In this paper well-conditioning of boundary value problems for systems of second order difference equa-tions is studied.First,a sufficient condition for the existence of a unique bounded solution (for large enough num...In this paper well-conditioning of boundary value problems for systems of second order difference equa-tions is studied.First,a sufficient condition for the existence of a unique bounded solution (for large enough number of steps) of an associated homogeneous system is given.Finally,a sufficient condition for well-condi-tioning,intrinsically related to the problem data is proposed.展开更多
In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference me...In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference methods. It is proved that the method has optimal order error estimate O(h3) in H1 norm. Finally, two examples show that the method is effective.展开更多
In this paper we study the forced oscillations of boundary value problems of a class of higher order functional partial differential equations.The principal tool is an everaging techniqe which enables one to establish...In this paper we study the forced oscillations of boundary value problems of a class of higher order functional partial differential equations.The principal tool is an everaging techniqe which enables one to establish oscillation in terms of related functional differential inequallities.展开更多
Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 ...Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 are established,where [φ(x)] =(|x |p-2x) with p > 1.Our result is new even when [φ(x)] = x in above problem,i.e.p = 2.Examples are presented to illustrate the effciency of the theorem in this paper.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
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, solutions of Riemann boundary value problems with nodes are extended to the case where they may have singularties of high order at the nodes. Moreover, further extension is discussed when the free term ...In this paper, solutions of Riemann boundary value problems with nodes are extended to the case where they may have singularties of high order at the nodes. Moreover, further extension is discussed when the free term of the problem involved also possesses singularities at the nodes. As an application, certain singular integral equation is discussed.展开更多
In this paper, we study the nonlinear second-order boundary value problem of delay differential equation.. Without the assumption of the nonnegativity of f, we still obtain the existence of the positive solution.
The author employs the method of upper and lower solutions together with the monotone iterative technique to obtain the existence theorem of minimal and maximal solutions for a boundary value problem of second order i...The author employs the method of upper and lower solutions together with the monotone iterative technique to obtain the existence theorem of minimal and maximal solutions for a boundary value problem of second order impulsive differential equation.展开更多
In this paper, we discuss the existence of positive solutions of the secondorder delay boundary value problems. By applying the fixed-point theorem in a cone, we show the existence of at least one positive solution wi...In this paper, we discuss the existence of positive solutions of the secondorder delay boundary value problems. By applying the fixed-point theorem in a cone, we show the existence of at least one positive solution with singularity and the superlinear semipositone. As an demonstrate our results. application, we also give some examples todemonstrate our results.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
By employing the fixed point theorem of cone expansion and compression of norm type, we investigate the existence of positive solutions of generalized Sturm-Liouville boundary value problems for a nonlinear singular d...By employing the fixed point theorem of cone expansion and compression of norm type, we investigate the existence of positive solutions of generalized Sturm-Liouville boundary value problems for a nonlinear singular differential equation with a parameter. Some sufficient conditions for the existence of positive solutions are established. In the last section, an example is presented to illustrate the applications of our main results.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
The nonlinear two-point boundary value problem(TPBVP)is converted to a new form of the problem including integral terms.Then the combination of weak-form integral equation method(WFIEM)and special functions create a r...The nonlinear two-point boundary value problem(TPBVP)is converted to a new form of the problem including integral terms.Then the combination of weak-form integral equation method(WFIEM)and special functions create a robust and applicable numerical scheme to solve the problem.To display the accuracy of our method,some examples are investigated.Also,the fractional Boussinesq-like equation involving the β-derivative has been considered that describes the propagation of small amplitude long capillary-gravity waves on the surface of shallow water.展开更多
An iterative process of positive solution for BVP w'+h(t)f(w)=0, w(0)=w(1)= 0 is established, where h(t) is allowed to changes sign on [0,1]. The process starts from a simple function.
In this paper,we have proposed a numerical method for Singularly Perturbed Boundary Value Problems(SPBVPs)of convection-diffusion type of third order Ordinary Differential Equations(ODEs)in which the SPBVP is reduced ...In this paper,we have proposed a numerical method for Singularly Perturbed Boundary Value Problems(SPBVPs)of convection-diffusion type of third order Ordinary Differential Equations(ODEs)in which the SPBVP is reduced into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions.The numerical method combines boundary value technique,asymptotic expansion approximation,shooting method and finite difference scheme.In order to get a numerical solution for the derivative of the solution,the domain is divided into two regions namely inner region and outer region.The shooting method is applied to the inner region while standard finite difference scheme(FD)is applied for the outer region.Necessary error estimates are derived for the method.Computational efficiency and accuracy are verified through numerical examples.The method is easy to implement and suitable for parallel computing.展开更多
基金This work has been partially supported by the "Generalitat Valenciana" grant GV1118/93the Spanish D. G. I. C. Y.T. grant PB93-0381
文摘In this paper well-conditioning of boundary value problems for systems of second order difference equa-tions is studied.First,a sufficient condition for the existence of a unique bounded solution (for large enough number of steps) of an associated homogeneous system is given.Finally,a sufficient condition for well-condi-tioning,intrinsically related to the problem data is proposed.
