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Existence of solution and positive solution of BVP for nonlinear third-order dynamic equation. (English) Zbl 1099.34022
Summary: We are concerned with the following third-order two-point boundary value problem on time scale $$\mathbb{T}$$ $\begin{cases} u^{\Delta\Delta\Delta} (t)+f\bigl(t,u(t), u^{\Delta\Delta}(t)\bigr)=0,\;t\in\bigl[a,\rho(b)\bigr],\\ u(a)=A,\;u\bigl( \sigma^2(b)\bigr)=B,\;u^{\Delta\Delta}(a)=C. \end{cases}$ Some existence criteria for a solution and for a positive solution are established by using Leray-Schauder’s fixed-point theorem. Our main conditions are local. An example is included, too.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 39A10 Additive difference 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] R.P. Agarwal, M. Bohner, W.T. Li, Nonoscillation and oscillation theory for functional differential equations, Pure and Applied Mathematics Series, Marcel Dekker, New York, 2004. · Zbl 1068.34002 [3] Agarwal, R.P.; Bohner, M.; Wong, P., Sturm – liouville eigenvalue problems on time scales, Appl. math. comput., 99, 153-166, (1999) · Zbl 0938.34015 [4] Agarwal, R.P.; O’Regan, D., Nonlinear boundary value problems on time scales, Nonlinear anal., 44, 527-535, (2001) · Zbl 0995.34016 [5] Anderson, D., Solutions to second-order three-point problems on time scales, J. differential equations appl., 8, 673-688, (2002) · Zbl 1021.34011 [6] 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 [7] Bohner, M.; Peterson, A., Dynamic equations on time scales, (2001), Springer New York · Zbl 1021.34005 [8] Bohner, M.; Peterson, A., Advances in dynamic equations on time scales, (2003), Birkhauser Boston · Zbl 1025.34001 [9] Cabada, A.; Lois, S., Existence of solution for discontinuous third order boundary value problems, J. comput. appl. math., 110, 105-114, (1999) · Zbl 0936.34015 [10] 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 [11] Erbe, L.; Hilger, S., Sturmian theory on measure chains, Differential equation dyn. systems, 1, 223-246, (1993) · Zbl 0868.39007 [12] Erbe, L.; Peterson, A., Positive solutions for a nonlinear differential equation on a measure chain, Math. comput. modelling, 32, 5-6, 571-585, (2000) · Zbl 0963.34020 [13] GulsanTopal, S., Second-order periodic boundary value problems on time scales, Comput. math. appl., 48, 637-648, (2004) · Zbl 1068.34016 [14] Henderson, J., Double solutions of impulsive dynamic boundary value problems on a time scale, J. differential equations appl., 8, 345-356, (2002) · Zbl 1003.39019 [15] 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 [16] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001 [17] Kaymakcalan, B.; Lakshmikanthan, V.; Sivasundaram, S., Dynamic systems on measure chains, (1996), Kluwer Academic Publishers Boston · Zbl 0869.34039 [18] Sun, J.P., A new existence theorem for right focal boundary value problems on a measure chain, Appl. math. lett., 18, 41-47, (2005) · Zbl 1074.34017 [19] 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
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