### Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications

DOI: http://dx.doi.org/10.12775/TMNA.2016.005

#### Abstract

#### Keywords

#### References

A.R. Aftabizadeh, Y.K. Huang and J. Wiener, Bounded solutions for differential equations with reflection of the argument, J. Math. Anal. Appl. 135 (1988), 31-37.

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM. Rev. 18 (1976), 620-709.

D. Andrade and T.F. Ma, Numerical solutions for a nonlocal equation with reflection of the argument, Neural Parallel Sci. Comput. 10 (2002), 227-233.

M. Brokate and A. Friedman, Optimal design for heat conduction problems with hysteresis, Siam J. Control and Optimization 27-4 (1989), 697-717.

A. Cabada, The method of lower and upper solutions for third - order periodic boundary value problems, J. Math. Anal. Appl. 195 (1995), 568-589.

A. Cabada and F.A.F. Tojo, Comparison results for first order linear operators with reflection and periodic boundary value conditions, Nonlinear Anal. 78 (2013), 32-46.

A. Cabada and F.A.F. Tojo, Solutions of the first order linear equation with reflection and general linear conditions, Global Journal of Mathematical Sciences (GJMS) 2:1 (2013).

A. Cabada and F.A.F. Tojo, Existence results for a linear equation with reflection, non-constant coefficient and periodic boundary conditions, J. Math. Anal. Appl. 412 (2013), 529-546.

A. Cabada, G. Infante and F.A.F. Tojo, Nontrivial solutions of perturbed Hammerstein integral equations with reflections, Bound. Value Probl. 2013:86 (2013).

A. Cabada, R.L. Pouso and F.L. Minhos, Extremal solutions to fourth-order functional boundary value problems including multipoint conditions, Nonlinear Anal. Real World Appl. 10 (2009), 2157-2170.

S. Campbell and J.W. Macki, Control of the temperature at one end of a rod, Math. Comput. Modelling 32 (2000), 825-842.

W. W. Coblentz, Emissivity of straight and helical filaments of tungsten, Bulletin of the Bureau of Standards 14 (1918), 115-131.

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

H. Fan and R. Ma, Loss of positivity in a nonlinear second order ordinary differential equations, Nonlinear Anal. 71 (2009), 437-444.

R. Figueroa and R.L. Pouso, Minimal and maximal solutions to second-order boundary value problems with state-dependent deviating arguments, Bull. Lond. Math. Soc. 43 (2011), 164-174.

A. Friedman and L. S. Jiang, Periodic solutions for a thermostat control problem, Comm. in Partial Differential Equations 13 (1988), 515-550.

D. Franco, G. Infante and D. O'Regan, Positive and nontrivial solutions for the Urysohn integral equation, Acta Math. Sin. 22 (2006), 1745-1750.

D. Franco, G. Infante and D. O'Regan, Nontrivial solutions in abstract cones for Hammerstein integral systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 14 (2007), 837-850.

C.S. Goodrich, On nonlinear boundary conditions satisfying certain asymptotic behavior, Nonlinear Anal. 76 (2013), 58-67.

P. Guidotti and S. Merino, Gradual loss of positivity and hidden invariant cones in a scalar heat equation, Differential Integral Equations 13 (2000), 1551-1568.

D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, 1988.

Ch.P. Gupta, Existence and uniqueness theorems for boundary value problems involving reflection of the argument, Nonlinear Anal. 11 (1987), 1075-1083.

Ch.P. Gupta, Two-point boundary value problems involving reflection of the argument, Internat. J. Math. Math. Sci. 10 (1987), 361-371.

G. Infante, Eigenvalues of some non-local boundary-value problems, Proc. Edinb. Math. Soc. 46 (2003), 75-86.

G. Infante, Positive solutions of some nonlinear BVPs involving singularities and integral BCs, Discrete Contin. Dyn. Syst. Series S 1 (2008), 99-106.

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

G. Infante and P. Pietramala, Nonlocal impulsive boundary value problems with solutions that change sign, CP1124, Mathematical Models in Engineering, Biology, and Medicine, Proceedings of the International Conference on Boundary Value Problems, (A. Cabada, E. Liz and J.J. Nieto, eds.) (2009), 205-213.

G. Infante and P. Pietramala, Perturbed Hammerstein integral inclusions with solutions that change sign, Comment. Math. Univ. Carolin. 50 (2009), 591-605.

G. Infante and P. Pietramala, A cantilever equation with nonlinear boundary conditions, Electron. J. Qual. Theory Differ. Equ., Spec. Ed. I, No. 15 (2009), 1-14.

G. Infante and J.R.L. Webb, Three point boundary value problems with solutions that change sign, J. Integral Equations Appl. 15, (2003), 37-57.

G. Infante and J.R.L. Webb, Nonzero solutions of Hammerstein integral equations with discontinuous kernels, J. Math. Anal. Appl. 272 (2002), 30-42.

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.

