Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups

In this paper we consider the evolution equation (DE) (d/dt) u (t) Au (t), t (O, T) for an -quasi-dissipative" operator A in a Banach space X, from the viewpoint of difference approximation. We introduce a notion of " DS-limit solution" of (DE) and discuss the construction of the DS-limit solutions. We also give a generation theorem of nonlinear semigroups through difference approxima- tion. Recently several authors have treated the evolution equation (DE) from the view-point of difference approximation. The result of Crandall and Liggett 7 is the first fundamental one in this direction. Kenmochi and Oharu 9 extended the result in 7 to the case where the difference scheme for (DE) permits errors. Takahashi 16, 17 formulated a more general approximate difference scheme and determined the conditions under which the solution of the difference scheme converges. In this paper we introduce a notion of w-quasi-dissipative operator as a generalization of -dissipative operator. We consider the approximate difference scheme for the Cauchy problem for (DE) under the same formulation as in 17. Our first purpose is to give a convergence theorem for difference approximation and to improve the result in 17. At the same time, it is shown that the limit function of solutions of difference approximation is uni- quely determinded by the initial condition and is independent of the choice of difference scheme. Hence we shall call the limit function a DS-limit solution of the Cauchy problem. Recently B\'enilan 2 has introduced the notion of " integral solution " and "bonne solution" and investigated properties of bonne solutions. Our second purpose is to investigate basic properties of DS-limit solutions and to study the relationship between those solutions. Difference approximation of Cauchy problems for quasi-dissipative operators 641The results mentioned above can be considered from the view point of the theory of nonlinear semigroups. Our third purpose is to discuss the gen- eration of a nonlinear semigroup associated with a given operator A. Our fourth purpose is to give a sufficient condition which assures genera- tion of nonlinear semigroups through the difference approximation. Many authors have already treated the generation of semigroups. Our results imply the results of Crandall and Liggett 7 and Martin 12. Also we give a simple application of our results to the continuous perturbation of m-dissipative operators. are equivalent when the duality map is single-valued but not so in general. (See Example 1. 1 at the end of this section. ) Let S X and A be an (-quasi) -dissipative operator in X. Then we say that A is maximal (-quasi) -dissipative on S if any (-quasi) -dissipative exten- sion of A coincides with A on S. If A is a dissipative operator such that R (I- A) =X for all >0, then we say that A is m-dissipative. It is well known that if A is a dissipative operator such that R (I-₀A) =X for some ₀>0, then A is n -dissipative. We refer to 8 for other properties of m-dissipative operators. An -quasi-dissipative operator A in X can be regarded as an -quasi- dissipative operator in x**. Therefore, we can associate with A an operator A in x**such that d is an extension of A, D (A) (A) and cjl is maximal -quasi-dissipative on D (A) in X^**. We call such a a maximal (^**) extension of A. (See 17. ) Let X₀ X. A one parameter family \T (t) ;t 0\ of operators from X₀ into itself is called a semigroup of type on X₀ if it has the following pro- perties: (i) for x, y X₀ and t 0, T (t) x-T (t) y e^ t x-y ; (ii) T (O) x=x for x X₀ and T (t+s) =T (t) T (s) for t, s 0 ; (iii) for each x X₀, T (t) x is strongly continuous in t 0. In the following, we prepare some estimates which will play a central role in later argument. LEMMA 1. 1. Let A be an operator in X and a real number. Then the following three conditions are equivalent: (i) A is -quasi-dissipative; (ii) for any x₈, y₈ A (i=1, 2) and, >0, (+-) x₁-x₂ x₁-x₂- y₁+ x₂-x₁- y₂ ; (iii) for any x₈, y₈ A (i=1, 2) and >0, Furthermore, in these cases we have (iv) for any x, y A, u D (A) and >0, (1-) x-u x-u- y+ Au. PROOF. Suppose that A is -quasi-dissipative. Let x₈, y₈ A (i=1, 2) and, >0. By definition, there exist f F (x₁-x₂) and g F (x₂-x₁) such that y₁, f+ y₂, g x₁-x₂^2 Therefore, we