Quasilinear vector differential equations with maximal monotone terms and nonlinear boundary conditions

• Autorzy: Ralf Bader , Nikolaos S. Papageorgiou
• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

Dirichlet; maximal monotone operator; Yosida approximation; monotone operator; resolvent operator; measurable selection; demicontinuous operator; Neumann and periodic problems; coercive operator; projection theorem

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2000
• Tom: 73
• Numer: 1
• Strony: 69-92

Daty

wydano

2000

otrzymano

1999-07-20

Twórcy

autor

Ralf Bader

• Center for Mathematical Sciences, Munich University of Technology (TUM), Arcisstr. 21, D-80333 München, Germany

autor

Nikolaos S. Papageorgiou

• Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece

Bibliografia

• [1] R. Adams, Sobolev Spaces, Academic Press, New York, 1975.
• [2] L. Boccardo, P. Drábek, D. Giachetti and M. Kučera, Generalization of Fredholm alternative for nonlinear differential operators, Nonlinear Anal. 10 (1986), 1083-1103.
• [3] H. Dang and S. F. Oppenheimer, Existence and uniqueness results for some nonlinear boundary value problems, J. Math. Anal. Appl. 198 (1996), 35-48.
• [4] M. del Pino, M. Elgueta and R. Manásevich, A homotopic deformation along p of a Leray-Schauder degree result and existence for $(|u'|^{p-2} u')' + f(t,u) = 0$, u(0) = u(T) = 0, p>1, J. Differential Equations 80 (1989), 1-13.
• [5] M. del Pino, R. Manásevich and A. Murúa, Existence and multiplicity of solutions with prescribed period for a second order quasilinear o.d.e., Nonlinear Anal. 18 (1992), 79-92.
• [6] C. Fabry and D. Fayyad, Periodic solutions of second order differential equations with a p-Laplacian and asymmetric nonlinearities, Rend. Istit. Mat. Univ. Trieste 24 (1992), 207-227.
• [7] Z. Guo, Boundary value problems of a class of quasilinear ordinary differential equations, Differential Integral Equations 6 (1993), 705-719.
• [8] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.
• [9] E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer, New York, 1965.
• [10] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, Kluwer, Dordrecht, 1997.
• [11] D. Kandilakis and N. S. Papageorgiou, Neumann problem for a class of quasilinear ordinary differential equations, Atti Sem. Mat. Fis. Univ. Modena, to appear.
• [12] R. Manásevich and J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. Differential Equations 145 (1998), 367-393.
• [13] M. Marcus and V. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rational Mech. Anal. 45 (1972), 294-320.
• [14] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989.
• [15] E. Zeidler, Nonlinear Functional Analysis and its Applications I, Springer, New York, 1985.
• [16] E. Zeidler, Nonlinear Functional Analysis and its Applications II, Springer, New York, 1990.