×

Multi-parameter fourth order impulsive integral boundary value problems with one-dimensional \(m\)-Laplacian and deviating arguments. (English) Zbl 1316.34068

The authors are interested in a fourth order with \(m\)-Laplacian functional differential equation with impulsive integral boundary values. By the use of fixed point theory, they investigate the problem in the both cases of advanced and delayed argument. They discuss also the non existence of positive solutions.

MSC:

34K10 Boundary value problems for functional-differential equations
34K45 Functional-differential equations with impulses
47N20 Applications of operator theory to differential and integral equations

References:

[1] Kuang, Y: Delay Differential Equations: With Applications in Population Dynamics. Academic Press, Boston (1993) · Zbl 0777.34002
[2] Yan, JR: Existence and global attractivity of positive periodic solution for an impulsive Lasota-Wazewska model. J. Math. Anal. Appl. 279, 111-120 (2003) · Zbl 1032.34077 · doi:10.1016/S0022-247X(02)00613-3
[3] Gyöi, I, Ladas, G: Oscillation Theorem of Delay Differential Equations with Applications. Clarendon, Oxford (1991) · Zbl 0780.34048
[4] Lalli, BS, Zhang, BG: On a periodic delay population model. Q. Appl. Math. 52, 35-42 (1994) · Zbl 0788.92022
[5] Gopalsamy, K, Kulenović, MRS, Ladas, G: Environmental periodicity and time delays in a food-limited population model. J. Math. Anal. Appl. 147, 545-555 (1990) · Zbl 0701.92021 · doi:10.1016/0022-247X(90)90369-Q
[6] Zhang, XM, Feng, MQ: Transformation technique, fixed point theorem and positive solutions for second-order impulsive differential equations with deviating arguments. Adv. Differ. Equ. 2014, 312 (2014) · Zbl 1347.34103 · doi:10.1186/1687-1847-2014-312
[7] Nieto, JJ, Rodríguez-López, R: Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations. J. Math. Anal. Appl. 318, 593-610 (2006) · Zbl 1101.34051 · doi:10.1016/j.jmaa.2005.06.014
[8] Yan, JR, Shen, JH: Impulsive stabilization of functional differential equations by Lyapunov-Razumikhin functions. Nonlinear Anal. 37, 245-255 (1999) · Zbl 0951.34049 · doi:10.1016/S0362-546X(98)00045-5
[9] Li, JL, Shen, JH: New comparison results for impulsive functional differential equations. Appl. Math. Lett. 23, 487-493 (2010) · Zbl 1200.34098 · doi:10.1016/j.aml.2009.12.010
[10] Yang, XX, Shen, JH: Nonlinear boundary value problems for first-order impulsive functional differential equations. Appl. Math. Comput. 189, 1943-1952 (2007) · Zbl 1125.65074 · doi:10.1016/j.amc.2006.12.085
[11] Liu, YS: Periodic boundary value problems for first order functional differential equations with impulse. J. Comput. Appl. Math. 223, 27-39 (2009) · Zbl 1162.34050 · doi:10.1016/j.cam.2007.12.015
[12] Liu, YJ: Further results on periodic boundary value problems for nonlinear first-order impulsive functional differential equations. J. Math. Anal. Appl. 327, 435-452 (2007) · Zbl 1119.34062 · doi:10.1016/j.jmaa.2006.01.027
[13] He, ZM, Yu, JS: Periodic boundary value problem for first-order impulsive functional differential equations. J. Math. Anal. Appl. 272, 67-78 (2002) · Zbl 1016.34023 · doi:10.1016/S0022-247X(02)00133-6
[14] Ding, W, Han, MA, Mi, JR: Periodic boundary value problem for the second-order impulsive functional differential equations. Comput. Math. Appl. 50, 491-507 (2005) · Zbl 1095.34042 · doi:10.1016/j.camwa.2005.03.010
[15] Sun, JP, Wang, XQ: Monotone positive solutions for an elastic beam equation with nonlinear boundary conditions. Math. Probl. Eng. 2011, Article ID 609189 (2011) · Zbl 1235.74374
[16] Yao, QL: Positive solutions of nonlinear beam equations with time and space singularities. J. Math. Anal. Appl. 374, 681-692 (2011) · Zbl 1219.34033 · doi:10.1016/j.jmaa.2010.08.056
