We consider the analysis and numerical solution of a forward-backward boundary value problem. We provide some motivation, prove existence and uniqueness in a function class especially geared to the problem at hand, provide various energy estimates, prove a priori error estimates for the Galerkin method, and show the results of some numerical computations.

We study the sequence $u_n$, which is solution of $-div\left(a(x,𝔻u_n)\right)+{\Phi }^{\text{'}\text{'}}\left(\right|u_n\left|\right)u_n=f_n+g_n$ in $\Omega$ an open bounded set of ${𝐑}^N$ and $u_n=0$ on $\partial \Omega$, when $f_n$ tends to a measure concentrated on a set of null Orlicz-capacity. We consider the relation between this capacity and the $N$-function $\Phi$, and prove a non-existence result.

We study the sequence un, which is solution of $-\mathrm{div}\left(a(x,\nabla u_n)\right)+{\Phi }^{\text{'}\text{'}}\left(\right|u_n\left|\right)u_n=f_n+g_n$ in Ω an open bounded set of RN and un= 0 on ∂Ω, when fn tends to a measure concentrated on a set of null Orlicz-capacity. We consider the relation between this capacity and the N-function Φ, and prove a non-existence result.

We prove some quantitatively sharp estimates concerning the Δ₂ and ∇₂ conditions for functions which generalize known ones. The sharp forms arise in the connection between Orlicz space theory and the theory of elliptic partial differential equations.

Recently there has been an increasing interest in studying $p\left(t\right)$-Laplacian equations, an example of which is given in the following form $$\left(\right|u^{\text{'}}{\left(t\right)|}^{p\left(t\right)-2}{u}^{\text{'}}{\left(t\right))}^{\text{'}}+c\left(t\right){\left|u\left(t\right)\right|}^{q\left(t\right)-2}u\left(t\right)=0,t>0.$$ In particular, the first study of sufficient conditions for oscillatory solution of $p\left(t\right)$-Laplacian equations was made by Zhang (2007), but to our knowledge, there has not been a paper which gives the oscillatory conditions by utilizing Riccati inequality. Therefore, we establish sufficient conditions for oscillatory solution of nonlinear differential equations with...