# American Institute of Mathematical Sciences

2015, 2015(special): 248-257. doi: 10.3934/proc.2015.0248

## Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays

 1 Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand, Thailand

Received  August 2014 Revised  May 2015 Published  November 2015

We investigate the existence of fixed points for a very general class of cyclic implicit contractive set-valued operators. We also point out that this class contains an important case of ordered contractions. As an application, we show the solvability of delayed fractional integral inclusion problems.
Citation: Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248
##### References:
 [1] A. El-Sayed and A. Ibrahim, Multivalued fractional differential equations,, Applied Mathematics and Computation, 68 (1995). [2] A.-G. Ibrahim and A. M. El-Sayed, Definite integral of fractional order for set-valued functions.,, J. Fractional Calc., 11 (1997), 81. [3] A. M. El-Sayed and A.-G. Ibrahim, Set-valued integral equations of fractional-orders,, Applied Mathematics and Computation, 118 (2001). [4] N. Ahmed and K. Teo, Optimal control of distributed parameter systems., North Holland, (1981). [5] N. Ahmed and X. Xiang, Existence of solutions for a class of nonlinear evolution equations with nonmonotone perturbations,, Nonlinear Analysis: Theory, 22 (1994). [6] Y. Ling and S. Ding, A class of analytic functions defined by fractional derivation.,, J. Math. Anal. Appl., 186 (1994), 504. [7] D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation.,, J. Math. Anal. Appl., 204 (1996), 609. [8] A. Kilbas and J. Trujillo, Differential equations of fractional order: Methods, results and problems. I.,, Appl. Anal., 78 (2001), 1. [9] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales.,, Fundamenta math., 3 (1922), 133. [10] W. Kirk, P. Srinivasan, and P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions.,, Fixed Point Theory, 4 (2003), 79. [11] V. Popa, Fixed point theorems for mappings in d-complete topological spaces.,, Math. Morav., 6 (2002), 87. [12] I. Altun and H. Simsek, Some fixed point theorems on ordered metric spaces and application,, Fixed Point Theory and Applications, 2010 (2010). [13] H. K. Nashine, Z. Kadelburg, and P. Kumam, Implicit-relation-type cyclic contractive mappings and applications to integral equations,, Abstract and Applied Analysis, 2012 (2012).

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##### References:
 [1] A. El-Sayed and A. Ibrahim, Multivalued fractional differential equations,, Applied Mathematics and Computation, 68 (1995). [2] A.-G. Ibrahim and A. M. El-Sayed, Definite integral of fractional order for set-valued functions.,, J. Fractional Calc., 11 (1997), 81. [3] A. M. El-Sayed and A.-G. Ibrahim, Set-valued integral equations of fractional-orders,, Applied Mathematics and Computation, 118 (2001). [4] N. Ahmed and K. Teo, Optimal control of distributed parameter systems., North Holland, (1981). [5] N. Ahmed and X. Xiang, Existence of solutions for a class of nonlinear evolution equations with nonmonotone perturbations,, Nonlinear Analysis: Theory, 22 (1994). [6] Y. Ling and S. Ding, A class of analytic functions defined by fractional derivation.,, J. Math. Anal. Appl., 186 (1994), 504. [7] D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation.,, J. Math. Anal. Appl., 204 (1996), 609. [8] A. Kilbas and J. Trujillo, Differential equations of fractional order: Methods, results and problems. I.,, Appl. Anal., 78 (2001), 1. [9] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales.,, Fundamenta math., 3 (1922), 133. [10] W. Kirk, P. Srinivasan, and P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions.,, Fixed Point Theory, 4 (2003), 79. [11] V. Popa, Fixed point theorems for mappings in d-complete topological spaces.,, Math. Morav., 6 (2002), 87. [12] I. Altun and H. Simsek, Some fixed point theorems on ordered metric spaces and application,, Fixed Point Theory and Applications, 2010 (2010). [13] H. K. Nashine, Z. Kadelburg, and P. Kumam, Implicit-relation-type cyclic contractive mappings and applications to integral equations,, Abstract and Applied Analysis, 2012 (2012).
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