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### Open Access Journals

$u_i(t)=int_0^Tg_i(t,s)[a_i(s,u_1(s),u_2(s),...,u_n(s))+b_i(s,u_1(s),u_2(s),...,u_n(s))]ds,$ $t \in [0,T],$ $1<=i<=n,$

where $T>0$ is fixed
and the nonlinearities $a_i(t,u_1,u_2,\cdots,u_n)$ can be *
singular* at $t=0$ and $u_j=0$ where $j\in\{1,2,\cdots,n\}.$
Criteria are established for the existence of * fixed-sign
solutions* $(u_1^*,u_2^*,\cdots,u_n^*)$ to the above system, i.e.,
$\theta_iu_i^*(t)\geq 0$ for $t\in [0,T]$ and $1\leq i\leq n,$ where
$\theta_i\in \{1,-1\}$ is fixed. We also include an example to
illustrate the usefulness of the results obtained.

$u^(''')_ i (t) = f_i(t, u_1(\phi_1(t)), u_2(\phi_2(t)), · · · , u_n(\phi_n(t))), t \in [a, b]$ $u_i(a) = u^'_i(z_i) = 0$, \gamma_i u_i(b) + \delta_iu^('')_i (b) = 0$

where $i$ = 1, 2, · · · , $n$, $1/2 (a + b) < z_i < b, \gamma_i > 0$, and $\phi_i$ are deviating arguments. By using some fixed point theorems, we establish the existence of one or more *fixed-sign* solutions $u = (u_1, u_2, · · · , u_n)$ for the system, i.e., for each 1 $<=$ $i$ $<=$ $n$, $\theta_iui(t) >= 0$ for $t \in [a, b]$, where $\theta_i \in$ {1,−1} is fixed. An example is also presented to illustrate the results obtained.

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