A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities

Let x be a real reflexive locally uniformly convex banach space with locally uniformly convex dual space x. let t:xdt→2x be maximal monotone of type γdφ (i.e.,there exist d≥0 and a nondecreasing function φ:0,∞→0,∞ with φ(0)=0 such that 〈v,x-y〉≥-dx-φy for all xdt,vtx,and yx),l:x⊃d(l)→x be linear,surjective,and closed such that l-1:x→x is compact,and c:x→x be a bounded demicontinuous operator. a new degree theory is developed for operators of the type l+t+c. the surjectivity of l can be omitted provided that rl is closed,l is densely defined and self-adjoint,and x=h,a real hilbert space. the theory improves the degree theory of berkovits and mustonen for l+c,where c is bounded demicontinuous pseudomonotone. new existence theorems are provided. in the case when l is monotone,a maximality result is included for l and l+t. the theory is applied to prove existence of weak solutions in x=l20,t;h01ω of the nonlinear equation given by ∂u/∂t-i=1n(∂/∂xi)aix,u,u+hx,u,u=fx,t,x,tqt; ux,t=0,x,t∂qt; and ux,0=ux,t,xω,where >0,qt=ω×0,t,∂qt=∂ω×0,t,aix,u,u=∂/∂xiρx,u,u+aix,u,u (i=1,2,.,n),hx,u,u=-δu+gx,u,u,ω is a nonempty,bounded,and open subset of rn with smooth boundary,and ρ,ai,g:ω×r×rn→r satisfy suitable growth conditions. in addition,a new existence result is given concerning existence of weak solutions for nonlinear wave equation with nonmonotone nonlinearity. © 2016 teffera m. asfaw.