On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay
For each x0∈[0,2π) and k∈N, we obtain some existence theorems of periodic solutions to the two-point boundary value problem u′′(x)+k2u(x-x0)+g(x,u(x-x0))=h(x) in (0,2π) with u(0)-u(2π)=u′(0)-u′(2π)=0 when g:(0,2π)×R→R is a Caratheodory function which grows linearly in u as u→∞, and h∈L1(0,2π) may sa...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2018-01-01
|
| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2018/5321314 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849307550034427904 |
|---|---|
| author | Nai-Sher Yeh |
| author_facet | Nai-Sher Yeh |
| author_sort | Nai-Sher Yeh |
| collection | DOAJ |
| description | For each x0∈[0,2π) and k∈N, we obtain some existence theorems of periodic solutions to the two-point boundary value problem u′′(x)+k2u(x-x0)+g(x,u(x-x0))=h(x) in (0,2π) with u(0)-u(2π)=u′(0)-u′(2π)=0 when g:(0,2π)×R→R is a Caratheodory function which grows linearly in u as u→∞, and h∈L1(0,2π) may satisfy a generalized Landesman-Lazer condition (1+sign(β))∫02πh(x)v(x)dx<∫v(x)>0gβ+(x)vx1-βdx+∫v(x)<0gβ-(x)vx1-βdx for all v∈N(L)\{0}. Here N(L) denotes the subspace of L1(0,2π) spanned by sinkx and coskx, -1<β≤0, gβ+(x)=lim infu→∞(gx,uu/u1-β), and gβ-(x)=lim infu→-∞(gx,uu/u1-β). |
| format | Article |
| id | doaj-art-1e65f89842e04d5d969c1fb97c932c71 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-1e65f89842e04d5d969c1fb97c932c712025-08-20T03:54:43ZengWileyAbstract and Applied Analysis1085-33751687-04092018-01-01201810.1155/2018/53213145321314On Solvability Theorems of Second-Order Ordinary Differential Equations with DelayNai-Sher Yeh0Department of Mathematics, Fu Jen Catholic University, Xinzhuang District, New Taipei City 24205, TaiwanFor each x0∈[0,2π) and k∈N, we obtain some existence theorems of periodic solutions to the two-point boundary value problem u′′(x)+k2u(x-x0)+g(x,u(x-x0))=h(x) in (0,2π) with u(0)-u(2π)=u′(0)-u′(2π)=0 when g:(0,2π)×R→R is a Caratheodory function which grows linearly in u as u→∞, and h∈L1(0,2π) may satisfy a generalized Landesman-Lazer condition (1+sign(β))∫02πh(x)v(x)dx<∫v(x)>0gβ+(x)vx1-βdx+∫v(x)<0gβ-(x)vx1-βdx for all v∈N(L)\{0}. Here N(L) denotes the subspace of L1(0,2π) spanned by sinkx and coskx, -1<β≤0, gβ+(x)=lim infu→∞(gx,uu/u1-β), and gβ-(x)=lim infu→-∞(gx,uu/u1-β).http://dx.doi.org/10.1155/2018/5321314 |
| spellingShingle | Nai-Sher Yeh On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay Abstract and Applied Analysis |
| title | On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay |
| title_full | On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay |
| title_fullStr | On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay |
| title_full_unstemmed | On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay |
| title_short | On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay |
| title_sort | on solvability theorems of second order ordinary differential equations with delay |
| url | http://dx.doi.org/10.1155/2018/5321314 |
| work_keys_str_mv | AT naisheryeh onsolvabilitytheoremsofsecondorderordinarydifferentialequationswithdelay |