Constant-sign solutions for systems of singular integral equations of Hammerstein type

We consider the system of Hammerstein integral equations u"i(t)[email protected]!"0^Tg"i(t,s)f"i(s,u"1(s)[email protected]"1(s),u"2(s)[email protected]"2(s),...,u"n(s)[email protected]"n(s))ds,[email protected]?[0,T],[email protected][email protected]?n where T>0 is fixed, @r"i's are given functions and the nonlinearities f"i(t,x"1,x"2,...,x"n) can be singular at t=0 and x"j=0 where [email protected]?{1,2,...,n}. Criteria are offered for the existence of constant-sign solutions, i.e., @q"iu"i(t)>=0 for [email protected]?[0,T] and [email protected][email protected]?n, where @q"[email protected]?{1,-1} is fixed. The tools used are a nonlinear alternative of Leray-Schauder type, Krasnosel'skii's fixed point theorem in a cone and Schauder's fixed point theorem. We also include examples and applications to illustrate the usefulness of the results obtained.

[1]  Ravi P. Agarwal,et al.  Constant-Sign Solutions of a System of Integral Equations with Integrable Singularities , 2007 .

[2]  K. Q. Lan,et al.  Multiple positive solutions of Hammerstein integral equations and applications to periodic boundary value problems , 2004, Appl. Math. Comput..

[3]  Daniel Franco,et al.  COLLISIONLESS ORBITS OF SINGULAR AND NONSINGULAR DYNAMICAL SYSTEMS , 2006 .

[4]  Pedro J. Torres,et al.  Weak singularities may help periodic solutions to exist , 2007 .

[5]  Najla Yazidi,et al.  Monotone method for singular Neumann problem , 2002 .

[6]  Ravi P. Agarwal,et al.  Constant-sign solutions of a system of integral equations: The semipositone and singular case , 2005 .

[7]  Maria Meehan,et al.  Existence Theory for Nonlinear Integral and Integrodifferential Equations , 1998 .

[8]  Jifeng Chu,et al.  Positive solutions of Neumann problems with singularities , 2008 .

[9]  Jifeng Chu,et al.  Positive periodic solutions of Hill’s equations with singular nonlinear perturbations , 2008 .

[10]  Alberto Cabada,et al.  A Positive Operator Approach to the Neumann Problem for a Second Order Ordinary Differential Equation , 1996 .

[11]  Irena Rachůnková,et al.  Existence of Nonnegative and Nonpositive Solutions for Second Order Periodic Boundary Value Problems , 2001 .

[12]  Ravi P. Agarwal,et al.  Constant-Sign Solutions for Systems of Fredholm and Volterra Integral Equations: The Singular Case , 2008 .

[13]  M. A. Delpino,et al.  Infinitely Many T-Periodic Solutions for a Problem Arising in Nonlinear Elasticity , 1993 .

[14]  Manuel del Pino,et al.  T-periodic solutions for some second order differential equations with singularities , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[15]  Patrick Habets,et al.  Upper and Lower Solutions for a Generalized Emden-Fowler Equation , 1994 .

[16]  Liu Hut-zhao.  Existence of Positive Solutions to Second Order Neumann Boundary Value Problems , 2000 .

[17]  Pedro J. Torres,et al.  Existence and Uniqueness of Elliptic Periodic Solutions of the Brillouin Electron Beam Focusing System , 2000 .

[18]  Ravi P. Agarwal,et al.  Nonlinear integral equations singular in the dependent variable , 2007, Appl. Math. Lett..

[19]  Wenjie Gao,et al.  Singular nonlinear boundary value problems arising in boundary layer theory , 1999 .

[20]  Meirong Zhang,et al.  A relationship between the periodic and the Dirichlet BVPs of singular differential equations , 1998, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[21]  Meirong Zhang,et al.  Periodic solutions of second order non-autonomous singular dynamical systems , 2007 .

[22]  Ravi P. Agarwal,et al.  Opial's and wirtinger's type discrete inequalities in two independent variables , 1992 .

[23]  Kunquan Lan,et al.  Positive Solutions of Semilinear Differential Equations with Singularities , 1998 .

[24]  Ravi P. Agarwal,et al.  Existence Criteria for Singular Boundary Value Problems with Sign Changing Nonlinearities , 2002 .

[25]  A. Callegari,et al.  Some singular, nonlinear differential equations arising in boundary layer theory , 1978 .

[26]  Ravi P. Agarwal,et al.  A Note on a Singular Integral Equation Arising in a Problem in Communications , 2001 .

[27]  Sergio Solimini,et al.  On forced dynamical systems with a singularity of repulsive type , 1990 .

[28]  D. O’Regan,et al.  Positive Solutions of Differential, Difference and Integral Equations , 1998 .

[29]  Pedro J. Torres,et al.  Applications of Schauder's fixed point theorem to singular differential equations , 2007 .

[30]  Ravi P. Agarwal,et al.  Constant‐sign solutions for singular systems of Fredholm integral equations , 2010 .

[31]  Denis Bonheure,et al.  Forced singular oscillators and the method of lower and upper solutions , 2003 .

[32]  Salvador Villegas,et al.  Liapunov-type inequalities and Neumann boundary value problems at resonance , 2005 .

[33]  Patrick Habets,et al.  Periodic-solutions of Some Lienard Equations With Singularities , 1990 .

[34]  A. Callegari,et al.  A Nonlinear Singular Boundary Value Problem in the Theory of Pseudoplastic Fluids , 1980 .

[35]  Ravi P. Agarwal,et al.  Multiple positive solutions of singular problems by variational methods , 2005 .

[36]  Alberto Cabada,et al.  Monotone method for the Neumann problem with lower and upper solutions in the reverse order , 2001, Appl. Math. Comput..

[37]  Leo F. Boron,et al.  Positive solutions of operator equations , 1964 .

[38]  Meirong Zhang,et al.  Multiplicity of positive periodic solutions to superlinear repulsive singular equations , 2005 .

[39]  Meirong Zhang,et al.  Periodic Solutions of Equations of Emarkov-Pinney Type , 2006 .

[40]  Gennaro Infante,et al.  Nonzero solutions of Hammerstein integral equations with discontinuous kernels , 2002 .

[41]  Francesca Faraci,et al.  Solutions of Hammerstein Integral Equations via a Variational Principle , 2003 .

[42]  Pedro J. Torres,et al.  Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem , 2003 .

[43]  Sergio Solimini,et al.  On periodic solutions of nonlinear differential equations with singularities , 1987 .