The Existence of Positive Solutions for a Fourth-Order Difference Equation with Sum Form Boundary Conditions
We consider the fourth-order difference equation: Δ(z(k+1)Δ3u(k-1))=w(k)f(k,u(k)), k∈{1,2,…,n-1} subject to the boundary conditions: u(0)=u(n+2)=∑i=1n+1g(i)u(i), aΔ2u(0)-bz(2)Δ3u(0)=∑i=3n+1h(i)Δ2u(i-2), aΔ2u(n)-bz(n+1)Δ3u(n-1)=∑i=3n+1h(i)Δ2u(i-2), where a,b>0 and Δu(k)=u(k+1)-u(k) for k∈{0,1,…,n...
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/578672 |
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| author | Yanping Guo Xuefei Lv Yude Ji Yongchun Liang |
| author_facet | Yanping Guo Xuefei Lv Yude Ji Yongchun Liang |
| author_sort | Yanping Guo |
| collection | DOAJ |
| description | We consider the fourth-order difference equation: Δ(z(k+1)Δ3u(k-1))=w(k)f(k,u(k)), k∈{1,2,…,n-1} subject to the boundary conditions: u(0)=u(n+2)=∑i=1n+1g(i)u(i), aΔ2u(0)-bz(2)Δ3u(0)=∑i=3n+1h(i)Δ2u(i-2), aΔ2u(n)-bz(n+1)Δ3u(n-1)=∑i=3n+1h(i)Δ2u(i-2), where a,b>0 and Δu(k)=u(k+1)-u(k) for k∈{0,1,…,n-1}, f:{0,1,…,n}×[0,+∞)→[0,+∞) is continuous. h(i) is nonnegative i∈{2,3,…,n+2}; g(i) is nonnegative for i∈{0,1,…,n}. Using fixed point theorem of cone expansion and compression of norm type and Hölder’s inequality, various existence, multiplicity, and nonexistence results of positive solutions for above problem are derived, which extends and improves some known recent results. |
| format | Article |
| id | doaj-art-6eca737b92314551aeddbe6d384da230 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-6eca737b92314551aeddbe6d384da2302025-02-03T06:00:56ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/578672578672The Existence of Positive Solutions for a Fourth-Order Difference Equation with Sum Form Boundary ConditionsYanping Guo0Xuefei Lv1Yude Ji2Yongchun Liang3School of Electrical Engineering, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, ChinaCollege of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, ChinaCollege of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, ChinaSchool of Electrical Engineering, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, ChinaWe consider the fourth-order difference equation: Δ(z(k+1)Δ3u(k-1))=w(k)f(k,u(k)), k∈{1,2,…,n-1} subject to the boundary conditions: u(0)=u(n+2)=∑i=1n+1g(i)u(i), aΔ2u(0)-bz(2)Δ3u(0)=∑i=3n+1h(i)Δ2u(i-2), aΔ2u(n)-bz(n+1)Δ3u(n-1)=∑i=3n+1h(i)Δ2u(i-2), where a,b>0 and Δu(k)=u(k+1)-u(k) for k∈{0,1,…,n-1}, f:{0,1,…,n}×[0,+∞)→[0,+∞) is continuous. h(i) is nonnegative i∈{2,3,…,n+2}; g(i) is nonnegative for i∈{0,1,…,n}. Using fixed point theorem of cone expansion and compression of norm type and Hölder’s inequality, various existence, multiplicity, and nonexistence results of positive solutions for above problem are derived, which extends and improves some known recent results.http://dx.doi.org/10.1155/2014/578672 |
| spellingShingle | Yanping Guo Xuefei Lv Yude Ji Yongchun Liang The Existence of Positive Solutions for a Fourth-Order Difference Equation with Sum Form Boundary Conditions Abstract and Applied Analysis |
| title | The Existence of Positive Solutions for a Fourth-Order Difference Equation with Sum Form Boundary Conditions |
| title_full | The Existence of Positive Solutions for a Fourth-Order Difference Equation with Sum Form Boundary Conditions |
| title_fullStr | The Existence of Positive Solutions for a Fourth-Order Difference Equation with Sum Form Boundary Conditions |
| title_full_unstemmed | The Existence of Positive Solutions for a Fourth-Order Difference Equation with Sum Form Boundary Conditions |
| title_short | The Existence of Positive Solutions for a Fourth-Order Difference Equation with Sum Form Boundary Conditions |
| title_sort | existence of positive solutions for a fourth order difference equation with sum form boundary conditions |
| url | http://dx.doi.org/10.1155/2014/578672 |
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