Advertisement

An existence theorem for implicit differential equations in a Banach space

  • 48 Accesses

  • 9 Citations

Summary

In this paper it is proved an existence theorem for the implicit differential equation F(t, x, x′)=0; x(t0)=x0 on a Banach space where it is supposed that F satisfies certain Lipschitz condition in x′ and is α-Lipschitz or α-Lipschitz in x.

References

  1. [1]

    A. Ambrosetti,Proprietà spettrali di certi operatori lineari non compatti, Rend. Sem. Mat. Univ. Padova,42 (1969), pp. 189–200.

  2. [2]

    A. Ambrosetti,Un teorema di esistenza per le equazioni differenziali negli spazi di Banach, Rend. Sem. Mat. Univ. Padova,39 (1967), pp. 349–361.

  3. [3]

    C. Corduneanu,Equazioni differenziali negli spazi di Banach. Teoremi di esistenza e prolungabilità, Rend. dell'Accad. dei Lincei,23 (1957).

  4. [4]

    G. Darbo,Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova,24 (1955), pp. 84–92.

  5. [5]

    K. Kuratowski,Topologie Pantswowe Wydaw Naukowe, Warsaw, 1958.

  6. [6]

    G. Pulvirenti,Equazioni differenziali in forma implicita in un ospazio di Banach, Annali di Mat. Pura ed Appl., (4)56 (1961), pp. 177–191.

  7. [7]

    J. Schauder,Der fixpunktsatz in Funktionalräumen, Studia Math.,2 (1930), pp. 171–180.

  8. [8]

    G. Sell,A characterization of smooth α-Lipschitz mappings on a Hilbert space, Ath. Accad. Naz. Lincei, Rend. Cl. Sci. fis. mat. nat.,52 (1972), pp. 668–674.

  9. [9]

    J. R. L. Webb,Remarks on K-set contractions, Bolletino U.M.I., (4)4 (1971), pp. 614–629.

  10. [10]

    J. R. L. Webb,A fixed point theorem and applications to functional equations in Banach spaces, Bollettino U.M.I., (4)4 (1971), pp. 755–788.

  11. [11]

    J. R. L. Webb,On a characterization of k-set contractions, Atti Accad. Naz. Lincei, Rend. Cl. Sci. fis., mat. nat.,50 (1971), pp. 686–689.

Download references

Author information

Additional information

Entrata in Redazione il 25 maggio 1977.

This research was done while the authors was visiting at the University of Minnesota.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Benavides, T.D. An existence theorem for implicit differential equations in a Banach space. Annali di Matematica 118, 119–130 (1978). https://doi.org/10.1007/BF02415125

Download citation

Keywords

  • Differential Equation
  • Banach Space
  • Existence Theorem
  • Lipschitz Condition
  • Implicit Differential Equation