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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
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