# American Institute of Mathematical Sciences

November  2013, 33(11&12): 5507-5519. doi: 10.3934/dcds.2013.33.5507

## Positive solutions of nonlinear equations via comparison with linear operators

 1 School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QW, United Kingdom

Received  June 2011 Revised  March 2012 Published  May 2013

We discuss positive solutions of problems that arise from nonlinear boundary value problems in the particular situation where the nonlinear term $f(t,u)$ depends explicitly on $t$ and this dependence is crucial. We give new fixed point index results using comparisons with linear operators. These prove new results on existence of positive solutions under some conditions which can be sharp.
Citation: Jeffrey R. L. Webb. Positive solutions of nonlinear equations via comparison with linear operators. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5507-5519. doi: 10.3934/dcds.2013.33.5507
##### References:
 [1] K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985. Reprinted: Dover Publications 2010, ISBN 13: 9780486474410 [2] L. Erbe, Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems, Math. Comput. Modelling, 32 (2000), 529-539. doi: 10.1016/S0895-7177(00)00150-3. [3] J. R. Graef and L. Kong, Existence results for nonlinear periodic boundary-value problems, Proc. Edinb. Math. Soc., 52 (2009), 79-95. doi: 10.1017/S0013091507000788. [4] D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones," Academic Press, 1988. [5] M. S. Keener and C. C. Travis, Positive cones and focal points for a class of $n$th order differential equations, Trans. Amer. Math. Soc., 237 (1978), 331-351. doi: 10.2307/1997625. [6] L. Kong and Q. Kong, Higher order boundary value problems with nonhomogeneous boundary conditions, Nonlinear Anal., 72 (2010), 240-261. doi: 10.1016/j.na.2009.06.050. [7] L. Kong and J. S. W. Wong, Positive solutions for higher order multi-point boundary value problems with nonhomogeneous boundary conditions, J. Math. Anal. Appl., 367 (2010), 588-611. doi: 10.1016/j.jmaa.2010.01.063. [8] M. A. Krasnosel'skiĭ, "Positive Solutions of Operator Equations," P. Noordhoff Ltd. Groningen, 1964. [9] M. A. Krasnosel'skiĭ and P. P. Zabreĭko, "Geometrical Methods of Nonlinear Analysis," Springer, Berlin, 1984. doi: 10.1007/978-3-642-69409-7. [10] K. Q. Lan, Multiple positive solutions of semilinear differential equations with singularities, J. Lond. Math. Soc. (2), 63 (2001), 690-704. doi: 10.1112/S002461070100206X. [11] K. Q. Lan, Multiple positive solutions of Hammerstein integral equations with singularities, Differential Equations Dynam. Syst., 8 (2000), 175-192. [12] K. Q. Lan, Multiple positive solutions of conjugate boundary value problems with singularities, Appl. Math. Comput., 147 (2004), 461-474. doi: 10.1016/S0096-3003(02)00739-7. [13] K. Q. Lan, Multiple positive solutions of Hammerstein integral equations and applications to periodic boundary value problems, Appl. Math. Comput., 154 (2004), 531-542. doi: 10.1016/S0096-3003(03)00733-1. [14] K. Q. Lan, Multiple eigenvalues for singular Hammerstein integral equations with applications to boundary value problems, J. Comput. Appl. Math., 189 (2006), 109-119, doi: 10.1016/j.cam.2005.03.029. [15] K. Q. Lan, Eigenvalues of semi-positone Hammerstein integral equations and applications to boundary value problems, Nonlinear Anal., 71 (2009), 5979-5993. doi: 10.1016/j.na.2009.05.022. [16] K. Q. Lan and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations, 148 (1998), 407-421. doi: 10.1006/jdeq.1998.3475. [17] B. Liu, L. Liu and Y. Wu, Positive solutions for a singular second-order three-point boundary value problem, Appl. Math. Comput., 196 (2008), 532-541. doi: 10.1016/j.amc.2007.06.013. [18] R. Ma and L. Ren, Positive solutions for nonlinear $m$-point boundary value problems of Dirichlet type via fixed-point index theory, Appl. Math. Lett., 16 (2003), 863-869. doi: 10.1016/S0893-9659(03)90009-7. [19] R. H. Martin, "Nonlinear Operators and Differential Equations in Banach Spaces," Wiley, New York, 1976. [20] R. D. Nussbaum, Periodic solutions of some nonlinear integral equations, Dynamical systems, (Proc. Internat. Sympos., Univ. Florida, Gainesville, Fla., 1976), 221-249. Academic Press, New York, (1977). [21] Y. Sun, L. Liu, J. Zhang and R. P. Agarwal, Positive solutions of singular three-point boundary value problems for second-order differential equations, J. Comput. Appl. Math., 230 (2009), 738-750. doi: 10.1016/j.cam.2009.01.003. [22] J. R. L. Webb, Solutions of nonlinear equations in cones and positive linear operators, J. Lond. Math. Soc. (2), 82 (2010), 420-436. doi: 10.1112/jlms/jdq037. [23] J. R. L. Webb, A class of positive linear operators and applications to nonlinear boundary value problems, Topol. Methods Nonlinear Anal., 39 (2012), 221-242. [24] J. R. L. Webb and K. Q. Lan, Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type, Topol. Methods Nonlinear Anal., 27 (2006), 91-116. [25] J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems involving integral conditions, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 45-67. doi: 10.1007/s00030-007-4067-7. [26] J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: A unified approach, J. Lond. Math. Soc. (2), 74 (2006), 673-693. doi: 10.1112/S0024610706023179. [27] J. R. L. Webb and G. Infante, Nonlocal boundary value problems of arbitrary order, J. Lond. Math. Soc. (2), 79 (2009), 238-258. doi: 10.1112/jlms/jdn066. [28] J. R. L. Webb, G. Infante and D. Franco, Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 427-446. doi: 10.1017/S0308210506001041. [29] G. Zhang and J. Sun, Positive solutions of $m$-point boundary value problems, J. Math. Anal. Appl., 291 (2004), 406-418. doi: 10.1016/j.jmaa.2003.11.034.

