scholarly journals Existence of the Mild Solutions for Delay Fractional Integrodifferential Equations with Almost Sectorial Operators

2012 ◽  
Vol 2012 ◽  
pp. 1-22
Author(s):  
Fang Li
Keyword(s):  
Banach Space ◽  
Existence Theorem ◽  
Integrodifferential Equation ◽  
Separable Banach Space ◽  
Measure Of Noncompactness ◽  
Mild Solutions ◽  
Integrodifferential Equations ◽  
Resolvent Operators ◽  
Sectorial Operators ◽  
Almost Sectorial Operators

This paper is concerned with the existence of mild solutions for the fractional integrodifferential equations with finite delay and almost sectorial operators in a separable Banach spaceX. We obtain existence theorem for mild solutions to the above-mentioned equations, by means of measure of noncompactness and the resolvent operators associated with almost sectorial operators. As an application, the existence of mild solutions for some integrodifferential equation is obtained.

scholarly journals Existence Theory for Integrodifferential Equations and Henstock-Kurzweil Integral in Banach Spaces

2007 ◽  
Vol 2007 ◽  
pp. 1-12 ◽  
Author(s):  
Aneta Sikorska-Nowak
Keyword(s):  
Banach Space ◽  
Boundary Conditions ◽  
Banach Spaces ◽  
Integrodifferential Equation ◽  
Measure Of Noncompactness ◽  
Existence Theorems ◽  
Integrodifferential Equations ◽  
Existence Theory

We prove existence theorems for the integrodifferential equationx'(t)=f(t,x(t),∫0tk(t,s,x(s))ds),x(0)=x0,t∈Ia=[0,a],a>0, wheref,k,xare functions with values in a Banach spaceEand the integral is taken in the sense of HL. Additionally, the functionsfandksatisfy certain boundary conditions expressed in terms of the measure of noncompactness.

scholarly journals An Existence Result for Neutral Delay Integrodifferential Equations with Fractional Order and Nonlocal Conditions

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
Fang Li ◽  
Gaston M. N'Guérékata
Keyword(s):  
Banach Space ◽  
Existence Theorem ◽  
Fractional Order ◽  
Existence Result ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Integrodifferential Equations ◽  
Neutral Delay ◽  
Measures Of Noncompactness ◽  
Condensing Maps

We study the existence of mild solutions of a class of neutral delay integrodifferential equations with fractional order and nonlocal conditions in a Banach spaceX. An existence result on the mild solution is obtained by using the theory of the measures of noncompactness and the theory of condensing maps. Two examples are given to illustrate the existence theorem.

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.

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 THE SET OF SOLUTIONS OF INTEGRODIFFERENTIAL EQUATIONS IN BANACH SPACES

2008 ◽  
Vol 78 (3) ◽  
pp. 507-522 ◽  
Author(s):  
RAVI P. AGARWAL ◽  
DONAL O’REGAN ◽  
ANETA SIKORSKA-NOWAK
Keyword(s):  
Banach Space ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Integrodifferential Equation ◽  
Integrodifferential Equations ◽  
All Solutions ◽  
Image Position

AbstractIn this paper, we first prove an existence theorem for the integrodifferential equation (*)where f,k,x are functions with values in a Banach space E and the integral is taken in the sense of Henstock–Kurzweil–Pettis. In the second part of the paper we show that the set S of all solutions of the problem (*) is compact and connected in (C(Id,E),ω), where $I_{d} \subset I_{a} $.

scholarly journals A Nonlocal Cauchy Problem for Fractional Integrodifferential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Fang Li ◽  
Jin Liang ◽  
Tzon-Tzer Lu ◽  
Huan Zhu
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Measure Of Noncompactness ◽  
Mild Solutions ◽  
Integrodifferential Equations ◽  
The Fixed Point Theorem ◽  
Nonlocal Cauchy Problem ◽  
Condensing Maps

This paper is concerned with a nonlocal Cauchy problem for fractional integrodifferential equations in a separable Banach spaceX. We establish an existence theorem for mild solutions to the nonlocal Cauchy problem, by virtue of measure of noncompactness and the fixed point theorem for condensing maps. As an application, the existence of the mild solution to a nonlocal Cauchy problem for a concrete integrodifferential equation is obtained.

scholarly journals Existence of Mild Solutions for a Semilinear Integrodifferential Equation with Nonlocal Initial Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Carlos Lizama ◽  
Juan C. Pozo
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Integrodifferential Equation ◽  
Measure Of Noncompactness ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Hausdorff Measure Of Noncompactness ◽  
Nonlocal Initial Conditions ◽  
Closed Operators ◽  
Point Argument

Using Hausdorff measure of noncompactness and a fixed-point argument we prove the existence of mild solutions for the semilinear integrodifferential equation subject to nonlocal initial conditionsu′(t)=Au(t)+∫0tB(t-s)u(s)ds+f(t,u(t)),t∈[0,1],u(0)=g(u), whereA:D(A)⊆X→X, and for everyt∈[0,1]the mapsB(t):D(B(t))⊆X→Xare linear closed operators defined in a Banach spaceX. We assume further thatD(A)⊆D(B(t))for everyt∈[0,1], and the functionsf:[0,1]×X→Xandg:C([0,1];X)→XareX-valued functions which satisfy appropriate conditions.

scholarly journals Existence Results for an Impulsive Neutral Fractional Integrodifferential Equation with Infinite Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Alka Chadha ◽  
Dwijendra N. Pandey
Keyword(s):  
Banach Space ◽  
Mild Solution ◽  
Hausdorff Measure ◽  
Integrodifferential Equation ◽  
Measure Of Noncompactness ◽  
Infinite Delay ◽  
Existence Results ◽  
Hausdorff Measure Of Noncompactness ◽  
Arbitrary Banach Space ◽  
Fractional Integrodifferential Equation

We consider an impulsive neutral fractional integrodifferential equation with infinite delay in an arbitrary Banach spaceX. The existence of mild solution is established by using solution operator and Hausdorff measure of noncompactness.

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.

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