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Positive solutions to semi-positone second-order three-point problems on time scales. (English) Zbl 1188.34119
Summary: Using a fixed point theorem of generalized cone expansion and compression we establish the existence of at least two positive solutions for the nonlinear semi-positone three-point boundary value problem on time scales $$u^{\Delta\nabla}(t)+\lambda f(t,u(t))=0,\quad u(a)=0,\quad \alpha u(\eta)=u(T).$$ Here $t\in[a,T]_{\Bbb T}$, where $\Bbb T$ is a time scale, $\alpha>0$, $\eta\in(a,\rho(T))_{\Bbb T}$, $\alpha(\eta-a)<T-a$, and the parameter $\lambda >0$ belongs to a certain interval. These results are new for difference equations as well as for general time scales. An example is provided for differential, difference, and $q$-difference equations.

##### MSC:
 34N05 Dynamic equations on time scales or measure chains 34B10 Nonlocal and multipoint boundary value problems for ODE 34B18 Positive solutions of nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations
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##### References:
 [1] Anderson, D. R.: Taylor polynomials for nabla dynamic equations on time scales, Panamerican math. J. 12, No. 4, 17-27 (2002) · Zbl 1026.34011 [2] Anderson, D. R.: Solutions to second-order three-point problems on time scales, J. difference equations appl. 8, No. 8, 673-688 (2002) · Zbl 1021.34011 · doi:10.1080/10236919021000000726 [3] Anderson, D. R.: Existence of solutions for nonlinear multi-point problems on time scales, Dynam. syst. Appl. 15, 21-34 (2006) · Zbl 1104.34018 [4] Anderson, D. R.; Ma, R. Y.: Second-order n-point eigenvalue problems on time scales, Adv. difference equ., 1-17 (2006) [5] Anderson, D. R.; Wong, P. J. Y.: Positive solutions for second-order semipositone problems on time scales, Computers math. Appl. 58, 281-291 (2009) · Zbl 1189.34167 · doi:10.1016/j.camwa.2009.02.033 [6] Atici, F. M.; Guseinov, G. Sh.: On Green’s functions and positive solutions for boundary value problems on time scales, J. comput. Appl. math. 141, 75-99 (2002) · Zbl 1007.34025 · doi:10.1016/S0377-0427(01)00437-X [7] Bohner, M.; Peterson, A.: Dynamic equations on time scales, an introduction with applications, (2001) · Zbl 0978.39001 [8] Dacunha, J. J.; Davis, J. M.; Singh, P. K.: Existence results for singular three-point boundary value problems on time scales, J. math. Anal. appl. 295, 378-391 (2004) · Zbl 1069.34012 · doi:10.1016/j.jmaa.2004.02.049 [9] Erbe, L. H.; Peterson, A. C.: Positive solutions for a nonlinear differential equation on a measure chain, Math. computer modelling 32, 571-585 (2000) · Zbl 0963.34020 · doi:10.1016/S0895-7177(00)00154-0 [10] Erbe, L. H.; Peterson, A. C.: Green’s functions and comparison theorems for differential equations on measure chains, Dynam. continuous, discrete impulsive systems 6, 121-137 (1999) · Zbl 0938.34027 [11] Luo, H.: Positive solutions to singular multi-point dynamic eigenvalue problems with mixed derivatives, Nonlinear anal. 70, 1679-1691 (2009) · Zbl 1168.34014 · doi:10.1016/j.na.2008.02.051 [12] Ma, R. Y.: Positive solutions of a nonlinear three-point boundary-value problem, Electronic J. Differential eqs. 1999, No. 34, 1-8 (1999) · Zbl 0926.34009 [13] Ma, R. Y.: Multiplicity of positive solutions for second-order three-point boundary-value problems, Computers math. Appl. 40, 193-204 (2000) · Zbl 0958.34019 · doi:10.1016/S0898-1221(00)00153-X [14] Ma, R. Y.: Positive solutions for second-order three-point boundary-value problems, Appl. math. Letters 14, 1-5 (2001) · Zbl 0989.34009 [15] Zhai, C. B.: Positive solutions for semi-positone three-point boundary value problems, J. comput. Appl. math. 228, 279-286 (2009) · Zbl 1182.34036 · doi:10.1016/j.cam.2008.09.019 [16] Zhang, G. W.; Sun, J. X.: A generalization of the cone expansion and compression fixed point theorem and applications, Nonlinear anal. 67, 579-586 (2007) · Zbl 1127.47050 · doi:10.1016/j.na.2006.06.003