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$\phi $-Laplacian; $L^1$-Carath\'eodory function; Schauder fixed-point Theorem.
We study the existence of solutions of the system $$ \begin {cases} (\phi _1(u_1'(t)))'= f_1(t,u_1(t),u_2(t),u'_1(t),u_2'(t)),\qquad \text {a.e. $t\in [0,T]$}, (\phi _2(u_2'(t)))'= f_2(t,u_1(t),u_2(t),u'_1(t),u_2'(t)),\qquad \text {a.e. $t\in [0,T]$}, \end {cases} $$ submitted to nonlinear coupled boundary conditions on $[0,T]$ where $\phi _1,\phi _2\colon (-a, a)\rightarrow \mathbb {R}$, with $0 < a < +\infty $, are two increasing homeomorphisms such that $\phi _1(0) = \phi _2(0) = 0$, and $f_i:[0,T]\times \mathbb {R}^{4}\rightarrow \mathbb {R}$, $i\in \{1,2\}$ are two $L^1$-Carathéodory functions. Using some new conditions and Schauder fixed point Theorem, we obtain solvability result.
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