Properties of integral solutions of differential inclusions with m- accretive operators
Let X be a separable Banach space and let us consider the following differential inclusions: \[ (1)\;x'(t)\in -Ax(t)+F(t,x(t)),\;(2)\;x'(t)\in -Ax(t)+\overline{co}F(t,x(t)) \] with initial condition \(x_ 0\in \overline{D(A)}\). Here \(A:D(A)\subset X\mapsto 2^ X\) is an m- accretive operator, i.e. \(<y_ 1-y_ 2,x_ 1-x_ 2>_+\geq 0\) for every \(x_ i\in D(A)\), \(y_ i\in Ax_ i\), \(i=1,2\) and \(I+A\) is surjective \((<y,x>_+=\sup \{<y,x'>|\) \(x'\in j(x)\}\) where j denotes the duality map). Integral solution of (1) is a pair \((x(.),f(.)), x(.)\in C([0,T],\overline{D(A)})\), \(x(0)=x_ 0\), \(f(.)\in L_ 1([0,T],X)\), \(f(t)\in F(t,x(t))\) for almost all \(t\in [0,T]\) such that \[ \| x(t)-u\|^ 2\leq \| x(s)-u\|^ 2+2\int^{t}_{s}<f(\tau)-v,x(\tau)-u>_+d\tau \text{ for } every\quad u\in D(A),\;v\in Ax. \] It is assumed that the set-valued map \(F:[0,T]\times \overline{D(A)}\mapsto 2^ X\) satisfies the following conditions: 1) The values of \(F(t,x)\) are closed bounded sets. 2) For every \(x\in \overline{D(A)}\), \(F(\cdot,x)\) is measurable and there exists an integrable function k(t)\(\geq 0\) such that for almost all \(t\in [0,T]\) and every \(x,y\in \overline{D(A)}\), \(dist(F(t,x),F(t,y))\leq k(t)\| x-y\|\). Here dist denotes the Hausdorff metrics. 3) There exists \(y\in \overline{D(A)}\) and an integrable function n(t)\(\geq 0\) such that dist(F(t,y),0)\(\leq n(t)\) for almost all \(t\in [0,T].\) By \(H_ F(x_ 0)\) and \(H_{\overline{co}F}(x_ 0)\) we denote all integral solutions of (1) and (3). The following is the main theorem of the paper. Theorem. Let \(X'\) be strictly convex and let the semigroup S(t), \(t\geq 0\), given by A be compact. Let the set-valued map \(F(t,x)\) have compact values and satisfy the assumptions given above. Then \(H_{\overline{co}F}(x_ 0)\) is a compact subset of \(C([0,T],X)\) and \(H_{\overline{co}F}(x_ 0)=\overline{H_ F(x_ 0)}.\) The closure is taken in the space \(C([0,T],X)\). The statement of the theorem is also proved under slightly modified assumptions.
scientific article; zbMATH DE number 4214401
scientific article