Well-posedness of Third Order Differential Equations in Hölder Continuous Function Spaces

Abstract

In this paper, by using operator-valued ${\\dot{C}}^{\\unicode[STIX]{x1D6FC}}$\n -Fourier multiplier results on vector-valued Holder continuous function spaces, we give a characterization of the $C^{\\unicode[STIX]{x1D6FC}}$\n -well-posedness for the third order differential equations $au^{\\prime \\prime \\prime }(t)+u^{\\prime \\prime }(t)=Au(t)+Bu^{\\prime }(t)+f(t)$\n , ( $t\\in \\mathbb{R}$\n ), where $A,B$\n are closed linear operators on a Banach space $X$\n such that $D(A)\\subset D(B)$\n , $a\\in \\mathbb{C}$\n and $0<\\unicode[STIX]{x1D6FC}<1$\n .