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Multiple unbounded positive solutions for three-point BVPs with sign-changing nonlinearities on the positive half-line. (English) Zbl 1195.34042

The authors consider the following second-order nonlinear three-point boundary value problem on the positive half-line
\[ -x''+cx'+\lambda x=f(t,x(t), x'(t)), \quad t\in (0,\infty), \]
\[ x(0)-\alpha x(\eta)=a_0, \quad \lim_{t\to \infty}\frac{x'(t)}{re^{rt}}=b_0, \]
where \(a_0, b_0\) are nonnegative real numbers, \(\alpha\geq 0,\) \(\eta>0,\) \(c,\lambda\) are real positive constants, \(r\in (0,c),\) and \(f: (0,\infty)\times [0,\infty)\times {\mathbb R}\to {\mathbb R}\) is a Carathéodory function which may change sign. Under general polynomial growth conditions on \(f\) the existence of nontrivial single and multiple unbounded positive solutions is proved via fixed point theorems in a cone in a special weighted Banach space.

MSC:

34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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[1] Agarwal, R.P., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge Tracts in Mathematics, vol. 141. Cambridge University Press, Cambridge (2001) · Zbl 0960.54027
[2] Agarwal, R.P., O’Regan, D.: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht (2001)
[3] Agarwal, R.P., O’Regan, D., Wong, P.J.Y.: Positive Solutions of Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht (1999)
[4] Aris, O.R.: The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts. Clarendon, Oxford (1975) · Zbl 0315.76052
[5] Bai, J.V.: Existence and uniqueness of nonlinear boundary value problems on infinity intervals. J. Math. Anal. Appl. 147, 127–133 (1990)
[6] Bai, Z., Ge, W.: Existence of three positive solutions for second-order boundary value problems. Comput. Math. Appl. 48, 699–707 (2004) · Zbl 1066.34019 · doi:10.1016/j.camwa.2004.03.002
[7] Baily, N.T.J.: The Mathematical Theory of Infectious Diseases. Griffin, London (1975) · Zbl 0329.10020
[8] Britton, N.F.: Reaction-Diffusion Equations and their Applications to Biology. Academic Press, New York (1986) · Zbl 0602.92001
[9] Corduneanu, C.: Integral Equations and Stability of Feedback Systems. Academic Press, New York (1973) · Zbl 0273.45001
[10] Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985) · Zbl 0559.47040
[11] Djebali, S.: Travelling wave solutions to a reaction-diffusion system arising in epidemiology. Nonlinear Anal. Ser. B, Real World Appl. 2(4), 417–442 (2001) · Zbl 1017.92029 · doi:10.1016/S0362-546X(99)00287-4
[12] Djebali, S.: Traveling wave solutions to a reaction-diffusion system from combustion theory. Nonlinear Stud. 11(4), 603–626 (2004) · Zbl 1083.34022
[13] Djebali, S., Kavian, O., Moussaoui, T.: Qualitative properties and existence of solutions for a generalized Fisher-like equation. Int. J. Pure Appl. Math. Sci. (to appear) · Zbl 1301.34020
[14] Djebali, S., Mebarki, K.: Existence results for a class of BVPs on the positive half-line. Commun. Appl. Nonlinear Anal. 14(2), 13–31 (2007) · Zbl 1129.34017
[15] Djebali, S., Mebarki, K.: Multiple positive solutions for singular BVPs on the positive half-line. Comput. Math. Appl. 55(12), 2940–2952 (2008) · Zbl 1142.34316 · doi:10.1016/j.camwa.2007.11.023
[16] Djebali, S., Mebarki, K.: On the singular generalized Fisher-like equation with derivative depending nonlinearity. Appl. Math. Comput. (2008). doi: 10.1016/j.amc.2008.08.009 · Zbl 1183.34039
[17] Djebali, S., Moussaoui, T.: A class of second order BVPs on infinite intervals. Electron. J. Qual. Theory Differ. Equ. 4, 1–19 (2006) · Zbl 1134.34018
