Boundary value problems for integro-differential and singular higher-order differential equations

We investigate third-order strongly nonlinear differential equations of the type ( Phi ( k ( t ) u '' ( t ) ) ) ' = f ( t , u ( t ) , u ' ( t ) , u '' ( t ) ) , a.e. on [ 0 , T ] , \left(\Phi \left(k\left(t){u}<^>{<^>{\prime\prime} }\left(t)))<^>{\prime} =f\left(t,u\left(t),u<^>{\prime} \left(t),{u}<^>{<^>{\prime\prime} }\left(t)),\hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}\left[0,T], where Phi \Phi is a strictly increasing homeomorphism, and the non-negative function k k may vanish on a set of measure zero. Using the upper and lower solution method, we prove existence results for some boundary value problems associated with the aforementioned equation. Moreover, we also consider second-order integro-differential equations like ( Phi ( k ( t ) v ' ( t ) ) ) ' = f t , integral 0 t v ( s ) d s , v ( t ) , v ' ( t ) , a.e. on [ 0 , T ] , \left(\Phi \left(k\left(t)v<^>{\prime} \left(t)))<^>{\prime} =f\left(t,\underset{0}{\overset{t}{\int }}v\left(s){\rm{d}}s,v\left(t),v<^>{\prime} \left(t)\right),\hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}\left[0,T], for which we provide existence results for various types of boundary conditions, including periodic, Sturm-Liouville, and Neumann-type conditions.