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##### References:
 [1] K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985. Reprinted: Dover Publications 2010, ISBN 13: 9780486474410 [2] L. Erbe, Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems, Math. Comput. Modelling, 32 (2000), 529-539. doi: 10.1016/S0895-7177(00)00150-3. [3] J. R. Graef and L. Kong, Existence results for nonlinear periodic boundary-value problems, Proc. Edinb. Math. Soc., 52 (2009), 79-95. doi: 10.1017/S0013091507000788. [4] D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones," Academic Press, 1988. [5] M. S. Keener and C. C. Travis, Positive cones and focal points for a class of $n$th order differential equations, Trans. Amer. Math. Soc., 237 (1978), 331-351. doi: 10.2307/1997625. [6] L. Kong and Q. Kong, Higher order boundary value problems with nonhomogeneous boundary conditions, Nonlinear Anal., 72 (2010), 240-261. doi: 10.1016/j.na.2009.06.050. [7] L. Kong and J. S. W. Wong, Positive solutions for higher order multi-point boundary value problems with nonhomogeneous boundary conditions, J. Math. Anal. Appl., 367 (2010), 588-611. doi: 10.1016/j.jmaa.2010.01.063. [8] M. A. Krasnosel'skiĭ, "Positive Solutions of Operator Equations," P. Noordhoff Ltd. Groningen, 1964. [9] M. A. Krasnosel'skiĭ and P. P. Zabreĭko, "Geometrical Methods of Nonlinear Analysis," Springer, Berlin, 1984. doi: 10.1007/978-3-642-69409-7. [10] K. Q. Lan, Multiple positive solutions of semilinear differential equations with singularities, J. Lond. Math. Soc. (2), 63 (2001), 690-704. doi: 10.1112/S002461070100206X. [11] K. Q. Lan, Multiple positive solutions of Hammerstein integral equations with singularities, Differential Equations Dynam. Syst., 8 (2000), 175-192. [12] K. Q. Lan, Multiple positive solutions of conjugate boundary value problems with singularities, Appl. Math. Comput., 147 (2004), 461-474. doi: 10.1016/S0096-3003(02)00739-7. [13] K. Q. Lan, Multiple positive solutions of Hammerstein integral equations and applications to periodic boundary value problems, Appl. Math. Comput., 154 (2004), 531-542. doi: 10.1016/S0096-3003(03)00733-1. [14] K. Q. Lan, Multiple eigenvalues for singular Hammerstein integral equations with applications to boundary value problems, J. Comput. Appl. Math., 189 (2006), 109-119, doi: 10.1016/j.cam.2005.03.029. [15] K. Q. Lan, Eigenvalues of semi-positone Hammerstein integral equations and applications to boundary value problems, Nonlinear Anal., 71 (2009), 5979-5993. doi: 10.1016/j.na.2009.05.022. [16] K. Q. Lan and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations, 148 (1998), 407-421. doi: 10.1006/jdeq.1998.3475. [17] B. Liu, L. Liu and Y. Wu, Positive solutions for a singular second-order three-point boundary value problem, Appl. Math. Comput., 196 (2008), 532-541. doi: 10.1016/j.amc.2007.06.013. [18] R. Ma and L. Ren, Positive solutions for nonlinear $m$-point boundary value problems of Dirichlet type via fixed-point index theory, Appl. Math. Lett., 16 (2003), 863-869. doi: 10.1016/S0893-9659(03)90009-7. [19] R. H. Martin, "Nonlinear Operators and Differential Equations in Banach Spaces," Wiley, New York, 1976. [20] R. D. Nussbaum, Periodic solutions of some nonlinear integral equations, Dynamical systems, (Proc. Internat. Sympos., Univ. Florida, Gainesville, Fla., 1976), 221-249. Academic Press, New York, (1977). [21] Y. Sun, L. Liu, J. Zhang and R. P. Agarwal, Positive solutions of singular three-point boundary value problems for second-order differential equations, J. Comput. Appl. Math., 230 (2009), 738-750. doi: 10.1016/j.cam.2009.01.003. [22] J. R. L. Webb, Solutions of nonlinear equations in cones and positive linear operators, J. Lond. Math. Soc. (2), 82 (2010), 420-436. doi: 10.1112/jlms/jdq037. [23] J. R. L. Webb, A class of positive linear operators and applications to nonlinear boundary value problems, Topol. Methods Nonlinear Anal., 39 (2012), 221-242. [24] J. R. L. Webb and K. Q. Lan, Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type, Topol. Methods Nonlinear Anal., 27 (2006), 91-116. [25] J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems involving integral conditions, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 45-67. doi: 10.1007/s00030-007-4067-7. [26] J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: A unified approach, J. Lond. Math. Soc. (2), 74 (2006), 673-693. doi: 10.1112/S0024610706023179. [27] J. R. L. Webb and G. Infante, Nonlocal boundary value problems of arbitrary order, J. Lond. Math. Soc. (2), 79 (2009), 238-258. doi: 10.1112/jlms/jdn066. [28] J. R. L. Webb, G. Infante and D. Franco, Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 427-446. doi: 10.1017/S0308210506001041. [29] G. Zhang and J. Sun, Positive solutions of $m$-point boundary value problems, J. Math. Anal. Appl., 291 (2004), 406-418. doi: 10.1016/j.jmaa.2003.11.034.
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