EXISTENCE OF MINIMAL AND MAXIMAL SOLUTIONS FOR A QUASILINEAR DIFFERENTIAL EQUATION WITH NONLOCAL BOUNDARY CONDITION ON THE HALF-LINE

TitleEXISTENCE OF MINIMAL AND MAXIMAL SOLUTIONS FOR A QUASILINEAR DIFFERENTIAL EQUATION WITH NONLOCAL BOUNDARY CONDITION ON THE HALF-LINE
Publication TypeJournal Article
Year of Publication2016
AuthorsDERHAB, MOHAMMED, MEKNI, HAYAT
Secondary TitleCommunications in Applied Analysis
Volume20
Issue2
Start Page209
Pagination13
Date Published06/2016
Type of Workscientific: mathematics
ISSN1083-2564
AMS34B10, 34B15
Abstract

This work is concerned with the construction of minimal and maximal solutions for a second order quasilinear differential equation on the half-line with nonlocal boundary condition and without condition at infinity, where the nonlinearity is a continuous function depending on the first derivative of the unknown function. We also give an example to illustrate our results.

URLhttp://www.acadsol.eu/en/articles/20/2/3.pdf
DOI10.12732/caa.v20i2.3
Short TitleEXISTENCE OF MINIMAL AND MAXIMAL SOLUTIONS
Refereed DesignationRefereed
Full Text

