| Multiple Solutions for Discrete First-Order Periodic Problems |
| Received:October 25, 2012 Revised:January 13, 2014 |
| Key Words:
discrete first-order periodic problems Landesman-Lazer type condition Leray-Schauder degree bifurcation existence.
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| Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.11326127; 11101335), the Science Research Project of Gansu University (Grant No.2013A-001) and NWNU-LKQN-11-23. |
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| Abstract: |
| Let $T>1$ be an integer, $\mathbb{T}=\{0,1,2,\ldots,T-1\}$. This paper is concerned with the existence of periodic solutions of the discrete first-order periodic boundary value problems $$\Delta u(t)-a(t)u(t)=\lambda u(t)+f(u(t-\tau(t)))-h(t),~~t\in\mathbb{T},$$ $$u(0)=u(T),$$ where $\Delta u(t)=u(t+1)-u(t)$, $a:\mathbb{T}\to\mathbb{R}$ and satisfies $\prod^{T-1}_{t=0}(1+a(t))=1$, $\tau:\mathbb{T}\to\mathbb{Z}$ $t-\tau(t)\in\mathbb{T}$ for $t\in\mathbb{T}$, $f:\mathbb{R}\to\mathbb{R}$ is continuous and satisfies Landesman-Lazer type condition and $h:\mathbb{T}\to\mathbb{R}$. The proofs of our main results are based on the Rabinowitz's global bifurcation theorem and Leray-Schauder degree. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2014.03.009 |
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