Nonlinear evolution operators in Banach spaces.

*(English)*Zbl 0567.47047Let X be a real Banach space, and
\[
(DE)_ s\quad u'(t)\in A_ tu(t),\quad s<t<T,
\]
with initial conditions \(u(s)=x\), \(x\in \overline{D(A_ s)}\); wherein \(0\leq s<T\), u(\(\cdot)\) stands for an X- valued unknown function on the interval [s,T], u’(\(\cdot)\) represents a derivative of u, and \(\{A_ t: t\in [0,T]\}\) is a given family of nonlinear, possibly multivalued operators in X.

From the text: ”The present paper is devoted to the study of two problems. The first aim of this paper is to introduce two notions of generalized solutions to \((DE)_ s\) such that the associated solution operators form an evolution operator. Firstly we consider the case in which approximate differential or difference equations for \((DE)_{s'}\) can be derived and there exist limits of solutions of those approximate equations.”

”On the other hand,... we employ a certain type of t-dependence of the operator \(A_ t\) in order to extend the notion of Benilan’s integral solution to the time-dependent case under consideration, and then show that given an initial-value \(x\in D(A_ s)\), the associated weak solution is uniquely determined in the class of such extended integral solutions.”

”The next problem is to discuss the generation of an evolution operator that provides generalized solutions of the initial value problem (for \((DE)_ s)\). In 1967, T. Kato [J. Math. Soc. Jap. 19, 508-520 (1967; Zbl 0163.383)] gave a general theory of nonlinear evolution operators in Banach spaces with uniformly convex duals and, in 1972, M. G. Crandall and A. Pazy [Isr. J. Math. 11, 57-94 (1972; Zbl 0249.34049)] established a generation theory in general Banach spaces.”

The authors’ second aim is to discuss the construction of evolution operators which furnish the integral solutions of \((DE)_ s\) and extend the results of Crandall and Pazy [ibid.], L. C. Evans [Isr. J. Math. 26, No.1, 1-42 (1977; Zbl 0349.34043)] and others.

From the text: ”The present paper is devoted to the study of two problems. The first aim of this paper is to introduce two notions of generalized solutions to \((DE)_ s\) such that the associated solution operators form an evolution operator. Firstly we consider the case in which approximate differential or difference equations for \((DE)_{s'}\) can be derived and there exist limits of solutions of those approximate equations.”

”On the other hand,... we employ a certain type of t-dependence of the operator \(A_ t\) in order to extend the notion of Benilan’s integral solution to the time-dependent case under consideration, and then show that given an initial-value \(x\in D(A_ s)\), the associated weak solution is uniquely determined in the class of such extended integral solutions.”

”The next problem is to discuss the generation of an evolution operator that provides generalized solutions of the initial value problem (for \((DE)_ s)\). In 1967, T. Kato [J. Math. Soc. Jap. 19, 508-520 (1967; Zbl 0163.383)] gave a general theory of nonlinear evolution operators in Banach spaces with uniformly convex duals and, in 1972, M. G. Crandall and A. Pazy [Isr. J. Math. 11, 57-94 (1972; Zbl 0249.34049)] established a generation theory in general Banach spaces.”

The authors’ second aim is to discuss the construction of evolution operators which furnish the integral solutions of \((DE)_ s\) and extend the results of Crandall and Pazy [ibid.], L. C. Evans [Isr. J. Math. 26, No.1, 1-42 (1977; Zbl 0349.34043)] and others.

Reviewer: S.L.Singh

##### MSC:

47H20 | Semigroups of nonlinear operators |

47E05 | General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX) |

47J05 | Equations involving nonlinear operators (general) |

47H06 | Nonlinear accretive operators, dissipative operators, etc. |