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).


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