The existence of solution for boundary value problems for differential equations with deviating arguments and p-Laplacian

We consider a boundary value problem for a differential equation with deviating arguments and p-Laplacian: ${\displaystyle -\left({\varphi }_p(x\prime )\right)\prime +\frac{d}{dt}\nabla F\left(x\right)+g(t,x\left(t\right),x(\delta \left(t\right))}$, x'(t), x'(τ(t))) = 0, t ∈ [0,1]; ${\displaystyle x\left(t\right)=\underline{\phi }\left(t\right),}$ t ≤ 0; ${\displaystyle x\left(t\right)=\overline{\phi }\left(t\right)}$, t ≥ 1. An existence result is obtained with the help of the Leray-Schauder degree theory, with no restriction on the damping forces d/dt grad F(x).