Existence of solutions for fractional Langevin equation involving generalized Caputo derivative with periodic boundary conditions

2020 ◽  
Author(s):  
Amita Devi ◽  
Anoop Kumar
Keyword(s):  
Boundary Conditions ◽  
Langevin Equation ◽  
Caputo Derivative ◽  
Periodic Boundary ◽  
Existence Of Solutions ◽  
Periodic Boundary Conditions ◽  
Fractional Langevin Equation

scholarly journals Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper Semicontinuity

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 55
Author(s):  
Valeria Marraffa ◽  
Bianca Satco
Keyword(s):  
Boundary Conditions ◽  
Differential Inclusions ◽  
Periodic Boundary ◽  
Existence Of Solutions ◽  
Compact Convex ◽  
Upper Semicontinuity ◽  
Periodic Boundary Conditions ◽  
Set Valued Mapping ◽  
Periodic Problems ◽  
The Right

We are studying first order differential inclusions with periodic boundary conditions where the Stieltjes derivative with respect to a left-continuous non-decreasing function replaces the classical derivative. The involved set-valued mapping is not assumed to have compact and convex values, nor to be upper semicontinuous concerning the second argument everywhere, as in other related works. A condition involving the contingent derivative relative to the non-decreasing function (recently introduced and applied to initial value problems by R.L. Pouso, I.M. Marquez Albes, and J. Rodriguez-Lopez) is imposed on the set where the upper semicontinuity and the assumption to have compact convex values fail. Based on previously obtained results for periodic problems in the single-valued cases, the existence of solutions is proven. It is also pointed out that the solution set is compact in the uniform convergence topology. In particular, the existence results are obtained for periodic impulsive differential inclusions (with multivalued impulsive maps and finite or possibly countable impulsive moments) without upper semicontinuity assumptions on the right-hand side, and also the existence of solutions is derived for dynamic inclusions on time scales with periodic boundary conditions.

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