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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)

MSC:

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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