G. Infante and J.R.L. Webb, Nonlinear nonlocal boundary value problems and perturbed Hammerstein integral equations, Proc. Edinb. Math. Soc. 49 (2006), 637-656.

T. Jankowski, Solvability of three point boundary value problems for second order differential equations with deviating arguments, J. Math. Anal. Appl. 312 (2005), 620-636.

T. Jankowski, Multiple solutions for a class of boundary-value problems with deviating arguments and integral boundary conditions, Dynam. Systems Appl. 19 (2010), 179-188.

T. Jankowski, Nonnegative solutions to nonlocal boundary value problems for systems of second-order differential equations dependent on the first-order derivatives, Nonlinear Anal. 87 (2013), 83-101.

T. Jankowski, Positive solutions to second-order differential equations with dependence on the first-order derivative and nonlocal boundary conditions, Bound. Value Probl. 2013:8 (2013).

T. Jankowski, Positive solutions to Sturm-Liouville problems with nonlocal boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 144A (2014), 119-138.

T. Jankowski and W. Szatanik, Second-order differential equations with deviating arguments, Bound. Value Probl. 2006, Art. ID 23092, 15 pp.

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.

G.L. Karakostas and P.Ch. Tsamatos, Multiple positive solutions of some Fredholm integral equations arisen from non-local boundary-value problems, Electron. J. Differential Equations 2002, No. 30, 17 pp.

I. Karatsompanis and P.K. Palamides, Polynomial approximation to a non-local boundary value problem, Comput. Math. Appl. 60 (2010), 3058-3071.

M.A. Krasnosel'skii and P.P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin, (1984).

K.Q. Lan, Multiple positive solutions of Hammerstein integral equations with singularities, Differential Equations and Dynamical Systems 8 (2000), 175-195.

T.F. Ma, E.S. Miranda and M.B. de Souza Cortes, A nonlinear differential equation involving reflection of the argument, Arch. Math. (Brno) 40 (2004), 63-68.

R. Ma, A survey on nonlocal boundary value problems, Appl. Math. E-Notes 7 (2001), 257-279.

R.H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York, (1976).

J.J. Nieto and J. Pimentel, Positive solutions of a fractional thermostat model, Bound. Value Probl. 2013:5 (2013).

S.K. Ntouyas, Nonlocal initial and boundary value problems: a survey, Handbook of Differential Equations: Ordinary Differential Equations. Vol. II, 461-557, Elsevier B.V., Amsterdam, 2005.

P. Pietramala, A note on a beam equation with nonlinear boundary conditions, Bound. Value Probl. (2011), Art. ID 376782, 14 pp.

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.

D. O'Regan, Existence results for differential equations with reflection of the argument, J. Austral. Math. Soc. Ser. A 57 (1994), 237-260.

D. O'Regan and M. Zima, Leggett-Williams norm-type fixed point theorems for multivalued mappings, Appl. Math. Comput. 187 (2007), 1238-1249.

P.K. 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.

D. Piao, Pseudo almost periodic solutions for differential equations involving reflection of the argument, J. Korean Math. Soc. 41 (2004), 747-754.

D. Piao, Periodic and almost periodic solutions for differential equations with reflection of the argument, Nonlinear Anal. 57 (2004), 633-637.

D. Piao and Na Xin, Bounded and almost periodic solutions for second order differential equation involving reflection of the argument, arXiv:1302.0616 [math.CA].

A. Stikonas, 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.

W. Szatanik, Quasi-solutions for generalized second order differential equations with deviating arguments, J. Comput. Appl. Math. 216 (2008), 425-434.

W. Szatanik, Minimal and maximal solutions for integral boundary value problems for the second order differential equations with deviating arguments, Dynam. Systems Appl. 19 (2010), 87-96.

P.J. Torres, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnosel'skii fixed point theorem, J. Differential Equations 190 (2003), 643-662.

K. Ujihara, Reflectivity of metals at high temperatures, J. Appl. Phys. 43 (1972), 2376-2383.

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

J.R.L. Webb, Optimal constants in a nonlocal boundary value problem, Nonlinear Anal. 63 (2005), 672-685.

J.R.L. Webb, Existence of positive solutions for a thermostat model, Nonlinear Anal. Real World Appl. 13 (2012), 923-938.

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.

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.

J.R.L. Webb and G. Infante, Nonlocal boundary value problems of arbitrary order, J. London Math. Soc. 79 (2009), 238-258.

J.R.L. Webb and M. Zima, Multiple positive solutions of resonant and non-resonant nonlocal boundary value problems, Nonlinear Anal. 71 (2009), 1369-1378.

W.M. Whyburn, Differential equations with general boundary conditions, Bull. Amer. Math. Soc. 48 (1942), 692-704.

J. Wiener and A.R. Aftabizadeh, Boundary value problems for differential equations with reflection of the argument, Internat. J. Math. Math. Sci. 8 (1985), 151-163.

### Refbacks

- There are currently no refbacks.