A fractional \(q\)-difference equation with integral boundary conditions and comparison theorem. (English) Zbl 1401.39010
Summary: In this article, we mainly prove the existence of extremal solutions for a fractional \(q\)-difference equation involving Riemann-Lioville type fractional derivative with integral boundary conditions. A comparison theorem under weak conditions is also build, and then we apply the comparison theorem, monotone iterative technique and lower-upper solution method to prove the existence of extremal solutions. Moreover, we can construct two iterative schemes approximating the extremal solutions of the fractional \(q\)-difference equation with integral boundary conditions. In the last section, a simple example is presented to illustrate the main result.
MSC:
| 39A13 | Difference equations, scaling (\(q\)-differences) |
| 33D05 | \(q\)-gamma functions, \(q\)-beta functions and integrals |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
Keywords:
fractional \(q\)-difference equation; comparison theorem; integral boundary conditions; extremal solutions; iterative methodReferences:
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