Existence theory of fractional order three-dimensional differential system at resonance

This paper deals with three-dimensional differential system of nonlinear fractional order problem D-0(alpha+) upsilon(%) = f(%, omega(%), omega l(%), omega'0(%), ..., omega(n-1)(%)), % is an element of (0, 1), D-0(beta+)nu(%) = g(%, upsilon(%), upsilon 0(%), upsilon 0f(%), ..., upsilon(n-1)(%)), % is an element of (0, 1), D-0+(gamma)omega(%) = h(%, nu(%), nu'(%), nu"(%), ..., nu(n-1)(%)), % is an element of (0, 1), with the boundary conditions, upsilon(0) = upsilon 0(0) = ... = upsilon(n-2)(0) = 0, upsilon(n-1)(0) = upsilon(n-1)(1), nu(0) = nu'(0) = ... = nu(n-2)(0) = 0, nu(n-1)(0) = nu(n-1)(1), omega(0) = omega r(0) = ... = omega(n-2)(0) = 0, omega(n-1)(0) = omega(n-1)(1), where D-0+(alpha) , D-beta(0+), D-0+(gamma) are the standard Caputo fractional derivative, n - 1 < alpha, beta, gamma <= n, n >= 2 and we derive sufficient conditions for the existence of solutions to the fraction order three-dimensional differential system with boundary value problems via Mawhin's coincidence degree theory, and some new existence results are obtained. Finally, an illustrative example is presented.