Positive solutions for nonlinear systems of third-order generalized sturm-liouville boundary value problems with $(p_1,p_2,ldots,p_n)$-laplacian

In this work, byemploying the Leggett-Williams fixed point theorem, we study theexistence of at least three positive solutions of boundary valueproblems for system of third-order ordinary differential equationswith $(p_1,p_2,ldots,p_n)$-Laplacianbegin{eqnarray*}left { begin{array}{ll} (phi_{p_i}(u_i''(t)))'  +  a_i(t) f_i(t,u_1(t), u_2(t), ldots, u_n(t)) =0 hspace{1cm} 0  leq t leq 1, alpha_i u_i(0) - beta_i u_i'(0) = mu_{i1} u_i(xi_i),hspace{0.2cm} gamma_i u_i(1) + delta_i u_i'(1) = mu_{i2} u_i(eta_i), hspace{0.5cm}  u_i''(0) = 0,end{array} right.end{eqnarray*}where $ phi_{p_i}(s) = |s|^{p_i-2}s,$, are $p_i$-Laplacianoperators, $p_i > 1, 0 < xi_i < 1, 0 < eta_i < 1$ and $mu_{i1},mu_{i2}> 0$ for $i = 1,2, ldots,n$.