On the oscillation of solutions of Ψ-Hilfer generalized proportional fractional differential equations

Document Type : Regular research papers

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1 Department of Mathematics, Periyar University, Salem - 636011, Tamil Nadu, India.

2 Department of Mathematics, Periyar University, Salem, Tamil Nadu, India

Abstract

In this piece of work, we examined the oscillatory behavior of the Ψ-Hilfer generalized proportional fractional differential equations. In the literature, several oscillation criteria are established using various kinds of fractional operators such as Conformable, Hadamard, Ψ-Riemann Liouville, Ψ-Caputo and generalized proportional fractional derivative interms of Riemann Liouville and Caputo settings. Under various conditions, we established some sufficient conditions for the solutions of the above problem to be oscillatory via the Ψ-Hilfer generalized proportional fractional derivative with respect to another function. With the help of Young’s inequality and the Volterra integral equation, we developed new oscillation criteria for the above problem to be oscillatory. As a result of this study, we generalized and regained some existing results in the literature because of the suitable selection of the kernel Ψ. Also, we provided two examples to demonstrate the usefulness of our findings. In particular, one of the example shows that the equation has a non-oscillatory solution when the hypothesis fails.

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