Existence of solutions of semilinear differential equations with nonlocal conditions in Banach spaces (Q2724893)

Here, the authors consider the semilinear differential equation in a Banach space NEWLINE\[NEWLINE{du(t)}/{dt}+Au(t)=f(u(t)),\qquad t\in(0,b],\tag{1}NEWLINE\]NEWLINE with the nonlocal condition NEWLINE\[NEWLINEu(0)+g(t_1,t_2,\dots,t_p,u(t_1),\dots,u(t_p))=u_0\tag{2}NEWLINE\]NEWLINE and \(0\leq t_0<t_1<\cdots<t_p\leq b\). Under suitable assumptions, they prove that problem (1)--(2) has a unique, local and strong solution. Moreover, an extension theorem for solutions to this problem is also established. The proofs are based on the analytic semigroups and the contraction principle. Recall that one of the first papers in which the existence of solutions to the evolution equation with nonlocal conditions in a Banach space was given by \textit{L. Byszewski} [J. Math. Anal. Appl. 162, No.~2, 494-505 (1991; Zbl 0748.34040)].