Existence of Solution to a Quadratic Functional Integro-Differential Fractional Equation

Authors

  • B. D. Karande Department of Mathematics, Maharashtra Udaygirimahavidyalaya, Udgir, S.R.T.M.U. Nanded, Maharashtra
  • S. N. Kondekar Department of Mathematics, Degloor College Degloor, S.R.T.M.U. Nanded, Maharashtra

DOI:

https://doi.org/10.26713/cma.v11i4.1425

Keywords:

Banach algebras, Integro-Differential equation, Existence result, Locally attractive solution, Extremal solution

Abstract

An algebraic fixed point theorem involving the two operators in a Banach algebra is used to prove the existence of solutions to fractional order quadratic functional integro-differential equation in (\mathcal{R}_+\). Also, we establish the locally attractivity results and extremal solutions along with suitable example.

Downloads

Download data is not yet available.

References

M. I. Abbas, On the existence of locally attractive solutions of a nonlinear quadratic Voltra integral equation of fractional order, Advances in Difference Equations, 2010 (2010), Article number 127093, 1 – 11, DOI: 10.1155/2010/127093.

R. P. Agarwal, Y. Zhou and Y. He, Existence of fractional neutral functional differential equations, Computers & Mathematics with Applications 59(3) (2010), 1095 – 1100, DOI: 10.1016/j.camwa.2009.05.010.

A. Babakhani and V. Daftardar-Gejii, Existence of positive solutions of nonlinear fractional differential equations, Journal of Mathematical Analysis and Applications 278 (2003), 434 – 442, DOI: 10.1016/S0022-247X(02)00716-3.

J. Banas and B. C. Dhage, Globally asymptotic stability of solutions of a functional integral equations, Nonlinear Analysis: Theory, Methods & Applications 69(7) (2008), 1945 – 1952, DOI: 10.1016/j.na.2007.07.038.

J. Banas and B. Rzepka, An application of measures of noncompactness in the study of asymptotic stability, Applied Mathematics Letters 16 (2003), 1 – 6, DOI: 10.1016/S0893-9659(02)00136-2.

S. Das, Functional Fractional Calculus for System Identification and Controls, Springer-Verlag, Berlin ” Heidelberg (2008), DOI: 10.1007/978-3-540-72703-3.

S. Das, Functional Fractional Calculus, Springer-Verlag, Berlin ” Heidelberg (2011), DOI: 10.1007/978-3-642-20545-3.

B. C. Dhage, A fixed point theorem in Banach algebras involving three operators with applications, Kyungpook Mathematical Journal 44 (2004), 145 – 155, URL: https://www.koreascience.or.kr/article/JAKO200410102428645.page.

B. C. Dhage, A non-linear alternative in Banach Algebras with applications to functional differential equations, Non-linear Functional Analysis and Applications 8 (2004), 563 – 575, URL: www.researchgate.net/publication/265332287.

B. C. Dhage, Fixed point theorems in ordered Banach algebras and applications, Pan-American Mathematical Journal 9 (1999), 93 – 102.

B. C. Dhage, On existence of extremal solutions of nonlinear functional integral equations in Banach algebras, Journal of Applied Mathematics and Stochastic Analysis 2004:3 (2004), 271 – 282, URL: http://emis.impa.br/EMIS/journals/HOA/JAMSA/Volume2004_3/282.pdf.

S. Djebali and Z. Sahnoun, Nonlinear alternatives of Schauder and Krasnosel'skij types with applications to Hammerstein integral equations in L1 spaces, Journal of Differential Equations 249 (2010), 2061 – 2075, DOI: 10.1016/j.jde.2010.07.013.

M. M. El Borai and M. I. Abbas, On some integro-differential equations of fractional orders involving caratheodory nonlinearities, International Journal of Modern Mathematics 2(1) (2007), 41 – 52, URL: https://www.researchgate.net/profile/Mahmoud_El-Borai/publication/228568004_On_some_integro-differential_equations_of_fractional_orders_involving_Caratheodory_nonlinearities/links/00b7d5234c7ba351e9000000/Onsome-integro-differential-equations-of-fractional-orders-involving-Caratheodorynonlinearities.pdf.

M. M. El Borai and M. I. Abbas, Solvability of an infinite system of singular integral equations, Serdica Mathematical Journal 33 (2007), 241 – 252, URL: https://www.researchgate.net/profile/Mohamed_Abbas36/publication/281776710_Solvability_of_an_infinite_system_of_singular_integral_equations_and_applications/links/5f32e4e0458515b729183001/Solvability-of-an-infinite-system-of-singular-integralequations-and-applications.pdf.

M. M. El Borai, W. G. El-Sayed and M. I. Abbas, Monotonic solutions of a class of quadratic singular integral equations of Voltra type, International Journal of Contemporary Mathematical Sciences 2 (2007), 89 – 102, URL: https://www.researchgate.net/publication/265938524.

A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer-Verlag, New York (2003), DOI: 10.1007/978-0-387-21593-8.

D. J. Guo and V. Lakshmikantham, Nonlinear problems in Abstract cones, Notes and Reports in Mathematics in Science and Engineering, Vol. 5, Academic press, Massachusetts (1988).

S. Heikkilä and V. Lakshmikantham, Monotone iterative Techniques for discontinuous nonlinear differential equations, in: Monographs and Textbooks in Pure and Applied Mathematics, Vol. 181, Marcel Dekker, New York (1994), URL: https://books.google.co.in/books?hl=en&lr=&id=GGjyWjJbFUoC&oi=fnd&pg=PP11&dq=35.%09S.

