞).展开更多
运用Mawhin重合度理论,讨论一类半直线上三阶多点边值问题{x″′(t)=f(t,x(t),x′(t),x″(t))+e(t),t∈[0,+∞),x(0)=m∑i=1α_(i)x(ξ_(i)),x(1)=n∑j=1β_(j)x(η_(j)),limt→+∞x′(t)=∑k=1lγkx″(ζk)在dim Ker L=3共振情形下解...运用Mawhin重合度理论,讨论一类半直线上三阶多点边值问题{x″′(t)=f(t,x(t),x′(t),x″(t))+e(t),t∈[0,+∞),x(0)=m∑i=1α_(i)x(ξ_(i)),x(1)=n∑j=1β_(j)x(η_(j)),limt→+∞x′(t)=∑k=1lγkx″(ζk)在dim Ker L=3共振情形下解的存在性,f:[0,1]×R→R满足S-Carathéodory条件,e∈L1[0,∞),α_(i),β_(j),γk∈R,0<ξ1<ξ2<…<ξm<+∞,0<η1<η2<…<ηn<+∞,0<ζ_(1)<ζ_(2)<…<ζ_(l)<+∞(m,n,l∈Z^(+)),并且满足下列条件:(C1)m∑i=1α_(i)=1,m∑i=1α_(i)ξi=0,m∑i=1α_(i)ξ2i=0,∑j=1nβ_(j)=1,∑j=1nβ_(j)η_(j)=1,∑j=1nβ_(j)η2j=1,∑k=1lγk=1;(C2)Δ=∣∣∣∣Q_(1)e−tQ_(1)te−tQ_(1)t2e−tQ_(2)e−tQ_(2)te−tQ_(2)t2e−tQ_(3)e−tQ_(3)te−tQ_(3)t2e−t∣∣∣∣:=∣∣∣∣a_(11)a_(21)a_(31)a_(12)a_(22)a_(32)a_(13)a_(23)a_(33)∣∣∣∣≠0,其中,Q_(1)y=m∑i=1α_(i)∫ξi_(0)∫^(s)_(0)∫^(τ)_(0)y(v)dv dτds,Q_(2)y=∑j=1nβ_(j)∫^(η_(j))_(0)∫^(s)_(0)∫^(τ)_(0)y(v)dv dτds,Q_(3)y=l∑k=1γk∫^(+∞)_(γk)y(s)ds。展开更多
Estimates of the type L1-L∞ for the Schrödinger Equation on the Line and on Half-Line with a regular potential V(x), express the dispersive nature of the Schrödinger Equation and are the essential e...Estimates of the type L1-L∞ for the Schrödinger Equation on the Line and on Half-Line with a regular potential V(x), express the dispersive nature of the Schrödinger Equation and are the essential elements in the study of the problems of initial values, the asymptotic times for large solutions and Scattering Theory for the Schrödinger equation and non-linear in general;for other equations of Non-linear Evolution. In general, the estimates Lp-Lp' express the dispersive nature of this equation. And its study plays an important role in problems of non-linear initial values;likewise, in the study of problems nonlinear initial values;see [1] [2] [3]. On the other hand, following a series of problems proposed by V. Marchenko [4], that we will name Marchenko’s formulation, and relate it to a generalized version of Theorem 1 given in [1], the main theorem (Theorem 1) of this article provides a transformation operator W?that transforms the Reduced Radial Schrödinger Equation (RRSE) (whose main characteristic is the addition a singular term of quadratic order to a regular potential V(x)) in the Schrödinger Equation on Half-Line (RSEHL) under W. That is to say;W?eliminates the singular term of quadratic order of potential V(x) in the asymptotic development towards zero and adds to the potential V(x) a bounded term and a term exponentially decrease fast enough in the asymptotic development towards infinity, which continues guaranteeing the uniqueness of the potential V(x) in the condition of the infinity boundary. Then the L1-L∞ estimates for the (RRSE) are preserved under the transformation operator , as in the case of (RSEHL) where they were established in [3]. Finally, as an open question, the possibility of extending the L1-L∞ estimates for the case (RSEHL), where added to the potential V(x) an analytical perturbation is mentioned.展开更多
文摘In this work, we are concerned with the existence and multiplicity of positive solutions for singular boundary value problems on the half-line. Two problems from epi- demiology and combustion theory set on the positive half-line are investigated upper and lower solution techniques combined with fixed point index on cones in priate Banach spaces. The results complement recent ones in the literature. We use appropriate Banach spaces. The results complement recent ones in the literature.
文摘We show an example of a bounded potential on the half-line obtained as the image of an Inverse Transformation Operator of the Bessel singular potential of the Reduced Radial Schrödinger Equation, and show us the Estimates of the A(α) amplitude.
文摘This paper investigates existence of positive solutions of singular sub-linear boundary value problems on a half-line. Necessary and sufficient conditions for the existence of positive continuous solutions or smooth solutions on [0, ∞] are given by constructing new lower and upper solutions.
文摘We consider solving integral equations of the second kind defined on the half-line [0, infinity) by the preconditioned conjugate gradient method. Convergence is known to be slow due to the non-compactness of the associated integral operator. In this paper, we construct two different circulant integral operators to be used as preconditioners for the method to speed up its convergence rate. We prove that if the given integral operator is close to a convolution-type integral operator, then the preconditioned systems will have spectrum clustered around 1 and hence the preconditioned conjugate gradient method will converge superlinearly. Numerical examples are given to illustrate the fast convergence.
