On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions

Abstract

We investigate sufficient conditions for existence and uniqueness of solutions for a coupled system of fractional order hybrid differential equations (HDEs) with multi-point hybrid boundary conditions given by

D-omega(x(t)/H(t, x(t), z(t))) = -K-1 (t, x(t), z(t)), omega epsilon (2, 3],

D-epsilon(z(t)/G(t, x(t), z(t))) = -K-2 (t, x(t), z(t)), epsilon epsilon(2, 3]

x(t)/H(t, x(t), z(t))vertical bar(t=1) = 0, D-mu(x(t)/H(t, x(t), z(t)))vertical bar(t=delta 1) =0, x((2))(0) = 0

z(t)/G(t, x(t), z(t))vertical bar(t=1) = 0, D-nu(z(t)/G(t, x(t), z(t)))vertical bar(t=delta 2) =0, z((2))(0) = 0

where t epsilon [0, 1], delta(1), delta(2), mu, upsilon epsilon (0, 1), and D-omega, D-epsilon, D-mu and D-upsilon are Caputo's fractional derivatives of order omega, is an element of, mu and nu, respectively, K-1, K-2 epsilon C([0, 1] x R x R, R) and G, H epsilon C([0, 1] x R x R, R - {0}). We use classical results due to Dhage and Banach's contraction principle (BCP) for the existence and uniqueness of solutions. For applications of our results, we include examples.