Previous |  Up |  Next


Title: A new approach for solving nonlinear BVP's on the half-line for second order equations and applications (English)
Author: Matucci, Serena
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 140
Issue: 2
Year: 2015
Pages: 153-169
Summary lang: English
Category: math
Summary: We present a new approach to solving boundary value problems on noncompact intervals for second order differential equations in case of nonlocal conditions. Then we apply it to some problems in which an initial condition, an asymptotic condition and a global condition is present. The abstract method is based on the solvability of two auxiliary boundary value problems on compact and on noncompact intervals, and uses some continuity arguments and analysis in the phase space. As shown in the applications, Kneser-type properties of solutions on compact intervals and a priori bounds of solutions on noncompact intervals are key ingredients for the solvability of the problems considered, as well as the properties of principal solutions of an associated half-linear equation. The application of this method leads to some new existence results, which complement and extend some previous ones in the literature. (English)
Keyword: global solution
Keyword: nonlocal boundary value problem
Keyword: noncompact interval
Keyword: continuous dependence of solution
Keyword: fixed point theorem
Keyword: principal solution
MSC: 34A40
MSC: 34B10
MSC: 34B15
MSC: 34B18
MSC: 34B40
idZBL: Zbl 06486931
idMR: MR3368491
DOI: 10.21136/MB.2015.144323
Date available: 2015-06-30T12:15:38Z
Last updated: 2020-07-29
Stable URL:
Reference: [1] Andres, J., Górniewicz, L.: Topological Fixed Point Principles for Boundary Value Problems.Topological Fixed Point Theory and Its Applications 1 Kluwer Academic Publishers, Dordrecht (2003). Zbl 1029.55002, MR 1998968
Reference: [2] Cecchi, M., Došlá, Z., Marini, M.: Principal solutions and minimal sets of quasilinear differential equations.Dyn. Syst. Appl. 13 (2004), 221-232. Zbl 1123.34026, MR 2140874
Reference: [3] Chanturiya, T. A.: Monotone solutions of a system of nonlinear differential equations.Ann. Pol. Math. 37 Russian (1980), 59-70. MR 0574996
Reference: [4] Coppel, W. A.: Stability and Asymptotic Behavior of Differential Equations.D. C. Heath and Company VIII Boston (1965). Zbl 0154.09301, MR 0190463
Reference: [5] Došlá, Z., Marini, M., Matucci, S.: Positive solutions of nonlocal continuous second order BVP's.Dyn. Syst. Appl. 23 (2014), 431-446. Zbl 1312.34057, MR 3241888
Reference: [6] Došlá, Z., Marini, M., Matucci, S.: On some boundary value problems for second order nonlinear differential equations.Math. Bohem. 137 (2012), 113-122. Zbl 1265.34113, MR 2978257
Reference: [7] Došlá, Z., Marini, M., Matucci, S.: A boundary value problem on a half-line for differential equations with indefinite weight.Commun. Appl. Anal. 15 (2011), 341-352. Zbl 1244.34045, MR 2867356
Reference: [8] Došlý, O., Řehák, P.: Half-Linear Differential Equations.North-Holland Mathematics Studies 202 Elsevier, Amsterdam (2005). Zbl 1090.34001, MR 2158903
Reference: [9] Elbert, Á., Kusano, T.: Principal solutions of non-oscillatory half-linear differential equations.Adv. Math. Sci. Appl. 8 (1998), 745-759. Zbl 0914.34031, MR 1657164
Reference: [10] Erbe, L. H., Wang, H.: On the existence of positive solutions of ordinary differential equations.Proc. Am. Math. Soc. 120 (1994), 743-748. Zbl 0802.34018, MR 1204373, 10.1090/S0002-9939-1994-1204373-9
Reference: [11] Franco, D., Infante, G., Perán, J.: A new criterion for the existence of multiple solutions in cones.Proc. R. Soc. Edinb., Sect. A, Math. 142 (2012), 1043-1050. Zbl 1264.47059, MR 2981023, 10.1017/S0308210511001016
Reference: [12] Gaudenzi, M., Habets, P., Zanolin, F.: An example of a superlinear problem with multiple positive solutions.Atti Semin. Mat. Fis. Univ. Modena 51 (2003), 259-272. Zbl 1221.34057, MR 2045073
Reference: [13] Hartman, P.: Ordinary Differential Equations.Birkhäuser, Boston (1982). Zbl 0476.34002, MR 0658490
Reference: [14] Kiguradze, I., Půža, B.: Boundary Value Problems for Systems of Linear Functional Differential Equations.Folia Fac. Sci. Natur. Univ. Masaryk. Brun. Math. 12 Masaryk University, Brno (2003). Zbl 1161.34300, MR 2001509
Reference: [15] Marini, M., Matucci, S.: A boundary value problem on the half-line for superlinear differential equations with changing sign weight.Rend. Ist. Mat. Univ. Trieste 44 (2012), 117-132. Zbl 1269.34027, MR 3019556
Reference: [16] Mirzov, J. D.: Asymptotic Properties of Solutions of Systems of Nonlinear Nonautonomous Ordinary Differential Equations.Folia Fac. Sci. Natur. Univ. Masaryk. Brun. Math. 14 Masaryk University, Brno (2004). Zbl 1154.34300, MR 2144761
Reference: [17] Motreanu, D., Rădulescu, V.: Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems.Nonconvex Optimization and Its Applications 67 Kluwer Academic Publishers, Dordrecht (2003). Zbl 1040.49001, MR 1985870, 10.1007/978-1-4757-6921-0_5
Reference: [18] Powers, D. L.: Boundary Value Problems and Partial Differential Equations.Elsevier/Academic Press, Amsterdam (2010). Zbl 1187.35001, MR 2584516
Reference: [19] Wang, J.: The existence of positive solutions for the one-dimensional $p$-Laplacian.Proc. Am. Math. Soc. 125 (1997), 2275-2283. Zbl 0884.34032, MR 1423340, 10.1090/S0002-9939-97-04148-8


Files Size Format View
MathBohem_140-2015-2_5.pdf 309.3Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo