Existence of solutions for \(n\)-dimensional fractional order BVP with \(\infty\)-point boundary conditions via the concept of measure of noncompactness. (English) Zbl 1487.34046
Summary: In this paper, using the concept of measure of noncompactness we present some results on the existence of \(n\)-fixed points for a class of operators. Also as an application, we derive the sufficient conditions for the existence of solutions for \(n\)-dimensional fractional order functional boundary value problems.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47H10 | Fixed-point theorems |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 45G15 | Systems of nonlinear integral equations |
Keywords:
\(n\)-fixed point; Caputo fractional derivative; fixed point theorem; functional differential equation; measure of noncompactness; \(n\)-fixed pointReferences:
| [1] | R. P. Agarwal, M. Meehan, and D. O’Regan, Fixed point theory and applications, Cambridge University Press, Cambridge, 2001. · Zbl 0960.54027 |
| [2] | A. Aghajani, Y. Jalilian, and J. J. Trujillo, On the existence of solutions of fractional integro-difierential equations, Fract. Calc. Appl. Anal. 15(1), (2012), 44-69. · Zbl 1279.45008 |
| [3] | A. Aghajani, and N. Sabzali, Existence of coupled fixed points via measure of noncompactness and applications, J. Nonl. Conv. Anal. 15(5), (2014), 941-952. · Zbl 1418.47003 |
| [4] | Z. Bai, and H. Lu, Positive solutions for a boundary value problem of nonlinear fractional differential equations, J. Math. Anal. Appl. 311(2), (2005) 495-505. · Zbl 1079.34048 |
| [5] | J. Banas, On measures of noncompactness in Banach spaces, Comment. Math. Univ. Carolinae 21 (1980), 131-143. · Zbl 0438.47051 |
| [6] | J. Banas, and K. Goebel, Measure of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 60, Marcel Dekker, New York, 1980. · Zbl 0441.47056 |
| [7] | M. Benchohra, J. R. Graef, and F. Z. Mostafai, Weak solutions for nonlinear fractional difierential equations on reflexive Banach spaces, Electron. J. Qual. Theory. 54 (2010), 1-10. · Zbl 1206.26006 |
| [8] | M. Borcut, Tripled fixed point theorems for monotone mappings in partially ordered metric spaces, Carpathian J. of Math. 28(2), (2012), 215-222. · Zbl 1289.54116 |
| [9] | P. Borisut, P. Kumam, I. Ahmed, and K. Sitthithakerngkiet, Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems, Symmetry, 11(6), (2019), 829. |
| [10] | L. Cadariu, L. Gavruta, and P. Gavruta, Weighted space method for the stability of some nonlinear equations. Appl Anal Discrete Math. 6(1), (2012), 126-139. · Zbl 1289.39054 |
| [11] | G. Darbo, Punti uniti in trasformazioni a codominio non compatto. Rend. Sem. Mat. Un. Padova. 24 (1955), 84-92. · Zbl 0064.35704 |
| [12] | M. Feng, X. Zhang, and W. Ge, New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions, Bound. Value Probl. 2011, Art. ID 720702, 20 pp. · Zbl 1214.34005 |
| [13] | A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies 204, Elsevier Science B. V., Amsterdam 2006. · Zbl 1092.45003 |
| [14] | K. Kuratowski, Sur les espaces complets, Fund. Math. 15(1930) 301-309. · JFM 56.1124.04 |
| [15] | H. Lee, and S. Kim, Multivariate coupled fixed point theorems on ordered partial metric spaces, J. Korean Math. Soc. 51 (2014), 1189-1207. · Zbl 1306.54047 |
| [16] | K. Li, J. Peng, and J. Gao, Nonlocal fractional semilinear Differential equations in separable Banach spaces, Electron. J. Differential Equations, 2013(7) (2013), 1-7. · Zbl 1292.26024 |
| [17] | J. Liang, Z. Liu, and X.Wang, Solvability for a couple system of nonlinear fractional differential equations in a Banach space, Fract. Calc. Appl. Anal. 16(1) (2013), 51-63. · Zbl 1312.34020 |
| [18] | I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999. · Zbl 0924.34008 |
| [19] | A. Saadi, and M. Benbachir, Positive solutions for three-point nonlinear fractional boundary value problems, E. J. Qualitative Theory of Diff. Equ. 3 (2011), 1-19. · Zbl 1206.26008 |
| [20] | J. Sabatier, O. P. Agarwal, and J. A. T. Machado, Advances in fractional calculus: Theoretical developments and applications in physics and engineering, Springer, Dordrecht, 2007. · Zbl 1116.00014 |
| [21] | F. J. Torres, Existence of a positive solution for a boundary value problem of a nonlinear fractional differential equation, BIMS 39(2), (2013), 307-323. · Zbl 1306.34016 |
| [22] | J.Wang, Y. Zhou, and M. Feckan, Abstract Cauchy problem for fractional differential equations, Nonlinear Dynam. DOI 10.1007/s11071-012-0452-9. |
| [23] | S. Zhang, Existence results of positive solutions to fractional differential equation with integral boundary conditions, Math. Bohem. 135(2), (2010), 299-317. · Zbl 1224.26025 |
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