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Global-in-time Gevrey regularity solutions for the functionalized Cahn-Hilliard equation
Existence theorems for generalized nonlinear quadratic integral equations via a new fixed point result
Department of Mathematics and Computer Science, University of Perugia, Perugia, 060123, Italy |
The existence of $ L^{2} $-nonnegative solutions for nonlinear quadratic integral equations on a bounded closed interval is investigated. Two existence results for different classes of functions are shown. As a consequence an existence theorem for the Chandrasekhar integral quadratic equation, well-known in theory of radiative transfer, is obtained. The aim is achieved by means of a new fixed point theorem for multimaps in locally convex linear topological spaces.
References:
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J. Banaś, M. Lecko and W. G. El-Sayed,
Existence theorems for some quadratic integral equations, J. Math. Anal. Appl., 222 (1998), 276-285.
doi: 10.1006/jmaa.1998.5941. |
[2] |
J. Banaś and A .Martinon,
Monotonic solutions of a quadratic integral equation of Volterra type, Comput. Math. Appl., 47 (2004), 271-279.
doi: 10.1016/S0898-1221(04)90024-7. |
[3] |
A. Bellour, D. O'Regan and M.-A. Taoudi,
On the existence of integrable solutions for a nonlinear quadratic integral equation, J. Appl. Math. Comput., 46 (2014), 67-77.
doi: 10.1007/s12190-013-0737-2. |
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V. C. Boffi and G. Spiga,
Nonlinear removal effects in time-dependent particle transport theory, Z. Angew. Math. Phys., 34 (1983), 347-357.
doi: 10.1007/BF00944855. |
[5] |
V. C. Boffi and G. Spiga,
An equation of Hammerstein type arising in particle transport theory, J. Math. Phys., 24 (1983), 1625-1629.
doi: 10.1063/1.525857. |
[6] | L. W. Busbridge, The Mathematics of Radiative Transfer, Cambridge Univesrity Press, Cambridge, 1960. Google Scholar |
[7] |
J. Caballero, D. O'Regan and K. Sadarangani,
On solutions of an integral equation related to traffic flow on unbounded domains, Arch. Math. (Basel), 82 (2004), 551-563.
doi: 10.1007/s00013-003-0609-3. |
[8] |
T. Cardinali, D. O'Regan and P. Rubbioni,
Mönch sets and fixed point theorems for multimaps in locally convex topological vector spaces, Fixed Point Theory, 18 (2017), 147-153.
doi: 10.24193/fpt-ro.2017.1.12. |
[9] |
T. Cardinali and F. Papalini,
Fixed point theorems for multifunctions in topological vector spaces, J. Math. Anal. Appl., 186 (1994), 769-777.
doi: 10.1006/jmaa.1994.1332. |
[10] |
S. Chandrasekhar, Radiative Transfer, Dover Publications Inc., New York, 1960. |
[11] |
Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic Publishers, Boston, MA, 2003.
doi: 10.1007/978-1-4419-9158-4. |
[12] |
A. M. A. El-Sayed, H. H. G. Hashem and E. A. A. Ziada,
Picard and Adomian methods for quadratic integral equation, Comput. Appl. Math., 29 (2010), 447-463.
doi: 10.1590/S1807-03022010000300007. |
[13] |
R. Figueroa and G. Infante, A Schauder-type theorem for discontinuous operators with applications to second-order BVPs, Fixed Point Theory Appl., (2016), 11 pp.
doi: 10.1186/s13663-016-0547-y. |
[14] |
S. C. Hu, M. Khavanin and W. Zhuang,
Integral equations arising in the kinetic theory of gases, Appl. Anal., 34 (1989), 261-266.
doi: 10.1080/00036818908839899. |
[15] |
Z. Q. Liu and S. M. Kang,
Existence of monotone solutions for a nonlinear quadratic integral equation of Volterra type, Rocky Mountain J. Math., 37 (2007), 1971-1980.
doi: 10.1216/rmjm/1199649833. |
[16] |
R. López Pouso, Schauder's fixed-point theorem: New applications and a new version for discontinuous operators, Bound. Value Probl., 2012 (2012), 14 pp.
