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Solvability of a fractional Cauchy problem based on modified fixed point results of non-compactness measures

1 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore-632014, TN, India
2 Department of Mathematics and Applied Mathematics, University of Johannesburg, Kingsway Campus, Auckland Park 2006, South Africa
3 Cloud Computing Center, University Malaya, Malaysia

Special Issues: Initial and Boundary Value Problems for Differential Equations

We study the solvability of a fractional Cauchy problem based on new development of fixed point theorem, where the operator is suggested to be non-compact on its domain. Moreover, we shall prove that the solution is bounded by a fractional entropy (entropy solution). For this purpose, we establish a collection of basic fixed point results, which generalizes and modifies some well known results. Our attention is toward the concept of a measure of non-compactness to generalize µ-set contractive condition, using three control functions.
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Keywords fractional calculus; fractional differential operator; fractional differential equation; fixed point theorem; measure of non-compactness; entropy solution

Citation: Hemant Kumar Nashine, Rabha W. Ibrahim. Solvability of a fractional Cauchy problem based on modified fixed point results of non-compactness measures. AIMS Mathematics, 2019, 4(3): 847-859. doi: 10.3934/math.2019.3.847

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