X. Hu and J. Yan, The global attractivity and asymptotic stability of solution of a nonlinear integral equation, Journal of Mathematical Analysis and Applications 321 (2006), 147 – 156, DOI: 10.1016/j.jmaa.2005.08.010.

B. D. Karande, Existence of uniform global locally attractive solutions for fractional order nonlinear random integral equation, Journal of Global Research in Mathematical Archives 1(8) (2013), 34 – 43.

B. D. Karande, Fractional order functional integro-differential equation in Banach algebras, Malaysian Journal of Mathematical Sciences 8(S) (2014), 1 – 16, URL: https://einspem.upm.edu.my/journal/fullpaper/vol8soct/1.%20B.D.Karande_ICMSS2013_.pdf.

B. D. Karande, Global attractively of solutions for a nonlinear functional integral equation of fractional order in Banach space, in AIP Conference Proceedings: "10th international Conference on Mathematical Problems in Engineering, Aerospace and Sciences”, 1637 (2014), 469 – 478, DOI: 10.1063/1.4904612.

A. A. Kilbas and J. J. Trujillo, Differential equations of fractional order: methods, results and problems ” I, Applicable Analysis 78 (2001), 153 – 192, DOI: 10.1080/00036810108840931.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam (2006), URL: https://www.sciencedirect.com/bookseries/north-holland-mathematics-studies/vol/204/suppl/C.

M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press Book, The Macmillan, New York (1964), URL: https://books.google.co.in/books/about/Topological_Methods_in_the_Theory_of_Non.html?id=T41BxgEACAAJ&redir_esc=y.

M. A. Krasnoselskii, Two remarks on the method of successive approximations, Uspekhi Matematicheskikh Nauk 10 (1955), 123 – 127, URL: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=7954&option_lang=eng.

V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Analysis: Theory, Methods & Applications 69 (2008), 2677 – 2682, DOI: 10.1016/j.na.2007.08.042.

H. Lu, S. Sun, D. Yang and H. Teng, Theory of fractional hybrid differential equations with linear perturbations of second type, Boundary Value Problems 2013 (2013), Article number 23, URL: https://boundaryvalueproblems.springeropen.com/articles/10.1186/1687-2770-2013-23.

J. A. T. Machado, V. Kiryakova and F. Mainardi, Recent history of fractional calculus, Communications in Nonlinear Science and Numerical Simulation 16 (2011), 1140 – 1153, DOI: 10.1016/j.cnsns.2010.05.027.

R. Magin, Fractional calculus models of complex dynamics in biological tissues, Computers & Mathematics with Applications 59 (2010), 1586 – 1593, DOI: 10.1016/j.camwa.2009.08.039.

R. Magin, M. Ortigueira, I. Podlubny and J. J. Trujillo, On the fractional signals and systems, Signal Processing 91 (2011), 350 – 371, DOI: 10.1016/j.sigpro.2010.08.003.

F. C. Merala, T. J. Roystona and R. Magin, Fractional calculus in viscoelasticity: an experimental study, Communications in Nonlinear Science and Numerical Simulation 15 (2010), 939 – 945, DOI: 10.1016/j.cnsns.2009.05.004.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New Yark (1993).

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1993).

I. Podlubny, Fractional differential equations, Vol. 198, in An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, 1st edition, Academic Press, URL: https://www.elsevier.com/books/fractional-differentialequations/podlubny/978-0-12-558840-9.

J. Sabatier, H. C. Nguyen, C. Farges, J. Y. Deletage, X. Moreau, F. Guillemard and B. Bavoux, Fractional models for thermal modeling and temperature estimation of a transistor junction, Advances in Difference Equations 2011 (2011), Article number 687363, DOI: 10.1155/2011/687363.

S. Samko, A. A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Amsterdam (1993), URL: https://nesic.site/documents/%E6%95%B0%E5%AD%A6/Fractional%20Integrals%20and%20Derivatives%3A%20Theory%20and%20Applications%20(1993).pdf.

M. R. SidiAmmi, El Hassan El Kinani, Delfim F.M. Torres, Existence and uniqueness of solutions to a functional Integr0-Differential Equation, Electronic Journal of Differential Equation 2012(103) (2012), 1 – 9, URL: http://emis.impa.br/EMIS/journals/EJDE/Volumes/Volumes/2012/103/sidi.pdf.

H. M. Srivastava and R. K. Saxena, Operators of fractional integration and their applications, Applied Mathematics & Computation 118 (2001), 1 – 52, DOI: 10.1016/s0096-3003(99)00208-8.

S. Swarup, Integral Equations, 14th edition, Krishna Prakashan Media (P) Ltd., Meerut (2006), URL: https://books.google.co.in/books?hl=en&lr=&id=OWwmt0z3MtEC&oi=fnd&pg=PA1&dq=Shanti+Swarup,+Integral+equations&ots=7sOUuQHeq8&sig=lM5XDH02A38ZEMMwI_hx6OVkELM#v=onepage&q=Shanti%20Swarup%2C%20Integral%20equations&f=false.

V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer-Verlag Berlin ” Heidelberg (2010), DOI: 10.1007/978-3-642-14003-7.

Downloads

Published

31-12-2020
CITATION

How to Cite

Karande, B. D., & Kondekar, S. N. (2020). Existence of Solution to a Quadratic Functional Integro-Differential Fractional Equation. Communications in Mathematics and Applications, 11(4), 635–650. https://doi.org/10.26713/cma.v11i4.1425

Issue

Section

Research Article