[18] Fisher, R.: The wave of advance of advantageous genes. Ann. Eugen. 7, 335–369 (1937) · JFM 63.1111.04
[19] Frigon, M.: Application de la Théorie de la Transversalité à des Problèmes Non Linéaires pour des Équations Différentielles Ordinaires, Dissertationes Mathematicae, Warszawa, vol. CCXCVI (1990)
[20] Guo, Y., Ge, W.: Positive solutions for three-point boundary value problems with dependence on the first order derivative. J. Math. Anal. Appl. 290, 291–301 (2004) · Zbl 1054.34025 · doi:10.1016/j.jmaa.2003.09.061
[21] Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, San Diego (1988) · Zbl 0661.47045
[22] Gupta, C.P.: A note on a second order three-point value problem. J. Math. Anal. Appl. 186, 277–281 (1994) · Zbl 0805.34017 · doi:10.1006/jmaa.1994.1299
[23] Gupta, C.P., Trofimchuk, S.I.: A sharper condition for the solvability of a three-point second-order boundary value problem. J. Math. Anal. Appl 205, 586–584 (1997) · Zbl 0874.34014 · doi:10.1006/jmaa.1997.5252
[24] Kermack, W.O., McKendric, A.G.: Contributions to the mathematical theory of epidemics. Proc. R. Soc. A 115, 700–721 (1927) · JFM 53.0517.01 · doi:10.1098/rspa.1927.0118
[25] Krasnosel’skii, M.A.: Positive Solutions of Operator Equations. Noordhoff, Groningen (1964)
[26] Li, F., Han, G.: Generalization for Amann’s and Legget-Williams three-solution theorems and applications. J. Math. Anal. Appl. 298, 638–654 (2004) · Zbl 1073.47055 · doi:10.1016/j.jmaa.2004.06.001
[27] Lian, H., Ge, W.: Existence of positive solutions for Sturm-Liouville boundary value problems on the half-line. J. Math. Anal. Appl. 321, 781–792 (2006) · Zbl 1104.34020 · doi:10.1016/j.jmaa.2005.09.001
[28] Lian, H., Ge, W.: Solvability for second-order three-point boundary value problems on a half-line. Appl. Math. Lett. 19, 1000–1006 (2006) · Zbl 1123.34307 · doi:10.1016/j.aml.2005.10.018
[29] Ma, R.: Positive solutions for second order three-point boundary value problems. Appl. Math. Anal. Lett. 14, 1–5 (2001) · Zbl 0989.34009 · doi:10.1016/S0893-9659(00)00102-6
[30] Murray, J.D.: Mathematical Biology. Biomathematics Texts, vol. 19. Springer, Berlin (1989)
[31] O’Regan, D., Yan, B., Agarwal, R.P.: Solutions in weighted spaces of singular boundary value problems on the half-line. J. Comput. Appl. Math. 205, 751–763 (2007) · Zbl 1124.34008 · doi:10.1016/j.cam.2006.02.055
[32] Przeradzki, B.: Travelling waves for reaction-diffusion equations with time depending nonlinearities. J. Math. Anal. Appl. 281, 164–170 (2003) · Zbl 1032.35089
[33] Tian, Y., Ge, W.: Positive solutions for multi-point boundary value problem on the half-line. J. Math. Anal. Appl. 325, 1339–1349 (2007) · Zbl 1110.34018 · doi:10.1016/j.jmaa.2006.02.075
[34] Tian, Y., Ge, W., Shan, W.: Positive solutions for three-point boundary value problem on the half-line. Comput. Math. Appl. 53, 1029–1039 (2007) · Zbl 1131.34019 · doi:10.1016/j.camwa.2006.08.035
[35] Yan, B., O’Regan, D., Agarwal, R.P.: Unbounded solutions for singular boundary value problems on the semi-infinite interval: Upper and lower solutions and multiplicity. J. Comput. Appl. Math. 197, 365–386 (2006) · Zbl 1116.34016 · doi:10.1016/j.cam.2005.11.010
[36] Zeidler, E.: Nonlinear Functional Analysis and its Applications. Fixed Point Theorems, vol. I. Springer, New York (1986) · Zbl 0583.47050
[37] Zima, M.: On positive solutions of boundary value problems on the half-line. J. Math. Anal. Appl. 259, 127–136 (2001) · Zbl 1003.34024 · doi:10.1006/jmaa.2000.7399
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