## Positive solutions of a second-order integral boundary value problem.(English)Zbl 1106.34014

The author uses a well-known fixed-point theorem to study the existence of positive solutions of the nonlinear integral boundary value problem
$-(au')' + b u = f(t, u),$
$(\cos \gamma_0) u(0) - (\sin \gamma_0) u'(0) = H_1 \left ( \int_0^1 \! u(\tau) \, d \alpha(\tau) \right ),$
$(\cos \gamma_1) u(1) + (\sin \gamma_1) u'(1) = H_2 \left ( \int_0^1 \! u(\tau) \, d \beta(\tau) \right ).$
The method used is an interesting variant of the traditional means to show the existence of positive solutions via cone theoretic techniques. The author considers the above boundary value problem as a perturbation of the boundary value problem $$-(au')' + b u = f(t, u), (\cos \gamma_0) u(0) - (\sin \gamma_0) u'(0) = 0, (\cos \gamma_1) u(1) + (\sin \gamma_1) u'(1) = 0$$. Using the Green function associated with the boundary value problem with homogeneous boundary conditions, the author defines a cone preserving operator and obtains many fixed-point theorems.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 47H10 Fixed-point theorems
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### References:

 [1] Deimling, K., Nonlinear Functional Analysis (1985), Springer: Springer Berlin · Zbl 0559.47040 [2] Feng, W.; Webb, J. R.L., Solvability of a $$m$$-point boundary value problems with nonlinear growth, J. Math. Anal. Appl., 212, 467-480 (1997) · Zbl 0883.34020 [3] Grabmüller, H., Integralgleichungen (für Mathematiker) (2000), Institut für Angewandte Mathematik, Vorlesung im Sommersemester, Friedrich-Alexander-Universität Erlangen-Nürnberg [4] Guo, D.; Laksmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press Boston · Zbl 0661.47045 [5] Guo, D.; Sun, J.; Liu, Z., Functional Analysis Approaches in the Theory of Nonlinear Ordinary Differential Equations (1995), Shandong Press of Science and Technology: Shandong Press of Science and Technology Jinan, (in Chinese) [6] Gupta, C. P., Solvability of a three-point nonlinear boundary value problem for a second-order ordinary differential equation, J. Math. Anal. Appl., 168, 540-551 (1992) · Zbl 0763.34009 [7] Gupta, C. P., A generalized multi-point boundary value problem for second-order ordinary differential equations, Appl. Math. Comput., 89, 133-146 (1998) · Zbl 0910.34032 [8] Henderson, J., Double solutions of three-point boundary-value problems for second-order differential equations, Electron. J. Differential Equations, 115, 1-7 (2004) · Zbl 1075.34013 [9] Il’in, V. A.; Moiseev, E. I., Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and difference aspects, Differential Equations, 23, 803-810 (1987) · Zbl 0668.34025 [10] Il’in, V. A.; Moiseev, E. I., Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator, Differential Equations, 23, 979-987 (1987) · Zbl 0668.34024 [11] Infante, G., Eigenvalues of some non-local boundary-value problems, Proc. Edinburgh Math. Soc., 46, 75-86 (2003) · Zbl 1049.34015 [12] Krasnoselskii, M. A.; Zabreiko, P. P., Geometrical Methods of Nonlinear Analysis (1984), Springer · Zbl 0546.47030 [13] Krein, M. G.; Rutman, M. A., Linear operators leaving invariant a cone in a Banach space, Transl. Amer. Math. Soc., 10, 199-325 (1962) · Zbl 0030.12902 [14] Liu, Z.; Li, F., Multiple positive solutions of nonlinear two-point boundary values, J. Math. Anal. Appl., 203, 610-625 (1996) · Zbl 0878.34016 [15] Ma, R., Positive solutions for a nonlinear three-point boundary value problem, Electron. J. Differential Equations, 1999, 1-8 (1999) · Zbl 0926.34009 [16] Ma, R., Existence theorems for a second-order $$m$$-point boundary value problem, J. Math. Anal. Appl., 211, 545-555 (1997) · Zbl 0884.34024 [17] Ma, R., Existence of positive solutions for superlinear semipositone $$m$$-point boundary value problems, Proc. Edinburgh Math. Soc., 46, 279-292 (2003) · Zbl 1069.34036 [18] Ma, R.; Wang, H., Positive solutions of nonlinear three-point boundary value problems, J. Math. Anal. Appl., 279, 1216-1227 (2003) [19] Ma, R.; Thompson, B., Positive solutions for nonlinear $$m$$-point eigenvalue problems, J. Math. Anal. Appl., 297, 24-37 (2004) · Zbl 1057.34011 [20] Ma, R.; Thompson, B., Global behavior of positive solutions of nonlinear three-point boundary value problems, Nonlinear Anal., 60, 685-701 (2005) · Zbl 1069.34016 [21] Moshinsky, M., Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas, Bol. Soc. Mat. Mexicana, 7, 1-25 (1950) [22] Palamides, P. K., Positive and monotone solutions of an $$m$$-point boundary value problem, Electron. J. Differential Equations, 18, 1-16 (2002) · Zbl 1023.34019 [23] Timoshenko, T., Theory of Elastic Stability (1971), McGraw-Hill: McGraw-Hill New York [24] Webb, J. R.L., Positive solutions of some three-point boundary value problems via fixed point index theory, Nonlinear Anal., 47, 4319-4332 (2001) · Zbl 1042.34527 [25] Yang, Z.; O’Regan, D., Positive solvability of systems of nonlinear Hammerstein integral equations, J. Math. Anal. Appl., 311, 600-614 (2005) · Zbl 1079.45005 [26] Zhang, G.; Sun, J., Positive solutions of $$m$$-point boundary value problems, J. Math. Anal. Appl., 291, 406-418 (2004) · Zbl 1069.34037
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