# Existence and Attractivity Theorems for Nonlinear Hybrid Fractional Integrodifferential Equations with Anticipation and Retardation

### Abstract

In this paper, we establish the existence and a global attractivity results for a nonlinear mixed quadratic and linearly perturbed hybrid fractional integrodifferential equation of second type involving the Caputo fractional derivative on unbounded intervals of real line with the mixed arguments of anticipations and retardation. The hybrid fixed point theorem of Dhage is used in the analysis of our nonlinear fractional integrodifferential problem. A positivity result is also obtained under certain usual natural conditions. Our hypotheses and claims have also been explained with the help of a natural realization.

### References

M. Cichón and H. A. H. Salem, “On the solutions of Caputo-Hadamard Pettis-type fractional differential equations”, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM vol. 113, no. 4, pp. 3031–3053, 2019.

M. Cichón and H. A. H. Salem, “On the lack of equivalence between differential and integral forms of the Caputo-type fractional problems”, J. Pseudo-Differ. Oper. Appl. DOI: 10.1007/s11868-020-00345-z. 2020.

J. Banas, B. C. Dhage, “Global asymptotic stability of solutions of a functional integral equations”, Nonlinear Analysis, vol. 69, pp.1945–1952, 2008.

T. A. Burton, T. Furumochi, “A note on stability by Schauder’s theorem”, Funkcialaj Ekvacioj, vol. 44, pp. 73–82, 2001.

K. Deimling, Nonlinear Functional Analysis, Springer Verlag, Berlin, 1985.

B. C. Dhage, “Local fixed point theory for the product of two operators in Banach algebras”, Math. Sci Res. Journal, vol. 9 no. 7, 373–381, 2003.

B. C. Dhage, “Local fixed point theory involving three operators in Banach algebras”, Topological Methods in Nonlinear Anal. vol. 24 , pp. 377–386, 2004.

B. C. Dhage, “A fixed point theorem in Banach algebras involving three operators with applications”, Kyungpook Math. J. vol. 44, pp. 145–155, 2004.

B. C. Dhage, “A nonlinear alternative in Banach algebras with applications to functional differential equations”, Nonlinear Funct. Anal. Appl. vol. 8, pp. 563–575, 2004.

B. C. Dhage, “A nonlinear alternative with applications to nonlinear perturbed differential equations”, Nonlinear Studies, vol. 13, no. 4, pp. 343–354, 2006.

B. C. Dhage, “Asymptotic stability of nonlinear functional integral equations via measures of noncompactness”, Comm. Appl. Nonlinear Anal. vol. 15, no. 2, pp. 89–101, 2008.

B. C. Dhage, “Local asymptotic attractivity for nonlinear quadratic functional integral equations”, Nonlinear Analysis, vol. 70, no. 5, pp. 1912–1922, 2009.

B. C. Dhage, “Global attractivity results for nonlinear functional integral equations via a Krasnoselskii type fixed point theorem”, Nonlinear Analysis, vol. 70 2485–2493, 2009.

B. C. Dhage, “Attractivity and positivity results for nonlinear functional integral equations via measures of noncompactness”, Differ. Equ. Appl., vol. 2, pp. 299–318, 2010.

B. C. Dhage, “Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations”, Differ. Equ. Appl., vol. 2 , pp 465–486, 2010.

B. C. Dhage, “Some characterizations of nonlinear first order differential equations on unbounded intervals”, Differ. Equ. Appl., vol. 2, pp. 151–162, 2010.

B. C. Dhage, “Some variants of two basic hybrid fixed point theorems of Krasnoselskii and Dhage with applications”, Nonlinear Studies, vol. 25, no. 3, 559–573, 2018.

B. C. Dhage, “Existence and attractivity theorems for nonlinear first order hybrid differential equations with anticipation and retardation”, Jn ̃a ̄n ̄abha, vol. 49, no. 2, pp. 45–63, 2019.

B. C. Dhage, “Global asymptotic attractivity and stability theorems for nonlinear Caputo fractional differential equations”, J. Frac. Cal. Appl., vol. 12, no. 1, pp. 223–237, 2021.

B. C. Dhage, “Existence and attractivity theorems for nonlinear hybrid fractional differential equations with anticipation and retardation”, J. Nonlinear Funct. Anal., vol. 2020, Article ID 47, pp. 1–18, 2020.

B. C. Dhage, S. B. Dhage, S. D. Sarkate, “Attractivity and existence results for hybrid differential equations with anticipation and retardation”, J. Math. Comput. Sci., vol. 4, no. 2, pp. 206–225, 2014.

B. C. Dhage, D. O’Regan, “A fixed point theorem in Banach algebras with applications to functional integral equations”, Funct. Diff. Equ., vol. 3, pp. 259–267, 2009.

A. Granas and J. Dugundji, Fixed Point Theory, Springer Verlag, New York, 2003.

A. Granas, R. B. Guenther and J. W. Lee, “Some general existence principles for Carath`eodory theory of nonlinear differential equations”, J. Math. Pures et Appl., vol. 70, pp. 153–196, 1991.

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Singapore, 2000.

X. Hu, J. Yan, “The global attractivity and asymptotic stability of solution of a nonlinear integral equation”, J. Math. Anal. Appl., vol. 321, pp. 147–156, 2006.

A. A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier: Amsterdam, The Netherlands, 2006.

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.

*CUBO, A Mathematical Journal*, vol. 22, no. 3, pp. 325–350, Dec. 2020.

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.