[17] O’Regan, D: Solvability of some fourth (and higher) order singular boundary value problems. J. Math. Anal. Appl. 161, 78-116 (1991) · Zbl 0795.34018 · doi:10.1016/0022-247X(91)90363-5
[18] Yang, B: Positive solutions for the beam equation under certain boundary conditions. Electron. J. Differ. Equ. 2005, 78 (2005) · Zbl 1075.34025
[19] Zhang, XG: Existence and iteration of monotone positive solutions for an elastic beam equation with a corner. Nonlinear Anal., Real World Appl. 10, 2097-2103 (2009) · Zbl 1163.74478 · doi:10.1016/j.nonrwa.2008.03.017
[20] Gupta, GP: Existence and uniqueness theorems for the bending of an elastic beam equation. Appl. Anal. 26, 289-304 (1988) · Zbl 0611.34015 · doi:10.1080/00036818808839715
[21] Agarwal, RP: On fourth-order boundary value problems arising in beam analysis. Differ. Integral Equ. 2, 91-110 (1989) · Zbl 0715.34032
[22] Bonanno, G, Bella, BD: A boundary value problem for fourth-order elastic beam equations. J. Math. Anal. Appl. 343, 1166-1176 (2008) · Zbl 1145.34005 · doi:10.1016/j.jmaa.2008.01.049
[23] Han, GD, Xu, ZB: Multiple solutions of some nonlinear fourth-order beam equations. Nonlinear Anal. TMA 68, 3646-3656 (2008) · Zbl 1145.34008 · doi:10.1016/j.na.2007.04.007
[24] Zhang, XG, Liu, LS: Positive solutions of fourth-order four-point boundary value problems with p-Laplacian operator. J. Math. Anal. Appl. 336, 1414-1423 (2007) · Zbl 1125.34018 · doi:10.1016/j.jmaa.2007.03.015
[25] Feng, MQ: Multiple positive solutions of four-order impulsive differential equations with integral boundary conditions and one-dimensional p-Laplacian. Bound. Value Probl. 2011, Article ID 654871 (2011) · Zbl 1206.47096 · doi:10.1186/1687-2770-2011-720702
[26] Cabada, A, Tersian, S: Existence and multiplicity of solutions to boundary value problems for fourth-order impulsive differential equations. Bound. Value Probl. 2014, 105 (2014) · Zbl 1319.34044 · doi:10.1186/1687-2770-2014-105
[27] Afrouzi, GA, Hadjian, A, Radulescu, VD: Variational approach to fourth-order impulsive differential equations with two control parameters. Results Math. 65, 371-384 (2014) · Zbl 1294.34031 · doi:10.1007/s00025-013-0351-5
[28] Sun, JT, Chen, HB, Yang, L: Variational methods to fourth-order impulsive differential equations. J. Appl. Math. Comput. 35, 323-340 (2011) · Zbl 1218.34029 · doi:10.1007/s12190-009-0359-x
[29] Xie, JL, Luo, ZG: Solutions to a boundary value problem of a fourth-order impulsive differential equation. Bound. Value Probl. 2013, 154 (2013) · Zbl 1297.34038 · doi:10.1186/1687-2770-2013-154
[30] Zhang, XM, Feng, MQ: Positive solutions for classes of multi-parameter fourth-order impulsive differential equations with one-dimensional singular p-Laplacian. Bound. Value Probl. 2014, 112 (2014) · Zbl 1307.34058 · doi:10.1186/1687-2770-2014-112
[31] Jankowski, T: Positive solutions of one-dimensional p-Laplacian boundary value problems for fourth-order differential equations with deviating arguments. J. Optim. Theory Appl. 149, 47-60 (2011) · Zbl 1221.34063 · doi:10.1007/s10957-010-9774-2
[32] Wang, HY: On the number of positive solutions of nonlinear systems. J. Math. Anal. Appl. 281, 287-306 (2003) · Zbl 1036.34032 · doi:10.1016/S0022-247X(03)00100-8
[33] Guo, DJ, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988) · Zbl 0661.47045
[34] Jankowski, T: Positive solutions to third-order impulsive Sturm-Liouville boundary value problems with deviating arguments and one-dimensional p-Laplacian. Dyn. Syst. Appl. 20, 575-586 (2011) · Zbl 1253.34057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.