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nonlinear boundary value problem; differential inclusion; measurable selector; Ky Fan’s fixed point theorem
The paper deals with the periodic boundary value problem (1) $L_4 x(t) + a(t)x(t) \in F(t,x(t))$, $t\in J= [a,b]$, (2) $L_i x(a)= L_i x(b)$, $i=0,1,2,3$, where $L_0x(t)= a_0x(t)$, $L_ix(t)=a_i(t)L_{i-1}x(t)$, $i=1,2,3,4$, $a_0(t)= a_4(t)=1$, $a_i(t)$, $i=1,2,3$ and $a(t)$ are continuous on $J$, $a(t)\geq 0$, $a_i(t)>0$, $i=1,2$, $a_1(t)= a_3(t)\cdot F(t,x): J \times R \to$\{nonempty convex compact subsets of $R$\}, $R= (-\infty , \infty )$. The existence of such periodic solution is proven via Ky Fan's fixed point theorem.
[1] Y. Kitamura: On nonoscillatory solutions of functional differential equations with a general deviating argument. Hiroshima Math. J. 8 (1978), 49-62. MR 0466865 | Zbl 0387.34048
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