Solvability of a Parametric Fractional-Order Integral Equation Using Advance Darbo G-Contraction Theorem
Abstract
:1. Introduction
2. Methods
- (i)
- (Regularity) if and only if W is relatively compact.
- (ii)
- The family is a nonempty and .
- (iii)
- (Monotonic) .
- (iv)
- (Invariant under closure) .
- (v)
- (Invariant under convex hull) .
- (vi)
- for all .
- (vii)
- (Generalized Cantor’s intersection theorem) If for is a decreasing sequence of closed subsets of E and then is nonempty.
- G is non-decreasing;
- For each sequence of positive real numbers if and only if ;
- There exists such that
- 1.
- for all
- 2.
- for all
- ϕ is non-decreasing;
- ϕ is right-continuous on ;
- ;
- for each .
- ψ is increasing;
- ψ is right-continuous on ;
- ;
- for each .
- (1)
- For every , the mapping defined by where is continuous and non-increasing function is an example of -function.
- (2)
- Foe each let us define by the rule is an example of -function.
3. Results
4. Discussion
4.1. Fractional Integral Equation
- (a)
- The function is a member of the space which has finite limit at infinity.
- (b)
- The function is continuous and , moreover there exist continuous function with such that the following inequality will satisfies;
- (c)
- The function is continuous and there exists a non-decreasing and continuous function and such that;
- (d)
- The function is uniformly continuous on for any , moreover, for any such that and the following equality hold:
- (e)
- The functions defined as , and are bounded on . The functions and are vanishes at infinity.
- (f)
- There exist a positive number and satisfying the inequalityandwhere .
4.2. Numerical Example
4.3. Convergence to the Fixed Point
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
The closed ball centered at x with radius r | |
The class of nonempty, bounded, closed and convex sets. | |
Measure of noncompactness. | |
Set of all real numbers. | |
Set of all positive real numbers. | |
Set of all positive integers. | |
Closer of set . | |
The family of all bounded subsets of the space E | |
The subfamily of consisting only relatively compact sets. | |
The convex hull and closed convex hull of respectively. |
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n | 0 | 1 | 2 | 3 | 4 | Error = | |
---|---|---|---|---|---|---|---|
1/4 | z(n) | 1 | 0 | 0.5 | 0.98829 | 1.04883 | 0.301 |
1/2 | z(n) | 1 | 0 | 0.5 | 0.908102 | 0.98557 | 0.41 |
3/4 | z(n) | 1 | 0 | 0.5 | 0.928555 | 1.01562 | 0.454 |
1 | z(n) | 1 | 0 | 0.5 | 0.941959 | 1.0437 | 0.464 |
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Nikam, V.; Gopal, D.; Ibrahim, R.W. Solvability of a Parametric Fractional-Order Integral Equation Using Advance Darbo G-Contraction Theorem. Foundations 2021, 1, 286-303. https://doi.org/10.3390/foundations1020021
Nikam V, Gopal D, Ibrahim RW. Solvability of a Parametric Fractional-Order Integral Equation Using Advance Darbo G-Contraction Theorem. Foundations. 2021; 1(2):286-303. https://doi.org/10.3390/foundations1020021
Chicago/Turabian StyleNikam, Vishal, Dhananjay Gopal, and Rabha W. Ibrahim. 2021. "Solvability of a Parametric Fractional-Order Integral Equation Using Advance Darbo G-Contraction Theorem" Foundations 1, no. 2: 286-303. https://doi.org/10.3390/foundations1020021
APA StyleNikam, V., Gopal, D., & Ibrahim, R. W. (2021). Solvability of a Parametric Fractional-Order Integral Equation Using Advance Darbo G-Contraction Theorem. Foundations, 1(2), 286-303. https://doi.org/10.3390/foundations1020021