Weak, regularized, and mild solutions to Itô integrated stochastic Cauchy problems in Hilbert spaces

2016 ◽  
pp. 113-195
Keyword(s):  
Hilbert Spaces ◽  
Mild Solutions ◽  
Cauchy Problems

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.

scholarly journals Almost Automorphic Solutions to Nonautonomous Stochastic Functional Integrodifferential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Li Xi-liang ◽  
Han Yu-liang
Keyword(s):  
Hilbert Spaces ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Stochastic Equations ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Almost Automorphic ◽  
Banach Contraction ◽  
Stability Results ◽  
Stochastic Functional

This paper concerns the square-mean almost automorphic solutions to a class of abstract semilinear nonautonomous functional integrodifferential stochastic evolution equations in real separable Hilbert spaces. Using the so-called “Acquistapace-Terreni” conditions and Banach contraction principle, the existence, uniqueness, and asymptotical stability results of square-mean almost automorphic mild solutions to such stochastic equations are established. As an application, square-mean almost automorphic solution to a concrete nonautonomous integro-differential stochastic evolution equation is analyzed to illustrate our abstract results.

Nonlocal problem for fractional stochastic evolution equations with solution operators

2016 ◽  
Vol 19 (6) ◽  
Author(s):  
Pengyu Chen ◽  
Xuping Zhang ◽  
Yongxiang Li
Keyword(s):  
Hilbert Spaces ◽  
Evolution Equations ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Nonlocal Problem ◽  
Growth Conditions ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Nonlocal Initial Conditions ◽  
Analysis Theory

AbstractIn this article, we are concerned with a class of fractional stochastic evolution equations with nonlocal initial conditions in Hilbert spaces. The existence of mild solutions is obtained under the situation that the nonlinear term satisfies some appropriate growth conditions by using fractional calculations, Schauder fixed point theorem, stochastic analysis theory,

scholarly journals Existence and Attractivity for Fractional Evolution Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Yong Zhou ◽  
Jia Wei He ◽  
Bashir Ahmad ◽  
Ahmed Alsaedi
Keyword(s):  
Fractional Derivative ◽  
Evolution Equations ◽  
Global Attractivity ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Cauchy Problems ◽  
Fractional Evolution Equations ◽  
Liouville Fractional Derivative

We study the existence and attractivity of solutions for fractional evolution equations with Riemann-Liouville fractional derivative. We establish sufficient conditions for the global attractivity of mild solutions for the Cauchy problems in the case that semigroup is compact.

scholarly journals Complex powers of nondensely defined operators

2011 ◽  
Vol 90 (104) ◽  
pp. 47-64 ◽  
Author(s):  
Marko Kostic
Keyword(s):  
Mild Solutions ◽  
Acute Angle ◽  
Analytic Semigroups ◽  
Cauchy Problems ◽  
Growth Order ◽  
Fractional Powers ◽  
Complex Powers ◽  
Cosine Functions ◽  
Abstract Cauchy Problems ◽  
Densely Defined

The power (?A)b, b ? C is defined for a closed linear operator A whose resolvent is polynomially bounded on the region which is, in general, strictly contained in an acute angle. It is proved that all structural properties of complex powers of densely defined operators with polynomially bounded resolvent remain true in the newly arisen situation. The fractional powers are considered as generators of analytic semigroups of growth order r > 0 and applied in the study of corresponding incomplete abstract Cauchy problems. In the last section, the constructed powers are incorporated in the analysis of the existence and growth of mild solutions of operators generating fractionally integrated semigroups and cosine functions.

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.

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