Lipschitz-Stabilität von Optimierungs- und Approximationsaufgaben / Lipschitz-Stability for Programming and Approximation Problems

Lyapunov Functions and Lipschitz Stability for Riemann–Liouville Non-Instantaneous Impulsive Fractional Differential Equations

Symmetry ◽

10.3390/sym13040730 ◽

2021 ◽

Vol 13 (4) ◽

pp. 730

Author(s):

Ravi Agarwal ◽

Snezhana Hristova ◽

Donal O’Regan

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lyapunov Functions ◽

Sufficient Conditions ◽

Initial Time ◽

Lipschitz Stability ◽

Comparison Results ◽

Liouville Fractional Derivative ◽

Fractional Differential ◽

Definition Of

In this paper a system of nonlinear Riemann–Liouville fractional differential equations with non-instantaneous impulses is studied. We consider a Riemann–Liouville fractional derivative with a changeable lower limit at each stop point of the action of the impulses. In this case the solution has a singularity at the initial time and any stop time point of the impulses. This leads to an appropriate definition of both the initial condition and the non-instantaneous impulsive conditions. A generalization of the classical Lipschitz stability is defined and studied for the given system. Two types of derivatives of the applied Lyapunov functions among the Riemann–Liouville fractional differential equations with non-instantaneous impulses are applied. Several sufficient conditions for the defined stability are obtained. Some comparison results are obtained. Several examples illustrate the theoretical results.

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An inverse problem of radiative potentials and initial temperatures in parabolic equations with dynamic boundary conditions

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2020-0067 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

El Mustapha Ait Ben Hassi ◽

Salah-Eddine Chorfi ◽

Lahcen Maniar

Keyword(s):

Inverse Problem ◽

Boundary Conditions ◽

Parabolic Equations ◽

Stability Result ◽

Stability Estimate ◽

Lipschitz Stability ◽

Dynamic Boundary Conditions ◽

Logarithmic Convexity ◽

Dynamic Boundary ◽

Logarithmic Stability

Abstract We study an inverse problem involving the restoration of two radiative potentials, not necessarily smooth, simultaneously with initial temperatures in parabolic equations with dynamic boundary conditions. We prove a Lipschitz stability estimate for the relevant potentials using a recent Carleman estimate, and a logarithmic stability result for the initial temperatures by a logarithmic convexity method, based on observations in an arbitrary subdomain.

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