Some functional differential equations

A. Pelczar

GDML_Books, (1973), p.

Résumé

CONTENTSIntroduction............................................................................................................................................................................... 5Chapter 0. PRELIMINARIES0.1. (Preliminary remarks and notation)............................................................................................................................. 90.2. (Notation — continuation).............................................................................................................................................. 100.3. (Notation and some definitions).................................................................................................................................. 100.4. (Statement of problems; definition of solutions of differential functional equations)......................................... 120.5. (Equivalence of problems: differential and integral; definition of solutions of integral equations).................. 14Chapter I. EXISTENCE AND UNIQUENESS OF SOLUTIONS AND THE CONVERGENCE OF SUCCESSIVEAPPROXIMATIONS IN COMPACT SETS1.1. Notation and definitions................................................................................................................................................. 171.2. Uniqueness...................................................................................................................................................................... 181.3. Existence and successive approximations................................................................................................................ 191.4. Existence without uniqueness...................................................................................................................................... 221.5. Some generalizations of the results from 1.2-1.4..................................................................................................... 231.6. Some supplementary remarks..................................................................................................................................... 24Chapter II. LOCAL AND GLOBAL EXISTENCE AND UNIQUENESS2.1. Notation and definitions................................................................................................................................................. 272.2. Union of solutions........................................................................................................................................................... 282.3. Global uniqueness.......................................................................................................................................................... 292.4. Definition of the condition (W) and some remarks................................................................................................... 312.6. Local existence of solutions.......................................................................................................................................... 322.6. Lemmas............................................................................................................................................................................ 332.7. Limits of solutions on the boundary............................................................................................................................. 362.8. Prolongations................................................................................................................................................................... 382.9. Global existence under the assumptions on uniqueness...................................................................................... 392.10. Global existence without uniqueness....................................................................................................................... 412.11. Global existence without uniqueness by the method of A. Bielecki, T. Dłotko and M. Kuczma...................... 432.12. Existence of solutions under the assumptions (Y) and (Ỹ)................................................................................... 462.13. Local convergence of successive approximations under the assumptions (V)............................................... 472.14. Remarks on some generalizations........................................................................................................................... 49Chapter III. CONTINUOUS DEPENDENCE OF SOLUTIONS ON GIVEN FUNCTIONS3.1. Continuous dependence on $λ, ψ, {φ^a}$............................................ 513.2. Continuous dependence on ƒ....................................................... 52

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