Abstract
In this paper, we consider a p-Laplacian eigenvalue boundary value problem involving both right Caputo and left Riemann-Liouville types fractional derivatives. To prove the existence of solutions, we apply the Schaefer’s fixed point theorem. Furthermore, we present the Lyapunov inequality for the corresponding problem.
Highlights
We prove the existence of solutions for an iterated fractional boundary value problem
−C Dbα− φp Daβ+u (t) + χ(t)φp (u(t)) = 0, a < t < b u(i) (a) = Daβ++iu (b) = 0, i = 0, 1, ..., n − 1, where n − 1 < α, β ≤ n, n ≥ 2, C Dbα− and Daβ+ refer to the right Caputo derivative and the left Riemann—Liouville derivative respectevely, φp(s) =
The problems generated by equations involving both left and right fractional derivatives, arise in the study of Euler—Lagrange equations for fractional problems of calculus of variations, see [4,14,15]
Summary
We prove the existence of solutions for an iterated fractional boundary value problem (1.1). U(i) (0) = D0β++iu (1) = 0 , i = 0, 1, ..., n − 1, we obtain Lyapunov type inequalities for the corresponding problem (1.2). The problems generated by equations involving both left and right fractional derivatives, arise in the study of Euler—Lagrange equations for fractional problems of calculus of variations, see [4,14,15]. This type of problems has been considered by many authors, see [1,2,3,5,6, 7,8,9,10,12,16]. The main results are Theorem 2, which establishes existence of solution for the eigenvalue problem for left and right fractional differential equation 1.1, and Theorem t-lyapunov where, we obtain a new Lyapunov type inequality for problem
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