On weak solutions of operator equations (Q1326934)

This article deals with the equation \[ \Lambda(y)+ A(y)=f \] where \(\Lambda: D(\Lambda) \subseteq \mathbb{X}\to \mathbb{X}^*\) is an operator for which \[ (\Lambda(y_ 1)- \Lambda(y_ 2), y_ 1-y_ 2)\geq- c_ \Lambda (R,\| y_ 1- y_ 2\|') \] with some function \(c_ \Lambda (r,\tau)\) and \(A\) is an operator from \(\mathbb{X}\) into \(\mathbb{X}^*\), for which \[ \lim_{\| y\|_ \mathbb{X}\to \infty, y\in D(\Lambda)} \| y\|_ \mathbb{X}^{-1} (\Lambda(y)+ A(y), y)= +\infty, \] \(\mathbb{X}\) is a reflexive Banach space with norm \(\|\cdot\|\), \(\|\cdot \|'\) is an auxiliary norm. Under some additional natural conditions the existence of weak solutions to this equation is proved and further some approximation and weak compactness properties of weak solution sets are described. Two examples of partial differential operators are presented for which the abstract results are applied.