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Nonlinear differential equations with Marchaud-Hadamard-type fractional derivative in the weighted space of summable functions. (English) Zbl 1132.26314
Summary: The paper is devoted to the study of a Cauchy-type problem for the nonlinear differential equation of fractional order $0<\alpha<1$, $$\aligned &( D^\alpha_{0+,\mu}y)(x)=f(x,y(x)), \\ &(x^\mu\cal J^{1-\alpha}_{0+,\mu}y)(0+)=b,\quad b\in\Bbb R,\endaligned$$ containing the Marchaud-Hadamard-type fractional derivative $(D^\alpha_{0+,\mu}y)(x)$, on the half-axis $\Bbb R^+=(0,+\infty)$ in the space $X^{p,\alpha}_{c,0}(\Bbb R_+)$ defined for $\alpha>0$ by $$X^{p,\alpha}_{c,0}(\Bbb R_+)=\{y\in X^p_c(\Bbb R_+): D^\alpha_{0+,\mu}y\in X^p_{c,0}(\Bbb R_+)\}.$$ Here $X^p_{c,0}(\Bbb R_+)$ is the subspace of $X^p_c(\Bbb R_+)$ of functions $g$ with compact support on infinity: $g(x)\equiv 0$ for large enough $x>R$. The equivalence of this problem and a nonlinear Volterra integral equation is established. The existence and uniqueness of the solution $y(x)$ of the above Cauchy-type problem is proved by using the Banach fixed point theorem. The solution in closed form of the above problem for the linear differential equation with $\{f(x,y(x))=\lambda y(x)+f(x)\}$ is constructed. The corresponding assertions for the differential equations with the Marchaud-Hadamard fractional derivative $(D_{0+}^\alpha y)(x)$ are presented. Examples are given.

26A33Fractional derivatives and integrals (real functions)
34K30Functional-differential equations in abstract spaces
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
45D05Volterra integral equations
47N20Applications of operator theory to differential and integral equations
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