Logarithmic Convexity, First Order Differential Inequalities and Some Applications

Howard Allen Levine
1970 Transactions of the American Mathematical Society  
Let, for / e [0, T) (T< oo), D(t) be a dense linear subspace of a Hubert space H, and let M(t) and N(t) be linear operators (possibly unbounded) mapping D(t) into H. Let/: [0, T)xH^ H. We give sufficient conditions on M, N and/in order to insure uniqueness and stability of solutions to (1) M(i)du¡dt = N(t)u+f(t,u), «(0) given. This problem is not in general well posed in the sense of Hadamard. We cite some examples of (1) from the literature. We also give some examples of the problem (2) M(t) ~
more » ... problem (2) M(t) ~ = N(t)u+f(t, u, ^, u(0), ^ (0) prescribed, for which questions of uniqueness and stability were discussed in a previous paper.
doi:10.2307/1995652 fatcat:yljopwq375glxnzrqhuy5sxexi