Existence of Positive Solutions For Nonlinear Fractional Problem On The Half Line
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Abstract
This paper deals with the dierential equations of fractional order onthe half-line. By Leggett-Williams theorem, we present recent results for theexistence of positive solutions for a Caputo fractional problem. An illustrativeexample is also presented.
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Belarbi, S.
(2015).
Existence of Positive Solutions For Nonlinear Fractional Problem On The Half Line.
Acta Mathematica Universitatis Comenianae, 84(1), 1-12.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/15/120
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References
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Math. Anal. 18(2), (2011), 235-244.
[2] B. Ahmad: Existence of solutions for irregular boundary value problems of nonlinear fractional dierential equations. Appl. Math. Lett. 23(4), (2010), 390-394.
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[6] Z. Bai, H. Lu: Positive solutions for boundary value problem of nonlinear fractional dierential equation, J. Math. Anal. Appl. 311(2), (2005), 495-505.
[7] J. Caballero, J. Harjani, K. Sadarangani: Positive solutions for a class of singular fractional boundary value problems. Comput. Math. Appl. 62(3), (2011), 1325-1332.
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[11] ER. Kaufmann, E. Mboumi: Positive solutions of a boundary value problem for a nonlinear fractional dierential equation. Electron J Qual Theory Di Equ. 3, (2008), 1-11.
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[13] R. W. Leggett and L. R. Williams: Multiple positive xed points of nonlinear operators on ordered Banach spaces. Inidana Univ. Math. J. 28, (1979), 673-688.
[14] H. Lian, H. Pang and W. Ge: Solvability for second-order three-point boundary value problems at resonance on a half-line. J. Math. Anal. Appl. 337, (2008), 1171-1181.
[15] S. Liang, J. Zhang: Positive solutions for boundary value problems of nonlinear fractional dierential equation, Nonlinear Analysis, 71, (2009), 5545-5550.
[16] X. Liu, M. Jia: Multiple solutions of nonlocal boundary-value problems for fractional dierential equations on the half-line. Electron. J. Qual. Theory Dier. Equ. 56, (2011).
[17] L. Liu, Z. Wang, Y. Wu: Multiple positive solutions of the singular boundary value problems for second-order dierential equations on the the halfline. Nonlinear Anal. 71, (2009), 2564-2575.
[18] K. Miller, B. Ross: An Introduction to the Fractional Calculus and Fractional Dierential Equations. Wiley, New York, (1993).
[19] D. O'Regan, M. Zima: Leggett-Williams norm-type theorems for coincidences. Arch. Math. (Basel) 87, (2006), 233-244.
[20] I. Podlubny: Fractional Dierential Equations, Mathematics in Science and Engineering. Academic Press New York, (1999).
[21] J.H. Wang, H.J. Xiang, Z.G. Liu: Positive Solutions for Three- point Boundary Values Problems of Nonlinear Fractional Dierential Equations with P-Laplacian. Far East Journal of Applied Mathematics, 37, (2009), 33-47.
[22] J. Wang: The existence of of positive solutions for the one-dimensional p-Laplacian. Proc. Am. Math. Soc. 125, (1997), 2275-2283.
[23] X. G. Zhang, L.S. Liu: Positive Solutions of Fourth-order Four-point Boundary Value Problems with P-Laplacian Operator. J. Math. Anal. Appl. 336, (2007), 1414-1423.
Math. Anal. 18(2), (2011), 235-244.
[2] B. Ahmad: Existence of solutions for irregular boundary value problems of nonlinear fractional dierential equations. Appl. Math. Lett. 23(4), (2010), 390-394.
[3] B. Ahmad , J. Nieto: Riemann-Liouville fractional integro-dierential equations with fractional nonlocal integral boundary conditions, Bound. Value Probl. 36, (2011), 9 pp.
[4] R.I. Avery: A generalization of the Leggett-Williams xed point theorem, MSR Hot-Line. 6, (1999), 9-14.
[5] Z.b. Bai, W.G. Ge: Existence of Three Positive Solutions for Second order Boundary Value Problem. Compt. Math. Appl., 48(2004), 699-707.
[6] Z. Bai, H. Lu: Positive solutions for boundary value problem of nonlinear fractional dierential equation, J. Math. Anal. Appl. 311(2), (2005), 495-505.
[7] J. Caballero, J. Harjani, K. Sadarangani: Positive solutions for a class of singular fractional boundary value problems. Comput. Math. Appl. 62(3), (2011), 1325-1332.
[8] D. Guo and V. Lakshmikantham: Nonlinear problems in abstract cones, Academic Press, San Diego, (1988).
[9] M.Houas, Z. Dahmani, M. Benbachir: New Results for a Boundary Value Problem for Dierential Equations of Arbitrary Order. IJMMS. (2013), 7(2): 195-211.
[10] M.Houas, Z. Dahmani: New Results for a Caputo Boundary Value Problem. American Journal of Computational and Applied Mathematics (2013), 3(3): 143-161.
[11] ER. Kaufmann, E. Mboumi: Positive solutions of a boundary value problem for a nonlinear fractional dierential equation. Electron J Qual Theory Di Equ. 3, (2008), 1-11.
[12] A. Kilbas, H. Srivastava, J.Trujillo: Theory and Applications of Fractional Dierential Equations. North-Holland Mathematics Studies, 204, Elsevier, Amsterdam, (2006).
[13] R. W. Leggett and L. R. Williams: Multiple positive xed points of nonlinear operators on ordered Banach spaces. Inidana Univ. Math. J. 28, (1979), 673-688.
[14] H. Lian, H. Pang and W. Ge: Solvability for second-order three-point boundary value problems at resonance on a half-line. J. Math. Anal. Appl. 337, (2008), 1171-1181.
[15] S. Liang, J. Zhang: Positive solutions for boundary value problems of nonlinear fractional dierential equation, Nonlinear Analysis, 71, (2009), 5545-5550.
[16] X. Liu, M. Jia: Multiple solutions of nonlocal boundary-value problems for fractional dierential equations on the half-line. Electron. J. Qual. Theory Dier. Equ. 56, (2011).
[17] L. Liu, Z. Wang, Y. Wu: Multiple positive solutions of the singular boundary value problems for second-order dierential equations on the the halfline. Nonlinear Anal. 71, (2009), 2564-2575.
[18] K. Miller, B. Ross: An Introduction to the Fractional Calculus and Fractional Dierential Equations. Wiley, New York, (1993).
[19] D. O'Regan, M. Zima: Leggett-Williams norm-type theorems for coincidences. Arch. Math. (Basel) 87, (2006), 233-244.
[20] I. Podlubny: Fractional Dierential Equations, Mathematics in Science and Engineering. Academic Press New York, (1999).
[21] J.H. Wang, H.J. Xiang, Z.G. Liu: Positive Solutions for Three- point Boundary Values Problems of Nonlinear Fractional Dierential Equations with P-Laplacian. Far East Journal of Applied Mathematics, 37, (2009), 33-47.
[22] J. Wang: The existence of of positive solutions for the one-dimensional p-Laplacian. Proc. Am. Math. Soc. 125, (1997), 2275-2283.
[23] X. G. Zhang, L.S. Liu: Positive Solutions of Fourth-order Four-point Boundary Value Problems with P-Laplacian Operator. J. Math. Anal. Appl. 336, (2007), 1414-1423.