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Existence of a solution and a positive solution of a boundary value problem for a nonlinear fourth-order dynamic equation. (English) Zbl 1152.34322
Summary: By using the Schauder fixed point theorem, we offer some existence criteria for a solution and a positive solution to the following fourth-order two-point boundary value problem on time scale $\Bbb T$: $$u^{\Delta\Delta\Delta\Delta}(t)= f(t,u(t), u^{\Delta\Delta}(t))= 0, \quad t\in [a,\rho^2(b)],$$ $$u(a)=A, \quad u(\sigma^2(b))=B, \quad u^{\Delta\Delta}(a)= C, \quad u^{\Delta\Delta}(b)=D,$$ where $a,b\in\Bbb T$ and $a<b$.

##### MSC:
 34B15 Nonlinear boundary value problems for ODE 34B18 Positive solutions of nonlinear boundary value problems for ODE 39A10 Additive difference equations 47N20 Applications of operator theory to differential and integral equations
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##### References:
 [1] Agarwal, R. P.; Bohner, M.: Basic calculus on time scales and some of its applications, Results math. 35, 3-22 (1999) · Zbl 0927.39003 [2] Agarwal, R. P.; Bohner, M.; Wong, P.: Sturm--Liouville eigenvalue problems on time scales, Appl. math. Comput. 99, 153-166 (1999) · Zbl 0938.34015 · doi:10.1016/S0096-3003(98)00004-6 [3] Agarwal, R. P.; O’regan, D.: Nonlinear boundary value problems on time scales, Nonlinear anal. 44, 527-535 (2001) · Zbl 0995.34016 · doi:10.1016/S0362-546X(99)00290-4 [4] Anderson, D.: Solutions to second-order three-point problems on time scales, J. differential equations appl. 8, 673-688 (2002) · Zbl 1021.34011 · doi:10.1080/10236919021000000726 [5] Atici, E. 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 [6] Aftabizadeh, A. R.: Existence and uniqueness theorems for fourth-order boundary value problems, J. math. Anal. appl. 116, 415-426 (1986) · Zbl 0634.34009 · doi:10.1016/S0022-247X(86)80006-3 [7] Bohner, M.; Peterson, A.: Dynamic equations on time scales, (2001) · Zbl 0993.39010 [8] Chyan, C. J.; Henderson, J.: Eigenvalue problems for nonlinear differential equations on a measure chain, J. math. Anal. appl. 245, 547-559 (2000) · Zbl 0953.34068 · doi:10.1006/jmaa.2000.6781 [9] Del Pino, M. A.; Manasevich, R. F.: Existence for fourth-order boundary value problem under a two-parameter nonresonance condition, Proc. amer. Math. soc. 112, 81-86 (1991) · Zbl 0725.34020 · doi:10.2307/2048482 [10] Erbe, L.; Hilger, S.: Sturmian theory on measure chains, Differential equations dynam. Systems 1, 223-246 (1993) · Zbl 0868.39007 [11] Erbe, L.; Peterson, A.: Positive solutions for a nonlinear differential equation on a measure chain, Math. comput. Modelling 32, No. 5--6, 571-585 (2000) · Zbl 0963.34020 · doi:10.1016/S0895-7177(00)00154-0 [12] Topal, S. Gulsan: Second-order periodic boundary value problems on time scales, Comput. math. Appl. 48, 637-648 (2004) · Zbl 1068.34016 · doi:10.1016/j.camwa.2002.04.005 [13] Henderson, J.; Yin, W. K. C.: Existence of solutions for third-order boundary value problems on a time scale, Comput. math. Appl. 45, 1101-1111 (2003) · Zbl 1057.39011 · doi:10.1016/S0898-1221(03)00091-9 [14] Hilger, S.: Analysis on measure chains--a unified approach to continuous and discrete calculus, Results math. 18, 18-56 (1990) · Zbl 0722.39001 [15] Kaymakcalan, B.; Lakshmikanthan, V.; Sivasundaram, S.: Dynamic systems on measure chains, (1996) · Zbl 0869.34039 [16] Liu, B.: Positive solutions of fourth-order two point boundary value problems, Appl. math. Comput. 148, 407-420 (2004) · Zbl 1039.34018 · doi:10.1016/S0096-3003(02)00857-3 [17] Ma, R.: Positive solutions of fourth-order two point boundary value problems, Ann. differential equations 15, 305-313 (1999) · Zbl 0964.34021 [18] Sun, J. P.: Existence of solution and positive solution of BVP for nonlinear third-order dynamic equation, Nonlinear anal. 64, 629-636 (2006) · Zbl 1099.34022 · doi:10.1016/j.na.2005.04.046 [19] Usmani, R. A.: A uniqueness theorem for a boundary value problem, Proc. amer. Math. soc. 77, 327-335 (1979) · Zbl 0424.34019 · doi:10.2307/2042181 [20] Yang, Y.: Fourth-order two-point boundary value problem, Proc. amer. Math. soc. 104, 175-180 (1988) · Zbl 0671.34016 · doi:10.2307/2047481 [21] Yao, Q.: Solution and positive solution for a semilinear third-order two-point boundary value problem, Appl. math. Lett. 17, 1171-1175 (2004) · Zbl 1061.34012 · doi:10.1016/j.aml.2003.09.011