American Institute of Mathematical Sciences

February  2020, 25(2): 691-699. doi: 10.3934/dcdsb.2019261

Positive and increasing solutions of perturbed Hammerstein integral equations with derivative dependence

 Dipartimento di Matematica e Informatica, Università della Calabria, 87036 Arcavacata di Rende, Cosenza, Italy

Dedicated to Professor Juan J. Nieto on the occasion of his sixtieth birthday

Received  January 2019 Revised  March 2019 Published  February 2020 Early access  November 2019

We discuss the existence and non-existence of non-negative, non-decreasing solutions of certain perturbed Hammerstein integral equations with derivative dependence. We present some applications to nonlinear, second order boundary value problems subject to fairly general functional boundary conditions. The approach relies on classical fixed point index theory.

Citation: Gennaro Infante. Positive and increasing solutions of perturbed Hammerstein integral equations with derivative dependence. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 691-699. doi: 10.3934/dcdsb.2019261
References:
 [1] E. Alves, T. F. Ma and M. L. Pelicer, Monotone positive solutions for a fourth order equation with nonlinear boundary conditions, Nonlinear Anal., 71 (2009), 3834-3841.  doi: 10.1016/j.na.2009.02.051. [2] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM. Rev., 18 (1976), 620-709.  doi: 10.1137/1018114. [3] A. Cabada, An overview of the lower and upper solutions method with nonlinear boundary value conditions, Bound. Value Probl., 2011 (2011), Art. ID 893753, 18 pp. doi: 10.1155/2011/893753. [4] A. Cabada, G. Infante and F. A. F. Tojo, Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications, Topol. Methods Nonlinear Anal., 47 (2016), 265-287.  doi: 10.12775/TMNA.2016.005. [5] F. Cianciaruso, G. Infante and P. Pietramala, Solutions of perturbed Hammerstein integral equations with applications, Nonlinear Anal. Real World Appl., 33 (2017), 317-347.  doi: 10.1016/j.nonrwa.2016.07.004. [6] R. Conti, Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. Un. Mat. Ital., 22 (1967), 135-178. [7] H. Fan and R. Ma, Loss of positivity in a nonlinear second order ordinary differential equations, Nonlinear Anal., 71 (2009), 437-444.  doi: 10.1016/j.na.2008.10.117. [8] D. Franco, G. Infante and J. Perán, A new criterion for the existence of multiple solutions in cones, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1043-1050.  doi: 10.1017/S0308210511001016. [9] D. Franco, D. O'Regan and J. Perán, Fourth-order problems with nonlinear boundary conditions, J. Comput. Appl. Math., 174 (2005), 315-327.  doi: 10.1016/j.cam.2004.04.013. [10] H. Garai, L. K. Dey and A. Chanda, Positive solutions to a fractional thermostat model in Banach spaces via fixed point results, J. Fixed Point Theory Appl., 20 (2018), Art. 106, 24 pp. doi: 10.1007/s11784-018-0584-8. [11] C. S. Goodrich, On nonlocal BVPs with nonlinear boundary conditions with asymptotically sublinear or superlinear growth, Math. Nachr., 285 (2012), 1404-1421.  doi: 10.1002/mana.201100210. [12] C. S. Goodrich, Positive solutions to boundary value problems with nonlinear boundary conditions, Nonlinear Anal., 75 (2012), 417-432.  doi: 10.1016/j.na.2011.08.044. [13] C. S. Goodrich, On nonlinear boundary conditions satisfying certain asymptotic behavior, Nonlinear Anal., 76 (2013), 58-67.  doi: 10.1016/j.na.2012.07.023. [14] C. S. Goodrich, A note on semipositone boundary value problems with nonlocal, nonlinear boundary conditions, Arch. Math. (Basel), 103 (2014), 177-187.  