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Functional-differential equations with Riemann-Liouville integrals in the nonlinearities

Milan Medveď (2014)

Mathematica Bohemica

A sufficient condition for the nonexistence of blowing-up solutions to evolution functional-differential equations in Banach spaces with the Riemann-Liouville integrals in their right-hand sides is proved. The linear part of such type of equations is an infinitesimal generator of a strongly continuous semigroup of linear bounded operators. The proof of the main result is based on a desingularization method applied by the author in his papers on integral inequalities with weakly singular kernels....

Hamilton’s Principle with Variable Order Fractional Derivatives

Atanackovic, Teodor, Pilipovic, Stevan (2011)

Fractional Calculus and Applied Analysis

MSC 2010: 26A33, 70H25, 46F12, 34K37 Dedicated to 80-th birthday of Prof. Rudolf GorenfloWe propose a generalization of Hamilton’s principle in which the minimization is performed with respect to the admissible functions and the order of the derivation. The Euler–Lagrange equations for such minimization are derived. They generalize the classical Euler-Lagrange equation. Also, a new variational problem is formulated in the case when the order of the derivative is defined through a constitutive equation....

The global existence of mild solutions for semilinear fractional Cauchy problems in the α-norm

Rong-Nian Wang, De-Han Chen, Yan Wang (2012)

Annales Polonici Mathematici

We study the local and global existence of mild solutions to a class of semilinear fractional Cauchy problems in the α-norm assuming that the operator in the linear part is the generator of a compact analytic C₀-semigroup. A suitable notion of mild solution for this class of problems is also introduced. The results obtained are a generalization and continuation of some recent results on this issue.

Weighted fractional differential equations with infinite delay in Banach spaces

Qixiang Dong, Can Liu, Zhenbin Fan (2016)

Open Mathematics

This paper is devoted to the study of fractional differential equations with Riemann-Liouville fractional derivatives and infinite delay in Banach spaces. The weighted delay is developed to deal with the case of non-zero initial value, which leads to the unboundedness of the solutions. Existence and uniqueness results are obtained based on the theory of measure of non-compactness, Schaude’s and Banach’s fixed point theorems. As auxiliary results, a fractional Gronwall type inequality is proved,...

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