Multiple positive solutions of fourth-order impulsive differential equations with integral boundary conditions and one-dimensional \(p\)-Laplacian. (English) Zbl 1206.47096
Summary: By using the fixed point theory for a completely continuous operator, this paper investigates the existence of positive solutions for a class of fourth-order impulsive boundary value problems with integral boundary conditions and one-dimensional \(p\)-Laplacian. Moreover, we offer some interesting discussion of the associated boundary value problems. Upper and lower bounds for these positive solutions are also given, so our work is new.
MSC:
| 47N20 | Applications of operator theory to differential and integral equations |
| 47H10 | Fixed-point theorems |
| 34B37 | Boundary value problems with impulses for ordinary differential equations |
Keywords:
fixed point theory; existence of positive solutions; fourth-order impulsive boundary value problemsReferences:
| [1] | Lakshmikantham V, Baĭnov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Singapore; 1989:xii+273. · Zbl 0719.34002 · doi:10.1142/0906 |
| [2] | Benchohra M, Henderson J, Ntouyas S: Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications. Volume 2. Hindawi Publishing Corporation, New York, NY, USA; 2006:xiv+366. · Zbl 1130.34003 · doi:10.1155/9789775945501 |
| [3] | Ahmad B, Sivasundaram S: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Analysis: Hybrid Systems 2010, 4(1):134-141. 10.1016/j.nahs.2009.09.002 · Zbl 1187.34038 · doi:10.1016/j.nahs.2009.09.002 |
| [4] | Baĭnov DD, Simeonov PS: Systems with Impulse Effect. Ellis Horwood, Chichester, UK; 1989:255. · Zbl 0683.34032 |
| [5] | Samoĭlenko AM, Perestyuk NA: Impulsive Differential Equations. Volume 14. World Scientific, Singapore; 1995:x+462. · Zbl 0837.34003 |
| [6] | Yan J: Existence of positive periodic solutions of impulsive functional differential equations with two parameters. Journal of Mathematical Analysis and Applications 2007, 327(2):854-868. 10.1016/j.jmaa.2006.04.018 · Zbl 1114.34052 · doi:10.1016/j.jmaa.2006.04.018 |
| [7] | Feng M, Xie D: Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations. Journal of Computational and Applied Mathematics 2009, 223(1):438-448. 10.1016/j.cam.2008.01.024 · Zbl 1159.34022 · doi:10.1016/j.cam.2008.01.024 |
| [8] | Nieto JJ: Basic theory for nonresonance impulsive periodic problems of first order. Journal of Mathematical Analysis and Applications 1997, 205(2):423-433. 10.1006/jmaa.1997.5207 · Zbl 0870.34009 · doi:10.1006/jmaa.1997.5207 |
| [9] | Nieto JJ: Impulsive resonance periodic problems of first order. Applied Mathematics Letters 2002, 15(4):489-493. 10.1016/S0893-9659(01)00163-X · Zbl 1022.34025 · doi:10.1016/S0893-9659(01)00163-X |
| [10] | Guo D: Multiple positive solutions for first order nonlinear impulsive integro-differential equations in a Banach space. Applied Mathematics and Computation 2003, 143(2-3):233-249. 10.1016/S0096-3003(02)00356-9 · Zbl 1030.45009 · doi:10.1016/S0096-3003(02)00356-9 |
| [11] | Liu X, Guo D: Periodic boundary value problems for a class of second-order impulsive integro-differential equations in Banach spaces. Journal of Mathematical Analysis and Applications 1997, 216(1):284-302. 10.1006/jmaa.1997.5688 · Zbl 0889.45016 · doi:10.1006/jmaa.1997.5688 |
| [12] | Agarwal RP, O’Regan D: Multiple nonnegative solutions for second order impulsive differential equations. Applied Mathematics and Computation 2000, 114(1):51-59. 10.1016/S0096-3003(99)00074-0 · Zbl 1047.34008 · doi:10.1016/S0096-3003(99)00074-0 |
| [13] | Liu B, Yu J:Existence of solution of [InlineEquation not available: see fulltext.]-point boundary value problems of second-order differential systems with impulses. Applied Mathematics and Computation 2002, 125(2-3):155-175. 10.1016/S0096-3003(00)00110-7 · Zbl 1032.34024 · doi:10.1016/S0096-3003(00)00110-7 |
| [14] | Agarwal RP, Franco D, O’Regan D: Singular boundary value problems for first and second order impulsive differential equations. Aequationes Mathematicae 2005, 69(1-2):83-96. 10.1007/s00010-004-2735-9 · Zbl 1073.34025 · doi:10.1007/s00010-004-2735-9 |
