Uniqueness solution and stability results for singular fractional Riemann-Stieltjes integral boundary problems (Q6592017)

In this paper, the authors consider the following singular fractional boundary value problem with Riemann-Stieltjes integral boundary conditions: \N\[\N\begin{cases} D^{\eta}_{0^{+}}\delta(t)+p(t)f(t, \delta(t))+q(t)g(t,\delta(t))+r(t)\phi(t,\delta(t))=\omega,\\\N\delta^{(i)}(0)=0, ~i=0,1,\ldots,n-2, ~~ D^{\beta_1}_{0^{+}}\delta(1)=\tau\int_0^1k(s)D^{\beta_2}_{0^{+}}\delta(s)dA(s), \end{cases} \N\]\Nwhere \(D^{\eta}_{0^{+}},\) \(D^{\beta}_{0^{+}}\) are Riemann-Liouville fractional derivatives, \(\int_0^1k(s)D^{\beta_2}_{0^{+}}\delta(s)dA(s)\) is Riemann-Stieltjes integral, \(0<\beta<n-1<\eta\le n (n\ge 1),\) \(\eta-\beta-1>0,\) \(\tau>0,\) \(0<t<1\) and the nonlinear terms are singular both for time and space variables.\N\NExistence and uniqueness of solution, Hyers-Ulam stability and Hyers-Ulam-Rassias stability are studied. Examples illustrating the obtained results are also presented.