On a \(q\)-fractional variant of nonlinear Langevin equation of different orders
For given numbers \(\beta,\gamma\in(0,1)\), the authors study a boundary value problem for the \(q\)-fractional nonlinear Langevin equation of different orders with \(q\)-fractional antiperiodic boundary conditions in the form \[ {}^c\!D^{\beta}_q({}^c\!D^\gamma_q+\lambda)x(t)= \rho f(t,x(t))+\delta I^\zeta_qg(t,x(t)), \;\;0\leq t\leq 1, \;\;0<q<1, \] \[ x(0)=-x(1),\;\;{}^c\!D^\gamma_qx(0)=-{}^c\!D^\gamma_qx(1), \] where \({}^c\!D^{\beta}_q\) and \({}^c\!D^\gamma_q\) denote the Caputo type fractional \(q\)-derivative, \(I^\zeta_{q,0}(\cdot)=I^\zeta_{q}(\cdot)\) denotes the Riemann-Liouville integral with \(0<\zeta<1\), \(f\), \(g\) are given continuous functions, \(\lambda\neq0\), and \(\rho\), \(\delta\) are real constants. Some existence results for solutions of the given problem are established by means of classical tools of fixed point theory. An illustrative example is also presented.
scientific article; zbMATH DE number 6521985
scientific article