## First-order impulsive ordinary differential equations with advanced arguments.(English)Zbl 1122.34042

Consider the boundary value problem $x'(t)= f(t, x(t), x(\alpha(t))\quad\text{for }[0,T]\setminus \{t_1,t_2,\dots, t_m\},$
$\Delta x(t_k)= I_k(x(t_k))\quad\text{for }k= 1,\dots, m,\tag{$$*$$}$
$0= g(x(0), x(T)),$ where $$f$$, $$\alpha$$, $$g$$ and $$I_k$$ $$(1\leq k\leq m)$$ are continuous functions, $$0\leq t\leq\alpha(t)\leq T$$, $$\Delta x(t_k)= x(t^+_k)- x(t^-_k)$$.
Using the method of lower and upper solutions in reversed order and the method of coupled lower and upper solutions, the author establishes conditions guaranteeing the existence of a solution and of a quasi-solution to $$(*)$$, respectively.

### MSC:

 34K10 Boundary value problems for functional-differential equations 34K45 Functional-differential equations with impulses 34K07 Theoretical approximation of solutions to functional-differential equations
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### References:

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