| The theory of fractional differential equations has received extensive attention from more and more domestic and foreign scholars in recent decades,especially the fractional differential equation based on practical problems.More and more fractional differential equations are used to describe problems in the fields of thermodynamics,optical,fluid mechanics,materials and mechanical systems.Since the birth of the definition of q-difference operator(also known as quantum difference operator),q-calculus is an important bridge of math and physics.In recent years,many scholars have introduced q-calculus theory into equations.The journey of the new field of the scallet q-difference equation was opened.This paper mainly studies two types of boundary value problems of Riemann-Liouville fractional q-difference equations with p-Laptacian operators.One is the nonlinear high-order Riemann-Liouville fractional q-difference equation boundary value problem with p-Laplacian operator:The existence of two positive solutions to the problem is proved by using the fixed point theorem of cone extension and compression in functional form,and then the Leggett-Williams fixed point theorem is used to establish a related theory that there are at least three positive solutions to the problem.Finally,an example is used to prove the validity of the main results obtained.The other is the Riemann-Liouville fractional q-difference equation under the four-point boundary value condition with p-Laplacian operator:The existence of multiple positive solutions,The existence of two positive solutions of the q-difference equation is verified by a monotonic iterative method.Finally,an example is used to prove the validity of the main results obtained. |
