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Existence of solutions to singular integral equations

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  • We consider the system of integral equations

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

    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.

    Mathematics Subject Classification: Primary: 45B05, 45G15; Secondary: 45M20.


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