Preface ......................................................... 3
1 Fixed point results in Frechet and gauge spaces and
applications, by M. Frigon ................................... 9
Goal and outline of the lectures ............................. 9
2 Frechet and gauge spaces .................................... 11
2.1 Frechet spaces ......................................... 11
2.2 Gauge spaces ........................................... 18
3 Generalized contractions .................................... 25
3.1 Single-valued contractions in gauge spaces ............. 25
3.2 Homotopical invariance for single-valued maps .......... 31
3.3 Applications of results for contractions ............... 35
4 Multi-valued contractions ................................... 43
4.1 Multi-valued self-maps ................................. 43
4.2 Homotopical invariance for multi-valued maps ........... 45
5 Generalization of Caristi's Theorem ......................... 47
5.1 Generalization of Ekeland's variational Principle ...... 47
5.2 Inwardness type conditions for multi-valued maps ....... 51
6 Admissibly compact maps ..................................... 55
6.1 Admissibly compact self-maps .......................... 55
6.2 Homotopical admissibly compact maps .................... 59
6.3 Applications of admissibly compact maps ................ 61
7 Maps defined on cones ....................................... 69
7.1 Cone-compressing/extending type results ................ 70
7.2 Application to differential equations on [0, ∞) ........ 73
7.3 Fixed point results for maps on intervals in cones ..... 77
7.4 Results relying on mixed conditions in a cone .......... 80
8 Appendix: Multi-valued maps and fixed point index ........... 87
8.1 Multi-valued maps ...................................... 87
8.2 Fixed point index ...................................... 88
Bibliography ................................................ 89
9 A short course on positive solutions of systems of ODEs,
by G. Infante ............................................... 93
On this short course ........................................ 93
10 The Krasnosel'skii fixed point theorem ...................... 95
11 The fixed point index ...................................... 103
11.1 A non-existence result ................................ 110
12 Nonnegative solutions of systems of BVPs ................... 113
13 More general BCs ........................................... 119
13.1 A three-point problem ................................. 119
13.2 Nonlinear BCs ......................................... 123
14 Radial solutions of PDEs ................................... 131
14.1 Radial solutions of systems in annular domains ........ 132
14.2 Radial solutions in exterior domains .................. 134
Conclusions and further reading ............................ 136
Acknowledgments ............................................ 136
Bibliography ............................................... 137
15 Singular ø-Laplacians, by P. Jebelean ...................... 141
Goal and outline of the lectures ........................... 141
16 Radial solutions ........................................... 143
16.1 Dirichlet problem in the unit ball .................... 143
16.2 Dirichlet problem in an annular domain ................ 147
16.3 Neumann problem (I) ................................... 151
16.4 Neumann problem (II) .................................. 156
16.4.1 Preliminaries ....................................... 156
16.4.2 The problem and its variational formulation ......... 158
17 Non-radial solutions ....................................... 169
17.1 The Dirichlet problem - first results ................. 169
17.2 Adding further nonliniarities ......................... 175
18 Periodic solutions ......................................... 181
18.1 A relativistic pendulum system ........................ 181
18.2 Multiple critical orbits .............................. 184
Bibliography .................................................. 191
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