Multiple positive solutions of systems of Hammerstein integral equations with applications to fractional differential equations

Positive solutions of systems of Hammerstein integral equations are studied by using the theory of the fixed‐point index for compact maps defined on cones in Banach spaces. Criteria for the fixed‐point index of the Hammerstein integral operators being 1 or 0 are given. These criteria are generalizations of previous results on a single Hammerstein integral operator. Some of criteria are new and involve the first eigenvalues of the corresponding systems of linear Hammerstein operators. The existence and estimates of the first eigenvalues are given. Applications are given to systems of fractional differential equations with two‐point boundary conditions. The Green's functions of the boundary value problems are derived and their useful properties are provided. As illustrations, the existence of nonzero positive solutions of two specific such boundary value problems is studied.

[1]  Zengqin Zhao,et al.  Positive solutions of semi-positone Hammerstein integral equations and applications , 2012, Appl. Math. Comput..

[2]  Meng Zhang,et al.  Positive solutions for boundary value problems of nonlinear fractional differential equations , 2011, Appl. Math. Comput..

[3]  K. Lan.  Eigenvalues of semi-positone Hammerstein integral equations and applications to boundary value problems , 2009 .

[4]  Xiyou Cheng,et al.  Existence of positive solutions to systems of nonlinear integral or differential equations , 2009 .

[5]  Daqing Jiang,et al.  Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation , 2009 .

[6]  Gennaro Infante,et al.  Non‐local boundary value problems of arbitrary order , 2009 .

[7]  V. Lakshmikantham,et al.  Basic theory of fractional differential equations , 2008 .

[8]  V. Lakshmikantham,et al.  General uniqueness and monotone iterative technique for fractional differential equations , 2008, Appl. Math. Lett..

[9]  J. Webb.  Uniqueness of the principal eigenvalue in nonlocal boundary value problems , 2007 .

[10]  K. Lan,et al.  Optimal constants for two point boundary value problems , 2007 .

[11]  Wei Lin.  Global existence theory and chaos control of fractional differential equations , 2007 .

[12]  Gennaro Infante,et al.  Positive Solutions of Nonlocal Boundary Value Problems: A Unified Approach , 2006 .

[13]  Kunquan Lan,et al.  Multiple eigenvalues for singular Hammerstein integral equations with applications to boundary value problems , 2006 .

[14]  K. Lan,et al.  Multiple Positive Solutions of Semi‐Positone Sturm–Liouville Boundary Value Problems , 2006 .

[15]  J. Webb,et al.  Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type , 2006 .

[16]  Xiyou Cheng,et al.  Existence of positive solutions for a second-order ordinary differential system☆ , 2005 .

[17]  Donal O'Regan,et al.  Positive solvability of systems of nonlinear Hammerstein integral equations , 2005 .

[18]  Zhanbing Bai,et al.  Positive solutions for boundary value problem of nonlinear fractional differential equation , 2005 .

[19]  Varsha Daftardar-Gejji,et al.  Positive solutions of a system of non-autonomous fractional differential equations , 2005 .

[20]  Yongxiang Li.  Abstract existence theorems of positive solutions for nonlinear boundary value problems , 2004 .

[21]  Jingxian Sun,et al.  Positive solutions of m-point boundary value problems for a class of nonlinear fractional differential equations , 2004, Journal of Applied Mathematics and Computing.

[22]  Kunquan Lan,et al.  Multiple Positive Solutions of Semilinear Differential Equations with Singularities , 2001 .

[23]  Shuqin Zhang,et al.  The Existence of a Positive Solution for a Nonlinear Fractional Differential Equation , 2000 .

[24]  L. Erbe.  Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems , 2000 .

[25]  Zhang Zhitao,et al.  Existence of non-trivial solution for superlinear system of integral equations and its applications , 1999 .

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

[27]  Luigi Rodino,et al.  Existence and Uniqueness for a Nonlinear Fractional Differential Equation , 1996 .

[28]  Zhaoli Liu,et al.  Multiple Positive Solutions of Nonlinear Two-Point Boundary Value Problems , 1996 .

[29]  Sun Jingxian,et al.  Computation for Topological Degree and Its Applications , 1996 .

[30]  M. A. Krasnoselʹskii,et al.  Geometrical Methods of Nonlinear Analysis , 1984 .

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

[32]  H. Amann.  Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces , 1976 .

[33]  Klaus Schmitt,et al.  Nonzero solutions of boundary value problems for second order ordinary and delay-differential equations , 1972 .

[34]  Zhanbing Bai,et al.  EXISTENCE OF POSITIVE SOLUTIONS FOR SINGULAR FRACTIONAL DIFFERENTIAL EQUATIONS , 2008 .

[35]  E. Kaufmann,et al.  Positive solutions of a boundary value problem for a nonlinear fractional differential equation. , 2008 .

[36]  J. Henderson,et al.  Positive Solutions for Systems of nth Order Three-point Nonlocal Boundary Value Problems , 2007 .

[37]  Bai,et al.  POSITIVE SOLUTIONS OF m-POINT BOUNDARY-VALUE PROBLEMS , 2005 .

[38]  Ravi P. Agarwal,et al.  Constant-Sign Solutions of a System of Fredholm Integral Equations , 2004 .

[39]  W. Ames,et al.  Nonlinear problems in abstract cones , 1988 .

[40]  W. Perry,et al.  An iterative method for solution of a boundary value problem in non-Newtonian fluid flow , 1984 .

[41]  Roger D. Nussbaum,et al.  Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem , 1981 .

[42]  R. Leggett,et al.  Multiple positive fixed points of nonlinear operators on ordered Banach spaces , 1979 .

[43]  R. Nussbaum.  PERIODIC SOLUTIONS OF SOME NONLINEAR INTEGRAL EQUATIONS , 1977 .

[44]  M. Kreĭn,et al.  Linear operators leaving invariant a cone in a Banach space , 1950 .