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On a quadratic fractional Hammerstein-Volterra integral equation with linear modification of the argument. (English) Zbl 1228.45006
The authors examine a quadratic fractional Hammerstein-Volterra integral equation with a linear modification of the argument: $$ x(t)=a(t)+\frac{f(t,x(t))}{\Gamma(\alpha)} \int\limits_0^t \frac{k(t,\tau)u(\tau,x(\tau),x(\lambda\tau))}{(t-\tau)^{1-\alpha}}d\tau,\;\; t\in [0,1],\; 0<\alpha,\lambda<1,$$ where $a:[0,1]\to\mathbb R$, $f:[0,1]\times\mathbb R\to\mathbb R$, $k:[0,1]\times [0,1]\to\mathbb R$ and $u:[0,1]\times\mathbb R\times\mathbb R\to\mathbb R$ are functions satisfying suitable assumptions. Using a Darbo-type fixed point theorem and some techniques of the theory of measures of noncompactness, they derive the existence of a nonnegative continuous and nondecreasing solution to the above equation defined on $[0,1]$.

45G05Singular nonlinear integral equations
45M20Positive solutions of integral equations
47H08Measures of noncompactness and condensing mappings, $K$-set contractions, etc.
Full Text: DOI
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