## Approximation-solvability of Nonlinear Functional and Differential EquationsThis reference/text develops a constructive theory of solvability on linear and nonlinear abstract and differential equations - involving A-proper operator equations in separable Banach spaces, and treats the problem of existence of a solution for equations involving pseudo-A-proper and weakly-A-proper mappings, and illustrates their applications.;Facilitating the understanding of the solvability of equations in infinite dimensional Banach space through finite dimensional appoximations, this book: offers an elementary introductions to the general theory of A-proper and pseudo-A-proper maps; develops the linear theory of A-proper maps; furnishes the best possible results for linear equations; establishes the existence of fixed points and eigenvalues for P-gamma-compact maps, including classical results; provides surjectivity theorems for pseudo-A-proper and weakly-A-proper mappings that unify and extend earlier results on monotone and accretive mappings; shows how Friedrichs' linear extension theory can be generalized to the extensions of densely defined nonlinear operators in a Hilbert space; presents the generalized topological degree theory for A-proper mappings; and applies abstract results to boundary value problems and to bifurcation and asymptotic bifurcation problems.;There are also over 900 display equations, and an appendix that contains basic theorems from real function theory and measure/integration theory. |

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### Contents

PREFACE | 1 |

Further examples of Aproper mappings | 20 |

Aproperness and the constructive solvability of some | 34 |

Existence theorems and pseudoAproper mappings | 49 |

I1 EQUATIONS INVOLVING LINEAR APROPER | 55 |

Equations involving unbounded linear operators | 64 |

Generalized Fredholm alternative | 78 |

Constructive solvability of linear elliptic boundary | 86 |

GENERALIZED DEGREE FOR APROPER | 188 |

Singlevalued generalized degree for compact | 212 |

Some applications of the generalized degree | 219 |

Calculation of generalized degree | 237 |

Bifurcation and asymptotic bifurcation for equations | 250 |

SOLVABILITY OF PDEs AND ODEs | 265 |

Solvability of ordinary differential equations | 288 |

Approximationsolvability of periodic boundary value | 304 |

Stability of projective methods and Aproperness | 95 |

II1 FIXEDPOINT AND SURJECTIVITY THEOREMS | 100 |

Constructive fixedpoint theorems for Pycompact | 106 |

Constructive surjectivity theorems for PYcompact | 121 |

Approximationsolvability and solvability of equations | 127 |

Solvability of equations involving semilinear weakly | 160 |

Variational BPs and ABPs for PDEs and ODEs | 314 |

APPENDIX | 323 |

339 | |

NOTATION | 363 |

### Common terms and phrases

A-ball contractive A-proper mappings A-proper w.r.t. a-stable admissible scheme Anal applications approximation solvable w.r.t. assume asymptotic Banach space bifurcation bijective boundary value problem bounded sequence bounded set Brouwer degree Browder compact continuous mapping convergence Corollary D C X defined Definition degree theory demicontinuous denote differential equations duality map eigenvalue elliptic exist a subsequence finite finite-dimensional fixed point follows from Theorem Fredholm Friedrichs extension function G dD Galerkin method given Hence Hilbert space holds homeomorphism homotopy implies inequality invariance of domain Lemma Leray-Schauder Math monotone Petryshyn Pi-compact proof of Theorem Proposition prove Q.E.D. Remark reflexive reflexive space satisfies condition Section semilinear Sobolev Sobolev embedding theorem Sobolev space solution strongly subset subspace suffices to show Suppose surjectivity Theorem 2.1 uniquely approximation solvable weakly A-proper Xn,Pn xnj G