文章基本信息

标题：Some results on the fractional order Sturm-Liouville problems
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作者：Yuanfang Ru ; Fanglei Wang ; Tianqing An 等
期刊名称：Advances in Difference Equations
印刷版ISSN：1687-1839
电子版ISSN：1687-1847
出版年度：2017
卷号：2017
期号：1
页码：320
DOI：10.1186/s13662-017-1377-x
语种：English
出版社：Hindawi Publishing Corporation
摘要：In this work, we introduce some new results on the Lyapunov inequality, uniqueness and multiplicity results of nontrivial solutions of the nonlinear fractional Sturm-Liouville problems { D 0 + q ( p ( t ) u ′ ( t ) ) + Λ ( t ) f ( u ( t ) ) = 0 , 1 < q ≤ 2 , t ∈ ( 0 , 1 ) , α u ( 0 ) − β p ( 0 ) u ′ ( 0 ) = 0 , γ u ( 1 ) + δ p ( 1 ) u ′ ( 1 ) = 0 , $$\textstyle\begin{cases} D_{0^{+}}^{q} (p(t)u'(t))+\Lambda(t)f(u(t))=0,\quad1 < q\leq2, t\in (0,1), \\ \alpha u(0)-\beta p(0)u'(0)=0,\qquad\gamma u(1)+\delta p(1)u'(1)=0, \end{cases} $$ where α, β, γ, δ are constants satisfying 0 ≠ β γ + α γ ∫ 0 1 1 p ( τ ) d τ + α δ < + ∞ $0\neq \vert\beta\gamma+\alpha\gamma\int_(^)\frac){p(\tau)}\,d\tau +\alpha \delta\vert<+\infty$ , p ( ⋅ ) $p(\cdot)$ is positive and continuous on [ 0 , 1 ] $[0,1]$ . In addition, some existence results are given for the problem { D 0 + q ( p ( t ) u ′ ( t ) ) + Λ ( t ) f ( u ( t ) , λ ) = 0 , 1 < q ≤ 2 , t ∈ ( 0 , 1 ) , α u ( 0 ) − β p ( 0 ) u ′ ( 0 ) = 0 , γ u ( 1 ) + δ p ( 1 ) u ′ ( 1 ) = 0 , $$\textstyle\begin{cases} D_{0^{+}}^{q} (p(t)u'(t))+\Lambda(t)f(u(t),\lambda)=0,\quad1 < q\leq2, t\in (0,1), \\ \alpha u(0)-\beta p(0)u'(0)=0,\qquad\gamma u(1)+\delta p(1)u'(1)=0, \end{cases} $$ where λ ≥ 0 $\lambda\geq0$ is a parameter. The proof is based on the fixed point theorems and the Leray-Schauder nonlinear alternative for single-valued maps.
关键词：fractional differential equations ; Sturm-Liouville problems ; Lyapunov inequality ; fixed point theorem