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Existence of unique solution to switched fractional differential equations with p-Laplacian operator

In this paper, we study a class of nonlinear switched systems of fractional order with p-Laplacian operator. By applying a fixed point theorem for a concave operator on a cone, we obtain the existence and uniqueness of a positive solution for an integral boundary value problem with switched nonlinearity under some suitable assumptions. | Turk J Math (2015) 39: 864 – 871 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1503-25 Research Article Existence of unique solution to switched fractional differential equations with p-Laplacian operator Xiufeng GUO∗ College of Sciences, Hezhou University, Hezhou, Guangxi, P.R. China Received: 07.03.2015 • Accepted/Published Online: 16.05.2015 • Printed: 30.11.2015 Abstract: In this paper, we study a class of nonlinear switched systems of fractional order with p -Laplacian operator. By applying a fixed point theorem for a concave operator on a cone, we obtain the existence and uniqueness of a positive solution for an integral boundary value problem with switched nonlinearity under some suitable assumptions. An illustrative example is included to show that the obtained results are effective. Key words: Existence, positive solution, fractional-order switched system, integral boundary valued problems, p Laplacian operator 1. Introduction In this paper, we consider an integral boundary value problem (BVP for short) for fractional differential equations with switched nonlinearity and p -Laplacian operator: β D ϕ (Dα+ u(t)) = fσ(t) (t, u(t), D0γ+ u(t)), t ∈ J = [0, 1], 0+ p ∫0 1 u(0) = µ u(s)ds + λu(ξ), 0 α D0+ u(0) = κD0α+ u(η), ξ, η ∈ [0, 1], (1.1) where ϕp is a p -Laplacian operator, p > 1, ϕp is invertible, and (ϕp )−1 = ϕq , 1/p + 1/q = 1 , D0α+ , D0β+ denote the Caputo fractional derivative of order α, β . 0 0 is defined by ∫ t (t − s)α−1 f (s)ds. I0α+ f (t) = Γ(α) 0 865 GUO/Turk J Math Definition 2.2 ([16, 10]) The Riemann–Liouville derivative of order α > 0 for a function f : [0, +∞) → R can be written as ∫ x 1 dn f (s) L α D0+ f (x) = ds, n Γ(n − α) dx 0 (x − s)α−n+1 where n is the smallest integer greater than α . Definition 2.3 ([16, 10]) The Caputo fractional derivative of order α > 0 for a function f : [0, +∞) → R can be written as [ D0α+ f (x) = L Dα 0+ f (x) .