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# EXISTENCE OF SOLUTIONS FOR HYBRID FRACTIONAL PANTOGRAPH EQUATIONS

Mohamed Abdalla Darwish and Kishin Sadarangani
Applicable Analysis and Discrete Mathematics
Vol. 9, No. 1 (April 2015), pp. 150-167
Stable URL: http://www.jstor.org/stable/43666213
Page Count: 18
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## Abstract

In this paper, we study the existence of the hybrid fractional pantograph equation $\left\{ {_{x\left( 0 \right) = 0,}^{D_0^\alpha + \left[ {\frac{{x\left( t \right)}}{{f\left( {t,x\left( t \right),x\left( {\mu t} \right)} \right)}}} \right] = g\left( {t,x\left( t \right),x\left( {\sigma t} \right)} \right),0\langle t\langle 1,}} \right.$ where α, μ, σ ∊ (0,1) and $D_{{0^ + }}^\alpha$ denotes the Riemann-Liouville fractional derivative. The results are obtained using the technique of measures of noncompactness in the Banach algebras and a fixed point theorem for the product of two operators verifying a Darbo type condition. Some examples are provided to illustrate our results.

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