## On the fractional order $$m$$-point boundary value problem in reflexive Banach spaces and weak topologies.(English)Zbl 1176.34070

The author studies the existence of at least one pseudo solution of an $$m$$-point boundary value problem of Riemann-Liouville fractional-order differential equation of order $$a, n-1< a < n$$, $$n>2$$ or $$n = 2$$.

### MSC:

 34G20 Nonlinear differential equations in abstract spaces 26A33 Fractional derivatives and integrals 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
Full Text:

### References:

 [1] Arino, O.; Gautier, S.; Penot, T.P., A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations, Funkcial. ekvac., 27, 273-279, (1984) · Zbl 0599.34008 [2] Bai, Z.; Lü, H., Positive solutions for boundary value problem of nonlinear fractional differential equation, J. math. anal. appl., 311, 495-505, (2005) · Zbl 1079.34048 [3] Ball, J.M., Weak continuity properties of mapping and semi-groups, Proc. roy. soc. Edinburgh sect., A 72, 275-280, (1973-1974) [4] Bugajewski, D.; Szulfa, S., Kneser’s theorem for weak solutions of the Darboux problem in a Banach space, Nonlinear anal., 20, 2, 169-173, (1993) · Zbl 0776.34048 [5] Chandra, J.; Lakshmikantham, V.; Mitchell, A.R., Existence of solutions of boundary value problems for nonlinear second order systems in Banach spaces, Nonlinear anal., 2, 157-168, (1978) · Zbl 0385.34035 [6] Changci, P.; Wei, D.; Zhongli, W., Greens function and positive solutions of $$n$$-th order $$m$$-point boundary value problem, Appl. math. comput., 182, 1231-1239, (2006) · Zbl 1111.34024 [7] Chen, S.H.; Hu, T.; Chen, L., Existence results for $$n$$-point boundary value problem of second order ordinary differential equations, J. comput. appl. math., 180, 425-432, (2005) · Zbl 1069.34011 [8] Cichoń, M., Weak solutions of differential equations in Banach spaces, Discuss. math. differ. incl., 15, 5-14, (1995) · Zbl 0829.34051 [9] Diestel, J.; Uhl, J.J., () [10] M. Feng, W. Ge, Positive solutions for a class of boundary value problem with integral boundary conditions in Banach spaces, J. Comput. Appl. Math. (2007), doi:10.1016/j.cam.2007.11.003 [11] Geitz, R.F., Pettis integration, Proc. amer. math. soc., 82, 81-86, (1981) · Zbl 0506.28007 [12] Gupta, C.P., A generalized multi-point boundary value problem for second kind for a sturmliouville operator, Appl. math. comput., 89, 133-146, (1998) [13] Lian, W.; Wong, F.; Yeh, C., On the existence of positive solutions of nonlinear second order differential equations, Proc. amer. math. soc., 124, 1117-1126, (1996) · Zbl 0857.34036 [14] Liu, B., Positive solutions of a nonlinear four-point boundary value problems in Banach spaces, J. math. anal. appl., 305, 253-276, (2005) · Zbl 1073.34075 [15] Ma, R., Existence of solutions of nonlinear $$m$$-point boundary value problem, J. math. anal. appl., 256, 556-567, (2001) · Zbl 0988.34009 [16] Miller, K.S.; Ross, B., An introduction to the fractional calculus and differential equations, (1993), John Wiley New York · Zbl 0789.26002 [17] Mönch, H., Boundary value problems for nonlinear ordinary differential equation of second order in Banach spaces, Nonlinear anal., 4, 985-999, (1980) · Zbl 0462.34041 [18] Nakhushev, A.M., The sturm – liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms, Dokl. akad. nauk SSSR, 234, 308-311, (1977) · Zbl 0376.34015 [19] O’Regan, D., Weak solutions of ordinary differential equations in Banach spaces, Appl. math. lett., 12, 101-105, (1999) · Zbl 0933.34068 [20] Pettis, B.J., On integration in vector spaces, Trans. amer. math. soc., 44, 277-304, (1938) · Zbl 0019.41603 [21] Salem, H.A.H.; El-Sayed, A.M.A.; Moustafa, O.L., A note on the fractional calculus in Banach spaces, Stud. sci. math. hung., 42, 2, 115-130, (2005) · Zbl 1086.45004 [22] Samko, S.; Kilbas, A.; Marichev, O.L., Fractional integrals and drivatives, (1993), Gordon and Breach Science Publisher [23] Satco, B., Second order three boundary value problem in Banach spaces via Henstock and henstock – kurzweil – pettis integral, J. math. anal. appl., 332, 919-933, (2007) · Zbl 1127.34033 [24] Webb, T.R.L., Positive solutions of some three-point value problem via fixed point index theory, Nonlinear anal., 47, 4319-4332, (2001) · Zbl 1042.34527 [25] Szulfa, S., Boundary value problems for nonlinear ordinary differential equation of second order in Banach spaces, Nonlinear anal., 4, 1481-1487, (1984) · Zbl 0561.34048 [26] Zhao, Y.; Chena, H., Existence of multiple positive solutions for $$m$$-point boundary value problems in Banach spaces, J. comput. appl. math., 215, 1, 79-90, (2008) · Zbl 1147.34019
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.