| [15] | Lin X, Jiang D: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. Journal of Mathematical Analysis and Applications 2006, 321(2):501-514. 10.1016/j.jmaa.2005.07.076 · Zbl 1103.34015 · doi:10.1016/j.jmaa.2005.07.076 |
| [16] | Jankowski T: Positive solutions of three-point boundary value problems for second order impulsive differential equations with advanced arguments. Applied Mathematics and Computation 2008, 197(1):179-189. 10.1016/j.amc.2007.07.081 · Zbl 1145.34355 · doi:10.1016/j.amc.2007.07.081 |
| [17] | Jankowski T: Positive solutions to second order four-point boundary value problems for impulsive differential equations. Applied Mathematics and Computation 2008, 202(2):550-561. 10.1016/j.amc.2008.02.040 · Zbl 1255.34025 · doi:10.1016/j.amc.2008.02.040 |
| [18] | Feng M, Du B, Ge W:Impulsive boundary value problems with integral boundary conditions and one-dimensional [InlineEquation not available: see fulltext.]-Laplacian. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(9):3119-3126. 10.1016/j.na.2008.04.015 · Zbl 1169.34022 · doi:10.1016/j.na.2008.04.015 |
| [19] | Gallardo JM: Second-order differential operators with integral boundary conditions and generation of analytic semigroups. Rocky Mountain Journal of Mathematics 2000, 30(4):1265-1291. 10.1216/rmjm/1021477351 · Zbl 0984.34014 · doi:10.1216/rmjm/1021477351 |
| [20] | Karakostas GL, Tsamatos PCh: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems. Electronic Journal of Differential Equations 2002, 2002(30):1-17. · Zbl 0998.45004 |
| [21] | Lomtatidze A, Malaguti L: On a nonlocal boundary value problem for second order nonlinear singular differential equations. Georgian Mathematical Journal 2000, 7(1):133-154. · Zbl 0967.34011 |
| [22] | Corduneanu C: Integral Equations and Applications. Cambridge University Press, Cambridge, UK; 1991:x+366. · Zbl 0714.45002 · doi:10.1017/CBO9780511569395 |
| [23] | Agarwal RP, O’Regan D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:x+341. · Zbl 0988.34002 · doi:10.1007/978-94-010-0718-4 |
| [24] | Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems involving integral conditions. Nonlinear Differential Equations and Applications 2008, 15(1-2):45-67. 10.1007/s00030-007-4067-7 · Zbl 1148.34021 · doi:10.1007/s00030-007-4067-7 |
| [25] | Webb JRL, Infante G: Non-local boundary value problems of arbitrary order. Journal of the London Mathematical Society 2009, 79(1):238-258. · Zbl 1165.34010 · doi:10.1112/jlms/jdn066 |
| [26] | Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems: a unified approach. Journal of the London Mathematical Society 2006, 74(3):673-693. 10.1112/S0024610706023179 · Zbl 1115.34028 · doi:10.1112/S0024610706023179 |
| [27] | Ahmad, B.; Nieto, JJ, The monotone iterative technique for three-point second-order integrodifferential boundary value problems with [InlineEquation not available: see fulltext.]-Laplacian, No. 2007, 9 (2007) |
| [28] | Ahmad, B.; Nieto, JJ, Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions, No. 2009, 11 (2009) · Zbl 1167.45003 |
| [29] | Ahmad B, Alsaedi A, Alghamdi BS: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2008, 9(4):1727-1740. · Zbl 1154.34311 · doi:10.1016/j.nonrwa.2007.05.005 |
| [30] | Feng M, Ji D, Ge W: Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces. Journal of Computational and Applied Mathematics 2008, 222(2):351-363. 10.1016/j.cam.2007.11.003 · Zbl 1158.34336 · doi:10.1016/j.cam.2007.11.003 |
| [31] | Zhang X, Feng M, Ge W:Multiple positive solutions for a class of [InlineEquation not available: see fulltext.]-point boundary value problems. Applied Mathematics Letters 2009, 22(1):12-18. 10.1016/j.aml.2007.10.019 · Zbl 1163.34324 · doi:10.1016/j.aml.2007.10.019 |
| [32] | Feng M, Ge W:Positive solutions for a class of [InlineEquation not available: see fulltext.]-point singular boundary value problems. Mathematical and Computer Modelling 2007, 46(3-4):375-383. 10.1016/j.mcm.2006.11.009 · Zbl 1142.34012 · doi:10.1016/j.mcm.2006.11.009 |
