This paper deals with the existence and uniqueness of solution for the Cauchy problem of $ \varphi- $Caputo fractional evolution equations involving Volterra and Fredholm integral kernels. We derive a mild solution in terms of semigroup and construct a monotone iterative sequence for extremal solutions under a noncompactness measure condition of the nonlinearity. These results can be reduced to previous works with the classical Caputo fractional derivative. Furthermore, we give an example of initial-boundary value problem for the time-fractional parabolic equation to illustrate the application of the results.
Citation: Apassara Suechoei, Parinya Sa Ngiamsunthorn. Extremal solutions of $ \varphi- $Caputo fractional evolution equations involving integral kernels[J]. AIMS Mathematics, 2021, 6(5): 4734-4757. doi: 10.3934/math.2021278
This paper deals with the existence and uniqueness of solution for the Cauchy problem of $ \varphi- $Caputo fractional evolution equations involving Volterra and Fredholm integral kernels. We derive a mild solution in terms of semigroup and construct a monotone iterative sequence for extremal solutions under a noncompactness measure condition of the nonlinearity. These results can be reduced to previous works with the classical Caputo fractional derivative. Furthermore, we give an example of initial-boundary value problem for the time-fractional parabolic equation to illustrate the application of the results.
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Apassara Suechoei, Parinya Sa Ngiamsunthorn. Extremal solutions of $ \varphi- $Caputo fractional evolution equations involving integral kernels[J]. AIMS Mathematics, 2021, 6(5): 4734-4757. doi: 10.3934/math.2021278
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