doi: 10.1007/s00013-014-0678-5. [15] C. S. Goodrich, Semipositone boundary value problems with nonlocal, nonlinear boundary conditions, Adv. Differential Equations, 20 (2015), 117-142. [16] C. S. Goodrich, Radially symmetric solutions of elliptic PDEs with uniformly negative weight, Ann. Mat. Pura Appl., 197 (2018), 1585-1611.  doi: 10.1007/s10231-018-0738-8. [17] C. S. Goodrich, New Harnack inequalities and existence theorems for radially symmetric solutions of elliptic PDEs with sign changing or vanishing Green's function, J. Differential Equations, 264 (2018), 236-262.  doi: 10.1016/j.jde.2017.09.011. [18] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering, 5, Academic Press, Inc., Boston, 1988. [19] G. Infante, Nonlocal boundary value problems with two nonlinear boundary conditions, Commun. Appl. Anal., 12 (2008), 279-288. [20] G. Infante, A short course on positive solutions of systems of ODEs via fixed point index, Lecture Notes in Nonlinear Analysis (LNNA), 16 (2017), 93-140. [21] G. Infante, Nonzero positive solutions of a multi-parameter elliptic system with functional BCs, Topol. Methods Nonlinear Anal., 52 (2018), 665-675.  doi: 10.12775/TMNA.2017.060. [22] G. Infante and J. R. L. Webb, Loss of positivity in a nonlinear scalar heat equation, NoDEA Nonlinear Differential Equations Appl., 13 (2006), 249-261.  doi: 10.1007/s00030-005-0039-y. [23] G. Infante and J. R. L. Webb, Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations, Proc. Edinb. Math. Soc., 49 (2006), 637-656.  doi: 10.1017/S0013091505000532. [24] G. Kalna and S. McKee, The thermostat problem, TEMA Tend. Mat. Apl. Comput., 3 (2002), 15-29.  doi: 10.5540/tema.2002.03.01.0015. [25] G. Kalna and S. McKee, The thermostat problem with a nonlocal nonlinear boundary condition, IMA J. Appl. Math., 69 (2004), 437-462.  doi: 10.1093/imamat/69.5.437. [26] G. L. Karakostas, Existence of solutions for an n-dimensional operator equation and applications to BVPs, Electron. J. Differential Equations, 2014 (2014), 17pp. [27] G. L. Karakostas and P. Ch. Tsamatos, Existence of multiple positive solutions for a nonlocal boundary value problem, Topol. Methods Nonlinear Anal., 19 (2002), 109-121.  doi: 10.12775/TMNA.2002.007. [28] G. L. Karakostas and P. Ch. Tsamatos, Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems, Electron. J. Differential Equations, 2002 (2002), 17pp. [29] I. Karatsompanis and P. K. Palamides, Polynomial approximation to a non-local boundary value problem, Comput. Math. Appl., 60 (2010), 3058-3071.  doi: 10.1016/j.camwa.2010.10.006. [30] R. Ma, A survey on nonlocal boundary value problems, Appl. Math. E-Notes, 7 (2007), 257-279. [31] J. Mawhin, B. Przeradzki and K. Szymańska-Dȩbowska, Second order systems with nonlinear nonlocal boundary conditions, Electron. J. Qual. Theory Differ. Equ., 2018 2018, 1–11. doi: 10.14232/ejqtde.2018.1.56. [32] J. J. Nieto, Existence of a solution for a three-point boundary value problem for a second-order differential equation at resonance, Bound. Value Probl., 2013 (2013), 7pp.  doi: 10.1186/1687-2770-2013-130. [33] J. J. Nieto and J. Pimentel, Positive solutions of a fractional thermostat model, Bound. Value Probl., 2013 (2013), 11pp.  doi: 10.1186/1687-2770-2013-5. [34] S. K. Ntouyas, Nonlocal initial and boundary value problems: A survey, in Handbook of Differential Equations: Ordinary Differential Equations, 2, Elsevier B. V., Amsterdam, 2005,461–557. [35] P. Palamides, G. Infante and P. Pietramala, Nontrivial solutions of a nonlinear heat flow problem via Sperner's Lemma, Appl. Math. Lett., 22 (2009), 1444-1450.  