2009, 2009(Special): 818-827. doi: 10.3934/proc.2009.2009.818

Existence of solutions to singular integral equations

1. 

School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore

Received  July 2008 Revised  March 2009 Published  September 2009

We consider the system of integral equations


$u_i(t)=int_0^Tg_i(t,s)[a_i(s,u_1(s),u_2(s),...,u_n(s))+b_i(s,u_1(s),u_2(s),...,u_n(s))]ds,$   $t \in [0,T],$   $1<=i<=n,$

where $T>0$ is fixed and the nonlinearities $a_i(t,u_1,u_2,\cdots,u_n)$ can be singular at $t=0$ and $u_j=0$ where $j\in\{1,2,\cdots,n\}.$ Criteria are established for the existence of fixed-sign solutions $(u_1^*,u_2^*,\cdots,u_n^*)$ to the above system, i.e., $\theta_iu_i^*(t)\geq 0$ for $t\in [0,T]$ and $1\leq i\leq n,$ where $\theta_i\in \{1,-1\}$ is fixed. We also include an example to illustrate the usefulness of the results obtained.

Citation: 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
[1]

Patricia J.Y. Wong. On the existence of fixed-sign solutions for a system of generalized right focal problems with deviating arguments. Conference Publications, 2007, 2007 (Special) : 1042-1051. doi: 10.3934/proc.2007.2007.1042

[2]

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

[3]

J. Tyagi. Multiple solutions for singular N-Laplace equations with a sign changing nonlinearity. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2381-2391. doi: 10.3934/cpaa.2013.12.2381

[4]

Dongyan Li, Yongzhong Wang. Nonexistence of positive solutions for a system of integral equations on $R^n_+$ and applications. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2601-2613. doi: 10.3934/cpaa.2013.12.2601

[5]

G. C. Yang, K. Q. Lan. Systems of singular integral equations and applications to existence of reversed flow solutions of Falkner-Skan equations. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2465-2495. doi: 10.3934/cpaa.2013.12.2465

[6]

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

[7]

Daniela Giachetti, Francesco Petitta, Sergio Segura de León. Elliptic equations having a singular quadratic gradient term and a changing sign datum. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1875-1895. doi: 10.3934/cpaa.2012.11.1875

[8]

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

[9]

Liang Ding, Rongrong Tian, Jinlong Wei. Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1617-1625. doi: 10.3934/dcdsb.2018222

[10]

Huaiyu Zhou, Jingbo Dou. Classifications of positive solutions to an integral system involving the multilinear fractional integral inequality. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022070

[11]

Hui Guo, Tao Wang. A note on sign-changing solutions for the Schrödinger Poisson system. Electronic Research Archive, 2020, 28 (1) : 195-203. doi: 10.3934/era.2020013

[12]

Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151

[13]

Yavdat Il'yasov, Nadir Sari. Solutions of minimal period for a Hamiltonian system with a changing sign potential. Communications on Pure and Applied Analysis, 2005, 4 (1) : 175-185. doi: 10.3934/cpaa.2005.4.175

[14]

Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure and Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041

[15]

Oleksandr Boichuk, Victor Feruk. Boundary-value problems for weakly singular integral equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1379-1395. doi: 10.3934/dcdsb.2021094

[16]

Tiziana Cardinali, Paola Rubbioni. Existence theorems for generalized nonlinear quadratic integral equations via a new fixed point result. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1947-1955. doi: 10.3934/dcdss.2020152

[17]

Dumitru Motreanu. Three solutions with precise sign properties for systems of quasilinear elliptic equations. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 831-843. doi: 10.3934/dcdss.2012.5.831

[18]

Josep M. Olm, Xavier Ros-Oton. Existence of periodic solutions with nonconstant sign in a class of generalized Abel equations. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1603-1614. doi: 10.3934/dcds.2013.33.1603

[19]

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

[20]

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

 Impact Factor: 

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]