Existence of solutions and periodic solutions for nonlinear evolution inclusions
The authors establish two existence theorems for evolution inclusions: the first for a periodic problem and the second for a Cauchy problem. It is stated a preliminary surjectivity result. More exactly, if \(Y\) is a reflexive, strictly convex Banach space, \(L: D(L)\subset Y\to Y^*\) be a linear densely defined maximal monotone operator and \(T: Y\to P(Y^*)\) is a multivalued operator, which is bounded, generalized pseudomonotone with respect to \(D(L)\) and coercive, then \(R(L+T)= Y^*\). First, it is considered the following evolution inclusion: \[ \dot x(t)+A(t,x(t))+ F(t,x(t))\ni h(t),\quad \text{a.e. on }T= [0,b],\quad x(0)= x(b).\tag{1} \] Using the surjectivity result in the framework of an evolution triple of spaces, the authors show that problem (1) has a periodic solution. They prove that the Cauchy problem \[ \dot x(t)+ A(t,x(t))+ F(t,x(t))\ni h(t)\quad \text{a.e. on }T,\quad x(0)= x_0,\tag{2} \] has a solution \(x\in W_{pq}(T)\), where \(X\) is a dense subspace of a Hilbert space \(H\) and \(W_{pq}(T)= \{x\in L^p(T,X):\dot x\in L^1(T, X^*)\}\). These results are illustrated by two examples. First, it is considered the multivalued parabolic problem \[ {\partial x\over\partial t}-\Delta x+ r\sum^N_{k=1} (\sin x)D_kx+ u(t, z)= h(t,z)\quad \text{a.e. on }T\times Z\quad (Z\subset\mathbb{R}^N), \] \[ u(t,z)\in [f_1(t, z,x(t,z)), f_2(t,z,x(t,z))]\quad\text{a.e. on }T\times Z, \] \[ x(0,z)= x(b,z)\quad \text{a.e. on }Z,\quad x|_{T\times\Gamma}= 0, \] for which the existence of a periodic solution is shown. In the second example is presented the following optimal control problem for a system driven by a nonlinear parabolic equation \[ \int^b_0 \int_Z L(t,z,x(t, z), u(t,z)) dz dt\to \inf= m, \] \[ {\partial x\over\partial t}- \sum_{|\alpha|\leq m}(-1)^{|\alpha|} D^\alpha A_\alpha(t,z,\eta(x(t,z)))+f(t,z,x,\xi(x(t, z)))u(t,z)= h(t,z)\text{ a.e. on }T\times Z, \] \[ |u(t,z)|\leq M\quad\text{a.e. on }T\times Z,\quad u(\cdot,\cdot)\text{ measurable}. \] It is shown that this problem has an optimal solution.
scientific article; zbMATH DE number 1339490
scientific article