doi: 10.1016/j.aml.2009.03.014. [36] P. Pietramala, A note on a beam equation with nonlinear boundary conditions, Bound. Value Probl., 2011 (2011), Art. ID 376782, 14 pp. doi: 10.1155/2011/376782. [37] M. Picone, Su un problema al contorno nelle equazioni differenziali lineari ordinarie del secondo ordine, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (1908), 1-95. [38] A. Štikonas, A survey on stationary problems, Green's functions and spectrum of Sturm-Liouville problem with nonlocal boundary conditions, Nonlinear Anal. Model. Control, 19 (2014), 301-334.  doi: 10.15388/NA.2014.3.1. [39] J. R. L. Webb, Multiple positive solutions of some nonlinear heat flow problems, Discrete Contin. Dyn. Syst. (Suppl.), 2005,895–903. [40] J. R. L. Webb, Optimal constants in a nonlocal boundary value problem, Nonlinear Anal., 63 (2005), 672-685.  doi: 10.1016/j.na.2005.02.055. [41] J. R. L. Webb, Existence of positive solutions for a thermostat model, Nonlinear Anal. Real World Appl., 13 (2012), 923-938.  doi: 10.1016/j.nonrwa.2011.08.027. [42] J. R. L. Webb, Positive solutions of nonlinear differential equations with Riemann-Stieltjes boundary conditions, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), Paper No. 86, 13 pp. doi: 10.14232/ejqtde.2016.1.86. [43] J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: A unified approach, J. London Math. Soc., 74 (2006), 673-693.  doi: 10.1112/S0024610706023179. [44] W. M. Whyburn, Differential equations with general boundary conditions, Bull. Amer. Math. Soc., 48 (1942), 692-704.  doi: 10.1090/S0002-9904-1942-07760-3. [45] Z. Yang, Positive solutions to a system of second-order nonlocal boundary value problems, Nonlinear Anal., 62 (2005), 1251-1265.  doi: 10.1016/j.na.2005.04.030. [46] Z. Yang, Positive solutions of a second-order integral boundary value problem, J. Math. Anal. Appl., 321 (2006), 751-765.  doi: 10.1016/j.jmaa.2005.09.002. [47] J. Zhang, G. Zhang and H. Li, Positive solutions of second-order problem with dependence on derivative in nonlinearity under Stieltjes integral boundary condition, Electron. J. Qual. Theory Differ. Equ., 2018 (2018), 13pp.  doi: 10.14232/ejqtde.2018.1.4.

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References:
 [1] E. Alves, T. F. Ma and M. L. Pelicer, Monotone positive solutions for a fourth order equation with nonlinear boundary conditions, Nonlinear Anal., 71 (2009), 3834-3841.  doi: 10.1016/j.na.2009.02.051. [2] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM. Rev., 18 (1976), 620-709.  doi: 10.1137/1018114. [3] A. Cabada, An overview of the lower and upper solutions method with nonlinear boundary value conditions, Bound. Value Probl., 2011 (2011), Art. ID 893753, 18 pp. doi: 10.1155/2011/893753. [4] A. Cabada, G. Infante and F. A. F. Tojo, Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications, Topol. Methods Nonlinear Anal., 47 (2016), 265-287.  doi: 10.12775/TMNA.2016.005. [5] F. Cianciaruso, G. Infante and P. Pietramala, Solutions of perturbed Hammerstein integral equations with applications, Nonlinear Anal. Real World Appl., 33 (2017), 317-347.  doi: 10.1016/j.nonrwa.2016.07.004. [6] R. Conti, Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. Un. Mat. Ital., 22 (1967), 135-178. [7] H. Fan and R. Ma, Loss of positivity in a nonlinear second order ordinary differential equations, Nonlinear Anal., 71 (2009), 437-444.  