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

Abstract

We use a technique associated with measures of noncompactness to prove the existence of nondecreasing solutions to an integral equation with linear modification of the argument in the space C[0, 1]. In the last thirty years there has been a great deal of work in the field of differential equations with a modified argument. A special class is represented by the differential equation with affine modification of the argument which can be delay differential equations or differential equations with linear modifications of the argument. In this case we study the following integral equation$x{\left( t \right)} = a{\left( t \right)} + {\left( {Tx} \right)}{\left( t \right)}{\int_0^{\sigma {\left( t \right)}} {u{\left( {t,s,x{\left( s \right)},x{\left( {\lambda s} \right)}} \right)}ds} }\;0 < \lambda < 1$ which can be considered in connection with the following Cauchy problem x'(t) = u(t, s, x(t), x(λt)), t ∈ [0, 1], 0 < λ < 1 x(0) = u ₀.