Fixed point theory and integral equations


We consider Volterra equations of convolution type with kernels continuous on (0; T], allowing for a kernel with a singularity at zero, provided it is integrable on (0; 1). For such equations we may apply both Schauder's and Schaefer's theorems without having to mention the compactness or even the continuity of the mapping, since those properties will be immediately obtained from the form of the integral equation. Prototypes use kernels of the form C(t; s) = (t - s)q-1 where 0 < q < 1, as found throughout heat theory and in fractional differential equations of both Riemann-Liouville and Caputo type. We conclude with an example involving heat conduction.

Publication Title

Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis

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