doi: 10.1016/j.na.2008.10.117. [8] D. Franco, G. Infante and J. Perán, A new criterion for the existence of multiple solutions in cones, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1043-1050.  doi: 10.1017/S0308210511001016. [9] D. Franco, D. O'Regan and J. Perán, Fourth-order problems with nonlinear boundary conditions, J. Comput. Appl. Math., 174 (2005), 315-327.  doi: 10.1016/j.cam.2004.04.013. [10] H. Garai, L. K. Dey and A. Chanda, Positive solutions to a fractional thermostat model in Banach spaces via fixed point results, J. Fixed Point Theory Appl., 20 (2018), Art. 106, 24 pp. doi: 10.1007/s11784-018-0584-8. [11] C. S. Goodrich, On nonlocal BVPs with nonlinear boundary conditions with asymptotically sublinear or superlinear growth, Math. Nachr., 285 (2012), 1404-1421.  doi: 10.1002/mana.201100210. [12] C. S. Goodrich, Positive solutions to boundary value problems with nonlinear boundary conditions, Nonlinear Anal., 75 (2012), 417-432.  doi: 10.1016/j.na.2011.08.044. [13] C. S. Goodrich, On nonlinear boundary conditions satisfying certain asymptotic behavior, Nonlinear Anal., 76 (2013), 58-67.  doi: 10.1016/j.na.2012.07.023. [14] C. S. Goodrich, A note on semipositone boundary value problems with nonlocal, nonlinear boundary conditions, Arch. Math. (Basel), 103 (2014), 177-187.  doi: 10.1007/s00013-014-0678-5. [15] C. S. Goodrich, Semipositone boundary value problems with nonlocal, nonlinear boundary conditions, Adv. Differential Equations, 20 (2015), 117-142. [16] C. S. Goodrich, Radially symmetric solutions of elliptic PDEs with uniformly negative weight, Ann. Mat. Pura Appl., 197 (2018), 1585-1611.  doi: 10.1007/s10231-018-0738-8. [17] C. S. Goodrich, New Harnack inequalities and existence theorems for radially symmetric solutions of elliptic PDEs with sign changing or vanishing Green's function, J. Differential Equations, 264 (2018), 236-262.  doi: 10.1016/j.jde.2017.09.011. [18] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering, 5, Academic Press, Inc., Boston, 1988. [19] G. Infante, Nonlocal boundary value problems with two nonlinear boundary conditions, Commun. Appl. Anal., 12 (2008), 279-288. [20] G. Infante, A short course on positive solutions of systems of ODEs via fixed point index, Lecture Notes in Nonlinear Analysis (LNNA), 16 (2017), 93-140. [21] G. Infante, Nonzero positive solutions of a multi-parameter elliptic system with functional BCs, Topol. Methods Nonlinear Anal., 52 (2018), 665-675.  doi: 10.12775/TMNA.2017.060. [22] G. Infante and J. R. L. Webb, Loss of positivity in a nonlinear scalar heat equation, NoDEA Nonlinear Differential Equations Appl., 13 (2006), 249-261.  doi: 10.1007/s00030-005-0039-y. [23] G. Infante and J. R. L. Webb, Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations, Proc. Edinb. Math. Soc., 49 (2006), 637-656.  doi: 10.1017/S0013091505000532. [24] G. Kalna and S. McKee, The thermostat problem, TEMA Tend. Mat. Apl. Comput., 3 (2002), 15-29.  doi: 10.5540/tema.2002.03.01.0015. [25] G. Kalna and S. McKee, The thermostat problem with a nonlocal nonlinear boundary condition, IMA J. Appl. Math., 69 (2004), 437-462.  doi: 10.1093/imamat/69.5.437. [26] G. L. Karakostas, Existence of solutions for an n-dimensional operator equation and applications to BVPs, Electron. J. Differential Equations, 2014 (2014), 17pp. [27] G. L. Karakostas and P. Ch. Tsamatos, Existence of multiple positive solutions for a nonlocal boundary value problem, Topol. Methods Nonlinear Anal., 19 (2002), 109-121.  