| Title | EXISTENCE AND UNIQUENESS RESULTS FOR POSITIVE SOLUTIONS OF A NONLINEAR FRACTIONAL DIFFERENCE EQUATION |
| Publication Type | Journal Article |
| Year of Publication | 2015 |
| Authors | AWASTHI, PUSHPR, ERBE, LYNNH, PETERSON, ALLANC |
| Volume | 19 |
| Issue | 1 |
| Start Page | 61 |
| Pagination | 18 |
| Date Published | 2015 |
| ISSN | 1083-2564 |
| AMS | 39A10 |
| Abstract | In this paper we are concerned with the fractional self-adjoint equation $${∆^µ_{µ−1} (p∆x)(t) + q(t + µ − 1)x(t + µ − 1) = h(t), \ \ \ t ∈ \mathbb{N}_0, \ \ \ \ (0.1) }$$ where ${ 0 < µ ≤ 1, \ p : \mathbb{N}_{µ−1} → (0, ∞), \ q : \mathbb{N}_{µ−1} → [0, ∞), \ h : \mathbb{N}_0 → \mathbb{R} }$. Our sole reason for calling this equation self-adjoint is that when ${µ = 1}$ we get the well-studied second order self-adjoint difference equation [16, Chapters 6-9]. We will prove various results concerning the existence and uniqueness of positive solutions of the nonlinear fractional equation$${ ∆^µ_{µ−1} (p∆x)(t) + F(t, x(t + µ − 1)) = 0, \ t ∈ \mathbb{N}_0, \ \ \ \ \ \ \ (0.2) }$$ where ${ 0 < µ ≤ 1, \ p : \mathbb{N}_{µ−1} → (0, ∞)}$, and ${F : \mathbb{N}_0 × \mathbb{R} → [0, ∞) }$ by applying the Contraction Mapping Theorem. We also give examples illustrating our main results. |
| URL | http://www.acadsol.eu/en/articles/19/1/5.pdf |
| Refereed Designation | Refereed |
| Full Text | REFERENCES |