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doi: 10.3934/dcdss.2020152

Existence theorems for generalized nonlinear quadratic integral equations via a new fixed point result

Tiziana Cardinali and  Paola Rubbioni

Department of Mathematics and Computer Science, University of Perugia, Perugia, 060123, Italy

Dedicated to Professor Patrizia Pucci for her 65th birthday anniversary

Received  September 2018 Revised  October 2018 Published  November 2019

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.

Keywords: Quadratic integral equations, Chandrasekhar integral equations, fixed point theorems.
Mathematics Subject Classification: Primary: 45G10, 47H10; Secondary: 47N20, 45B05, 45D05.
Citation: Tiziana Cardinali, Paola Rubbioni. Existence theorems for generalized nonlinear quadratic integral equations via a new fixed point result. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020152
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Chandrasekhar, Radiative Transfer, Dover Publications Inc., New York, 1960.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

H. H. Schaefer, Topological Vector Spaces, Graduate Texts in Mathematics, Vol. 3. Springer-Verlag, New York-Berlin, 1971.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Chandrasekhar, Radiative Transfer, Dover Publications Inc., New York, 1960.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

H. H. Schaefer, Topological Vector Spaces, Graduate Texts in Mathematics, Vol. 3. Springer-Verlag, New York-Berlin, 1971.  Google Scholar

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