G-Metric Spaces via Fixed Point Techniques for Ψ-Contraction with Applications
Abstract
1. Mathematical Introduction
2. Basic Facts
- (1)
- if and only if ;
- (2)
- for all with ;
- (3)
- (symmetry in all three variables);
- (4)
- , for all .
- •
- If is small, then , and are close.
- •
- If is large, then at least one of the points is far from the others.
- •
- The function can be seen as a replacement for the usual metric , but the value depends on an additional reference point.
- •
- Triangle Representation: can be viewed as a measurement that establishes the “spread” of a triangle in space since it depends on three points at a time.
- •
- Deformation of Distances: Because G-metric spaces depend on a third point, they provide flexibility in measuring distances, unlike Euclidean spaces, where the distances between two points are fixed.
- •
- Curvature-Like Behavior: According to some interpretations, the G-metrics can capture curvature-like phenomena, in which the placement of extra points affects distances.
- (1)
- if , then ,
- (2)
- ,
- (3)
- .
- (1)
- a G-Cauchy sequence if, for any , there is such that for all , ;
- (2)
- a G-convergent sequence to if, for any , there is such that for all , .
- (1)
- is G-convergent to ϖ;
- (2)
- as ;
- (3)
- ;
- (4)
- as ;
- (5)
- as .
- (1)
- the sequence is G-Cauchy;
- (2)
- as .
- (1)
- ;
- (2)
- for , where stands for the n-th iterate of ψ;
- (3)
- ;
- (4)
- .
- (1)
- If , then we have Thus, , so condition (1) is satisfied.
- (2)
- . Thus, condition (2) is satisfied.
- (3)
- For every and , we haveThus, condition (3) is satisfied.
- (4)
- For every , we haveThus, condition (4) is satisfied.
- ,
- (order of the fractional derivative, ),
- (satisfying ).
3. Main Results
- (1)
- ⇒: If , then . Therefore, .⇐: If , then , which implies that . Hence, .
- (2)
- Since and .Clearly, .
- (3)
- The function is clearly symmetric because the supremum is taken over all the pairwise differences and reordering these differences does not change the result.
- (4)
- We need to check if the inequality holds for the given function:Now,Since the supremum represents the largest of the distances between the points, and the triangle inequality holds in , this inequality is valid.
4. Applications to Boundary Value Problems
- (a)
- indicates the displacement at position t,
- (b)
- indicates the applied force, which depend on both displacement and position,
- (c)
- the boundary conditions indicate the beam is fixed at both ends.
- (C1) there exists and for all , we have
- (C2) .
5. Application to Fredholm-Type Integral Equations
- (a)
- represents the temperature distribution at point in a heat conduction problem,
- (b)
- indicates the thermal conductivity coefficient of the material,
- (c)
- the kernel function represents the heat transfer kernel that describes how heat at point affects the temperature at point ,
- (d)
- the integral from t to r indicates the total accumulation of heat effects over the entire physical domain from point t to r,
- (e)
- the interval represents the physical length of the conducting material.
6. Application of Nonlinear Fractional Differential Equation
- (1)
- there exists a continuous function satisfying that
- (2)
- (a)
- Since is a self operator, and , then for any we have . Thus, , and hence is nonempty.
- (b)
- For any arbitrary element , the sequence , where is a self-operator. Therefore, , where , and hence is -closed.
- (c)
- It is only required to prove that is a generalized -contractive operator.Therefore,By using given condition , we obtain
Numerical Solution of Fractional Initial Value Problems
7. Conclusions and Future Work
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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n | 4 | 6 | 8 | 10 |
---|---|---|---|---|
Absolute Error |
Present Method | Predictor-Corrector Method [46] |
---|---|
1.95 | 1.9 | 1.85 | 1.8 | 1.75 | |
---|---|---|---|---|---|
[47] | |||||
Our Method |
n | 3 | 7 | 11 | 15 |
---|---|---|---|---|
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Albeladi, G.; Gamal, M.; Youssri, Y.H. G-Metric Spaces via Fixed Point Techniques for Ψ-Contraction with Applications. Fractal Fract. 2025, 9, 196. https://doi.org/10.3390/fractalfract9030196
Albeladi G, Gamal M, Youssri YH. G-Metric Spaces via Fixed Point Techniques for Ψ-Contraction with Applications. Fractal and Fractional. 2025; 9(3):196. https://doi.org/10.3390/fractalfract9030196
Chicago/Turabian StyleAlbeladi, Ghadah, Mohamed Gamal, and Youssri Hassan Youssri. 2025. "G-Metric Spaces via Fixed Point Techniques for Ψ-Contraction with Applications" Fractal and Fractional 9, no. 3: 196. https://doi.org/10.3390/fractalfract9030196
APA StyleAlbeladi, G., Gamal, M., & Youssri, Y. H. (2025). G-Metric Spaces via Fixed Point Techniques for Ψ-Contraction with Applications. Fractal and Fractional, 9(3), 196. https://doi.org/10.3390/fractalfract9030196