doi: 10.12775/TMNA.2002.007. [28] G. L. Karakostas and P. Ch. Tsamatos, Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems, Electron. J. Differential Equations, 2002 (2002), 17pp. [29] I. Karatsompanis and P. K. Palamides, Polynomial approximation to a non-local boundary value problem, Comput. Math. Appl., 60 (2010), 3058-3071.  doi: 10.1016/j.camwa.2010.10.006. [30] R. Ma, A survey on nonlocal boundary value problems, Appl. Math. E-Notes, 7 (2007), 257-279. [31] J. Mawhin, B. Przeradzki and K. Szymańska-Dȩbowska, Second order systems with nonlinear nonlocal boundary conditions, Electron. J. Qual. Theory Differ. Equ., 2018 2018, 1–11. doi: 10.14232/ejqtde.2018.1.56. [32] J. J. Nieto, Existence of a solution for a three-point boundary value problem for a second-order differential equation at resonance, Bound. Value Probl., 2013 (2013), 7pp.  doi: 10.1186/1687-2770-2013-130. [33] J. J. Nieto and J. Pimentel, Positive solutions of a fractional thermostat model, Bound. Value Probl., 2013 (2013), 11pp.  doi: 10.1186/1687-2770-2013-5. [34] S. K. Ntouyas, Nonlocal initial and boundary value problems: A survey, in Handbook of Differential Equations: Ordinary Differential Equations, 2, Elsevier B. V., Amsterdam, 2005,461–557. [35] P. Palamides, G. Infante and P. Pietramala, Nontrivial solutions of a nonlinear heat flow problem via Sperner's Lemma, Appl. Math. Lett., 22 (2009), 1444-1450.  doi: 10.1016/j.aml.2009.03.014. [36] P. Pietramala, A note on a beam equation with nonlinear boundary conditions, Bound. Value Probl., 2011 (2011), Art. ID 376782, 14 pp. doi: 10.1155/2011/376782. [37] M. Picone, Su un problema al contorno nelle equazioni differenziali lineari ordinarie del secondo ordine, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (1908), 1-95. [38] A. Štikonas, A survey on stationary problems, Green's functions and spectrum of Sturm-Liouville problem with nonlocal boundary conditions, Nonlinear Anal. Model. Control, 19 (2014), 301-334.  doi: 10.15388/NA.2014.3.1. [39] J. R. L. Webb, Multiple positive solutions of some nonlinear heat flow problems, Discrete Contin. Dyn. Syst. (Suppl.), 2005,895–903. [40] J. R. L. Webb, Optimal constants in a nonlocal boundary value problem, Nonlinear Anal., 63 (2005), 672-685.  doi: 10.1016/j.na.2005.02.055. [41] J. R. L. Webb, Existence of positive solutions for a thermostat model, Nonlinear Anal. Real World Appl., 13 (2012), 923-938.  doi: 10.1016/j.nonrwa.2011.08.027. [42] J. R. L. Webb, Positive solutions of nonlinear differential equations with Riemann-Stieltjes boundary conditions, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), Paper No. 86, 13 pp. doi: 10.14232/ejqtde.2016.1.86. [43] J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: A unified approach, J. London Math. Soc., 74 (2006), 673-693.  doi: 10.1112/S0024610706023179. [44] W. M. Whyburn, Differential equations with general boundary conditions, Bull. Amer. Math. Soc., 48 (1942), 692-704.  doi: 10.1090/S0002-9904-1942-07760-3. [45] Z. Yang, Positive solutions to a system of second-order nonlocal boundary value problems, Nonlinear Anal., 62 (2005), 1251-1265.  doi: 10.1016/j.na.2005.04.030. [46] Z. Yang, Positive solutions of a second-order integral boundary value problem, J. Math. Anal. Appl., 321 (2006), 751-765.  doi: 10.1016/j.jmaa.2005.09.002. [47] J. Zhang, G. Zhang and H. Li, Positive solutions of second-order problem with dependence on derivative in nonlinearity under Stieltjes integral boundary condition, Electron. J. Qual. Theory Differ. Equ., 2018 (2018), 13pp.  doi: 10.14232/ejqtde.2018.1.4.
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