## Positive solutions of singular Dirichlet boundary value problems with time and space singularities.(English)Zbl 1192.34027

The author studies the existence of positive solutions of a Dirichlet boundary value problem. The nonlinearity $$h(t,x,y)$$ involved in the differential equation can be singular in the time variable $$t$$, at $$t=0$$ and/or $$t=T$$, and can have a weak or strong singularity in the space variable, at $$x=0$$. The approach relies on regularization and sequential techniques combined with the method of lower and upper functions. The author also investigates the existence of a maximal positive solution for this boundary value problem. Some examples are given to illustrate the theory.

### MSC:

 34B16 Singular nonlinear boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
Full Text:

### References:

 [1] Rachůnková, I.; Staněk, S., Connections between types of singularities in differential equations and smoothness of solutions for Dirichlet BVPs, Dynam. contin. discrete impuls. syst. ser. A. math. anal., 10, 209-222, (2003) · Zbl 1043.34022 [2] Rachůnková, I.; Staněk, S.; Tvrdý, M., (), 607-723 [3] Agarwal, R.P.; O’Regan, D., Singular differential and integral equations with applications, (2003), Kluwer Dordrecht · Zbl 1027.34014 [4] Agarwal, R.P.; O’Regan, D., Semipositone Dirichlet boundary value problems with singular dependent nonlinearities, Huston J. math., 30, 297-308, (2004) · Zbl 1063.34013 [5] Agarwal, R.P.; O’Regan, D., (), 1-68 [6] Agarwal, R.P.; O’Regan, D.; Lakshmikantham, V., An upper and lower solution approach for nonlinear singular boundary value problems with $$y^\prime$$ dependence, Arch. inequal. appl., 1, 119-135, (2003) · Zbl 1046.34033 [7] Agarwal, R.P.; O’Regan, D.; Lakshmikantham, V.; Leela, S., Nonresonant singular boundary value problems with sign changing nonlinearities, Appl. math. comput., 167, 1236-1248, (2005) · Zbl 1086.34020 [8] Chu, J.; O’Regan, D., Multiplicity results for second order non-autonomous singular Dirichlet systems, Acta appl. math., 105, 323-338, (2009) · Zbl 1228.34037 [9] Kiguradze, I., Some optimal conditions for solvability of two-point singular boundary value problems, Funct. differ. equ., 10, 259-281, (2003) · Zbl 1062.34017 [10] Kiguradze, I.T.; Shekhter, B.L., Singular boundary value problems for second order ordinary differential equations, J. sov. math., 43, 2340-2417, (1988), Translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh. 30 (1987) 105-201 (in Russian) · Zbl 0782.34026 [11] Lomtatidze, A.; Tores, P., On a two-point boundary value problems for second order singular equations, Czechoslovak math. J., 53, 19-43, (2003) · Zbl 1023.34011 [12] Lü, H.; O’Regan, D.; Agarwal, R.P., Positive solutions of non-resonant singular boundary-value problems with a linear term, Proc. edinb. math. soc. II. ser., 50, 217-228, (2007) · Zbl 1128.34012 [13] Rachůnková, I.; Skryja, J., Singular Dirichlet boundary value problem for second order ODE, Georgian math. J., 14, 325-340, (2007) [14] Staněk, S., Positive solutions of singular positone Dirichlet boundary value problems, Math. comput. modelling, 33, 341-361, (2001) · Zbl 0996.34019 [15] Zhang, X.; Liu, L., Positive solutions of superlinear semipositone singular Dirichlet boundary value problems, J. math. anal. appl., 316, 525-537, (2006) · Zbl 1097.34019 [16] Zhou, W.S.; Cai, S.F., Positive solutions for a singular second order ordinary differential equations, Lobachevskii J. math., 24, 135-142, (2006) · Zbl 1124.34317 [17] Jiang, D.; Zhang, H., Nonuniform nonresonant singular Dirichlet boundary value problems for the one-dimensional $$p$$-Laplacian with sign changing nonlinearity, Nonlinear anal., 68, 1155-1168, (2008) · Zbl 1136.34017 [18] Rachůnková, I.; Skryja, J., Dirichlet problem with $$\phi$$-Laplacian and mixed singularities, Nonlinear oscil., 11, 81-95, (2008) [19] Yao, Z.; Zhou, W., Existence of positive solutions for the one-dimensional singular $$p$$-Laplacian, Nonlinear anal. TMA, 68, 2309-2318, (2008) · Zbl 1146.34022 [20] Zhou, W., Existence of positive solutions for a singular $$p$$-Laplacian Dirichlet problem, Electron. J. differential equations, 2008, 102, 1-6, (2008) [21] Astarita, C.; Marrucci, G., Principles of non-Newtonian fluid mechanics, (1974), McGraw-Hill · Zbl 0316.73001 [22] Aris, R., Introduction to the analysis of chemical reactors, (1965), Prentice-Hall Englewood Cliffs, NJ [23] Aris, R., The mathematical theory of diffusion and reaction of permeable catalysts, (1975), Clarendon Press Oxford · Zbl 0315.76051 [24] Bexley, J.V., A singular nonlinear boundary value problem: membrane response of a spherical cap, SIAM J. appl. math., 48, 497-505, (1988) · Zbl 0642.34014 [25] Callegari, A.; Nachman, A., A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. appl. math., 38, 275-282, (1980) · Zbl 0453.76002 [26] Callegari, A.; Nachman, A., Some singular nonlinear differential equations arising in boundary layer theory, J. math. anal. appl., 64, 96-105, (1978) · Zbl 0386.34026 [27] Hussaini, M.Y.; Lakin, W.D.; Nachman, A., On similarity solutions of a boundary layer problem with an upstream moving wall, SIAM J. appl. math., 47, 699-709, (1987) · Zbl 0634.76034 [28] Atkinson, C.; Bouillet, J.E., Some qualitative properties of solutions of a generalized diffusion equation, Proc. Cambridge philos. soc., 86, 495-510, (1979) · Zbl 0428.35046 [29] Esteban, J.R.; Vazques, J.L., On the equation to turbulent filtration in one-dimensional porous media, Nonlinear anal., 10, 1303-1325, (1986) · Zbl 0613.76102 [30] Thompson, H.B., Second order ordinary differential equations with fully nonlinear two point boundary conditions, Pacific J. math., 172, 255-277, (1996) · Zbl 0855.34024 [31] Thompson, H.B., Second order ordinary differential equations with fully nonlinear two point boundary conditions II, Pacific J. math., 172, 259-297, (1996) · Zbl 0862.34015 [32] Bartle, R.G., A modern theory of integration, (2001), AMS Providence, RI · Zbl 0968.26001 [33] Lang, S., Real and functional analysis, (1993), Springer New York, Inc. · Zbl 0831.46001 [34] Cabada, A.; Nieto, J.J., Extremal solutions of second order nonlinear periodic boundary value problems, Appl. math. comput., 40, 135-145, (1990) · Zbl 0723.65056 [35] Carl, S.; Heikkilä, S., () [36] Cherpion, M.; De Coster, C.; Habets, P., Monotone iterative method for boundary value problem, Differential integral equations, 12, 309-338, (1999) · Zbl 1015.34009 [37] Cid, J.A., On extremal fixed point in schauder’s theorem with application to differential equations, Bull. belg. math. soc. Simon stevin, 11, 15-20, (2004) · Zbl 1076.47044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.