American Institute of Mathematical Sciences

Advanced
July  2020, 13(7): 1947-1955. 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, 2020, 13 (7) : 1947-1955. 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

[1]

Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248
[2]

Xiaohui Yu. Liouville type theorems for singular integral equations and integral systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1825-1840. doi: 10.3934/cpaa.2016017
[3]

Natalia Skripnik. Averaging of fuzzy integral equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1999-2010. doi: 10.3934/dcdsb.2017118
[4]

William Rundell. Recovering an obstacle using integral equations. Inverse Problems & Imaging, 2009, 3 (2) : 319-332. doi: 10.3934/ipi.2009.3.319
[5]

Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925
[6]

Wenxiong Chen, Congming Li. Regularity of solutions for a system of integral equations. Communications on Pure & Applied Analysis, 2005, 4 (1) : 1-8. doi: 10.3934/cpaa.2005.4.1
[7]

Patricia J.Y. Wong. Existence of solutions to singular integral equations. Conference Publications, 2009, 2009 (Special) : 818-827. doi: 10.3934/proc.2009.2009.818
[8]

Roman Chapko, B. Tomas Johansson. Integral equations for biharmonic data completion. Inverse Problems & Imaging, 2019, 13 (5) : 1095-1111. doi: 10.3934/ipi.2019049
[9]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[10]

M. R. Arias, R. Benítez. Properties of solutions for nonlinear Volterra integral equations. Conference Publications, 2003, 2003 (Special) : 42-47. doi: 10.3934/proc.2003.2003.42
[11]

Diogo A. Gomes, Gabriele Terrone. Bernstein estimates: weakly coupled systems and integral equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 861-883. doi: 10.3934/cpaa.2012.11.861
[12]

Nakao Hayashi, Tohru Ozawa. Schrödinger equations with nonlinearity of integral type. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 475-484. doi: 10.3934/dcds.1995.1.475
[13]

Onur Alp İlhan. Solvability of some partial integral equations in Hilbert space. Communications on Pure & Applied Analysis, 2008, 7 (4) : 837-844. doi: 10.3934/cpaa.2008.7.837
[14]

Zhongying Chen, Bin Wu, Yuesheng Xu. Fast numerical collocation solutions of integral equations. Communications on Pure & Applied Analysis, 2007, 6 (3) : 643-666. doi: 10.3934/cpaa.2007.6.643
[15]

Congming Li, Jisun Lim. The singularity analysis of solutions to some integral equations. Communications on Pure & Applied Analysis, 2007, 6 (2) : 453-464. doi: 10.3934/cpaa.2007.6.453
[16]

Mingchun Wang, Jiankai Xu, Huoxiong Wu. On Positive solutions of integral equations with the weighted Bessel potentials. Communications on Pure & Applied Analysis, 2019, 18 (2) : 625-641. doi: 10.3934/cpaa.2019031
[17]

Nguyen Dinh Cong, Doan Thai Son. On integral separation of bounded linear random differential equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 995-1007. doi: 10.3934/dcdss.2016038
[18]

Yutian Lei, Chao Ma. Asymptotic behavior for solutions of some integral equations. Communications on Pure & Applied Analysis, 2011, 10 (1) : 193-207. doi: 10.3934/cpaa.2011.10.193
[19]

Phuong Le. Liouville theorems for an integral equation of Choquard type. Communications on Pure & Applied Analysis, 2020, 19 (2) : 771-783. doi: 10.3934/cpaa.2020036
[20]

Charles L. Epstein, Leslie Greengard, Thomas Hagstrom. On the stability of time-domain integral equations for acoustic wave propagation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4367-4382. doi: 10.3934/dcds.2016.36.4367

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (106)
  • HTML views (200)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences