On the Solvability of a Resonant Third-Order Integral m-Point Boundary Value Problem on the Half-Line

Abstract

In this work, the existence of at least one solution for the following third-order integral and $m$-point boundary value problem on the half-line at resonance ${\left(\rho \left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}\text{'}}=w\left(t,u\left(t\right),u^{\text{'}}\left(t\right),u^{\text{'}\text{'}}\left(t\right)\right),t\in [0,\infty ),u\left(0\right)={\sum }_{j=1}^m{\alpha }_j{\int }_0^{{\eta }_j}u\left(t\right)dt,u^{\text{'}}\left(0\right)=0,\underset{t⟶\infty }{\lim }{\left(\rho \left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}=0,$ will be investigated. The Mawhin’s coincidence degree theory will be used to obtain existence results while an example will be used to validate the result obatined.