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Positive solutions of nonlinear singular third-order two-point boundary value problem. (English) Zbl 1107.34019
Summary: We are concerned with the existence of single and multiple positive solutions to the nonlinear singular third-order two-point boundary value problem $u'''(t)+ \lambda a(t)f\bigl(u(t)\bigr)=0,\quad 0<t<1,\quad u(0)=u'(0)=u''(1)=0,$ where $$\lambda$$ is a positive parameter. Under various assumptions on $$a$$ and $$f$$, we establish intervals of the parameter $$\lambda$$ which yield the existence of at least one, at least two, and infinitely many positive solutions of the boundary value problem by using Krasnoselskii’s fixed-point theorem of cone expansion-compression type.

##### MSC:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34B24 Sturm-Liouville theory
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##### References:
  Agarwal, R.P.; Bohner, M.; Wong, P.J.Y., Positive solutions and eigenvalues of conjugate boundary value problems, Proc. Edinburgh math. soc., 42, 349-374, (1999) · Zbl 0934.34008  Agarwal, R.P.; O’Regan, D.; Wong, P.J.Y., Positive solutions of differential, difference, and integral equations, (1999), Kluwer Academic Boston, MA · Zbl 0923.39002  Anderson, D., Multiple positive solutions for a three-point boundary value problem, Math. comput. modelling, 27, 6, 49-57, (1998) · Zbl 0906.34014  Anderson, D.; Avery, R.I., Multiple positive solutions to a third-order discrete focal boundary value problem, Comput. math. appl., 42, 333-340, (2001) · Zbl 1001.39022  Cabada, A., The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems, J. math. anal. appl., 185, 302-320, (1994) · Zbl 0807.34023  Cabada, A., The method of lower and upper solutions for third-order periodic boundary value problems, J. math. anal. appl., 195, 568-589, (1995) · Zbl 0846.34019  Cabada, A.; Grossinbo, M.R.; Minhos, F., On the solvability of some discontinuous third order nonlinear differential equations with two point boundary conditions, J. math. anal. appl., 285, 174-190, (2003) · Zbl 1048.34033  Cabada, A.; Heikkilä, S., Extremality and comparison results for discontinuous third order functional initial-boundary value problems, J. math. anal. appl., 255, 195-212, (2001) · Zbl 0976.34009  Cabada, A.; Heikkilä, S., Uniqueness, comparison and existence results for third order initial-boundary value problems, Comput. math. appl., 41, 607-618, (2001) · Zbl 0991.34015  Cabada, A.; Lois, S., Existence of solution for discontinuous third order boundary value problems, J. comput. appl. math., 110, 105-114, (1999) · Zbl 0936.34015  Davis, J.M.; Henderson, J., Triple positive symmetric solutions for a lidstone boundary value problem, Differential equations dynam. systems, 7, 321-330, (1999) · Zbl 0981.34014  Erbe, L.H.; Wang, H., On the existence of positive solutions of ordinary differential equations, Proc. amer. math. soc., 120, 743-748, (1994) · Zbl 0802.34018  Gregus, M., Third order linear differential equations, Math. appl., (1987), Reidel Dordrecht · Zbl 0878.34025  Gregus, M., Two sorts of boundary-value problems of nonlinear third order differential equations, Arch. math., 30, 285-292, (1994) · Zbl 0819.34014  Grossinho, M.R.; Minhös, F., Existence result for some third order separated boundary value problems, Nonlinear anal., 47, 2407-2418, (2001) · Zbl 1042.34519  Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press San Diego, CA · Zbl 0661.47045  Krasnoselskii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen  Leggett, R.W.; Williams, L.R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana univ. math. J., 28, 673-688, (1979) · Zbl 0421.47033  Omari, P.; Trombetta, M., Remarks on the lower and upper solutions method for second- and third-order periodic boundary value problems, Appl. math. comput., 50, 1-21, (1992) · Zbl 0760.65078  Rachunkova, I., On some three-point problems for third-order differential equations, Math. bohem., 117, 98-110, (1992) · Zbl 0759.34020  Rusnak, J., Constructions of lower and upper solutions for a nonlinear boundary value problem of the third order and their applications, Math. slovaca, 40, 101-110, (1990) · Zbl 0731.34016  Rusnak, J., Existence theorems for a certain nonlinear boundary value problem of the third order, Math. slovaca, 37, 351-356, (1987) · Zbl 0631.34022  Senkyrik, M., Method of lower and upper solutions for a third-order three-point regular boundary value problem, Acta univ. palack. olomuc. fac. rerum natur. math., 31, 60-70, (1992) · Zbl 0769.34021  Senkyrik, M., Existence of multiple solutions for a third-order three-point regular boundary value problem, Math. bohem., 119, 113-321, (1994) · Zbl 0805.34018  Yao, Q., The existence and multiplicity of positive solutions for a third-order three-point boundary value problem, Acta math. appl. sinica, 19, 1, 117-122, (2003) · Zbl 1048.34031  Yosida, K., Functional analysis, (1978), Springer-Verlag Berlin · Zbl 0217.16001  Zhao, W., Existence and uniqueness of solutions for third order nonlinear boundary value problems, Tohoku math. J., 44, 2, 545-555, (1992) · Zbl 0774.34019
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