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.

SEMILINEAR NONLOCAL PROBLEMS WITHOUT THE ASSUMPTIONS OF COMPACTNESS IN BANACH SPACES

2010 ◽  
Vol 08 (02) ◽  
pp. 211-225 ◽  
Author(s):  
XINGMEI XUE
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Differential System ◽  
Hausdorff Measure ◽  
Measure Of Noncompactness ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Nonlocal Problems ◽  
Hausdorff Measure Of Noncompactness ◽  
Nonlocal Initial Conditions

In this paper, we study the semilinear differential equations with nonlocal initial conditions in the separable Banach spaces. We derive conditions expressed in terms of the Hausdorff measure of noncompactness under which the mild solutions exit. For illustration, a partial integral differential system is worked out.

scholarly journals Existence of the Mild Solution for Impulsive Semilinear Differential Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Alka Chadha ◽  
Dwijendra N. Pandey
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Mild Solution ◽  
Measure Of Noncompactness ◽  
Existence Of Solutions ◽  
Hausdorff Measure Of Noncompactness ◽  
Condensing Operator ◽  
Semilinear Differential Equation

We study the existence of solutions of impulsive semilinear differential equation in a Banach space X in which impulsive condition is not instantaneous. We establish the existence of a mild solution by using the Hausdorff measure of noncompactness and a fixed point theorem for the convex power condensing operator.

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 Exact Controllability for Hilfer Fractional Differential Inclusions Involving Nonlocal Initial Conditions

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Jun Du ◽  
Wei Jiang ◽  
Denghao Pang ◽  
Azmat Ullah Khan Niazi
Keyword(s):  
Fixed Point ◽  
Differential Inclusions ◽  
Measure Of Noncompactness ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Exact Controllability ◽  
Fractional Differential Inclusions ◽  
Nonlocal Initial Conditions ◽  
Fractional Differential System ◽  
Fractional Differential

The exact controllability results for Hilfer fractional differential inclusions involving nonlocal initial conditions are presented and proved. By means of the multivalued analysis, measure of noncompactness method, fractional calculus combined with the generalized Mo¨nch fixed point theorem, we derive some sufficient conditions to ensure the controllability for the nonlocal Hilfer fractional differential system. The results are new and generalize the existing results. Finally, we talk about an example to interpret the applications of our abstract results.

On Nonlinear Second Order Volterra-Fredholm Functional Integrodifferential Equation with Nonlocal Condition in Banach Spaces

2015 ◽  
Vol 54 (1) ◽  
pp. 75-96
Author(s):  
Machindra B. Dhakne ◽  
Poonam S. Bora
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Integrodifferential Equation ◽  
Measure Of Noncompactness ◽  
Nonlocal Condition ◽  
Existence Of Solution ◽  
Second Order ◽  
Fredholm Functional

Abstract Our purpose in this paper is to study the existence of solution of nonlinear second order mixed functional integrodifferential equation with nonlocal condition in Banach space by employing two different techniques namely the Darbo-Sadovskii's fixed point theorem with Hausdorff's measure of noncompactness and the Leray Schauder Alternative.

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 of solutions for some nonlinear delay integrodifferential equations with nonlocal initial conditions

2012 ◽  
Vol 45 (1) ◽  
Author(s):  
Zuomao Yan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Existence Of Solutions ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Integrodifferential Equations ◽  
Resolvent Operators ◽  
First Order ◽  
Nonlocal Initial Conditions ◽  
Nonlinear Delay

AbstractThe main purpose of this paper is the existence of mild solutions for a class of first-order nonlinear delay integrodifferential equations with nonlocal initial conditions in Banach spaces. We show that the solutions are given by the application of the theory of resolvent operators and the Sadovskii’s fixed point theorem. An example is presented in the end to show the applications of the obtained results.

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 Semilinear Volterra integrodifferential equations with nonlocal initial conditions

1999 ◽  
Vol 4 (2) ◽  
pp. 127-139 ◽  
Author(s):  
Sergiu Aizicovici ◽  
Mark Mckibben
Keyword(s):  
Banach Space ◽  
Global Existence ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Integrodifferential Equations ◽  
Cauchy Problems ◽  
Nonlocal Initial Conditions

We establish the global existence of mild solutions to a class of nonlocal Cauchy problems associated with semilinear Volterra integrodifferential equations in a Banach space.

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.

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