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

Liu, Xinzhi

J. Appl. Math. Simulation 2, No. 3, 185-198 (1989).

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

Keywords:

impulsive integro-differential equation; nonlinear boundary condition; upper and lower solutions; monotone iterative technique; multiple solutions; periodic boundary value problems