Positive Solutions to Singular and Delay Higher-Order Differential Equations on Time Scales
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Research Article Positive Solutions to Singular and Delay Higher-Order Differential Equations on Time Scales Liang-Gen Hu,1 Ti-Jun Xiao,2 and Jin Liang3 1
Department of Mathematics, University of Science and Technology of China, Hefei 230026, China School of Mathematical Sciences, Fudan University, Shanghai 200433, China 3 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China 2
Correspondence should be addressed to Jin Liang, [email protected] Received 21 March 2009; Accepted 1 July 2009 Recommended by Juan Jos´e Nieto We are concerned with singular three-point boundary value problems for delay higher-order dynamic equations on time scales. Theorems on the existence of positive solutions are obtained by utilizing the fixed point theorem of cone expansion and compression type. An example is given to illustrate our main result. Copyright q 2009 Liang-Gen Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction In this paper, we are concerned with the following singular three-point boundary value problem BVP for short for delay higher-order dynamic equations on time scales: −1n uΔ t wtft, ut − c, 2n
ut ψt, uΔ a − βi1 uΔ 2i
t ∈ a − c, a,
2i1
a αi1 uΔ ,
γi1 uΔ uΔ b, 2i
t ∈ a, b,
2i
2i
1.1
0 ≤ i ≤ n − 1,
where c ∈ 0, b − a/2, ∈ a, b, βi ≥ 0, 1 < γi < b − a βi / − a βi , 0 ≤ αi < b − γi γi − 1a − βi /b − , i 1, 2, . . . , n and ψ ∈ Ca − c, a. The functional w : a, b → 0, ∞ is continuous and f : a, b × 0, ∞ → 0, ∞ is continuous. Our
2
Boundary Value Problems
nonlinearity w may have singularity at t a and/or t b, and f may have singularity at u 0. To understand the notations used in 1.1, we recall the following definitions which can be found in 1, 2. a A time scale T is a nonempty closed subset of the real numbers R. T has the topology that it inherits from the real numbers with the standard topology. It follows that the jump operators σ, ρ : T → T, σt inf{τ ∈ T : τ > t},
ρt sup{τ ∈ T : τ < t}
1.2
supplemented by inf ∅ : sup T and sup ∅ : inf T are well defined. The point t ∈ T is left-dense, left-scattered, right-dense, right-scattered if ρt t, ρt < t, σt t, σt < t, respectively. If T has a left-scattered maximum t1 right-scattered minimum t2 , define Tk T − {t1 } Tk T − {t2 }; otherwise, set Tk T Tk T. By an interval a, b we always mean the intersection of the real interval a, b with the given time scale, that is, a, b ∩ T. Other types of intervals are defined similarly. b For a function f : T → R and t ∈ Tk , the Δ-derivative of f at t, denoted by f Δ t, is the number provided it exists with the property that, given any ε > 0, there is a neighborhood U ⊂ T of t such that fσt − fs − f Δ tσt − s ≤ ε|σt − s|,
∀s
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