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  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 & Continuous Dynamical Systems - B, 2020, 25 (2) : 691-699. doi: 10.3934/dcdsb.2019261
References:
[1]

E. AlvesT. 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.  Google Scholar

[2]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM. Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.  Google Scholar

[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.  Google Scholar

[4]

A. CabadaG. 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.  Google Scholar

[5]

F. CianciarusoG. 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.  Google Scholar

[6]

R. Conti, Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. Un. Mat. Ital., 22 (1967), 135-178.   Google Scholar

[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.  Google Scholar

[8]

D. FrancoG. 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.  Google Scholar

[9]

D. FrancoD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

C. S. Goodrich, Semipositone boundary value problems with nonlocal, nonlinear boundary conditions, Adv. Differential Equations, 20 (2015), 117-142.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

G. Infante, Nonlocal boundary value problems with two nonlinear boundary conditions, Commun. Appl. Anal., 12 (2008), 279-288.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

G. L. Karakostas, Existence of solutions for an n-dimensional operator equation and applications to BVPs, Electron. J. Differential Equations, 2014 (2014), 17pp.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[30]

R. Ma, A survey on nonlocal boundary value problems, Appl. Math. E-Notes, 7 (2007), 257-279.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

P. PalamidesG. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[39]

J. R. L. Webb, Multiple positive solutions of some nonlinear heat flow problems, Discrete Contin. Dyn. Syst. (Suppl.), 2005,895–903.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[47]

J. ZhangG. 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.  Google Scholar

show all references

References:
[1]

E. AlvesT. 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.  Google Scholar

[2]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM. Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.  Google Scholar

[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.  Google Scholar

[4]

A. CabadaG. 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.  Google Scholar

[5]

F. CianciarusoG. 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.  Google Scholar

[6]

R. Conti, Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. Un. Mat. Ital., 22 (1967), 135-178.   Google Scholar

[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.  Google Scholar

[8]

D. FrancoG. 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.  Google Scholar

[9]

D. FrancoD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

C. S. Goodrich, Semipositone boundary value problems with nonlocal, nonlinear boundary conditions, Adv. Differential Equations, 20 (2015), 117-142.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

G. Infante, Nonlocal boundary value problems with two nonlinear boundary conditions, Commun. Appl. Anal., 12 (2008), 279-288.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

G. L. Karakostas, Existence of solutions for an n-dimensional operator equation and applications to BVPs, Electron. J. Differential Equations, 2014 (2014), 17pp.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[30]

R. Ma, A survey on nonlocal boundary value problems, Appl. Math. E-Notes, 7 (2007), 257-279.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

P. PalamidesG. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[39]

J. R. L. Webb, Multiple positive solutions of some nonlinear heat flow problems, Discrete Contin. Dyn. Syst. (Suppl.), 2005,895–903.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[47]

J. ZhangG. 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.  Google Scholar

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