### Abstract

This paper is concerned with semilinear tow-point boundary value problems of the form -(p(x)u′)′ + f(x, u) = 0, a ≤ x ≤ b, α_{0}u(a) - α_{1}u′(a) = α, β_{1}u′(b) + β,_{1}u′(b) = β, α_{i} ≥ 0, β_{i}≥ 0, i = 0, 1, α_{0}+α_{1} > 0, β_{0}+β _{1} > 0, α_{0}+β_{0} > 0. Under the assumption inf f_{u} > -λ_{1}, where λ_{1} is the smallest eigenvalue of u = -(pu′)′ with the boundary conditions, unique existence theorems of solution for the continuous problem and a discretized system with not necessarily uniform nodes are given as well as error estimates. The results generalize three theorems of Lees for u″ = f(x, u), 0 ≤ x ≤ 1, u(0) = α, u(1) = β.

Original language | English |
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Pages (from-to) | 293-313 |

Number of pages | 21 |

Journal | Japan Journal of Industrial and Applied Mathematics |

Volume | 23 |

Issue number | 3 |

Publication status | Published - 2006 Oct |

### Fingerprint

### Keywords

- Discretization
- Error estimates
- Existence of solution
- Theorems of Lees
- Tow-point boundary value problems

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**On three theorems of lees for numerical treatment of semilinear two-point boundary value problems.** / Yamamoto, Tetsuro; Oishi, Shinichi.

Research output: Contribution to journal › Article

*Japan Journal of Industrial and Applied Mathematics*, vol. 23, no. 3, pp. 293-313.

}

TY - JOUR

T1 - On three theorems of lees for numerical treatment of semilinear two-point boundary value problems

AU - Yamamoto, Tetsuro

AU - Oishi, Shinichi

PY - 2006/10

Y1 - 2006/10

N2 - This paper is concerned with semilinear tow-point boundary value problems of the form -(p(x)u′)′ + f(x, u) = 0, a ≤ x ≤ b, α0u(a) - α1u′(a) = α, β1u′(b) + β,1u′(b) = β, αi ≥ 0, βi≥ 0, i = 0, 1, α0+α1 > 0, β0+β 1 > 0, α0+β0 > 0. Under the assumption inf fu > -λ1, where λ1 is the smallest eigenvalue of u = -(pu′)′ with the boundary conditions, unique existence theorems of solution for the continuous problem and a discretized system with not necessarily uniform nodes are given as well as error estimates. The results generalize three theorems of Lees for u″ = f(x, u), 0 ≤ x ≤ 1, u(0) = α, u(1) = β.

AB - This paper is concerned with semilinear tow-point boundary value problems of the form -(p(x)u′)′ + f(x, u) = 0, a ≤ x ≤ b, α0u(a) - α1u′(a) = α, β1u′(b) + β,1u′(b) = β, αi ≥ 0, βi≥ 0, i = 0, 1, α0+α1 > 0, β0+β 1 > 0, α0+β0 > 0. Under the assumption inf fu > -λ1, where λ1 is the smallest eigenvalue of u = -(pu′)′ with the boundary conditions, unique existence theorems of solution for the continuous problem and a discretized system with not necessarily uniform nodes are given as well as error estimates. The results generalize three theorems of Lees for u″ = f(x, u), 0 ≤ x ≤ 1, u(0) = α, u(1) = β.

KW - Discretization

KW - Error estimates

KW - Existence of solution

KW - Theorems of Lees

KW - Tow-point boundary value problems

UR - http://www.scopus.com/inward/record.url?scp=33845987554&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33845987554&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:33845987554

VL - 23

SP - 293

EP - 313

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

SN - 0916-7005

IS - 3

ER -