Existence and uniqueness of solutions to a class of operator equations (Q1206081)

Let \(X\) be a Hilbert space over the real or complex base field \(K\). The author considers existence and uniqueness of the solutions of the equation \[ \Phi(x):= x-\sigma- \sum^ m_{j=1} \widetilde A_ j\varphi_ j(x)= 0,\;\Phi: X^ n\to X^ n,\;\sigma\in X^ n, \] where \(\widetilde A_ j: X^ n\to X^ n\) is a linear operator defined by some matrix \(A_ j\in K^{n\times n}\) and \(\varphi_ j: X^ n\to X^ n\) \((j=1,\dots,m)\) are continuous mappings satisfying the conditions \[ \text{Re}(\widetilde D_ j(\varphi_ j(y)- \varphi_ j(z)),\;q-z)\leq (y-z,\;\widetilde\Lambda_ j (y- z))\;\forall y,z\in X^ n \] with \(D_ j\), \(\Lambda_ j\in K^{n\times n}\), \(\Lambda_ j= \Lambda^*_ j\).