Monotone and Concave Positive Solutions to a Boundary Value Problem for Higher-Order Fractional Differential Equation
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چکیده
and Applied Analysis 3 Definition 2.2 see 12 . The Caputo fractional derivative for a function y : 0,∞ → R can be written as D 0 y t 1 Γ n − α ∫ t 0 t − s n−α−1y n s ds, 2.2 where n α 1, α denotes the integer part of real number α. According to the definitions of fractional calculus, we can obtain that the fractional integral and the Caputo fractional derivative satisfy the following Lemma. Lemma 2.3 see 13 . Assume that u ∈ C 0, 1 and ρ ∈ m − 1, m , m ∈ N and v ∈ C1 0, 1 . Then, for t ∈ 0, 1 , a D 0 I ρ 0 v t v t , b I 0 D ρ 0 u t u t − ∑m−1 k 0 u k 0 /k! t, c limt→ 0 D ρ 0 u t limt→ 0 I ρ 0 u t 0. Definition 2.4. Let E be a real Banach space over R. A nonempty convex closed set p ⊂ E is said to be a cone, provided that a au ∈ P, for all u ∈ P, a 0, b u,−u ∈ P, implies u 0. Definition 2.5. Let E be a real Banach space and P ⊂ E a cone. A function φ : P → 0,∞ is called a nonnegative continuous concave functional if φ is continuous and φ ( λx 1 − λ y λφ x 1 − λ φy, 2.3 for all x, y ∈ P and 0 λ 1. Lemma 2.6 see 14 . Let E be a Banach space, K ⊆ E a cone in E, and Ω1, Ω2 two bounded open subsets of E with 0 ∈ Ω1 and Ω1 ⊂ Ω2. Suppose that T : K ∩ Ω2 \ Ω1 → K is continuous and completely continuous such that either i ‖Tu‖ ‖u‖ for u ∈ K ∩ ∂Ω1, ‖Tu‖ ‖u‖ for u ∈ K ∩ ∂Ω2, or ii ‖Tu‖ ‖u‖ for u ∈ K ∩ ∂Ω1, ‖Tu‖ ‖u‖ for u ∈ K ∩ ∂Ω2 holds. Then, T has a fixed point in K ∩ Ω2 \Ω1 . Let b, d, r > 0 be constants, Pr {u ∈ P : ‖u‖ < r}, P φ, b, d {u ∈ P : b φ u , ‖u‖ d}. 4 Abstract and Applied Analysis Lemma 2.7 see 15 . Let P be a cone in real Banach space E. Let T : Pc → Pc be a completely continuous map and φ a nonnegative continuous concave functional on P such that φ u ‖u‖, for all u ∈ Pc. Suppose that there exist constants a, b, d with 0 < a < b < d c such that i { u ∈ Pφ, b, d : φ u > b/ ∅, φ Tu > b ∀u ∈ P ( φ, b, d ) , ii ‖Tu‖ < a ∀u ∈ Pa, iii φ Tu > b, ∀u ∈ Pφ, b, c with ‖Tu‖ > d. 2.4 Then, T has at least three fixed points u1, u2, and u3 satisfying ‖u1‖ < a, b < φ u2 , ‖u3‖ > a, φ u3 < b. 2.5 Lemma 2.8. Assume that f t, u ∈ C 0, 1 × 0, ∞ , 0, ∞ , then u ∈ C 0, 1 be a solution of fractional boundary value problem 1.1 if and only if u ∈ C 0, 1 is a solution of integral equation u t ∫1 0 G t, s f s, u s ds, 2.6
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تاریخ انتشار 2014