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Analysis of nonlinear integral equations with Erdélyi-Kober fractional operator. (English) Zbl 1298.45011
This paper initiates the investigation of the following nonlinear integral equation with Erdélyi-Kober fractional operator
\[ y(t)=a(t)+\frac{b(t)}{\Gamma(\alpha)}\int\limits_{0}^{t}(t^{\beta}-s^{\beta})^{\alpha-1}s^{\gamma}f(s,y(s))ds,\qquad t\in J=[0,T],\quad T>0, \]
where \(a,b:J\to \mathbb R\), \(f:J\times \mathbb R\to \mathbb R\) and \(\alpha,\beta,\gamma\) are positive parameters.
Using 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 Q.-H. Ma and J. Pečarić [J. Math. Anal. Appl. 341, No. 2, 894–905 (2008; Zbl 1142.26015)].
The local stability of the solutions for the generalized equation
\[ y(t)=a(t)+\frac{b(t)}{\Gamma(\alpha)}\int\limits_{0}^{t}(t^{\beta}-s^{\beta})^{\alpha-1}s^{\gamma}g(t,s,y(s))ds, \qquad t\in \mathbb R_{+}=[0,\infty), \]
is 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.”

45G10 Other nonlinear integral equations
34A08 Fractional ordinary differential equations
Zbl 1142.26015
Full Text: DOI
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