## Abstract

Let X and Y be real normed spaces with an admissible scheme Γ = {E_{n}, V_{n}; F_{n}, W_{n}} and T: X → 2^{Y} A-proper with respect to Γ such that dist(y, A(x)) < kc(∥ x ∥) for all y in T(x) with ∥ x ∥ ≥ R for some R > 0 and k > 0, where c: R^{+} → R^{+} is a given function and A: X → 2^{Y} a suitable possibly not A-proper mapping. Under the assumption that either T or A is odd or that (u, Kx) ≥ 0 for all u in T(x) with ∥ x ∥ ≥ r > 0 and some K: X → Y^{*}, we obtain (in a constructive way) various generalizations of the first Fredholm theorem. The unique approximation-solvability results for the equation T(x) = f with T such that T(x) - T(y) ε{lunate} A(x - y) for x, y in X or T is Fréchet differentiable are also established. The abstract results for A-proper mappings are then applied to the (constructive) solvability of some boundary value problems for quasilinear elliptic equations. Some of our results include the results of Lasota, Lasota-Opial, Hess, Nečas, Petryshyn, and Babuška.

Original language | English (US) |
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Pages (from-to) | 468-502 |

Number of pages | 35 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 65 |

Issue number | 2 |

DOIs | |

State | Published - Sep 1978 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics