A functional calculus for almost sectorial operators and applications to abstract evolution equations

2002 ◽  
Vol 2 (1) ◽  
pp. 41-68 ◽  
Author(s):  
F. Periago ◽  
B. Straub
Keyword(s):  
Functional Calculus ◽  
Evolution Equations ◽  
Sectorial Operators ◽  
Abstract Evolution Equations ◽  
Almost Sectorial Operators

Attractivity for fractional evolution equations with almost sectorial operators

2018 ◽  
Vol 21 (3) ◽  
pp. 786-800 ◽  
Author(s):  
Yong Zhou
Keyword(s):  
Evolution Equations ◽  
Sufficient Conditions ◽  
Cauchy Problems ◽  
Fractional Evolution Equations ◽  
Integer Order ◽  
Sectorial Operators ◽  
Almost Sectorial Operators ◽  
Globally Attractive

Abstract In this paper, we initiate the question of the attractivity of solutions for fractional evolution equations with almost sectorial operators. We establish sufficient conditions for the existence of globally attractive solutions for the Cauchy problems in cases that semigroup is compact as well as noncompact. Our results essentially reveal certain characteristics of solutions for fractional evolution equations, which are not possessed by integer order evolution equations.

scholarly journals Mild Solutions for Impulsive Integro-Differential Equations Involving Hilfer Fractional Derivative with almost Sectorial Operators

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 313
Author(s):  
Kulandhaivel Karthikeyan ◽  
Panjaiyan Karthikeyan ◽  
Nichaphat Patanarapeelert ◽  
Thanin Sitthiwirattham
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Jump Conditions ◽  
Hilfer Fractional Derivative ◽  
Sectorial Operators ◽  
Almost Sectorial Operators ◽  
Suitable Definition ◽  
Definition Of

In this manuscript, we establish the mild solutions for Hilfer fractional derivative integro-differential equations involving jump conditions and almost sectorial operator. For this purpose, we identify the suitable definition of a mild solution for this evolution equations and obtain the existence results. In addition, an application is also considered.

scholarly journals Existence and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Pallavi Bedi ◽  
Anoop Kumar ◽  
Thabet Abdeljawad ◽  
Zareen A. Khan ◽  
Aziz Khan
Keyword(s):  
Banach Space ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Control Function ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Existence Results ◽  
Fractional Evolution Equations ◽  
Sectorial Operators ◽  
Almost Sectorial Operators

Abstract In this article, we are concerned with the existence of mild solutions and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators and nonlocal conditions. The existence results are obtained by first defining Green’s function and approximate controllability by specifying a suitable control function. These results are established with the help of Schauder’s fixed point theorem and theory of almost sectorial operators in a Banach space. An example is also presented for the demonstration of obtained results.

Non-autonomous semilinear evolution equations with almost sectorial operators

2008 ◽  
Vol 8 (4) ◽  
pp. 631-659 ◽  
Author(s):  
Alexandre N. Carvalho ◽  
Tomasz Dlotko ◽  
Marcelo J. D. Nascimento
Keyword(s):  
Evolution Equations ◽  
Sectorial Operators ◽  
Almost Sectorial Operators ◽  
Semilinear Evolution Equations

Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

2021 ◽  
Vol 19 (1) ◽  
pp. 111-120
Author(s):  
Qinghua Zhang ◽  
Zhizhong Tan
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Nonlinear Operator ◽  
Evolution Equations ◽  
Bounded Function ◽  
Inhomogeneous Term ◽  
Linear Evolution ◽  
Abstract Evolution Equations ◽  
Linear Evolution Equation

Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

scholarly journals Asymptotically almost periodic solutions of fractional relaxation inclusions with Caputo derivatives

2018 ◽  
Vol 104 (118) ◽  
pp. 23-41 ◽  
Author(s):  
Marko Kostic
Keyword(s):  
Fixed Point ◽  
Periodic Solutions ◽  
Fixed Point Theorems ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Periodic Coefficients ◽  
Sectorial Operators ◽  
Almost Sectorial Operators ◽  
Fractional Relaxation ◽  
Asymptotically Almost Periodic

We analyze asymptotically almost periodic solutions for a class of (semilinear) fractional relaxation inclusions with Stepanov almost periodic coefficients. As auxiliary tools, we use subordination principles, fixed point theorems and the well known results on the generation of infinitely differentiable degenerate semigroups with removable singularities at zero. Our results are well illustrated and seem to be not considered elsewhere even for fractional relaxation equations with almost sectorial operators.

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