Nonlinear Second Order Scalar Differential Inclusion Involving a Singular Φ−Laplacian Operator, With Nonlinear General Boundary Conditions

In this paper, we study the following second order scalar differential inclusion: bzxΦz′x′∈Azx+Gx,zx,z′x a.e on Λ=0,α under nonlinear general boundary conditions incorporating a large number of boundary problems including Dirichlet, Neumann, Neumann–Steklov, Sturm–Liouville, and periodic problems. By means of approximate problems, we succeed in establishing existence results for the main problem. We establish the existence of solutions when the approximate problem admits two sign conditions or a single sign condition and a single lower solution or a single sign condition and a single upper solution. Our proofs combine the method of lower and upper solutions, sign conditions, the analysis of multifunctions, and Yosida’s approximation.