# 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

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topMieczysław 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

top- [1] J. Appell, Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator, J. Math. Anal. Appl. 83 (1981), 251-263. Zbl0495.45007
- [2] J. Appell, Misure di non compattezza in spazi ideali, Rend. Sc. Instituto Lombardo A 119 (1985), 175-186.
- [3] O. Arino, S. Gautier and J. P. Penot, A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations, Funkcialaj Ekvac. 27 (1984), 273-279. Zbl0599.34008
- [4] J.M. Ball, Weak continuity properties of mappings and semi-groups Proc. Royal Soc. Edinbourgh Sect. A 72 (1979), 275-280.
- [5] M. Cichoń, On bounded weak solutions of a nonlinear differential equation in Banach spaces, Functiones et Approximatio 21 (1992), 27-35. Zbl0777.34041
- [6] M. Cichoń, Weak solutions of differential equations in Banach spaces, Discuss. Math. Diff. Incl. 15 (1995), 5-14.
- [7] J. Diestel and J.J. Uhl Jr., Vector Measures, Math. Surveys, Amer. Math. Soc., Providence, Rhode Island (15) (1977).
- [8] I. Kubiaczyk, On a fixed point for weakly sequentially continuous mappings, Discuss. Math. - Diff. Incl. 15 (1995), 15-20. Zbl0832.47046
- [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] 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] D. O'Regan, Integral equations in reflexive Banach spaces and the weak topologies, Proc. AMS 124 (1996), 607-614. Zbl0844.45009
- [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] S. Szufla, On the Hammerstein integral equation with weakly singular kernel, Funkcialaj Ekvac. 34 (1991), 279-285. Zbl0753.45011
- [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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