REFERENCES

[1] R. P. Agarwal and D. O’Regan, An infinite interval problem arising in circularly symmetric
deformations of shallow membrane caps, Internat. J. Non-Linear Mech. 39 (2004), 779-784.
[2] R. P. Agarwal and D. O’Regan, Boundary value problems of nonsingular type of the semiinfinite
interval, Tohoku Math. J. 51 (1999), 391-397.
[3] R .P. Agarwal and D. O’Regan, Boundary value problems on the half line in the theory of
colloids, Math. Probl. Eng. 8 (2002), 143-150.
[4] R. P. Agarwal and D. O’Regan, Infinite interval problem arising in non-linear mechanics and
non-Newtonian fluid flows, Internat. J. Non-Linear Mech. 38 (2003), 1369-1376.
[5] R. P. Agarwal and D. O’Regan, Infinite Interval Problems for Differential, Difference and
integral equations, Kluwer Academic Publishers, Dordrecht, 2001.
[6] R. P. Agarwal and D. O’Regan, Infinite Interval problems modeling phenomena which arise in
the theory of plasma and electrical potential theory, Stud. Appl. Math.111 (2003), 339-358.
[7] P. Amster and A. Deboli, A Neumann boundary value problem on an unbounded interval,
Electron. J. Differential Equations 90 (2008), 1-5.
[8] I. Bachir and H. Mˆaagli, existence and uniqueness for superlinear second-order differential
equations on the half-line, Electron. J. Differential Equations 08 (2015), 1-14.
[9] J. V. Baxley, Existence and uniqueness for nonlinear boundary value problems on infinite
intervals, J. Math. Anal. Appl. 147 (1990), 122-133.
[10] F. A. Berezin and M. A. Shubin, The Schr¨odinger Equation, Kluwer Academic Publishers,
Dordrecht, 1991
[11] H. Brezis, Semilinear Equations in R
N without condition at infinity, Appl. Math. Optim. 12
(1984), 271-282.
[12] M. Derhab, Existence of minimal and maximal solutions for a quasilinear elliptic equation with
integral boundary value conditions, Electron. J. Qual. Theory. Differ. Equ. 6 ( 2011), 1-18.
[13] S. Djebali and S. Zahar, Bounded solutions for a derivative dependent boundary value problem
on the half-line, Dynam. Systems Appl. 19 (2010), 545-556.
[14] P. W. Eloe, The quasilinearization method on an unbounded domain, Proc. Amer. Math. Soc.
131 (2002), 1481-1488.
[15] P. W. Eloe, L. J. Grimm and J. Mashburn, A boundary value problem on an unbounded
domain, Differential Equations and Dynamical Systems 8 (2000), 125-140.
[16] P. W. Eloe, E. R. Kaufmann and C. C. Tisdell, Multiple solutions of a boundary value problem
on an unbounded domain, Dynam. Systems Appl. 15 (2006), 53-64.
[17] L. Erbe and K. Schmitt, On radial solutions of some semilinear elliptic equations, Differential
and Integral Equations 1 (1988), 71-78.
[18] M. Frigon and D. O’Regan, Existence theory of compact and noncompact intervals, Comm.
Appl. Nonlinear Anal. 2 (1995), 75-82.
[19] J. Graham-Eagle, Monotone methods for semilinear elliptic equations in unbounded domains,
J. Math. Anal. Appl. 137 (1989), 122-131.
[20] A. Granas, R. B. Guenther, J. W. Lee and D. O’regan, Boundary value problems on infinite
intervals and semiconductor devices, J. Math. Anal. Appl. 116 (1986), 335-348.
[21] O. A. Gross, The boundary value problem on an infinite interval: existence, uniqueness, and
asymptotic behavior of bounded solutions to a class of nonlinear second order differential
equations, J. Math. Anal. Appl. 7 (1963), 100-109.
[22] J. Jeong, C. H. Kim and E. K. Lee, Solvability for nonlocal boundary value problems on a half
line with dim(Ker L)= 2, Bound. Value Probl. 2014 (2014), 11 pages.
[23] N. Kawano, E. Yanagida, and S. Yotsutani, Structure theorems for positive radial solutions to
div 
|Du|
m−2 Du
+ K (|x|) u
q = 0 in R
n, J. Math. Soc. Japan 45 (1993), 719-742.
[24] R. E. Kidder, Unsteady flow of gas through a semi-infinite porous medium, J. Appl. Mech. 24
(1957), 329-332.
[25] G. S. Ladde, V. Lakshmikantham and A. S. Vatsla, Monotone Iterative Techniques for Nonlinear
Differential Equations, Pitman Publishing Co., Boston, 1985.
[26] C. G. Kim, Solvability of multi-point boundary value problems on the half-line, J. Nonlinear
Sci. Appl. 5 (2012), 27-33.
[27] I. Kuzin and S. Pokhozhaev, Entire Solutions of Semilinear Elliptic Equations, Birkh¨auser,
Basel, 1997.
[28] G. I. Laptev, Existence of solutions of certain quasilinear elliptic equations in R
n without
conditions at infinity, Journal of Mathematical Sciences 150 (2008), 2384-2394.
[29] B. Liu, L. Liu and Y. Wu, Multiple solutions of singular three-point boundary value problems
on [0, +∞), Nonlinear Anal. 70 (2009), 3348-3357.
[30] B. Liu, L. Liu and Y. Wu, Unbounded solutions for three-point boundary value problems with
nonlinear boundary conditions on [0, +∞), Nonlinear Anal. 73 (2010), 2923-2932.
[31] Y. Liu, Monotone iteration method for differential equations involving integral boundary conditions
on the half line, Appl. Anal. 92 (2013), 72-95.
[32] E. I. Moiseev and G. O. Vafodorova, On the uniqueness of the solution of the first two boundary
value problems for the heat equation without initial conditions, Differ. Uravn. 46 (2010), 1465–
1471.
[33] F. H. Murray, On certain linear differential equations of the second order, Ann. of Math.24
(1922), 69-88.
[34] T. Y. Na, Computational Methods in Engineering Boundary Value Problems, Academic Press,
New York, 1979.
[35] Z. Nehari, On a nonlinear differential equation arising in nuclear physics, Proc. Roy. Irish.
Acad. 62 (1961-1963), 117-135.
[36] E. S. Noussair, On semilinear elliptic boundary value problems in unbounded domains, J.
Differential Equations 41 (1981), 482-495.
[37] E. S. Noussair, On the existence of solutions of nonlinear elliptic boundary value problems, J.
Differential Equations 34 (1979), 334-348.
[38] A. Ogata, On bounded positive solutions of nonlinear elliptic boundary value problems in an
exterior domains, Funkcial. Ekvac. 17 (1974), 207-222.
[39] D. O’Regan and R. Precup, Positive solutions of nonlinear systems with p-Laplacian on finite
and semi-infinite intervals, Positivity 11 (2007), 537-548.
[40] C. V. Pao, Nonlinear elliptic boundary value problems in unbounded domains, Nonlinear Anal.
18 (1992), 759-774.
[41] C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992.
[42] R. Sta´nczy, Bounded solutions for nonlinear elliptic equations in unbounded domains, J. Appl.
Anal. 6 (2000), 129-138.
[43] A. Varma and N. R. Amundson, Maximal and minimal solutions, effectiveness factors for
chemical reaction in porous catalysts. Chem. Eng. Sci. 28 (1973), 91-104.
[44] V. Volpert, Elliptic Partial Differential Equations. Volume 1: Fredholm Theory of Elliptic
Problems in Unbounded domains, Birkh¨auser, Basel, 2011.
[45] V. Volpert, Elliptic Partial Differential Equations. Volume 2: Reaction-Diffusion Equations,
Birkh¨auser, Basel, 2014.
[46] P. K. Wong, Existence and asymptotic behavior of proper solutions of class of second-order
nonlinear differential equations, Pacific J. Math. 13 (1963), 737-760.
[47] A. Yang and W. Ge, Positive solutions for second-order boundary value problem with integral
boundary conditions at resonance on a half-line, Journal of inequalities in pure and applied
mathematics, 10 (2009), 10 pages.
[48] F. Yoruk and N. A. Hamal, Existence results for nonlinear boundary value problems with
integral boundary conditions on an infinite interval, Bound. Value Probl. 2012 (2012), 17
pages.
[49] T. Yoshizawa, Note on the non-increasing solutions of y
′′ = f (x, y, y′
), Mem. College Sci.
Univ. Kyoto Ser. A Math. 27 (1952), 152-163.