## Existence and nonexistence results for positive solutions of an integral boundary value problem.(English)Zbl 1104.34017

This paper is concerned with the integral boundary value problem:
\begin{aligned} -(au')'+bu&= g(t)f(t,u),\quad 0<t<1,\\ (\cos\gamma_0)u(0)-(\sin\gamma_0)u'(0)&= \int_0^1u(\tau)\,d\alpha(\tau),\\ (\cos\gamma_1)u(1)+(\sin\gamma_1)u'(1)&= \int_0^1u(\tau)\,d\beta(\tau), \end{aligned} with $$a\in C^1([0,1],(0,+\infty))$$, $$b\in C([0,1],\mathbb{R}^+)$$ and $$f\in C([0,1]\times\mathbb{R}^+,\mathbb{R}^+)$$, $$g\in C([0,1],\mathbb{R}^+)$$. The functions $$\alpha$$ and $$\beta$$ are right continuous on $$[0,1),$$ left continuous at $$t=1,$$ and nondecreasing on $$[0,1],$$ with $$\alpha(0)=\beta(0)=0;$$ and $$\gamma_0,\,\gamma_1\in[0,\frac{\pi}{2}].$$ The integrals on the right-hand side are Riemann-Stieltjies integrals of $$u$$ with respect to $$\alpha$$ and $$\beta,$$ respectively. The problem encompasses Sturm-Liouville two-point boundary value problems and multipoint eigenvalue problems.
The author proves two existence results of at least one positive solution, one existence result of at least two positive solutions and a nonexistence result of positive solutions. A set of hypotheses on the nonlinear term $$f$$ are enunciated to get existence results. The proofs are based on the properties of the Green function and on the study of the spectral radius of an auxiliary linear operator; then, the author mainly appeals to the Krein-Rutman theorem, the Krasnosel’ski fixed-point theorem in cones of Banach spaces and a lemma by the author [Nonlinear Anal. Theory Methods Appl. 62, No. 7 (A), 1251–1265 (2005; Zbl 1089.34022)].
The main theorems extend previous results obtained by R. Ma and B. Thompson [J. Math. Anal. Appl. 297, No. 1, 24–37 (2004; Zbl 1057.34011)] and Z. Yang [J. Math. Anal. Appl. 321, No. 2, 751–765 (2006; Zbl 1106.34014)].
An example is given to illustrate the existence results. Finally, the author explains how the method used in this paper may be applied to obtain positive solutions for similar boundary value problems with nonlinear integral conditions.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations

### Keywords:

integral boundary value problem; positive solution

### Citations:

Zbl 1089.34022; Zbl 1057.34011; Zbl 1106.34014
Full Text:

### References:

 [1] Deimling, K., Nonlinear functional analysis, (1985), Springer Berlin, Heidelberg, New York, Tokyo · Zbl 0559.47040 [2] 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 [3] II’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, Differ. equ., 23, 803-810, (1987) [4] II’in, V.A.; Moiseev, E.I., Nonlocal boundary value problem of the second kind for a sturm – liouville operator, Differ. equ., 23, 8, 979-987, (1987) · Zbl 0668.34024 [5] Karakostas, G.L.; Tsamatos, P.Ch., Existence of multiple positive solutions for a nonlocal boundary value problem, Topol. methods nonlinear anal., 19, 109C121, (2002) · Zbl 1028.34023 [6] Karakostas, G.L.; Tsamatos, P.Ch., Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems, Electron. J. differential equations, 30, 1-17, (2002) · Zbl 0998.45004 [7] Karakostas, G.L.; Tsamatos, P.Ch., Sufficient conditions for the existence of nonnegative solutions of a nonlocal boundary value problem, Appl. math. lett., 15, 401C407, (2002) · Zbl 1028.34023 [8] Krasnoselski, M.A.; Zabreiko, P.P., Geometrical methods of nonlinear analysis, (1984), Springer [9] Krein, M.G.; Rutman, M.A., Linear operators leaving invariant a cone in a Banach space, Transl. AMS, 10, 199-325, (1962) · Zbl 0030.12902 [10] Liu, Z.; Li, F., Multiple positive solutions of nonlinear two-point boundary values, J. math. anal. appl., 203, 610-625, (1996) · Zbl 0878.34016 [11] Ma, R., Positive solutions for a nonlinear three-point boundary value problem, Electron. J. differential equations, 34, 1-8, (1999) [12] Ma, R., Existence theorems for a second order $$m$$-point boundary value problem, J. math. anal. appl., 211, 545-555, (1997) · Zbl 0884.34024 [13] Ma, R., Existence of positive solutions for superlinear semipositone $$m$$-point boundary value problems, Proc. edinb. math. soc., 46, 279-292, (2003) · Zbl 1069.34036 [14] Ma, R.; Wang, H., Positive solutions of nonlinear three-point boundary value problems, J. math. anal. appl., 279, 1216-1227, (2003) [15] Ma, R.; Thompson, B., Positive solutions for nonlinear $$m$$-point eigenvalue problems, J. math. anal. appl., 297, 24-37, (2004) · Zbl 1057.34011 [16] Ma, R., Nonlocal problems for nonlinear differential equations, (2004), Science Press Beijing, (in Chinese) [17] 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 [18] 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 [19] Webb, J.R.L., Optimal constants in a nonlocal boundary value problem, Nonlinear anal., (2005), (in press) · Zbl 1153.34320 [20] Yang, Z., Positive solutions to a system of second-order nonlocal boundary value problems, Nonlinear anal., 62, 1251-1265, (2005) · Zbl 1089.34022 [21] Z. Yang, Positive solutions of a second-order integral boundary value problem, J. Math. Anal. Appl. (in press) · Zbl 1106.34014 [22] 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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