Analysis of nonlinear integral equations with Erdélyi-Kober fractional operator (Q446060)

This paper initiates the investigation of the following nonlinear integral equation with Erdélyi-Kober fractional operator NEWLINENEWLINE\[NEWLINE y(t)=a(t)+\frac{b(t)}{\Gamma(\alpha)}\int\limits_{0}^{t}(t^{\beta}-s^{\beta})^{\alpha-1}s^{\gamma}f(s,y(s))ds,\qquad t\in J=[0,T],\quad T>0,NEWLINE\]NEWLINE NEWLINEwhere \(a,b:J\to \mathbb R\), \(f:J\times \mathbb R\to \mathbb R\) and \(\alpha,\beta,\gamma\) are positive parameters.NEWLINENEWLINEUsing the Schauder fixed point theorem, the authors obtain the existence results of the solution for this equation with some chosen parameters \(\alpha,\beta\) and \(\gamma\). Then the uniqueness is derived by virtue of a weakly singular integral inequality due to \textit{Q.-H. Ma} and \textit{J. Pečarić} [J. Math. Anal. Appl. 341, No. 2, 894--905 (2008; Zbl 1142.26015)].NEWLINENEWLINEThe local stability of the solutions for the generalized equation NEWLINENEWLINE\[NEWLINE y(t)=a(t)+\frac{b(t)}{\Gamma(\alpha)}\int\limits_{0}^{t}(t^{\beta}-s^{\beta})^{\alpha-1}s^{\gamma}g(t,s,y(s))ds, \qquad t\in \mathbb R_{+}=[0,\infty), NEWLINE\]NEWLINE NEWLINEis also studied. ``Meanwhile, three certain solutions sets \(Y_{K,\sigma }\), \(Y_{1,\lambda }\) and \(Y_{1,1}\), which tend to zero at an appropriate rate \(t^{ - \nu }\), \(0 < \nu = \sigma\) (or \(\lambda\) or 1) as \(t\to +\infty \), are constructed and local stability results of the solutions are obtained based on these sets respectively under some suitable conditions. Two examples are given to illustrate the results.''