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Existence of symmetric positive solutions for a boundary value problem with integral boundary conditions. (English) Zbl 1221.34062

The author studies the existence of symmetric positive solutions for the nonlinear nonlocal boundary problem

${\left(g\left(t\right){x}^{\text{'}}\left(t\right)\right)}^{\text{'}}+w\left(t\right)f\left(t,x\left(t\right)\right)=0,$
$ax\left(0\right)-b\underset{t\to {0}^{+}}{lim}g\left(t\right){x}^{\text{'}}\left(t\right)={\int }_{0}^{1}h\left(s\right)x\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds,\phantom{\rule{1.em}{0ex}}ax\left(1\right)+b\underset{t\to {1}^{-}}{lim}g\left(t\right){x}^{\text{'}}\left(t\right)={\int }_{0}^{1}h\left(s\right)x\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds,$

where $a,b>0$, $g\in {C}^{1}\left(\left[0,1\right],\left(0,\infty \right)\right)$, $w\in {L}^{p}\left(0,1\right)$ and $h\in {L}^{1}\left(\left(0,1\right),\left(0,\infty \right)\right)$ are symmetric on $\left[0,1\right]$, respectively; $f:\left[0,1\right]×\left[0,\infty \right)\to \left[0,\infty \right)$ is continuous and $f\left(t,x\right)=f\left(1-t,x\right)$. The main tool is the theory of fixed point index.

MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B10 Nonlocal and multipoint boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations
References:
 [1] Csavinszky, P.: Universal approximate solution of the Thomas–Fermi equation for ions, Phys. rev. A. 8, 1688-1701 (1973) [2] Granas, A.; Guenther, R. B.; Lee, J. W.: Nonlinear boundary value problems for ordinary differential equations, Dissertationes math. (1985) · Zbl 0615.34010 [3] Granas, A.; Guenther, R. B.; Lee, J. W.: A note on the Thomas–Fermi equations, Z. angew. Math. mech. 61, 240-241 (1981) [4] Luning, C. D.; Perry, W. L.: Positive solutions of negative exponent generalized Emden–Fowler boundary value problems, SIAM J. Math. anal. 12, 874-879 (1981) · Zbl 0478.34021 · doi:10.1137/0512073 [5] Wong, J. S. W.: On the generalized Emden–Fowler equations, SIAM rev. 17, 339-360 (1975) · Zbl 0295.34026 · doi:10.1137/1017036 [6] Lan, K.; Webb, J. R. L.: Positive solutions of semilinear differential equations with singularities, J. differential equations 148, 407-421 (1998) · Zbl 0909.34013 · doi:10.1006/jdeq.1998.3475 [7] Liu, L.; Sun, Y.: Positive solutions of singular boundary value problems of differential equations, Acta math. Sci. 25A, No. 4, 554-563 (2005) · Zbl 1110.34308 [8] 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 · doi:10.1016/S0362-546X(01)00547-8 [9] Yao, Q.: Existence and iteration of n symmetric positive solutions for a singular two-point boundary value problem, Comput. math. Appl. 47, 1195-1200 (2004) · Zbl 1062.34024 · doi:10.1016/S0898-1221(04)90113-7 [10] Henderson, J.; Thompson, H. B.: Multiple symmetric positive solutions for a second order boundary value problem, Proc. amer. Math. soc. 128, 2373-2379 (2000) · Zbl 0949.34016 · doi:10.1090/S0002-9939-00-05644-6 [11] Li, F.; Zhang, Y.: Multiple symmetric nonnegative solutions of second-order ordinary differential equations, Appl. math. Lett. 17, 261-267 (2004) · Zbl 1060.34009 · doi:10.1016/S0893-9659(04)90061-4 [12] Avery, R. I.; Henderson, J.: Three symmetric positive solutions for a second-order boundary value problem, Appl. math. Lett. 13, 1-7 (2000) · Zbl 0961.34014 · doi:10.1016/S0893-9659(99)00177-9 [13] Kosmatov, N.: Symmetric solutions of a multi-point boundary value problem, J. math. Anal. appl. 309, 25-36 (2005) · Zbl 1085.34011 · doi:10.1016/j.jmaa.2004.11.008 [14] Sun, Y.: Optimal existence criteria for symmetric positive solutions to a three-point boundary value problem, Nonlinear anal. 66, 1051-1063 (2007) · Zbl 1114.34022 · doi:10.1016/j.na.2006.01.004 [15] Cannon, J. R.: The solution of the heat equation subject to the specification of energy, Quart. appl. Math. 21, No. 2, 155-160 (1963) · Zbl 0173.38404 [16] Ionkin, N. I.: Solution of a boundary value problem in heat conduction theory with nonlocal boundary conditions, Differential equations 13, 294-304 (1977) [17] Chegis, R. Yu.: Numerical solution of a heat conduction problem with an integral boundary condition, Litovsk. mat. Sb. 24, 209-215 (1984) · Zbl 0578.65092 [18] Boucherif, A.: Second-order boundary value problems with integral boundary conditions, Nonlinear anal. 70, 364-371 (2009) · Zbl 1169.34310 · doi:10.1016/j.na.2007.12.007 [19] Infante, G.: Eigenvalues and positive solutions of odes involving integral boundary conditions, Discrete contin. Dyn. syst., No. Suppl., 436-442 (2005) · Zbl 1169.34311 [20] Yang, Z.: Positive solutions of a second order integral boundary value problem, J. math. Anal. appl. 321, 751-765 (2006) · Zbl 1106.34014 · doi:10.1016/j.jmaa.2005.09.002 [21] Ahmad, B.; Alsaedi, A.; Alghamdi, B. S.: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions, Nonlinear anal. RWA 9, 1727-1740 (2008) · Zbl 1154.34311 · doi:10.1016/j.nonrwa.2007.05.005 [22] Ahmad, B.; Alsaedi, A.: Existence of approximate solutions of the forced Duffing equation with discontinuous type integral boundary conditions, Nonlinear anal. RWA 10, 358-367 (2009) · Zbl 1154.34314 · doi:10.1016/j.nonrwa.2007.09.004 [23] Feng, M.; Du, B.; Ge, W.: Impulsive boundary value problems with integral boundary conditions and one-dimensional p-Laplacian, Nonlinear anal. 70, 3119-3126 (2009) · Zbl 1169.34022 · doi:10.1016/j.na.2008.04.015 [24] Feng, M.; Zhang, X.; Ge, W.: New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions, Bound. value probl. 2011 (2011) [25] Li, Y.; Zhang, T.: Multiple positive solutions of fourth-order impulsive differential equations with integral boundary conditions and one-dimensional p-Laplacian, Bound. value probl. 2011 (2011) · Zbl 1206.47096 · doi:10.1155/2011/654871 [26] Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones, (1988)