Boundary value problems for strongly nonlinear equations under a Wintner-Nagumo growth condition

Abstract We study the following strongly nonlinear differential equation: ( a ( t , x ( t ) ) Φ ( x ′ ( t ) ) ) ′ = f ( t , x ( t ) , x ′ ( t ) ) , a.e. in [ 0 , T ] $$\bigl(a \bigl(t,x(t) \bigr)\Phi\bigl(x'(t) \bigr) \bigr)'= f \bigl(t,x(t),x'(t) \bigr), \quad\text{a.e. in } [0,T] $$ subjected to various boundary conditions including, as particular cases, the classical Dirichlet, periodic, Neumann and Sturm-Liouville problems. We adopt the method of lower and upper solutions requiring a weak form of a Wintner-Nagumo growth condition.