A Hadamard Fractional Boundary Value Problem on an Infinite Interval at Resonance
Abstract
1. Introduction
2. Auxiliary Results
- (I1)
- The functions and are bounded variation functions satisfying the resonant conditions of (), (), and . In addition, (given in (9)).
- (I2)
- The function is a Carathéodory function that satisfies the following conditions
- (a)
- is Lebesgue measurable for each ;
- (b)
- is continuous (in the last n variables) for a.e. ;
- (c)
- There exist functions, i.e., , , and , a.e. such that
- and
- , and
- and
- , and
- For , we have
- For , we find
- For , we have
- For , we find
- (a)
- ;
- (b)
- .
- (a)
- is a continuous operator;
- (b)
- is a bounded set;
- (c)
- is a continuous operator;
- (d)
- is a relatively compact set in .
- (a)
- We obtain
- (b)
- First, we observe that there exists such that for all . Then, according to and (20), we obtain
- (c)
- We remark that is a continuous operator on . In addition, we find
- (d)
- First, we prove that is bounded. In fact, for any , we have
3. Main Results
- (I3 )
- There exist positive constants and such that for all , if one of the following conditions is satisfied, then either or :for any ;for any .
- (I4)
- There exists a positive constant r such that for any satisfying or and , then either
- (I5)
- (i)
- for any ;
- (ii)
- for any
4. Examples
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Tudorache, A.; Luca, R. A Hadamard Fractional Boundary Value Problem on an Infinite Interval at Resonance. Fractal Fract. 2025, 9, 378. https://doi.org/10.3390/fractalfract9060378
Tudorache A, Luca R. A Hadamard Fractional Boundary Value Problem on an Infinite Interval at Resonance. Fractal and Fractional. 2025; 9(6):378. https://doi.org/10.3390/fractalfract9060378
Chicago/Turabian StyleTudorache, Alexandru, and Rodica Luca. 2025. "A Hadamard Fractional Boundary Value Problem on an Infinite Interval at Resonance" Fractal and Fractional 9, no. 6: 378. https://doi.org/10.3390/fractalfract9060378
APA StyleTudorache, A., & Luca, R. (2025). A Hadamard Fractional Boundary Value Problem on an Infinite Interval at Resonance. Fractal and Fractional, 9(6), 378. https://doi.org/10.3390/fractalfract9060378