The purpose of this paper is to study the incomplete Cauchy problem \[ (P)\quad u''(t)\in Au(t),\quad 0<t<\infty,\quad u(0)=x,\quad \sup \{| u(t)|:\quad t\geq 0\}<\infty, \] where A is a nonlinear (possibly discontinuous and set-valued) m-accretive operator in a Banach space (X,\(| \cdot |)\). It is natural to consider such a problem because the corresponding complete Cauchy problem is not well posed, even if A is linear. Problem (P) is also of interest because of its relationship to fractional powers of linear operators and to interpolation theory, as well as to certain problems in probability and in optimization theory. It has applications to (nonlinear) partial differential equations and to variational inequalities.
In Section 2 we establish the existence of solutions to (P) and to a related boundary value problem. Next we provide a connection between these two problems. These theorems are preceded by several differentiation and convergence lemmata, as well as by several lemmata in accretive operator theory. They are followed by a proposition in the geometry of Banach spaces. In Section 3 we use the results of Section 2 to construct a class of nonlinear semigroups with some remarkable properties. We show, for example, that these semigroups have a smoothing effect on initial data. This fact, as well as the estimate \(| u'(t)| \leq c/t\), are reminiscent of the properties of the nonlinear semigroups generated by subdifferentials in Hilbert space and of linear analytic semigroups. These results, previously known only in Hilbert space, provide a partial solution to the problem raised by H. Brézis [Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (1973; Zbl 0252.47055)] and V. Barbu [Nonlinear semigroups and differential equations in Banach spaces (1976; Zbl 0328.47035)] in their books. In addition to regularity results, Section 3 also contains two theorems on the asymptotic behavior of these semigroups. To each such semigroup there corresponds, by the second author J. Funct. Anal. 36, 147-168 (1980; Zbl 0437.47048)], a unique m-accretive operator \(A_{1/2}\) which generates it via the exponential formula. When A is a linear, \(A_{1/2}\) coincides with the square root of A. In contrast with the linear case, we do not know how to construct \(A_{1/2}\) directly from A. Nevertheless, we are able to show that \(A_{1/2}\) inherits some of the properties of A. We conclude this section with a convergence result.
In Section 4 we study a difference inclusion which is the discrete analog of the differential inclusion (P). We prove existence theorems as well as several results on the asymptotic behavior of the solutions to this difference inclusion. The last section, Section 5, is devoted to a brief discussion of the corresponding quasi-autonomous problem. More details concerning this problem can be found in our paper [in Trends in the theory and practice of nonlinear analysis, Proc. 6th Int. Conf., Arlington/Tex. 1984, 387-392 (1985; Zbl 0568.34049)]. Although our main interest is in special or general Banach spaces, some of our results seem to be new even in Hilbert space. The Banach space setting has required new ideas in the proofs of the other results as well. Some of the results presented here were previously announced [S. Reich, Proc. Symp. Pure Math. 45, 307-324 (1986)].