scholarly journals Asymptotic Behavior of Solutions to Abstract Stochastic Fractional Partial Integrodifferential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Zhi-Han Zhao ◽  
Yong-Kui Chang ◽  
Juan J. Nieto
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Integrodifferential Equation ◽  
Fixed Point Theorems ◽  
Mild Solutions ◽  
Integrodifferential Equations ◽  
Linear Operators ◽  
Asymptotic Behavior Of Solutions ◽  
Almost Automorphic ◽  
Partial Integrodifferential Equation

The existence of asymptotically almost automorphic mild solutions to an abstract stochastic fractional partial integrodifferential equation is considered. The main tools are some suitable composition results for asymptotically almost automorphic processes, the theory of sectorial linear operators, and classical fixed point theorems. An example is also given to illustrate the main theorems.

scholarly journals Existence and asymptotic behavior of solutions for neutral stochastic partial integrodifferential equations with infinite delays

2016 ◽  
Vol 16 (06) ◽  
pp. 1650014 ◽  
Author(s):  
Mamadou Abdoul Diop ◽  
Tomás Caraballo ◽  
Mahamat Mahamat Zene
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Resolvent Operators ◽  
Infinite Delays ◽  
Fixed Point Principle ◽  
Exponentially Stable

In this work we study the existence, uniqueness and asymptotic behavior of mild solutions for neutral stochastic partial integrodifferential equations with infinite delays. To prove the results, we use the theory of resolvent operators as developed by R. Grimmer [13], as well as a version of the fixed point principle. We establish sufficient conditions ensuring that the mild solutions are exponentially stable in [Formula: see text]th-moment. An example is provided to illustrate the abstract results.

scholarly journals Impulsive Fractional Semilinear Integrodifferential Equations with Nonlocal Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Xue Wang ◽  
Bo Zhu
Keyword(s):  
Fixed Point ◽  
Resolvent Operator ◽  
Fixed Point Theorems ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Semigroup Theory ◽  
Nonlocal Conditions ◽  
Integrodifferential Equations ◽  
Existence Theory ◽  
Nonlocal Initial Conditions

This paper is devoted to a class of impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions. Based on the semigroup theory and some fixed point theorems, the existence theory of PC-mild solutions is established under the condition of compact resolvent operator. Furthermore, the uniqueness of PC-mild solutions is proved in the case of the noncompact resolvent operator.

scholarly journals Existence Results for Fractional Semilinear Integrodifferential Equations of Mixed Type with Delay

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Xue Wang ◽  
Bo Zhu
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Mixed Type ◽  
Fixed Point Theorems ◽  
Measure Of Noncompactness ◽  
Mild Solutions ◽  
Existence Results ◽  
Integrodifferential Equations ◽  
Resolvent Operators ◽  
Equations Of Mixed Type

In this paper, we discuss a class of fractional semilinear integrodifferential equations of mixed type with delay. Based on the theories of resolvent operators, the measure of noncompactness, and the fixed point theorems, we establish the existence and uniqueness of global mild solutions for the equations. An example is provided to illustrate the application of our main results.

scholarly journals Existence Theorems for Fractional Semilinear Integrodifferential Equations with Noninstantaneous Impulses and Delay

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Bo Zhu ◽  
Minhui Zhu
Keyword(s):  
Fixed Point ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fixed Point Theorems ◽  
Existence Theorems ◽  
Mild Solutions ◽  
Semigroup Theory ◽  
Integrodifferential Equations ◽  
Partial Differential ◽  
Noninstantaneous Impulses

In this paper, we consider a class of fractional semilinear integrodifferential equations with noninstantaneous impulses and delay. By the semigroup theory and fixed point theorems, we establish various theorems for the existence of mild solutions for the problem. An example involving partial differential equations with noninstantaneous impulses is given to show the application of our main results.

