Existence theorem for the Hammerstein integral equation

Mieczysław Cichoń; Ireneusz Kubiaczyk

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (1996)

  • Volume: 16, Issue: 2, page 171-177
  • ISSN: 1509-9407

Abstract

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In this paper we prove an existence theorem for the Hammerstein integral equation x ( t ) = p ( t ) + λ ∫ I K ( t , s ) f ( s , x ( s ) ) d s , where the integral is taken in the sense of Pettis. In this theorem continuity assumptions for f are replaced by weak sequential continuity and the compactness condition is expressed in terms of the measures of weak noncompactness. Our equation is considered in general Banach spaces.

How to cite

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Mieczysław Cichoń, and Ireneusz Kubiaczyk. "Existence theorem for the Hammerstein integral equation." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 16.2 (1996): 171-177. <http://eudml.org/doc/275952>.

@article{MieczysławCichoń1996,
abstract = {In this paper we prove an existence theorem for the Hammerstein integral equation $x(t) = p(t) + λ ∫_I K(t,s)f(s,x(s))ds$, where the integral is taken in the sense of Pettis. In this theorem continuity assumptions for f are replaced by weak sequential continuity and the compactness condition is expressed in terms of the measures of weak noncompactness. Our equation is considered in general Banach spaces.},
author = {Mieczysław Cichoń, Ireneusz Kubiaczyk},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {integral equations; measures of weak noncompactness; Banach spaces; weak continuity; Hammerstein integral equation; Banach space; weak measure of non-compactness},
language = {eng},
number = {2},
pages = {171-177},
title = {Existence theorem for the Hammerstein integral equation},
url = {http://eudml.org/doc/275952},
volume = {16},
year = {1996},
}

TY - JOUR
AU - Mieczysław Cichoń
AU - Ireneusz Kubiaczyk
TI - Existence theorem for the Hammerstein integral equation
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 1996
VL - 16
IS - 2
SP - 171
EP - 177
AB - In this paper we prove an existence theorem for the Hammerstein integral equation $x(t) = p(t) + λ ∫_I K(t,s)f(s,x(s))ds$, where the integral is taken in the sense of Pettis. In this theorem continuity assumptions for f are replaced by weak sequential continuity and the compactness condition is expressed in terms of the measures of weak noncompactness. Our equation is considered in general Banach spaces.
LA - eng
KW - integral equations; measures of weak noncompactness; Banach spaces; weak continuity; Hammerstein integral equation; Banach space; weak measure of non-compactness
UR - http://eudml.org/doc/275952
ER -

References

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  6. [6] M. Cichoń, Weak solutions of differential equations in Banach spaces, Discuss. Math. Diff. Incl. 15 (1995), 5-14. 
  7. [7] J. Diestel and J.J. Uhl Jr., Vector Measures, Math. Surveys, Amer. Math. Soc., Providence, Rhode Island (15) (1977). 
  8. [8] I. Kubiaczyk, On a fixed point for weakly sequentially continuous mappings, Discuss. Math. - Diff. Incl. 15 (1995), 15-20. Zbl0832.47046
  9. [9] A.R. Mitchell and Ch. Smith, An existence theorem for weak solutions of differential equations in Banach spaces, in: 'Nonlinear Equations in Abstract Spaces', ed. V. Lakshmikantham, Academic Press (1978), 387-404. 
  10. [10] H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlin. Anal. Th. Meth. Appl. 4 (1980), 985-999. Zbl0462.34041
  11. [11] D. O'Regan, Integral equations in reflexive Banach spaces and the weak topologies, Proc. AMS 124 (1996), 607-614. Zbl0844.45009
  12. [12] S. Szufla, On the application of measure of noncompactness to existence theorems, Rend Sem. Mat. Univ. Padova, 75 (1986), 1-14. Zbl0589.45007
  13. [13] S. Szufla, On the Hammerstein integral equation with weakly singular kernel, Funkcialaj Ekvac. 34 (1991), 279-285. Zbl0753.45011
  14. [14] M. Talagrand, Pettis integral and measure theory, Memoires Amer. Math. Soc., Amer. Math. Soc., Providence, Rhode Island 51 (307) (1984). 

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