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

Existence Theorems ◽

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.

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

Fasciculi Mathematici ◽

10.1515/fascmath-2015-0005 ◽

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 ◽

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.

Nonlinear Integrodifferential Equations of Mixed Type in Banach Spaces

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2007/65947 ◽

2007 ◽

Vol 2007 ◽

pp. 1-14 ◽

Cited By ~ 1

Author(s):

Aneta Sikorska-Nowak ◽

Grzegorz Nowak

Keyword(s):

Banach Space ◽

Mixed Type ◽

Existence Theorems ◽

Continuous Functions ◽

Measure Of Weak Noncompactness ◽

Equations Of Mixed Type ◽

Weakly Sequentially Continuous ◽

Equation Of Mixed Type

We prove two existence theorems for the integrodifferential equation of mixed type:x'(t)=f(t,x(t),∫0tk1(t,s)g(s,x(s))ds,∫0ak2(t,s)h(s,x(s))ds),x(0)=x0, where in the first part of this paperf, g, h, xare functions with values in a Banach spaceEand integrals are taken in the sense of Henstock-Kurzweil (HK). In the second partf, g, h, xare weakly-weakly sequentially continuous functions and integrals are taken in the sense of Henstock-Kurzweil-Pettis (HKP) integral. Additionally, the functionsf, g, h, xsatisfy some conditions expressed in terms of the measure of noncompactness or the measure of weak noncompactness.

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

Abstract and Applied Analysis ◽

10.1155/2012/729615 ◽

2012 ◽

Vol 2012 ◽

pp. 1-22

Author(s):

Fang Li

Keyword(s):

Banach Space ◽

Existence Theorem ◽

Separable Banach Space ◽

Mild Solutions ◽

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.

THE SET OF SOLUTIONS OF INTEGRODIFFERENTIAL EQUATIONS IN BANACH SPACES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972708000944 ◽

2008 ◽

Vol 78 (3) ◽

pp. 507-522 ◽

Cited By ~ 4

Author(s):

RAVI P. AGARWAL ◽

DONAL O’REGAN ◽

ANETA SIKORSKA-NOWAK

Keyword(s):

Banach Space ◽

Existence Theorem ◽

Banach Spaces ◽

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} $.

Existence of Solutions for Fractional Impulsive Integrodifferential Equations in Banach Spaces

International Journal of Differential Equations ◽

10.1155/2016/5648798 ◽

2016 ◽

Vol 2016 ◽

pp. 1-11

Author(s):

Haide Gou ◽

Baolin Li

Keyword(s):

Banach Space ◽

Fixed Point ◽

Banach Spaces ◽

Fixed Point Theorems ◽

Evolution Equations ◽

Existence Of Solutions ◽

Nonlocal Conditions ◽

Fractional Evolution Equations

We investigate the existence of solutions for a class of impulsive fractional evolution equations with nonlocal conditions in Banach space by using some fixed point theorems combined with the technique of measure of noncompactness. Our results improve and generalize some known results corresponding to those obtained by others. Finally, two applications are given to illustrate that our results are valuable.

Controllability for some nonlinear impulsive partial functional integrodifferential systems with infinite delay in Banach spaces

Open Journal of Mathematical Analysis ◽

10.30538/psrp-oma2020.0069 ◽

2020 ◽

Vol 4 (2) ◽

pp. 104-115

Author(s):

Khalil Ezzinbi ◽

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Banach Spaces ◽

Resolvent Operator ◽

Infinite Delay ◽

Sufficient Conditions ◽

Mönch Fixed Point Theorem ◽

The Resolvent Operator

This work concerns the study of the controllability for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces. We give sufficient conditions that ensure the controllability of the system by supposing that its undelayed part admits a resolvent operator in the sense of Grimmer, and by making use of the measure of noncompactness and the Mönch fixed-point Theorem. As a result, we obtain a generalization of the work of K. Balachandran and R. Sakthivel (Journal of Mathematical Analysis and Applications, 255, 447-457, (2001)) and a host of important results in the literature, without assuming the compactness of the resolvent operator. An example is given for illustration.

An Application Of The Theory Of Scale Of Banach Spaces

Annales Mathematicae Silesianae ◽

10.1515/amsil-2015-0005 ◽

2015 ◽

Vol 29 (1) ◽

pp. 51-59

Author(s):

Łukasz Dawidowski

Keyword(s):

Banach Space ◽

Cauchy Problem ◽

Boundary Conditions ◽

Heat Equation ◽

Banach Spaces ◽

Abstract Cauchy Problem ◽

Initial Boundary ◽

Scale Of Banach Spaces ◽

The Cauchy Problem ◽

Selection Of

AbstractThe abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem ut − Δu = u|u|s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.

Fixed point theorems of discontinuous increasing operators and applications to nonlinear integro-differential equations

Nagoya Mathematical Journal ◽

10.1017/s0027763000007303 ◽

2000 ◽

Vol 158 ◽

pp. 73-86

Author(s):

Jinqing Zhang

Keyword(s):

Banach Space ◽

Fixed Point ◽

Differential Equations ◽

Fixed Points ◽

Banach Spaces ◽

Fixed Point Theorems ◽

Existence Theorems ◽

Maximal And Minimal Solutions

AbstractIn this paper, we obtain some new existence theorems of the maximal and minimal fixed points for discontinuous increasing operators in C[I,E], where E is a Banach space. As applications, we consider the maximal and minimal solutions of nonlinear integro-differential equations with discontinuous terms in Banach spaces.

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

International Journal of Differential Equations ◽

10.1155/2014/780636 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 5

Author(s):

Alka Chadha ◽

Dwijendra N. Pandey

Keyword(s):

Banach Space ◽

Mild Solution ◽

Hausdorff Measure ◽

Fractional Integrodifferential Equation

Infinite Delay ◽

Existence Results ◽

Hausdorff Measure Of Noncompactness ◽

Arbitrary Banach Space ◽

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.

Weak solutions of differential equations in Banach spaces

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700003993 ◽

1986 ◽

Vol 33 (3) ◽

pp. 407-418 ◽

Cited By ~ 8

Author(s):

Nikolaos S. Papageorgiou

Keyword(s):

Banach Space ◽

Cauchy Problem ◽

Vector Field ◽

Banach Spaces ◽

Existence Result ◽

Continuous Vector ◽

Weakly Continuous ◽

Continuous Vector Field ◽

The Cauchy Problem

We consider the Cauchy problem x (t) = f (t,x (t)), x (0) = x0 in a nonreflexive Banach space X and for f: T × X → X a weakly continuous vector field. Using a compactness hypothesis involving a weak measure of noncompactness we prove an existence result that generalizes earlier theorems by Chow-Shur, Kato and Cramer-Lakshmikantham-Mitchell.