Abstract
This paper is concerned with the nonlinear fractional differential equations with initial values. Applying some new contraction conditions to a new-found Banach space \(L_{p,\alpha ;\psi }\), we give a refinement of some important inequalities in the theorems appeared in the literature. Our main work, in this paper, is in two themes. First, the contraction constant which so far was strictly less than one is extended so that it covers the case when it equals one. Second, the continuity condition of the nonlinear term is relaxed. The former is a new approach, while the latter has been done in our previous works and we extend those methods, here. This method may constitute a step forward in helping us to overcome the obstacles, in some practical aims. We state two examples, in this regard, in each direction. We also express two real problems of physics applications which can be converted to the problems to which we could apply our method.
Similar content being viewed by others
References
Ahmad, B., Nieto, J.J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58, 1838–1843 (2009)
Baghani, O.: On fractional Langevin equation involving two fractional orders. Commun. Nonlinear. Sci. Numer. Simul. 42, 675–681 (2017)
Baghani, O.: Solving state feedback control of fractional linear quadratic regulator systems using triangular functions. Commun. Nonlinear Sci. Numer. Simul. 73, 319–337 (2019)
Caputo, M.: Linear models of dissipation whose \(Q\) is almost frequency independent, part II. Geophys. J. R. Astr. Soc. 13, 529–539 (1967)
Debnath, L., Bhatta, D.: Integral Transforms and their Applications. Chapman & Hall/CRC, Taylor & Francis Group, New York (2007)
Deng, J., Ma, L.: Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations. Appl. Math. Lett. 23, 676–680 (2010)
Deng, J., Deng, Z.: Existence of solutions of initial value problems for nonlinear fractional differential equations. Appl. Math. Lett. 32, 6–12 (2014)
Folland, G.B.: Real Analysis: Modern Techniques and Their Applications. Wiley, New York (1999)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific Publishing, Singapore (2000)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Kosmatov, N.: Integral equations and initial value problems for nonlinear differential equations of fractional order. Nonlinear Anal. 70, 2521–2529 (2009)
Kosmatov, N.: Integral equations of fractional order in Lebesgue spaces. Fract. Calc. Appl. Anal. 19, 665–675 (2016)
Lakshmikantham, V.: Theory of fractional functional differential equations. Nonlinear Anal. 69, 3337–3343 (2008)
Nabavi Sales, S.M.S., Baghani, O.: On multi-singular integral equations involving \(n\) weakly singular kernels. Filomat 32, 1323–1333 (2018)
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. CRC Press, Philadelphia (1993)
Vivek, D., Baghani, O., Kanagarajan, K.: Theory of hybrid fractional differential equations with complex order. Sahand Commun. Math. Anal. 15, 65–76 (2019)
Vivek, D., Baghani, O., Kanagarajan, K.: Existence results for hybrid fractional differential equations with Hilfer fractional derivative. Casp. J. Math. Sci. 9, 294–304 (2020)
Wei, Z., Li, Q., Chea, J.: Initial value problems for fractional differential equations involving Riemann–Liouville sequential fractional derivative. J. Math. Anal. Appl. 367, 260–272 (2010)
Yu, T., Deng, K., Luo, M.: Existence and uniqueness of solutions of initial value problems for nonlinear Langevin equation involving two fractional orders. Commun. Nonlinear Sci. Numer. Simul. 19, 1661–1668 (2014)
Zhou, W.X., Chu, Y.D.: Existence of solutions for fractional differential equations with multi-point boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 17, 1142–1148 (2012)
Acknowledgements
The authors would like to express their deepest gratitude to the anonymous referee for his (her) useful comments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Baghani, O., Sales, S.M.S.N. Existence, uniqueness, and relaxation results in initial value type problems for nonlinear fractional differential equations. Anal.Math.Phys. 11, 16 (2021). https://doi.org/10.1007/s13324-020-00471-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13324-020-00471-3
Keywords
- Caputo fractional derivative
- Initial value problem
- Iterative method
- Existence and uniqueness
- \(L_{p, \alpha ;\psi }\) space
- Geraghty’s contraction condition
- Jensen’s inequality