Existence results for solutions of integral boundary value problems on time scales. (English) Zbl 1277.34123
Summary: This paper deals with the existence of solutions for integral boundary value problems on time scales. We provide sufficient conditions for the existence of solutions by using the Schauder fixed point theorem in a cone. Existence result for this problem is also given by the method of upper and lower solutions.
MSC:
| 34N05 | Dynamic equations on time scales or measure chains |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
References:
| [1] | Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results in Mathematics, 18, 1-2, 18-56 (1990) · Zbl 0722.39001 |
| [2] | Bohner, M.; Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications, x+358 (2001), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 0978.39001 · doi:10.1007/978-1-4612-0201-1 |
| [3] | Bohner, M.; Peterson, A., Advances in Dynamic Equations on Time Scales, xii+348 (2003), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 1025.34001 · doi:10.1007/978-0-8176-8230-9 |
| [4] | Davidson, F. A.; Rynne, B. P., The formulation of second-order boundary value problems on time scales, Advances in Difference Equations, 2006 (2006) · Zbl 1139.39026 · doi:10.1155/ADE/2006/31430 |
| [5] | Erbe, L.; Peterson, A., Green’s functions and comparison theorems for differential equations on measure chains, Dynamics of Continuous, Discrete and Impulsive Systems, 6, 1, 121-137 (1999) · Zbl 0938.34027 |
| [6] | Khan, R. A., The generalized method of quasilinearization and nonlinear boundary value problems with integral boundary conditions, Electronic Journal of Qualitative Theory of Differential Equations, 10, 1-9 (2003) · Zbl 1055.34033 |
| [7] | Yang, Z., Existence and nonexistence results for positive solutions of an integral boundary value problem, Nonlinear Analysis: Theory, Methods & Applications, 65, 8, 1489-1511 (2006) · Zbl 1104.34017 · doi:10.1016/j.na.2005.10.025 |
| [8] | Ahmad, B.; Alsaedi, A.; Alghamdi, B. S., Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions, Nonlinear Analysis: Real World Applications, 9, 4, 1727-1740 (2008) · Zbl 1154.34311 · doi:10.1016/j.nonrwa.2007.05.005 |
| [9] | Atici, F. M.; Guseinov, G. Sh., On Green’s functions and positive solutions for boundary value problems on time scales, Journal of Computational and Applied Mathematics, 141, 1-2, 75-99 (2002) · Zbl 1007.34025 · doi:10.1016/S0377-0427(01)00437-X |
| [10] | Batten, G. W., Second-order correct boundary conditions for the numerical solution of the mixed boundary problem for parabolic equations, Mathematics of Computation, 17, 405-413 (1963) · Zbl 0133.38601 · doi:10.1090/S0025-5718-1963-0156476-6 |
| [11] | Cannon, J. R.; Esteva, S. P.; van der Hoek, J., A Galerkin procedure for the diffusion equation subject to the specification of mass, SIAM Journal on Numerical Analysis, 24, 3, 499-515 (1987) · Zbl 0677.65108 · doi:10.1137/0724036 |
| [12] | Kamynin, L. I., A boundary value problem in the theory of heat conduction with boundary condition, USSR Computational Mathematics and Mathematical Physics, 4, 6, 33-59 (1964) · Zbl 0206.39801 · doi:10.1016/0041-5553(64)90080-1 |
| [13] | Bouziani, A.; Benouar, N.-E., Mixed problem with integral conditions for a third order parabolic equation, Kobe Journal of Mathematics, 15, 1, 47-58 (1998) · Zbl 0921.35068 |
| [14] | Denche, M.; Marhoune, A. L., Mixed problem with integral boundary condition for a high order mixed type partial differential equation, Journal of Applied Mathematics and Stochastic Analysis, 16, 1, 69-79 (2003) · Zbl 1035.35085 · doi:10.1155/S1048953303000054 |
| [15] | Gallardo, J. M., Second-order differential operators with integral boundary conditions and generation of analytic semigroups, The Rocky Mountain Journal of Mathematics, 30, 4, 1265-1292 (2000) · Zbl 0984.34014 · doi:10.1216/rmjm/1021477351 |
| [16] | Jankowski, T., Differential equations with integral boundary conditions, Journal of Computational and Applied Mathematics, 147, 1, 1-8 (2002) · Zbl 1019.34006 · doi:10.1016/S0377-0427(02)00371-0 |
| [17] | Kartynnik, A. V., A three-point mixed problem with an integral condition with respect to the space variable for second-order parabolic equations, Differential Equations, 26, 9, 1160-1166 (1990) · Zbl 0729.35053 |
| [18] | Liu, L.; Hao, X.; Wu, Y., Positive solutions for singular second order differential equations with integral boundary conditions, Mathematical and Computer Modelling, 57, 3-4, 836-847 (2013) · Zbl 1305.34040 · doi:10.1016/j.mcm.2012.09.012 |
| [19] | Zhang, X.; Ge, W., Symmetric positive solutions of boundary value problems with integral boundary conditions, Applied Mathematics and Computation, 219, 8, 3553-3564 (2012) · Zbl 1311.34054 · doi:10.1016/j.amc.2012.09.037 |
| [20] | Brykalov, S. A., A second-order nonlinear problem with two-point and integral boundary conditions, Georgian Mathematical Journal, 1, 3, 243-249 (1994) · Zbl 0807.34021 |
| [21] | Krasnosel’skii, M. A., Positive Solutions of Operator Equations (1964), Groningen, The Netherlands: P. Noordhoff Ltd., Groningen, The Netherlands · Zbl 0121.10604 |
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