Existence of solutions and periodic solutions for nonlinear evolution inclusions
type: Item
instance of: scholarly article
title: Existence of solutions and periodic solutions for nonlinear evolution inclusions
zbMATH Open document ID: 0931.34043
DOI: 10.1007/BF02857308
author: Nikolaos S. Papageorgiou
author: Francesca Papalini
author: Francesca Renzacci
publication date: 2000-02-23T00:00:00Z
review text: 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 t…
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 34G20
zbMATH DE Number: 1339490
zbMATH Keywords: nonlinear evolution inclusions
zbMATH Keywords: periodic solutions
zbMATH Keywords: Cauchy problem
zbMATH Keywords: generalized pseudomonotone operators
reviewed by: Marian Ivanovici
MaRDI profile type: Publication
cites work: On multivalued evolution equations in Hilbert spaces
cites work: Necessary and sufficient conditions for L1-strong- weak lower semicontinuity of integral functionals
cites work: Periodic solutions of semilinear equations of evolution of compact type
cites work: Q5665783
cites work: EXISTENCE OF PERIODIC SOLUTIONS FOR NONLINEAR EQUATIONS OF EVOLUTION
cites work: Q5641474
cites work: Q3998720
cites work: Evolutions governed by m-accretive plus compact operators
cites work: Monotone trajectories of differential inclusions and functional differential inclusions with memory
cites work: Integrals, conditional expectations, and martingales of multivalued functions
cites work: Existence of nontrivial solutions of semilinear elliptic equations
cites work: Existence of Periodic Solutions for Nonlinear Evolution Equations in Hilbert Spaces
cites work: Q4280640
cites work: Periodic solutions of nonlinear evolution inclusions
cites work: Q5890172
cites work: Convergence theorems for Banach space valued integrable multifunctions
cites work: On the set of solutions of a class of nonlinear evolution inclusions
cites work: Q4834500
cites work: Q3857336
cites work: Periodic solutions of semilinear evolution equations
cites work: Discontinuous Semilinear Differential Equations and Multiple Valued Maps
cites work: Nonlinear evolution equations in Banach spaces
cites work: The nonlinear version of Pazy's local existence theorem
cites work: An existence result for a class of nonlinear evolution equations in Banach spaces
cites work: Periodic Solutions for Nonlinear Evolution Equations in a Banach Space
cites work: Q3992492
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 1339490