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Nonlinear multivalued boundary value problems

100%
Bader R., Papageorgiou N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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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.
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Quasilinear vector differential equations with maximal monotone terms and nonlinear boundary conditions

100%
Bader R., Papageorgiou N. S.
Annales Polonici Mathematici
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2000
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tom 73
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nr 1
69-92
EN
We consider a quasilinear vector differential equation which involves the p-Laplacian and a maximal monotone map. The boundary conditions are nonlinear and are determined by a generally multivalued, maximal monotone map. We prove two existence theorems. The first assumes that the maximal monotone map involved is everywhere defined and in the second we drop this requirement at the expense of strengthening the growth hypothesis on the vector field. The proofs are based on the theory of operators of monotone type and on the Leray-Schauder fixed point theorem. At the end we present some special cases (including the classical Dirichlet, Neumann and periodic problems), which illustrate the general and unifying features of our work.
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Multi-valued superpositions

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Appell J., Thái N. H., De Pascale E., Zabreĭko P. P.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS Introduction.......................................................................................................... 5 1. Multifunctions and selections............................................................................... 7  1. Multifunctions and selections.................................................................. 7  2. Continuous multifunctions and selections........................................... 9  3. Measurable multifunctions and selections............................................ 16 2. Multifunctions of two variables............................................................................... 19  4. Carathéodory multifunctions and selections......................................... 19  5. The Scorza Dragoni property..................................................................... 25  6. Implicit function theorems......................................................................... 32 3. The superposition operator................................................................................... 33  7. The superposition operator in the space S........................................... 34  8. The superposition operator in ideal spaces......................................... 39  9. The superposition operator in the space C........................................... 47 4. Closures and convexifications.............................................................................. 49  10. Strong closures........................................................................................ 49  11. Convexifications....................................................................................... 52  12. Weak closures.......................................................................................... 56 5. Fixed points and integral inclusions..................................................................... 59  13. Fixed point theorems for multi-valued operators................................ 60  14. Hammerstein integral inclusions........................................................ 63  15. A reduction method................................................................................... 68 6. Applications............................................................................................................... 72  16. Applications to elliptic systems.............................................................. 72  17. Applications to nonlinear oscillations................................................. 75  18. Applications to relay problems.............................................................. 78 References.................................................................................................................... 81 Index of symbols........................................................................................................... 93 Index of terms................................................................................................................ 95
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