scholarly journals Ulam-Hyers Stability and Ulam-Hyers-Rassias Stability for Fuzzy Integrodifferential Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Nguyen Ngoc Phung ◽  
Bao Quoc Ta ◽  
Ho Vu
Keyword(s):  
Fixed Point ◽  
Successive Approximation ◽  
Integrodifferential Equation ◽  
Approximation Method ◽  
Fixed Point Method ◽  
Integrodifferential Equations ◽  
Point Method ◽  
Successive Approximation Method ◽  
Rassias Stability

In this paper, we establish the Ulam-Hyers stability and Ulam-Hyers-Rassias stability for fuzzy integrodifferential equations by using the fixed point method and the successive approximation method.

scholarly journals On Almost Automorphic Mild Solutions for Nonautonomous Stochastic Evolution Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Jing Cui ◽  
Litan Yan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Banach Fixed Point Theorem ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Almost Automorphic ◽  
Composition Theorem

We consider a class of nonautonomous stochastic evolution equations in real separable Hilbert spaces. We establish a new composition theorem for square-mean almost automorphic functions under non-Lipschitz conditions. We apply this new composition theorem as well as intermediate space techniques, Krasnoselskii fixed point theorem, and Banach fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions. Some known results are generalized and improved.

Asymptotic behavior of mild solutions for nonlinear fractional difference equations

2018 ◽  
Vol 21 (2) ◽  
pp. 527-551 ◽  
Author(s):  
Zhinan Xia ◽  
Dingjiang Wang
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Difference Equations ◽  
Mild Solutions ◽  
Contraction Mapping Principle ◽  
Fractional Difference ◽  
Alternative Theorem ◽  
Fractional Difference Equations ◽  
Banach Contraction ◽  
Asymptotically Periodic

AbstractIn this paper, we establish some sufficient criteria for the existence, uniqueness of discrete weighted pseudo asymptotically periodic mild solutions and asymptotic behavior for nonlinear fractional difference equations in Banach space, where the nonlinear perturbation is Lipschitz type, or non-Lipschitz type. The results are a consequence of application of different fixed point theorems, namely, the Banach contraction mapping principle, Leray-Schauder alternative theorem and Matkowski’s fixed point technique.

scholarly journals On Hybrid Type Nonlinear Fractional Integrodifferential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 984
Author(s):  
Faten H. Damag ◽  
Adem Kılıçman ◽  
Awsan T. Al-Arioi
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Lower Bounds ◽  
Fixed Point Theorems ◽  
Approximate Solutions ◽  
Integrodifferential Equations ◽  
Upper And Lower Bounds ◽  
Hybrid Type ◽  
Fractional Equations

In this paper, we introduce and investigate a hybrid type of nonlinear Riemann Liouville fractional integro-differential equations. We develop and extend previous work on such non-fractional equations, using operator theoretical techniques, and find the approximate solutions. We prove the existence as well as the uniqueness of the corresponding approximate solutions by using hybrid fixed point theorems and provide upper and lower bounds to these solutions. Furthermore, some examples are provided, in which the general claims in the main theorems are demonstrated.

scholarly journals Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces

2019 ◽  
Vol 2 (2) ◽  
pp. 18 ◽  
Author(s):  
Dimplekumar Chalishajar ◽  
Chokkalingam Ravichandran ◽  
Shanmugam Dhanalakshmi ◽  
Rangasamy Murugesu
Keyword(s):  
Fixed Point ◽  
Group Theory ◽  
Differential Equations ◽  
Banach Spaces ◽  
Fractional Order ◽  
Fixed Point Theorems ◽  
Mild Solutions ◽  
Order System ◽  
Semi Group ◽  
Non Local

In this paper, we establish the existence of piece wise (PC)-mild solutions (defined in Section 2) for non local fractional impulsive functional integro-differential equations with finite delay. The proofs are obtained using techniques of fixed point theorems, semi-group theory and generalized Bellman inequality. In this paper, we used the distributed characteristic operators to define a mild solution of the system. We also discussed the controversy related to the solution operator for the fractional order system using weak and strong Caputo derivatives. Examples are given to illustrate the theory.

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