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Boundary Conditions for 1-Set Contractions Maps in Banach Spaces

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Abstract

In this work, we have obtained some fixed point theorems for 1-set contractions under some boundary conditions such as the Leray–Schauder condition and the interior condition. Our results complement those obtained in Djebali and Hammache (Fixed Point Theory, to appear, 2014) and Garcia-Falset (Math. Nachr. 283(12):1736–1757, 2010) and improve (Petryshyn in Arch. Ration. Mech. Anal. 40:312–328 1971).

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References

  1. Agarwal, R.P., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications. In: Cambridge Tracts in Mathematics, vol. 141. Cambridge University Press, Cambridge (2001)

  2. Banas, J., Goebel, K.: Measure of noncompactness in Banach spaces. In: Math. Appl., vol. 57. Marcel Dekker, New York (1980)

  3. Barbu, V., Precupanu, T.: Convexity and Optimization in Banach Spaces. In: Springer Monographs in Mathematics (2012)

  4. Browder F.E.: Semicontractive and semiaccretive nonlinear mappings in Banach spaces. Bull. AMS 74, 660–665 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  5. Darbo G.: Punti uniti in transformationi a condominio non-compactto. Rend. Sem. Mat. Univ. Padova 24, 84–92 (1955)

    MathSciNet  MATH  Google Scholar 

  6. De Blasi F.S.: On a property of the unit sphere in Banach spaces. Bull. Math. Soc. Sci. Math. Roum. 21, 259–262 (1997)

    MathSciNet  MATH  Google Scholar 

  7. Djebali S., Hammache K.: Furi–Pera fixed point theorems in Banach algebras with applications. Acta Univ. Palacki. Olomuc. Fac. rer. nat. Math. 47, 55–75 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Djebali S., Hammache K.: Fixed point theorems for nonexpansive maps in Banach spaces. Nonlinear Anal. TMA 73, 3440–3449 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Djebali S., Hammache K.: Furi–Pera fixed point theorems for nonexpansive maps in Banach spaces. Fixed Point Theory 13(2), 461–474 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Djebali, S., Hammache, K.: Fixed point theorems for 1-set contractions in Banach spaces, Fixed Point Theory (2014, to appear)

  11. Duan H.-G., Xu S.-Y., Li G.: Fixed points of weakly 1-set contraction mappings. J. Korean Math. Soc. 45(6), 1725–1740 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Furi M., Pera P.: A continuation method on locally convex spaces and applications to ODE on noncompact intervals. Ann. Pol. Math. XLVII, 331–346 (1987)

    MathSciNet  MATH  Google Scholar 

  13. Garcia-Falset J.: Existence of fixed points for the sum of two operators. Math. Nachr. 283(12), 1736–1757 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Garcia-Falset J., Latrach K.: Krasnoselskii-type fixed-point theorems for weakly sequentially continuous map. Bull. Lond. Math. Soc. 44, 25–38 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Garcia-Falset J., Muñiz-Pérez O.: Fixed point theory for 1-set contractive and pseudocontractive mappings. Appl. Math. Comput. 219, 6843–6855 (2013)

    MathSciNet  MATH  Google Scholar 

  16. González C., Jimenéz-Melado A., Liorens-Fuster E.: A Mönch type fixed point theorem under the interior condition. J. Math. Anal. Appl. 352, 816–821 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jimenéz-Melado A., Morales C.H.: Fixed point theorems under the interior condition. Proc. AMS 134(2), 501–507 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Latrach K., Taoudi M.A., Zeghal A.: Some fixed point theorems of Schauder and Krasnosel’skij type and applications to nonlinear transport equations. J. Differ. Equ. 221, 256–271 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Leray J., Schauder J.: Topologie et équations fonctionnelles. Ann. Sci. Éc. Norm. Sup. 51, 45–78 (1934)

    MathSciNet  MATH  Google Scholar 

  20. Li G.-Z.: The fixed-point index and the fixed-point theorems of 1-set-contraction mapping. Proc. AMS 104(4), 1163–1170 (1988)

    Article  MATH  Google Scholar 

  21. Li G.-Z., Xu S.-Y., Duan H.-G.: Fixed point theorems of 1-set contractive operators in Banach spaces. Appl. Math. Lett. 19, 403–412 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu L.-S.: Approximation theorems and fixed point theorems for various classes of 1-set contractive mappings in Banach spaces. Acta Math. Sin. Eng. Ser. 17(1), 103–112 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mönch H.: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. TMA 4, 985–999 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nussbaum R.D.: The fixed point index and asymptotic fixed point theorems for k-set contractions. Bull. AMS 75, 490–495 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nussbaum R.D.: Degree theory for condensing maps. J. Math. Anal. Appl. 37, 741–766 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  26. O’Regan D.: Fixed-point theory for the sum of two operators. Appl. Math. Lett. 9(1), 1–8 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  27. Petryshyn W.V.: Structure of the fixed point sets of k-set contractions. Arch. Ration. Mech. Anal. 40, 312–328 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  28. Petryshyn W.V.: Remarks on condensing and k-set-contractive mappings. J. Math. Anal. Appl. 39, 717–741 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  29. Petryshyn W.V.: Fixed point theorems for various classes of 1-set-contractive and 1-ball-contractive mappings in Banach spaces. Trans. AMS 182, 323–3582 (1973)

    MathSciNet  MATH  Google Scholar 

  30. Reich S.: Fixed points of condensing functions. J. Math. Anal. Appl. 41, 460–467 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sadovskii B.N.: A fixed point principle. Funct. Anal. Appl. 1(2), 151–153 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shaini P.: Fixed points for 1-set contractions. Gen. Math. 19(2), 59–64 (2011)

    MathSciNet  MATH  Google Scholar 

  33. Smart, D.R.: Fixed Point Theorems. Cambridge University Press, London (1974)

  34. Zeidler, E.: Nonlinear Functional Analysis and its Applications. Fixed Point Theorems, vol. I. Springer, New York (1986)

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Authors and Affiliations

  1. Laboratoire “Théorie du Point Fixe et Applications”, École Normale Supérieure, B.P. 92, 16050, Kouba, Algiers, Algeria

    Smaïl Djebali & Karima Hammache

Authors
  1. Smaïl Djebali
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  2. Karima Hammache
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Correspondence to Karima Hammache.

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Djebali, S., Hammache, K. Boundary Conditions for 1-Set Contractions Maps in Banach Spaces. Mediterr. J. Math. 13, 1997–2010 (2016). https://doi.org/10.1007/s00009-015-0589-0

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  • DOI: https://doi.org/10.1007/s00009-015-0589-0

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