| [33] | Zhang X, Feng M, Ge W:Symmetric positive solutions for [InlineEquation not available: see fulltext.]-Laplacian fourth-order differential equations with integral boundary conditions. Journal of Computational and Applied Mathematics 2008, 222(2):561-573. 10.1016/j.cam.2007.12.002 · Zbl 1158.34012 · doi:10.1016/j.cam.2007.12.002 |
| [34] | Zhang X, Feng M, Ge W: Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(10):3310-3321. 10.1016/j.na.2007.09.020 · Zbl 1159.34020 · doi:10.1016/j.na.2007.09.020 |
| [35] | Yang Z: Positive solutions of a second-order integral boundary value problem. Journal of Mathematical Analysis and Applications 2006, 321(2):751-765. 10.1016/j.jmaa.2005.09.002 · Zbl 1106.34014 · doi:10.1016/j.jmaa.2005.09.002 |
| [36] | Ma R:Positive solutions for multipoint boundary value problem with a one-dimensional [InlineEquation not available: see fulltext.]-Laplacian. Computational & Applied Mathematics 2001, 42: 755-765. · Zbl 0987.34018 · doi:10.1016/S0898-1221(01)00195-X |
| [37] | Bai Z, Huang B, Ge W:The iterative solutions for some fourth-order [InlineEquation not available: see fulltext.]-Laplace equation boundary value problems. Applied Mathematics Letters 2006, 19(1):8-14. 10.1016/j.aml.2004.10.010 · Zbl 1092.34510 · doi:10.1016/j.aml.2004.10.010 |
| [38] | Liu B, Liu L, Wu Y: Positive solutions for singular second order three-point boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2007, 66(12):2756-2766. 10.1016/j.na.2006.04.005 · Zbl 1117.34021 · doi:10.1016/j.na.2006.04.005 |
| [39] | Zhang X, Liu L:A necessary and sufficient condition for positive solutions for fourth-order multi-point boundary value problems with [InlineEquation not available: see fulltext.]-Laplacian. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(10):3127-3137. 10.1016/j.na.2007.03.006 · Zbl 1143.34016 · doi:10.1016/j.na.2007.03.006 |
| [40] | Aftabizadeh AR: Existence and uniqueness theorems for fourth-order boundary value problems. Journal of Mathematical Analysis and Applications 1986, 116(2):415-426. 10.1016/S0022-247X(86)80006-3 · Zbl 0634.34009 · doi:10.1016/S0022-247X(86)80006-3 |
| [41] | Bai Z, Wang H: On positive solutions of some nonlinear fourth-order beam equations. Journal of Mathematical Analysis and Applications 2002, 270(2):357-368. 10.1016/S0022-247X(02)00071-9 · Zbl 1006.34023 · doi:10.1016/S0022-247X(02)00071-9 |
| [42] | Ma R, Wang H: On the existence of positive solutions of fourth-order ordinary differential equations. Applicable Analysis 1995, 59(1-4):225-231. · Zbl 0841.34019 |
| [43] | Liu L, Zhang X, Wu Y:Positive solutions of fourth order four-point boundary value problems with [InlineEquation not available: see fulltext.]-Laplacian operator. Journal of Mathematical Analysis and Applications 2007, 326(2):1212-1224. 10.1016/j.jmaa.2006.03.029 · Zbl 1113.34022 · doi:10.1016/j.jmaa.2006.03.029 |
| [44] | Kang P, Wei Z, Xu J: Positive solutions to fourth-order singular boundary value problems with integral boundary conditions in abstract spaces. Applied Mathematics and Computation 2008, 206(1):245-256. 10.1016/j.amc.2008.09.010 · Zbl 1169.34043 · doi:10.1016/j.amc.2008.09.010 |
| [45] | Webb JRL, Infante G, Franco D: Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions. Proceedings of the Royal Society of Edinburgh 2008, 138(2):427-446. · Zbl 1167.34004 |
| [46] | Ma H: Symmetric positive solutions for nonlocal boundary value problems of fourth order. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(3):645-651. 10.1016/j.na.2006.11.026 · Zbl 1135.34310 · doi:10.1016/j.na.2006.11.026 |
| [47] | Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275. · Zbl 0661.47045 |
| [48] | Eloe PW, Ahmad B:Positive solutions of a nonlinear [InlineEquation not available: see fulltext.]th order boundary value problem with nonlocal conditions. Applied Mathematics Letters 2005, 18(5):521-527. 10.1016/j.aml.2004.05.009 · Zbl 1074.34022 · doi:10.1016/j.aml.2004.05.009 |
| [49] | Hao, X.; Liu, L.; Wu, Y., Positive solutions for nonlinear [InlineEquation not available: see fulltext.]th-order singular nonlocal boundary value problems, No. 2007, 10 (2007) |
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