On a singular multi-point third-order boundary value problem on the half-line. (English) Zbl 1513.34091

This paper considers the following singular multi-point third-order boundary value problem posed on the half-line of the form \[ \begin{aligned} &-x'''(t)=f(t,x(t),x'(t)),\quad t>0,\\ &x(0)=\sum_{i=1}^{n_1}\alpha_ix(\xi_i),\\ &x'(0)=\sum_{i=1}^{n_2}\beta_ix'(\eta_i),\\ &\lim_{t\rightarrow \infty}x''(t)=0, \end{aligned} \] where the nonlinearity \(f\in C((0,\infty)\times[0,\infty)\times[0,\infty),[0,\infty))\) may be singular at \(t=0\) and there exist \(0<\alpha<\beta<\infty\) such that \(I_{\alpha,\beta}=\int_{\alpha}^{\beta}f(t,1+t^2,1+t)\textrm{d}t>0\); \(0\leq\alpha_j\leq\sum_{i=1}^{n_1}\alpha_i<1~(j=1,2,\dots,n_1)\), \(0<\xi_1<\xi_2<\cdots<\xi_{n_1}<\infty\); \(0\leq\beta_j\leq\sum_{i=1}^{n_2}\beta_i<1~(j=1,2,\dots,n_2)\), \(0<\eta_1<\eta_2<\cdots<\eta_{n_2}<\infty\).
Using the Krasnosel’skii fixed point theorem on cone compression and expansion, under upper and lower-homogeneity conditions for the nonlinearity \(f\), the authors established the existence of at least one positive solution to the above problem. Also, nonexistence results are proved under suitable a priori estimates. Two examples of applications are included to illustrate the existence theorems.
Reviewer: Minghe Pei (Jilin)


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
34B40 Boundary value problems on infinite intervals for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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[1] Agarwal, R. P.; O’Regan, D., Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht (2001) · Zbl 0988.34002
[2] Chen, S.; Zhang, Y., Singular boundary value problems on a half-line, J. Math. Anal. Appl. 195 (1995), 449-468 · Zbl 0852.34019
[3] Corduneanu, C., Integral Equations and Stability of Feedback Systems, Mathematics in Science and Engineering 104. Academic Press, New York (1973) · Zbl 0273.45001
[4] Djebali, S.; Mebarki, K., Multiple positive solutions for singular BVPs on the positive half-line, Comput. Math. Appl. 55 (2008), 2940-2952 · Zbl 1142.34316
[5] Djebali, S.; Saifi, O., Third order BVPs with \(\phi \)-Laplacian operators on \([0,+\infty)\), Afr. Diaspora J. Math. 16 (2013), 1-17 · Zbl 1283.34019
[6] Guo, D. J.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering 5. Academic Press, Boston (1988) · Zbl 0661.47045
[7] Liang, S.; Zhang, J., Positive solutions for singular third-order boundary value problem with dependence on the first order derivative on the half-line, Acta Appl. Math. 111 (2010), 27-43 · Zbl 1203.34038
[8] Liu, Y., Existence and unboundedness of positive solutions for singular boundary value problems on half-line, Appl. Math. Comput. 144 (2003), 543-556 · Zbl 1036.34027
[9] Wei, Z., A necessary and sufficient condition for the existence of positive solutions of singular super-linear \(m\)-point boundary value problems, Appl. Math. Comput. 179 (2006), 67-78 · Zbl 1166.34305
[10] Wei, Z., Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems, J. Appl. Math. Comput. 46 (2014), 407-422 · Zbl 1311.34052
[11] Yan, B.; O’Regan, D.; Agarwal, R. P., Unbounded positive solutions for second order singular boundary value problems with derivative dependence on infinite intervals, Funkc. Ekvacioj, Ser. Int. 51 (2008), 81-106 · Zbl 1158.34011
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