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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

Parin Chaipunya 1, and  Poom Kumam 1,

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.
Keywords: Cyclic operator, fixed point, fractional integral..
Mathematics Subject Classification: Primary: 47H09, 47H10; Secondary: 34A0.
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, 1, (1995), 15 - 25. Google Scholar

[2]

A.-G. Ibrahim and A. M. El-Sayed, Definite integral of fractional order for set-valued functions., J. Fractional Calc., 11, (1997), 81-87. Google Scholar

[3]

A. M. El-Sayed and A.-G. Ibrahim, Set-valued integral equations of fractional-orders, Applied Mathematics and Computation, 118, 1, (2001), 113 - 121. Google Scholar

[4]

N. Ahmed and K. Teo, Optimal control of distributed parameter systems. North Holland, 1981. Google Scholar

[5]

N. Ahmed and X. Xiang, Existence of solutions for a class of nonlinear evolution equations with nonmonotone perturbations, Nonlinear Analysis: Theory, Methods & Applications, 22, 1, (1994), 81 - 89. Google Scholar

[6]

Y. Ling and S. Ding, A class of analytic functions defined by fractional derivation., J. Math. Anal. Appl., 186, 2, (1994), 504-513. Google Scholar

[7]

D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation., J. Math. Anal. Appl., 204, 2 (1996), 609-625. Google Scholar

[8]

A. Kilbas and J. Trujillo, Differential equations of fractional order: Methods, results and problems. I., Appl. Anal., 78, 1-2 (2001), 153-192. Google Scholar

[9]

S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales., Fundamenta math., 3 (1922), 133-181. Google Scholar

[10]

W. Kirk, P. Srinivasan, and P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions., Fixed Point Theory, 4, 1 (2003), 79-89. Google Scholar

[11]

V. Popa, Fixed point theorems for mappings in d-complete topological spaces., Math. Morav., 6, (2002), 87-92. Google Scholar

[12]

I. Altun and H. Simsek, Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory and Applications, 2010, 1, 2010, 621469. Google Scholar

[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), 15. Google Scholar

show all references

References:
[1]

A. El-Sayed and A. Ibrahim, Multivalued fractional differential equations, Applied Mathematics and Computation, 68, 1, (1995), 15 - 25. Google Scholar

[2]

A.-G. Ibrahim and A. M. El-Sayed, Definite integral of fractional order for set-valued functions., J. Fractional Calc., 11, (1997), 81-87. Google Scholar

[3]

A. M. El-Sayed and A.-G. Ibrahim, Set-valued integral equations of fractional-orders, Applied Mathematics and Computation, 118, 1, (2001), 113 - 121. Google Scholar

[4]

N. Ahmed and K. Teo, Optimal control of distributed parameter systems. North Holland, 1981. Google Scholar

[5]

N. Ahmed and X. Xiang, Existence of solutions for a class of nonlinear evolution equations with nonmonotone perturbations, Nonlinear Analysis: Theory, Methods & Applications, 22, 1, (1994), 81 - 89. Google Scholar

[6]

Y. Ling and S. Ding, A class of analytic functions defined by fractional derivation., J. Math. Anal. Appl., 186, 2, (1994), 504-513. Google Scholar

[7]

D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation., J. Math. Anal. Appl., 204, 2 (1996), 609-625. Google Scholar

[8]

A. Kilbas and J. Trujillo, Differential equations of fractional order: Methods, results and problems. I., Appl. Anal., 78, 1-2 (2001), 153-192. Google Scholar

[9]

S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales., Fundamenta math., 3 (1922), 133-181. Google Scholar

[10]

W. Kirk, P. Srinivasan, and P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions., Fixed Point Theory, 4, 1 (2003), 79-89. Google Scholar

[11]

V. Popa, Fixed point theorems for mappings in d-complete topological spaces., Math. Morav., 6, (2002), 87-92. Google Scholar

[12]

I. Altun and H. Simsek, Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory and Applications, 2010, 1, 2010, 621469. Google Scholar

[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), 15. Google Scholar

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