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Existence of solutions of generalized fractional differential equation with nonlocal initial condition. (English) Zbl 07088846

The authors consider the IVP of the Katugampola fractional differential equation of order \(\alpha\in(0,1)\) and type \(\beta\in[0,1]\) having nonlocal initial conditions. The initial condition involves the Katugampola fractional integral with \(\rho >0\). The equation is given by \[(^\rho D^{\alpha,\beta}_{a+}x)(t)=f(t,x(t)),\] \[(^\rho I^{1-\gamma}_{a+}x)({a+})=\sum_{j=1}^{m}\eta_jx(\xi_j),\] where \((^\rho D^{\alpha,\beta}_{a+})\) is the generalized Katugampola fractional derivative of order \(\alpha\in(0,1)\), and \((^\rho I^{1-\gamma}_{a+})\) is the Katugampola fractional integral with \(\rho >0\), and is denoted by NIVP.
The authors study the existence of solution for the above problem using (i) Krasnosel’skii fixed point theorem and (ii) Schauder fixed point theorem. To achieve their goal first they establish an equivalence between the NIVP and a mixed type nonlinear Voltera integral equation given by \[x(t)=\frac{K}{\Gamma(\alpha)}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)^{\gamma -1}\sum_{j=1}^{m}n_j\int_{a}^{\xi_j}s^{\rho -1}\left(\frac{{\xi}^{\rho}-s^{\rho}}{\rho}\right)^{\alpha -1}f(s,x(s))ds\] \[+\frac{1}{\Gamma(\alpha)}\int_{a}^{t}s^{\rho -1}\left(\frac{t^{\rho}-s^{\rho}}{\rho}\right)^{\alpha -1}f(s,x(s))ds.\] They illustrate their results through examples.

MSC:

34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
47H10 Fixed-point theorems
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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