## Existence of nontrivial solutions for a nonlinear Sturm-Liouville problem with integral boundary conditions.(English)Zbl 1132.34022

The Sturm-Liouville problem with integral boundary value conditions $-(au')' + bu = g(t)f(t,u), \quad t \in (0,1),$
$(\cos \gamma_0) u(0) - (\sin \gamma_0) u'(0) = \int_0^1 u(\tau) \,d\alpha(\tau),$
$(\cos \gamma_1) u(1) + (\sin \gamma_1) u'(1) = \int_0^1 u(\tau) \,d\beta(\tau),$ is considered, where $$\int_0^1 u(\tau)\,d\alpha(\tau)$$ and $$\int_0^1 u(\tau) \,d\beta(\tau)$$ denote Riemann-Stieltjes integrals. Existence results of a nontrivial solution are established. The proofs are based on the Leray-Schauder degree theory.

### MSC:

 34B24 Sturm-Liouville theory 34B15 Nonlinear boundary value problems for ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations 47H11 Degree theory for nonlinear operators
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### References:

 [1] Feng, W.; Webb, J.R.L., Solvability of an $$m$$-point boundary value problem with nonlinear growth, J. math. anal. appl., 212, 467-480, (1997) · Zbl 0883.34020 [2] Gupta, C.P.; Ntouyas, S.K.; Tsamatos, P.Ch., Solvability of an $$m$$-point boundary value problem for second order ordinary differential equations, J. math. anal. appl., 189, 575-584, (1995) · Zbl 0819.34012 [3] Gupta, C.P.; Trofimchuk, S.I., A sharper condition for the solvability of a three-point second order boundary value problem, J. math. anal. appl., 205, 586-597, (1997) · Zbl 0874.34014 [4] Gupta, C.P.; Trofimchuk, S.I., Solvability of a multi-point boundary value problem and related a priori estimates, Can. appl. math. Q., 6, 45-60, (1998) · Zbl 0922.34014 [5] Gupta, C.P.; Trofimchuk, S.I., Existence of a solution of a three-point boundary value problem and the spectral radius of a related linear operator, Nonlinear anal., 34, 489-507, (1998) · Zbl 0944.34009 [6] Han, G.; Wu, Y., Nontrivial solutions of singular two-point boundary value problems with sign-changing nonlinear terms, J. math. anal. appl., 325, 1327-1338, (2007) · Zbl 1111.34019 [7] 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 [8] Karakostas, G.L.; Tsamatos, P.Ch., Existence of multiple positive solutions for a nonlocal boundary value problem, Topol. methods nonlinear anal., 19, 109-121, (2002) · Zbl 1028.34023 [9] 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 [10] Karakostas, G.L.; Tsamatos, P.Ch., Sufficient conditions for the existence of nonnegative solutions of a nonlocal boundary value problem, Appl. math. lett., 15, 401-407, (2002) · Zbl 1028.34023 [11] Krasnoselskii, M.A.; Zabreiko, B.P., Geometrical methods of nonlinear analysis, (1984), Springer [12] 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 [13] Ma, R., Nonlocal problems for nonlinear ordinary differential equations, (2004), Science Press Beijing, (in Chinese) [14] Ma, R.; O’Regan, D., Solvability of singular second order $$m$$-point boundary value problems, J. math. anal. appl., 301, 124-134, (2005) · Zbl 1062.34018 [15] Ma, R.; O’Regan, D., Nodal solutions for second-order $$m$$-point boundary value problems with nonlinearities across several eigenvalues, Nonlinear anal., 64, 1562-1577, (2006) · Zbl 1101.34006 [16] Sun, J.; Guo, G., Nontrivial solutions of singular superlinear sturm – liouville problems, J. math. anal. appl., 313, 518-536, (2006) · Zbl 1100.34019 [17] 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 [18] Webb, J.R.L., Optimal constants in a nonlocal boundary value problem, Nonlinear anal., 63, 672-685, (2005) · Zbl 1153.34320 [19] Webb, J.R.L.; Lan, K.Q., Eigenvalue criteria for existence of multiple positive solutions of local and nonlocal type, Topol. methods nonlinear anal., 27, 91-115, (2006) · Zbl 1146.34020 [20] Xu, X., Multiple sign-changing solutions for some $$m$$-point boundary value problems, Electron. J. differential equations, 89, 1-14, (2004) [21] Xu, X.; Sun, J., On sign-changing solution for some three-point boundary value problems, Nonlinear anal., 59, 491-505, (2004) · Zbl 1069.34019 [22] Yang, Z., Existence and nonexistence results for positive solutions of an integral boundary value problem, Nonlinear anal., 65, 1489-1511, (2006) · Zbl 1104.34017
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