基金Supported by the Fundamental Research Funds for the Central Universities (2011YXL044)the NNSF of China (11101385)
文摘In this paper, we are concerned with the existence of a second-order differential equation with m-point boundary conditions on the half line. The existence results are proved by the Leray-Scha¨uder point theorem in a special Banach space. Green function plays an important role in the proof and its discussion is very interesting. The results obtained generalize those in the previous references.
文摘面向“碳达峰、碳中和”目标,发展半波长输电技术可为构建以新能源为主体的新型电力系统提供能源保障。半波长技术因其输电距离远、经济性好等独特优势,在“西电东送”能源战略中有较好的应用前景。针对输电线路自然长度不足工频半波长现象以及固定调谐的不足,文章提出一种变阻抗型可控调谐方法,基于可控串补(thyristor controlled series compensator,TCSC)阻抗控制策略构造可微调的调谐电感,基于电容三级投切策略构造可分级的调谐电容。对可控调谐可行性、故障过电压特征以及串补过压抑制进行仿真分析。结果表明,可控调谐能够消除电气长度误差,使线路保持半波长特性,线路补偿效果与可控调谐长度相关。
文摘运用Mawhin重合度理论,讨论一类半直线上三阶多点边值问题(q(t)x″(t))′=f(t,x(t),x′(t),x″(t)),a.e.t∈[0,+∞);■(η■在dim Ker L=2共振情形下的可解性,获得了该边值问题至少存在一个解的充分条件.这里f:[0,1]×R^(3)→R满足L^(1)[0,+∞)-Carathéodory条件,αi,βj∈R(1≤i≤m,1≤j≤n),0<ξ_(1)<ξ_(2)<…<ξ_(m)<+∞,0<η_(1)<η_(2)<…<η_(n)<+∞(m,n∈Z+),q(t)>0,q(t)∈C[0,+∞)∩C^(2)(0,+∞),1/q(t)∈L^(1)[0,+∞).
文摘运用Mawhin重合度理论,讨论一类半直线上三阶多点边值问题{x″′(t)=f(t,x(t),x′(t),x″(t))+e(t),t∈[0,+∞),x(0)=m∑i=1α_(i)x(ξ_(i)),x(1)=n∑j=1β_(j)x(η_(j)),limt→+∞x′(t)=∑k=1lγkx″(ζk)在dim Ker L=3共振情形下解的存在性,f:[0,1]×R→R满足S-Carathéodory条件,e∈L1[0,∞),α_(i),β_(j),γk∈R,0<ξ1<ξ2<…<ξm<+∞,0<η1<η2<…<ηn<+∞,0<ζ_(1)<ζ_(2)<…<ζ_(l)<+∞(m,n,l∈Z^(+)),并且满足下列条件:(C1)m∑i=1α_(i)=1,m∑i=1α_(i)ξi=0,m∑i=1α_(i)ξ2i=0,∑j=1nβ_(j)=1,∑j=1nβ_(j)η_(j)=1,∑j=1nβ_(j)η2j=1,∑k=1lγk=1;(C2)Δ=∣∣∣∣Q_(1)e−tQ_(1)te−tQ_(1)t2e−tQ_(2)e−tQ_(2)te−tQ_(2)t2e−tQ_(3)e−tQ_(3)te−tQ_(3)t2e−t∣∣∣∣:=∣∣∣∣a_(11)a_(21)a_(31)a_(12)a_(22)a_(32)a_(13)a_(23)a_(33)∣∣∣∣≠0,其中,Q_(1)y=m∑i=1α_(i)∫ξi_(0)∫^(s)_(0)∫^(τ)_(0)y(v)dv dτds,Q_(2)y=∑j=1nβ_(j)∫^(η_(j))_(0)∫^(s)_(0)∫^(τ)_(0)y(v)dv dτds,Q_(3)y=l∑k=1γk∫^(+∞)_(γk)y(s)ds。
文摘Estimates of the type L1-L∞ for the Schrödinger Equation on the Line and on Half-Line with a regular potential V(x), express the dispersive nature of the Schrödinger Equation and are the essential elements in the study of the problems of initial values, the asymptotic times for large solutions and Scattering Theory for the Schrödinger equation and non-linear in general;for other equations of Non-linear Evolution. In general, the estimates Lp-Lp' express the dispersive nature of this equation. And its study plays an important role in problems of non-linear initial values;likewise, in the study of problems nonlinear initial values;see [1] [2] [3]. On the other hand, following a series of problems proposed by V. Marchenko [4], that we will name Marchenko’s formulation, and relate it to a generalized version of Theorem 1 given in [1], the main theorem (Theorem 1) of this article provides a transformation operator W?that transforms the Reduced Radial Schrödinger Equation (RRSE) (whose main characteristic is the addition a singular term of quadratic order to a regular potential V(x)) in the Schrödinger Equation on Half-Line (RSEHL) under W. That is to say;W?eliminates the singular term of quadratic order of potential V(x) in the asymptotic development towards zero and adds to the potential V(x) a bounded term and a term exponentially decrease fast enough in the asymptotic development towards infinity, which continues guaranteeing the uniqueness of the potential V(x) in the condition of the infinity boundary. Then the L1-L∞ estimates for the (RRSE) are preserved under the transformation operator , as in the case of (RSEHL) where they were established in [3]. Finally, as an open question, the possibility of extending the L1-L∞ estimates for the case (RSEHL), where added to the potential V(x) an analytical perturbation is mentioned.
