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

Abstract

We investigate third-order strongly nonlinear differential equations of the type (Φ(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 Φ\Phi is a strictly increasing homeomorphism, and the non-negative function kk 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 (Φ(k(t)v′(t)))′=ft,∫0tv(s)ds,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.

Keywords

• boundary value problems
• nonlinear differential operators
• φ-laplacian operator
• singular equation
• nagumo condition
• 34b15
• 34b24
• 34c25
• 34l30