基金heprojectissupportedbyNNSFofChina (No .1 9972 0 39) .
文摘In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference methods. It is proved that the method has optimal order error estimate O(h3) in H1 norm. Finally, two examples show that the method is effective.
文摘In this paper we study the forced oscillations of boundary value problems of a class of higher order functional partial differential equations.The principal tool is an everaging techniqe which enables one to establish oscillation in terms of related functional differential inequallities.
基金Supported by the Natural Science Foundation of Hunan Province（06JJ50008） Supported by the Natural Science Foundation of Guangdong Province（7004569）
文摘Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 are established,where [φ(x)] =(|x |p-2x) with p > 1.Our result is new even when [φ(x)] = x in above problem,i.e.p = 2.Examples are presented to illustrate the effciency of the theorem in this paper.
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.
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, solutions of Riemann boundary value problems with nodes are extended to the case where they may have singularties of high order at the nodes. Moreover, further extension is discussed when the free term of the problem involved also possesses singularities at the nodes. As an application, certain singular integral equation is discussed.
基金the Youth Research Foundation of Jiangxi University of Finance and Economics(No.04232015)the Technological Project Foundation of Jiangxi Province (Nos.GJJ08358GJJ08359)the Educational Reform Project Foundation of Jiangxi Province (No.JXJG07436)
文摘In this paper, we study the nonlinear second-order boundary value problem of delay differential equation.. Without the assumption of the nonnegativity of f, we still obtain the existence of the positive solution.
基金This work is supported by the National Natural Sciences Foundation of China(10471040) the Sciences Foundation of Shanxi (2005Z010) the Major Subject Foundation of Shanxi (20055024).
文摘The author employs the method of upper and lower solutions together with the monotone iterative technique to obtain the existence theorem of minimal and maximal solutions for a boundary value problem of second order impulsive differential equation.
文摘In this paper, we discuss the existence of positive solutions of the secondorder delay boundary value problems. By applying the fixed-point theorem in a cone, we show the existence of at least one positive solution with singularity and the superlinear semipositone. As an demonstrate our results. application, we also give some examples todemonstrate our results.
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 the National Natural Science Foundation of China (Grant No.10971046)the Natural Science Research Project of Anhui Province (Grant No.KJ2009B093)+1 种基金the Natural Science Foundation of Shandong Province(Grant No.ZR2009AM004)the Research Project of Bozhou Teachers College (Grant No.BSKY0805)
文摘By employing the fixed point theorem of cone expansion and compression of norm type, we investigate the existence of positive solutions of generalized Sturm-Liouville boundary value problems for a nonlinear singular differential equation with a parameter. Some sufficient conditions for the existence of positive solutions are established. In the last section, an example is presented to illustrate the applications of our main results.
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The nonlinear two-point boundary value problem(TPBVP)is converted to a new form of the problem including integral terms.Then the combination of weak-form integral equation method(WFIEM)and special functions create a robust and applicable numerical scheme to solve the problem.To display the accuracy of our method,some examples are investigated.Also,the fractional Boussinesq-like equation involving the β-derivative has been considered that describes the propagation of small amplitude long capillary-gravity waves on the surface of shallow water.
文摘An iterative process of positive solution for BVP w'+h(t)f(w)=0, w(0)=w(1)= 0 is established, where h(t) is allowed to changes sign on [0,1]. The process starts from a simple function.
文摘In this paper,we have proposed a numerical method for Singularly Perturbed Boundary Value Problems(SPBVPs)of convection-diffusion type of third order Ordinary Differential Equations(ODEs)in which the SPBVP is reduced into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions.The numerical method combines boundary value technique,asymptotic expansion approximation,shooting method and finite difference scheme.In order to get a numerical solution for the derivative of the solution,the domain is divided into two regions namely inner region and outer region.The shooting method is applied to the inner region while standard finite difference scheme(FD)is applied for the outer region.Necessary error estimates are derived for the method.Computational efficiency and accuracy are verified through numerical examples.The method is easy to implement and suitable for parallel computing.
