Existence of Solutions for a Nonlinear Dirichlet Problem Involving Gradient Dependent Lipschitz Convection Function

In this paper, our goal is to prove the existence of a weak solution (in ) for a fully nonlinear Dirichlet problem with a nonmonotone (e.g., Lipschitz) convection function F that depends on ∇ u , and a nonlinearity G that is not necessarily monotone and depends on the solution function u , and the higher order term is −ΔΓ( x , u ) − div a ( x , u , ∇ u ) provided that F , a , G , and Γ are Caratheòdory functions satisfying mild growth conditions. Here, Ω is a nonempty, bounded and open subset of R N with N ∈ Z + . We shall accomplish our goal by proving abstract existence results on the solvability of operator inclusion problems (in a Banach space X ) of the type and , where f ∈ X , is a β ‐expansive and m ‐accretive operator, is a L ‐Lipschitz continuous operator, is a compact operator and T : X ⟶ X is a β ‐expansive and continuous operator such that for some α > 0, we have 〈 T u − T v , j ( u − v )〉 ≥ − α ‖ u − v ‖ 2 for all ( u , v ) ∈ X 2 and j ( u − v ) ∈ J ( u − v ). The proofs are mainly based on the recent result on surjectivity of compact perturbation of β ‐expansive operator due to Asfaw. The abstract results and applications are new and give improvements (and/or generalizations) of those known results.