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The Existence of Positive Solutions for a Fourth‐Order Difference Equation with Sum Form Boundary Conditions

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Listed:
  • Yanping Guo
  • Xuefei Lv
  • Yude Ji
  • Yongchun Liang

Abstract

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(02)=u(n+)=∑i=1n+1g(i)u(i), aΔ2u(020)-bz()Δ3u()=∑i=3n+1h(i)Δ2u(i-2), aΔ2u(n)-bz(n+11)Δ3u(n-)=∑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.

Suggested Citation

  • Yanping Guo & Xuefei Lv & Yude Ji & Yongchun Liang, 2014. "The Existence of Positive Solutions for a Fourth‐Order Difference Equation with Sum Form Boundary Conditions," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:578672
    DOI: 10.1155/2014/578672
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    References listed on IDEAS

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    1. Jufang Wang & Changlong Yu & Yanping Guo, 2013. "Triple Positive Solutions of a Nonlocal Boundary Value Problem for Singular Differential Equations with p -Laplacian," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-7, March.
    2. Jufang Wang & Changlong Yu & Yanping Guo, 2013. "Triple Positive Solutions of a Nonlocal Boundary Value Problem for Singular Differential Equations with p‐Laplacian," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
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    Cited by:

    1. Yongyi Gu & Bingmao Deng & Jianming Lin, 2018. "Exact Traveling Wave Solutions to the (2 + 1)‐Dimensional Jaulent‐Miodek Equation," Advances in Mathematical Physics, John Wiley & Sons, vol. 2018(1).

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