The number of solutions for a class of nonlocal nonhomogeneous gradient operator equations

This paper deals with the number of solutions for a class of nonlocal nonhomogeneous gradient operator equations of the form a(I(u))I′(u)=fa(I(u))I′(u)=f, where I∈C1(X,R)I∈C1(X,R), XX is a reflexive Banach space, I(0)=0I(0)=0, II is even and strictly convex, I(u)‖u‖→+∞ as ‖u‖→∞‖u‖→∞, I′:X→X∗I′:X→X∗ is a bounded homeomorphism but is not necessarily homogeneous, a:(0,+∞)→Ra:(0,+∞)→R is continuous, f∈X∗∖{0}f∈X∗∖{0}. Some properties and examples of such a functional II are given. Some results on the number of solutions of the nonlocal equation are obtained.