Multiple positive solutions for first order nonlinear integro-differential equations in Banach spaces
Using the fixed point index theory some results on the existence of multiple positive solutions of the following problem are obtained: \[ (1)\quad u'(t)= f\bigl(t,u(t), (Tu)(t),(Su)(t)\bigr),\;t\in\mathbb{R}_+,\qquad (2)\quad u (\infty) =\beta u(0). \] The problem (1)--(2) is considered in a real Banach space \(E\) with a normal solid cone \(P\) defining a partial ordering in \(E\). It is assumed that \(f\in C(\mathbb{R}_+\times P\times P\times P,P)\), \(\beta>1\) and \[ (Tu)(t)=\int^t_0 k(t,s)u(s)ds, \quad(Su)(t)=\int^\infty_0 H(t,s)u(s)ds. \] Apart from this some additional conditions are assumed. The obtained results are illustrated by two interesting examples.
scientific article; zbMATH DE number 1902842