Fixed Point Theorems and Applications This book addresses fixed point theory a fascinating and far reaching field with applications in several areas of mathematics. The content is divided into two main parts. The first which is more theoretical develops the main abstract theorems on the existence and uniqueness of fixed points of maps. In turn the second part focuses on applications covering a large variety of significant results ranging from ordinary differential equations in Banach spaces to partial differential equations operator theory functional analysis measure theory and game theory. A final section containing 50 problems many of which include helpful hints rounds out the coverage. Intended for Master’s and PhD students in Mathematics or more generally mathematically oriented subjects the book is designed to be largely self contained although some mathematical background is needed: readers should be familiar with measure theory Banach and Hilbert spaces locally convex topological vector spaces and in general with linear functional analysis 1 The Banach contraction principle 7 2 The Boyd Wongtheorem 13 3 Further extensions of the contraction principle 16 4 Weak contractions 23 5 Contractions of ε type 29 6 Sequences of maps and ﬁxed points 36 7 Fixed points of non expansive maps 39 8 The Riesz mean ergodic theorem 42 9 The Brouwer ﬁxed point theorem 46 10 The Schauder Tychonoﬀ ﬁxed point theorem 50 11 Further consequences of the Schauder Tychonoﬀ theorem 55 12 TheMarkov Kakutani theorem 60 13 TheKakutani Ky Fan theorem 62 14 The implicit function theorem 70 15 Location of zeros 75 16 Ordinary diﬀerential equations in Banach spaces 78 17 The Lax Milgram lemma 89 18 An abstract elliptic problem 97 19 Semilinear evolution equations 101 20 An abstract parabolic problem 108 21 The invariant subspace problem 114 22 Measure preserving maps on compact Hausdorﬀ spaces 118 23 Invariant means on semigroups 120 24 Haar measures 123 25 Game theory 130 26 Problems