Existence of Nondecreasing and Continuous Solutions of an Integral Equation with Linear Modification of the Argument

<正> We use a technique associated with measures of noncompactness to prove the existenceof nondecreasing solutions to an integral equation with linear modification of the argument in thespace C[0,1].In the last thirty years there has been a great deal of work in the field of differentialequations with a modified argument.A special class is represented by the differential equation withaffine modification of the argument which can be delay differential equations or differential equationswith linear modifications of the argument.In this case we study the following integral equation x（t）=a（t）+（Tx）(t)∫0σ（t）u(t,s,x（s）,x（λs）)ds 0<λ<1 which can be considered in connection with thefollowing Cauchy problem x′（t）=u(t,s,x（t）,x（λt）),t ∈[0,1],0<λ<1 x（0）=u0.