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Positive solutions of singular third-order three-point boundary value problem. (English) Zbl 1074.34028
Summary: We investigate the existence of positive solutions for the nonlinear singular third-order three-point boundary value problem $u'''(t)-\lambda a(t) F(t, u(t))= 0,\quad 0< t< 1,\quad u(0)= u'(\eta)= u''(1)= 0,$ where $$\lambda$$ is a positive parameter and $$\eta\in[1/2, 1)$$ is a constant. By using a fixed-point theorem of cone expansion-compression type due to Krasnosel’skii, we establish various results on the existence of single and multiple positive solutions to the boundary value problem. Under various assumptions on the functions $$F$$ and $$a$$, we give explicitly the intervals for parameter $$\lambda$$ in which the existence of positive solutions is guaranteed. Especially, we allow the function $$a(t)$$ in the nonlinear term to have suitable singularities.

##### MSC:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
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##### References:
 [1] Anderson, D., Multiple positive solutions for a three-point boundary value problem, Math. comput. modelling, 27, 49-57, (1998) · Zbl 0906.34014 [2] 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 [3] Yao, Q., The existence and multiplicity of positive solutions for a third-order three-point boundary value problem, Acta math. appl. sinica, 19, 117-122, (2003) · Zbl 1048.34031 [4] Handerson, J.; Thompson, H.B., Multiple symmetric positive solutions for a second order boundary value problem, Proc. amer. math. soc., 128, 2373-2379, (2000) · Zbl 0949.34016 [5] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press San Diego · Zbl 0661.47045 [6] Krasnosel’skii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen · Zbl 0121.10604 [7] Yosida, K., Functional analysis, (1978), Springer-Verlag Berlin · Zbl 0217.16001
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