Existence of Solutions for Fractional Multi-Point Boundary Value Problems on an Infinite Interval at Resonance
Abstract
:1. Introduction
- (H1)
- (H2)
- There exist nonnegative functions such that and ,
- (H3)
- where
2. Preliminaries
- (i)
- for any
- (ii)
- for any
- (iii)
- .
3. Main Result
- (i)
- the functions from M are equicontinuous on any compact interval of
- (ii)
- the functions from M are equiconvergent at infinity.
- (H4)
- There exist positive constants A and B such that, for all , if one of the following conditions is satisfied:
- (i)
- for any ; (ii) for any ,
then either or - (H5)
- There exists a positive constant C such that, for every satisfying or , then eitherorThen boundary value problem (1) has at least one solution in X provided that
- (i)
- for any
- (ii)
- for any
- (H6)
- There exists a positive constant M such that, for each satisfying for all , we have either
- (H7)
- There exist positive constants G and such that, for every satisfying for all , we have eitherThen boundary value problem (1) has at least one solution in X provided that
4. Example
- (i)
- for , one has
- (ii)
- for , one gets
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Zhang, W.; Liu, W. Existence of Solutions for Fractional Multi-Point Boundary Value Problems on an Infinite Interval at Resonance. Mathematics 2020, 8, 126. https://doi.org/10.3390/math8010126
Zhang W, Liu W. Existence of Solutions for Fractional Multi-Point Boundary Value Problems on an Infinite Interval at Resonance. Mathematics. 2020; 8(1):126. https://doi.org/10.3390/math8010126
Chicago/Turabian StyleZhang, Wei, and Wenbin Liu. 2020. "Existence of Solutions for Fractional Multi-Point Boundary Value Problems on an Infinite Interval at Resonance" Mathematics 8, no. 1: 126. https://doi.org/10.3390/math8010126
APA StyleZhang, W., & Liu, W. (2020). Existence of Solutions for Fractional Multi-Point Boundary Value Problems on an Infinite Interval at Resonance. Mathematics, 8(1), 126. https://doi.org/10.3390/math8010126