Multiple positive solutions for a class of nonlinear integro-differential equations in Banach spaces
The author investigates the boundary value problem (BVP) of mixed type described by \[ \begin{gathered} -u^{(n)}(t)= f(t, u(t), u'(t),\dots, u^{(n-1)}(t), (Tu)(t), (Su)(t))\;\forall t\in J,\\ u^{(i)}(0)= \theta\,(i= 0,1,\dots, n-2),\quad u^{(n-1)}(\infty)= \theta\end{gathered} \] with \(J = [0,\infty)\), and \(f\) continuous in all its arguments on \(J\times P\times P\times\cdots\times P\), \(P\), where \(P\) stands for a cone in a real Banach space \(E\), while \[ (Tu)(t)= \int^t_0 K(t, s)u(s)\,ds,\quad (Su)(t)= \int^\infty_0 H(t,s) u(s)\,ds. \] After proving three auxuliary lemmas, the author states and proves two existence theorems for the above formulated BVP. The hypotheses are rather technical and cannot be reproduced here. The method of subsolutions and supersolutions is used. Fixed point theorems provide the tools.
higher-order integro-differential equation
method of subsolutions and supersolutions
nonlinear integro-differential equations
scientific article; zbMATH DE number 2105567