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1

On nonlinear, nonconvex evolution inclusions

100%
Papageorgiou N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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1995
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tom 15
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nr 1
29-42
EN
We consider a nonlinear evolution inclusion driven by an m-accretive operator which generates an equicontinuous nonlinear semigroup of contractions. We establish the existence of extremal integral solutions and we show that they form a dense, $G_δ$-subset of the solution set of the original Cauchy problem. As an application, we obtain "bang-bang"' type theorems for two nonlinear parabolic distributed parameter control systems.
2
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On nonlinear hemivariational inequalities

68%
Papageorgiou N., Smyrlis G.
Instytut Matematyczny Polskiej Akademii Nauk
EN
We conduct a detailed study of the existence theory for nonlinear hemivariational inequalities of second order. The problems under consideration are strongly nonlinear and not necessarily of variational nature. So we employ a variety of tools in order to solve them. More precisely, we use the general theory of nonlinear operators of monotone type, the method of upper-lower solutions, the multivalued Leray-Schauder principle, nonsmooth critical point theory coupled with Landesman-Lazer conditions and linking techniques and also truncation and penalization techniques. The problems that we examine involve Dirichlet boundary conditions; in the last section we also examine a problem with a nonhomogeneous and nonlinear Neumann boundary condition.
3
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Nonlinear multivalued boundary value problems

64%
Bader R., Papageorgiou N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2001
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tom 21
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nr 1
127-148
EN
In this paper, we study nonlinear second order differential inclusions with a multivalued maximal monotone term and nonlinear boundary conditions. We prove existence theorems for both the convex and nonconvex problems, when $domA ≠ ℝ^{N}$ and $domA = ℝ^{N}$, with A being the maximal monotone term. Our formulation incorporates as special cases the Dirichlet, Neumann and periodic problems. Our tools come from multivalued analysis and the theory of nonlinear monotone operators.
4
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On the existence of five nontrivial solutions for resonant problems with p-Laplacian

64%
Gasiński L., Papageorgiou N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2010
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tom 30
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nr 2
169-189
EN
In this paper we study a nonlinear Dirichlet elliptic differential equation driven by the p-Laplacian and with a nonsmooth potential. The hypotheses on the nonsmooth potential allow resonance with respect to the principal eigenvalue λ₁ > 0 of $(-Δₚ,W₀^{1,p}(Z))$. We prove the existence of five nontrivial smooth solutions, two positive, two negative and the fifth nodal.
5
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Optimal control of nonlinear evolution equations

64%
Papageorgiou N., Yannakakis N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2001
|
tom 21
|
nr 1
5-50
EN
In this paper, first we consider parametric control systems driven by nonlinear evolution equations defined on an evolution triple of spaces. The parametres are time-varying probability measures (Young measures) defined on a compact metric space. The appropriate optimization problem is a minimax control problem, in which the system analyst minimizes the maximum cost (risk). Under general hypotheses on the data we establish the existence of optimal controls. Then we pass to nonparametric systems, which are governed by nonlinear evolution equations with nonmonotone operators. We prove two existence results for such evolution inclusions, which are of independent interest and extend significantly the results existing in the literature. Then we solve time-optimal and Meyer-type optimization problems. In Section 5, we derive necessary conditions for saddle point optimality in the minimax control problem. We conclude the paper with three examples of distributed parameter control systems.
6

Periodic solutions for quasilinear vector differential equations with maximal monotone terms

64%
Kourogenis N., Papageorgiou N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
1997
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tom 17
|
nr 1-2
67-81
EN
We consider a quasilinear vector differential equation with maximal monotone term and periodic boundary conditions. Approximating the maximal monotone operator with its Yosida approximation, we introduce an auxiliary problem which we solve using techniques from the theory of nonlinear monotone operators and the Leray-Schauder principle. To obtain a solution of the original problem we pass to the limit as the parameter λ > 0 of the Yosida approximation tends to zero.
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