Nonlinear boundary value problems for first order impulsive integro- differential equations.(English)Zbl 0688.45015

Consider the impulsive integro-differential equation $$(1)\quad x'=f(t,x,Tx),$$ $$t\neq t_ k$$, $$t\in J$$, $$\Delta x|_{t=t_ k}=I_ k(x),$$ $$k=1,2,...,p$$, with the following nonlinear boundary condition (2) $$q(x(a),x(b))=0$$, where $$J=[a,b]$$, $$a<t_ 1<t_ 2<...<t_ p<b$$, f: $$J\times R\times R\to R$$ and $$I_ k: R\to R$$ for each $$k=1,2,...,p$$. T is a Volterra operator from PC[J,R] into PC[J,R], where $$PC[J,R]=\{y: J\to R:$$ y(t) is continuous at $$t\neq t_ k$$, $$y(t^-)$$ and $$y(t^+)$$ exist and $$y(t^-)=y(t)$$ for $$t=t_ k$$, $$k=1,2,...,p\}$$. Assume that T is continuous and monotone nondecreasing and for any bounded set $$A\subset PC[J,R],$$ TA is bounded. The author proves, with the help of upper and lower solutions, that the boundary value problem (1)-(2) has a solution lying between the upper and lower solutions. He also develops a monotone iterative technique and shows the existence of multiple solutions for a class of periodic boundary value problems.
Reviewer: Z.C.Wang

MSC:

 45J05 Integro-ordinary differential equations 45G10 Other nonlinear integral equations
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