On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam
Abstract
:1. Introduction
2. Preliminaries
2.1. q-Calculus
2.2. (p; q)-Calculus
- .
- .
- .
- .
2.3. Fixed Point Theory
- (a)
- E is --contraction if
- (b)
- E is -admissible if whenever .
- (c)
- is an endpoint of if .
- (d)
- has an approximate endpoint property if
- (e)
- is -admissible if for every and ,
- (f)
- is an --contraction if
- (1)
- E is -admissible on ;
- (2)
- s.t. ;
- (3)
- For every sequence in with , if for all , then for each .
- (1)
- ;
- (2)
- The continuous function is compact;
- (3)
- is contraction.
- 1.
- is -admissible;
- 2.
- for some and ;
- 3.
- For every sequence in with and for all , there is a subsequence of so that for each .
- 1.
- The upper semi-continuous function is so that and for all ;
- 2.
- is so that for each .
3. On the Generalized -Difference Navier Problem (4)
- For each and , we have
- Some exists s.t. ,
- For every sequence converging to , the inequality
- s.t. and ,
- and there is a non-decreasing function s.t. for all and each ,
4. On the Generalized -Difference Navier Problem (5)
- is bounded and integrable so that is measurable for each ;
- There is so that for each ;
- There is a sequence converging to so that the inequality
- For each and such that
- There is so that and . Here, is non-decreasing and upper semi-continuous;
- is bounded and integrable such that is measurable for each ;
5. Examples
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Etemad, S.; Ntouyas, S.K.; Stamova, I.; Tariboon, J. On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam. Fractal Fract. 2024, 8, 236. https://doi.org/10.3390/fractalfract8040236
Etemad S, Ntouyas SK, Stamova I, Tariboon J. On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam. Fractal and Fractional. 2024; 8(4):236. https://doi.org/10.3390/fractalfract8040236
Chicago/Turabian StyleEtemad, Sina, Sotiris K. Ntouyas, Ivanka Stamova, and Jessada Tariboon. 2024. "On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam" Fractal and Fractional 8, no. 4: 236. https://doi.org/10.3390/fractalfract8040236