doi: 10.1186/1687-2770-2012-92. |
[17] |
H. H. Schaefer, Topological Vector Spaces, Graduate Texts in Mathematics, Vol. 3. Springer-Verlag, New York-Berlin, 1971. |
show all references
References:
[1] |
J. Banaś, M. Lecko and W. G. El-Sayed,
Existence theorems for some quadratic integral equations, J. Math. Anal. Appl., 222 (1998), 276-285.
doi: 10.1006/jmaa.1998.5941. |
[2] |
J. Banaś and A .Martinon,
Monotonic solutions of a quadratic integral equation of Volterra type, Comput. Math. Appl., 47 (2004), 271-279.
doi: 10.1016/S0898-1221(04)90024-7. |
[3] |
A. Bellour, D. O'Regan and M.-A. Taoudi,
On the existence of integrable solutions for a nonlinear quadratic integral equation, J. Appl. Math. Comput., 46 (2014), 67-77.
doi: 10.1007/s12190-013-0737-2. |
[4] |
V. C. Boffi and G. Spiga,
Nonlinear removal effects in time-dependent particle transport theory, Z. Angew. Math. Phys., 34 (1983), 347-357.
doi: 10.1007/BF00944855. |
[5] |
V. C. Boffi and G. Spiga,
An equation of Hammerstein type arising in particle transport theory, J. Math. Phys., 24 (1983), 1625-1629.
doi: 10.1063/1.525857. |
[6] | L. W. Busbridge, The Mathematics of Radiative Transfer, Cambridge Univesrity Press, Cambridge, 1960. Google Scholar |
[7] |
J. Caballero, D. O'Regan and K. Sadarangani,
On solutions of an integral equation related to traffic flow on unbounded domains, Arch. Math. (Basel), 82 (2004), 551-563.
doi: 10.1007/s00013-003-0609-3. |
[8] |
T. Cardinali, D. O'Regan and P. Rubbioni,
Mönch sets and fixed point theorems for multimaps in locally convex topological vector spaces, Fixed Point Theory, 18 (2017), 147-153.
doi: 10.24193/fpt-ro.2017.1.12. |
[9] |
T. Cardinali and F. Papalini,
Fixed point theorems for multifunctions in topological vector spaces, J. Math. Anal. Appl., 186 (1994), 769-777.
doi: 10.1006/jmaa.1994.1332. |
[10] |
S. Chandrasekhar, Radiative Transfer, Dover Publications Inc., New York, 1960. |
[11] |
Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic Publishers, Boston, MA, 2003.
doi: 10.1007/978-1-4419-9158-4. |
[12] |
A. M. A. El-Sayed, H. H. G. Hashem and E. A. A. Ziada,
Picard and Adomian methods for quadratic integral equation, Comput. Appl. Math., 29 (2010), 447-463.
doi: 10.1590/S1807-03022010000300007. |
[13] |
R. Figueroa and G. Infante, A Schauder-type theorem for discontinuous operators with applications to second-order BVPs, Fixed Point Theory Appl., (2016), 11 pp.
doi: 10.1186/s13663-016-0547-y. |
[14] |
S. C. Hu, M. Khavanin and W. Zhuang,
Integral equations arising in the kinetic theory of gases, Appl. Anal., 34 (1989), 261-266.
doi: 10.1080/00036818908839899. |
[15] |
Z. Q. Liu and S. M. Kang,
Existence of monotone solutions for a nonlinear quadratic integral equation of Volterra type, Rocky Mountain J. Math., 37 (2007), 1971-1980.
doi: 10.1216/rmjm/1199649833. |
[16] |
R. López Pouso, Schauder's fixed-point theorem: New applications and a new version for discontinuous operators, Bound. Value Probl., 2012 (2012), 14 pp.
doi: 10.1186/1687-2770-2012-92. |
[17] |
H. H. Schaefer, Topological Vector Spaces, Graduate Texts in Mathematics, Vol. 3. Springer-Verlag, New York-Berlin, 1971. |
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