Existence and Global Asymptotic Behavior of Positive Solutions for Superlinear Singular Fractional Boundary Value Problems
Abstract
:1. Introduction
- (P1)
- .
- (P2)
- There exists a function satisfying some integrable conditions such that for each is nondecreasing onwhere for
- (P3)
- For each is nondecreasing on
- (i)
- (ii)
- .
- (iii)
- with
- (iv)
- For
- (v)
- For ,Note that (see, Proposition 4) if then .
- (vi)
- For and we letObserve that for all
- (C1)
- with for all .
- (C2)
- There exists with such that, for each is nondecreasing on .
- (C3)
- For each , whenever
2. Background Materials and Preliminaries
- (i)
- For
- (ii)
- For
- (i)
- From (18), for we haveUsing this fact and that for andwe deduce the required result with and .
- (ii)
- The results follows from the previous estimates and the fact that
- (i)
- (ii)
3. Proofs of the Existence Results
3.1. Properties of
- (i)
- (ii)
- (iii)
- (iv)
- (iii)
- The assertion is valid for . Assume thatTherefore, by using (29) and Fubini-Tonelli’s theorem, we deduce that
- (iv)
- Let and From part (i), we getSo, the series converges.Hence, we deduce by the dominated convergence theorem and Lemma 3 (iii), that
3.2. Proofs of the Existence of Solutions
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Aljarallah, E.; Bachar, I. Existence and Global Asymptotic Behavior of Positive Solutions for Superlinear Singular Fractional Boundary Value Problems. Fractal Fract. 2023, 7, 527. https://doi.org/10.3390/fractalfract7070527
Aljarallah E, Bachar I. Existence and Global Asymptotic Behavior of Positive Solutions for Superlinear Singular Fractional Boundary Value Problems. Fractal and Fractional. 2023; 7(7):527. https://doi.org/10.3390/fractalfract7070527
Chicago/Turabian StyleAljarallah, Entesar, and Imed Bachar. 2023. "Existence and Global Asymptotic Behavior of Positive Solutions for Superlinear Singular Fractional Boundary Value Problems" Fractal and Fractional 7, no. 7: 527. https://doi.org/10.3390/fractalfract7070527