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Abstract
In this paper, we discuss the existence of solutions for a stochastic initial value problem of Hyprid fractional dierential equations of Hadamard type given by 
 
 where HD is the Hadamard fractional derivative, and is the Hadamard fractional integral and be such that are investigated. The fractional calculus and stochastic analysis techniques are used to obtain the required results.
Highlights
We study the existence of solutions for a stochastic initial value problem of Hyprid fractional differential equations of Hadamard-type given by
We mean that the terms in the equation are perturbed either linearly or quadratically or through the combination of first and second types
Fractional calculus and fractional-order differential equations have been applied in many fields of science and engineering, such as physics ([4]-[5]), chemical ([20]-[21]), etc
Summary
We study the existence of solutions for a stochastic initial value problem of Hyprid fractional differential equations of Hadamard-type given by. Where HDα is the Hadamard fractional derivative, f : [1, T ] × L2(Ω) → L2(Ω) and b, σ : [1, T ] × L2(Ω) → L2(Ω), HJ (·) is the Hadamard fractional integral. We mean that the terms in the equation are perturbed either linearly or quadratically or through the combination of first and second types. Fractional calculus and fractional-order differential equations have been applied in many fields of science and engineering, such as physics ([4]-[5]), chemical ([20]-[21]), etc. Fractional differential equations involving Riemann-Liouville and Caputo-type fractional derivatives have extensively been studied by several researchers. A detailed description of the Hadamard derivative and integral can be found in (e.g., [2])
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