scholarly journals Existence of Mild Solutions to Nonlocal Fractional Cauchy Problems via Compactness

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Rodrigo Ponce
Keyword(s):  
Banach Spaces ◽  
Fractional Derivatives ◽  
Mild Solutions ◽  
Cauchy Problems ◽  
Families Of Operators ◽  
Resolvent Families

We obtain characterizations of compactness for resolvent families of operators and as applications we study the existence of mild solutions to nonlocal Cauchy problems for fractional derivatives in Banach spaces. We discuss here simultaneously the Caputo and Riemann-Liouville fractional derivatives in the cases0<α<1and1<α<2.

Weighted pseudo asymptotically Bloch periodic solutions to nonlocal Cauchy problems of integrodifferential equations in Banach spaces

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yong-Kui Chang ◽  
Jianguo Zhao
Keyword(s):  
Banach Spaces ◽  
Mild Solutions ◽  
Asymptotic Properties ◽  
Auxiliary Result ◽  
Appreciable Effect ◽  
Cauchy Problems ◽  
State Variable ◽  
Strict Contraction ◽  
Forced Term ◽  
Nonlocal Cauchy Problem

Abstract This paper is mainly concerned with some new asymptotic properties on mild solutions to a nonlocal Cauchy problem of integrodifferential equation in Banach spaces. Under some well-imposed conditions on the nonlocal Cauchy, the neutral and forced terms, respectively, we establish some existence results for weighted pseudo S-asymptotically (ω, k)-Bloch periodic mild solutions to the referenced equation on R + ${\mathbb{R}}_{+}$ by suitable superposition theorems. The results show that the strict contraction of the nonlocal Cauchy and the neutral terms with the state variable has an appreciable effect on the existence and uniqueness of such a solution compared with the forced term. As an auxiliary result, the existence of weighted pseudo S-asymptotically (ω, k)-Bloch periodic mild solutions is deduced under the sublinear growth condition on the force term with its state variable. The existence of weighted pseudo S-asymptotically ω-antiperiodic mild solution is also obtained as a special example.

scholarly journals Abstract degenerate fractional differential inclusions in Banach spaces

2017 ◽  
Vol 11 (1) ◽  
pp. 39-61 ◽  
Author(s):  
Marko Kostic
Keyword(s):  
Banach Spaces ◽  
Differential Inclusions ◽  
Linear Operators ◽  
Cauchy Problems ◽  
Fractional Differential Inclusions ◽  
Caputo Derivatives ◽  
Fractional Resolvent Families ◽  
Fractional Differential ◽  
Resolvent Families ◽  
Linear Degenerate

In the paper under review, we investigate a class of abstract degenerate fractional differential inclusions with Caputo derivatives. We consider subordinated fractional resolvent families generated by multivalued linear operators, which do have removable singularities at the origin. Semi-linear degenerate fractional Cauchy problems are also considered in this context.

Subordination principle for fractional diffusion-wave equations of Sobolev type

2020 ◽  
Vol 23 (2) ◽  
pp. 427-449
Author(s):  
Rodrigo Ponce
Keyword(s):  
Banach Space ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Wave Equations ◽  
Mild Solutions ◽  
Fractional Diffusion ◽  
Sobolev Type ◽  
Diffusion Wave ◽  
Fractional Differential ◽  
Resolvent Families

AbstractIn this paper we study subordination principles for fractional differential equations of Sobolev type in Banach space. With the help of the theory of Sobolev type resolvent families (known also as propagation family) as well as these subordination principles, we obtain the existence of mild solutions for this kind of equations. We study simultaneously the case 0 < α < 1 and 1 < α < 2 for the Caputo and Riemann-Liouville fractional derivatives.

Existence results of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Shengli Xie
Keyword(s):  
Differential Equations ◽  
Differential Evolution ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Infinite Delay ◽  
Mild Solutions ◽  
Existence Results ◽  
Order Case

AbstractIn this paper we prove the existence and uniqueness of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay in Banach spaces. We generalize the existence theorem for integer order differential equations to the fractional order case. The results obtained here improve and generalize many known results.

Semilinear fractional order integro-differential inclusions with infinite delay

2018 ◽  
Vol 25 (3) ◽  
pp. 317-327 ◽  
Author(s):  
Khalida Aissani ◽  
Mouffak Benchohra ◽  
Mohamed Abdalla Darwish
Keyword(s):  
Banach Spaces ◽  
Fractional Order ◽  
Differential Inclusions ◽  
Existence Of Solutions ◽  
Infinite Delay ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Multivalued Maps ◽  
Nonlinear Alternative

AbstractIn this paper, we study the existence of mild solutions for a class of semilinear fractional order integro-differential inclusions with infinite delay in Banach spaces. Sufficient conditions for the existence of solutions are derived by using a nonlinear alternative of Leray–Schauder type for multivalued maps due to Martelli. An example is given to illustrate the theory.

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