M. A. Krasnosel’skij asked under what condition there is an unique positive solution of the convex operator equation \(Ax=x\). Zhang Shisheng etc. gave the condition in which there exists a positive solution of a uniformly \(u_ 0\) convex operator equation. Moreover, in this paper the author proved the following theorem:
Let E be a real Banach space, P a normal cone; an operator \(A: P_{u_ 0}\to P_{u_ 0}\) is uniformly \(u_ 0\) convex and increasing in the meaning of \(CH_{u_ 0}\), if there are non-zero elements \(v_ 0\) and \(w_ 0\) such that \(v_ 0/\| v_ 0\|^ 2_{u_ 0}\leq A(v_ 0/\| v_ 0\|^ 2_{u_ 0}),\quad A(w_ 0/\| w_ 0\|^ 2_{u_ 0})\leq w_ 0/\| w_ 0\|^ 2_{u_ 0};\) then the operator equation \(Ax=x\) has a unique solution \(x^*\in P_{u_ 0}\) and, choosing \(y_ 0\in P_{u_ 0}\), \(y_ 0>0\), putting \(y_ n=By_{n-1}\), \(x_ n=A(y_{n-1}/\| y_{n-1}\|^ 2_{u_ 0})\), we have \(\| x_ n-x^*\| \to 0\).