Systems of essentially infinite-dimensional differential equations
Let \(F(Q, \mathbb{R}^n)\) be the Banach algebra of bounded functions on a set \(Q\) with values in \(\mathbb{R}^n\) and let \(X\) be a closed subalgebra of \(F(Q, \mathbb {R})\). For \(i \in \{1,\dots,n \}\) let \(S_i\) be a contraction \(C_0\)-semigroup in \(X\), \(S(t)u=(S_1(t)u_1, \dots,S_n(t)u_n)\) for \(u \in X^n\), such that there exists \(s >0\) for which \(S(s)=0,\; S_i(t)(uv)=S_i(t)u\,S_i(t)v\) for \(t \geq 0,\; u, v\in X\) and \(S_i(t)S_k(\tau)=S_k(\tau)S_i(t)\) for \(t, \tau \geq 0,\; i,k \in \{1,\dots,n\}\); the semigroup is generated by the operator \(S'(0)\), where \(S'(0)u=(S_1'(0)u_1,\dots,S_n'(0)u_n)\), and \(\text{dom}(S'(0))\) is dense in \(X^n\). Theorem 1. Let \(F: X^n \rightarrow X^n\) be such that there exists \(C>0\) for which \(|(F(u)-F(v))(x)| \leq C|(u-v)(x)|\) for every \(u, v \in X^n\) and \(x \in Q\). Then there is a unique \(u \in \text{dom}(S'(0)) \subset X^n\) such that \(S'(0)u=F(u)\). Theorem 2. If \(S_i=\Sigma\) and \(S_i'(0)=A\) for all \(i\in \{1,\dots,n\}\), \(a=(a_{i,k})_{i,k\in \{1,\dots,n\}}\), \(a_{i,k} \in X\), \(f \in X^n\), then \[ u\in \mathrm{dom}(S'(0)),\qquad u(x)=-\int_0^s \exp\left(\int_0^t(\Sigma(\tau)a)(x)d \tau\right) (S(t)f)(x)dt \] is the unique solution of \(S'(0)u+au=f\). Theorem 3. Let \(R>0\) and \(H\) a separable, infinite dimensional, real Hilbert space; let \(B_C(H)\) be the Banach space of self-adjoint bounded linear operators on \(H\); let \(Q_{n,d}\) be the set of the operators of \(B_C(H)\) whose rank does not exceed \(n \in \mathbb {N}\) and whose norm does not exceed \(d > 0\); let \(Z=\{u \in C^2(H,\mathbb{R})\), bounded on its support \(\subset \{x \in H: |x| \leq R \}\), such that \(u'': H \rightarrow B_C(H)\) is uniformly continuous and for every \(\epsilon > 0\) there exist \(K\) a compact subset of \(B_C(H)\), \(n\in \mathbb {N}\) and \(d > 0\) such that \(K + Q_{n,d}\) is an \(\varepsilon\)-net for \(\{u''(x): |x| \leq R\}\); let \(X\) be the closure of \(Z\) with respect to the sup-norm; for \(i \in \{1, \dots , n \}\) let \(j_i\) be a nonzero positive functional on \(B_C(H)\) such that all finite-rank operators belong to its kernel; let \(L_i: Z \rightarrow X,\; (L_i u)(x)= {1 \over 2} j_i (u''(x))\) and its closure \(\overline{L_i}\) generates a contraction \(C_0\)-semigroup on \(X\); \(\widetilde{L}u=(\overline{L_1} u_1, \dots, \overline{L_n} u_n)\) for \(u_i \in \text{dom}(\overline{L_i});\; f: H \times \mathbb{R}^n \rightarrow \mathbb{R}^n,\; f(\cdot,p) \in X^n\) for every \(p \in \mathbb{R}^n\), there exists \(C>0\) such that \(|f(x,p)-f(x,q)| \leq C|p-q|\) for every \(x \in H,\; p, q \in \mathbb{R}^n\). Then there is a unique solution in \(\text{dom}(\widetilde{L})\) of \(\widetilde{L}u(x)=f(x,u(x))\) for \(x \in H\). Moreover, the author gives an application to nonlinear higher-order systems and examples of application of Theorem 2 to essentially infinite dimensional elliptic operators described in Theorem 3 and to the classical theory of differential equations.
essentially infinite-dimensional differential equations
scientific article; zbMATH DE number 6110039