On the existence and uniqueness of a solution to a nonlinear differential equation with an unbounded operator coefficient in Banach space (Q2755242)

The author studies the existence and uniqueness of a bounded solution to the equation NEWLINE\[NEWLINEx'(t)=Ax(t)+ \sum_{k=2}^{\infty}b_{k}(x(t),x(t),\ldots, x(t))+y(t), \quad t\in \mathbb{R},NEWLINE\]NEWLINE where the operator \(A:B\to b\) satisfies the conditions: \(\sigma(A)\cap i\mathbb{R}=\emptyset\), \(\sigma(A)\) is a spectrum of \(A\); \(-A\) is the sectorial operator; \(b_{k}:B\to B, k\geq 2\), are \(k\)-linear bounded functions; \(B\) is a Banach space. Sufficient conditions for the existence of a bounded solution for a bounded right-hand side are obtained. Conditions for the existence of two different bounded solutions are presented.