FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

12.07.2015 Views
68 T. S. Aleroev, H. T. Aleroeva, Ning-Ming Nie, and Yi-Fa TangLet S be a Banach space, T : S ↦→ S be a mapping, and ‖ · ‖ denote thenorm of S.Definition 3.3 (see [25]). If there exists a constant ρ (0 ≤ ρ < 1)such that‖T x − T y‖ ≤ ρ‖x − y‖ (3.18)for any x, y ∈ S, then T is said to be a contractive mapping of S.Lemma 3.2 (Contractive Mapping Principle) (see [25]). If T isa contractive mapping of a Banach space S, then there exists a unique fixedpoint y ∈ S satisfying y = T y.2. Existence and Uniqueness of the Solutions for FBVPsIn this section, the existence and uniqueness of the solutions for FBVPs(3.7)–(3.8), (3.9)–(3.10), (3.1)–(3.2) and (3.3)–(3.4) are studied. Withoutloss of generality, only the case of homogeneous boundary conditions α =β = 0 in the above four kinds of FBVPs are considered.Lemma 3.3. (1) FBVP (3.1)–(3.2) is equivalent toy(t) =∫ baG(t, s)f(s, y(s)) ds, (3.19)where G(t, s) is called the fractional Green function defined as follows:⎧(t − a)(b − s)⎪⎨γ−1 (t − s)γ−1− , a ≤ s ≤ t ≤ b,(b − a)Γ(γ) Γ(γ)G(t, s) =(t − a)(b − s) ⎪⎩γ−1(3.20), a ≤ t ≤ s ≤ b.(b − a)Γ(γ)(2) FBVP (3.7)–(3.8) is equivalent toy(t) =∫ baĜ(t, s)f(s, y(s)) ds, (3.21)where Ĝ(t, s) is the fractional Green function defined as follows:⎧(t − a)⎪⎨γ−1 ( b − s) γ−1 (t − s) γ−1− , a ≤ s ≤ t ≤ b,Ĝ(t, s) =Γ(γ) b − a Γ(γ)(t − a) ⎪⎩γ−1 ( b − s) γ−1(3.22), a ≤ t ≤ s ≤ b.Γ(γ) b − a(3) FBVP (3.3)–(3.4) is equivalent toy(t) =∫ baG(t, s)g ( s, y(s), C a D θ sy(s) ) ds, (3.23)where G(t, s) is the fractional Green function defined in (3.20).

Boundary Value Problems for Differential Equations of Fractional Order 69(4) FBVP (3.9)–(3.10) is equivalent toy(t) =∫ baĜ(t, s)g ( s, y(s), RLa D θ sy(s) ) ds, (3.24)where Ĝ(t, s) is the fractional Green function defined in (3.22).Proof. We only give the proof for (1). The case for (3) is similar to that for(1). The proofs of (2) and (4) are referred to [33].According to Lemma 3.1(4), applying the operator J γ a to both sides ofthe equation in (3.1) yieldsy(t) − y(a) − y ′ (a)(t − a) + J γ a f(t, y(t)) = 0. (3.25)Since y(a) = y(b) = 0, from (3.25) one easily obtainsand theny ′ (a) =∫ by(t) = y ′ (a)(t − a) − J γ a f(t, y(t)) ===∫ ba∫ baa(b − s) γ−1f(s, y(s))ds, (3.26)(b − a)Γ(γ)(t − a)(b − s) γ−1∫ tf(s, y(s)) ds −(b − a)Γ(γ)aG(t, s)f(s, y(s)) ds.Conversely, applying the operator C a D γ t∫ bCa D γ t y(t) = C a D γ t=∫ baaG(t, s)f(s, y(s)) ds =(t − s) γ−1f(s, y(s)) ds =Γ(γ)to both sides of (3.19) yields(b − s) γ−1(b − a)Γ(γ) f(s, y(s)) dsC a D γ t (t − a) − C a D γ t J γ a f(t, y(t)) == −f(t, y(t)),and the homogeneous boundary condition is verified easily.LetP = C 0 [a, b],P 1 = C 1 [a, b] := { y(t) : y(t), y ′ (t) ∈ C 0 [a, b] } ,{}P 2 = C θ [a, b] := y(t) : y(t) ∈ C 0 [a, b], RLa Dt θ y(t) ∈ C 0 [a, b] .□

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