## Integral boundary value problems for first order impulsive functional differential equations.(English)Zbl 1156.34050

The paper studies the first-order functional differential equation
$x'(t)=f(t,x(t),x(\theta(t))),\quad t\not=t_k,\;t\in [0,T], \tag{1}$
where $$0<t_1<\dots<t_m<T$$, $$f$$ and $$\theta$$ are continuous with $$0\leq\theta(t)<t$$ for $$t\in[0,T]$$. Equation (1) is subject to impulsive conditions and integral boundary conditions
$\Delta x(t_k)=I_k(x(t_k)),\;k=1,\dots,m,\quad x(0)+\mu \int_0^Tx(s)ds=x(T). \tag{2}$
Here, $$\mu\leq 0$$, $$I_k$$ is continuous and $$\Delta x(t_k)=x(t_k^+)-x(t_k^-)$$ denotes a jump of $$x$$ at $$t_k$$.
The authors prove the existence of solutions of problem (1), (2) by using the lower and upper solutions method combined with the monotone iterative technique. In presence of a lower solution $$\alpha$$ and an upper solution $$\beta$$ with $$\alpha\leq \beta$$ and under further suitable conditions, they construct monotone sequences which converge to solutions of problem (1), (2).

### MSC:

 34K10 Boundary value problems for functional-differential equations 34K45 Functional-differential equations with impulses