scholarly journals Nonlinear Integrodifferential Equations of Mixed Type in Banach Spaces

2007 ◽  
Vol 2007 ◽  
pp. 1-14 ◽  
Author(s):  
Aneta Sikorska-Nowak ◽  
Grzegorz Nowak
Keyword(s):  
Banach Space ◽  
Integrodifferential Equation ◽  
Mixed Type ◽  
Measure Of Noncompactness ◽  
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.

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 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 Interaction of homoclinic solutions and Hopf points in functional differential equations of mixed type

2009 ◽  
Vol 139 (1) ◽  
pp. 123-155 ◽  
Author(s):  
Marc Georgi
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Homoclinic Orbit ◽  
Mixed Type ◽  
Functional Differential Equations ◽  
Homoclinic Orbits ◽  
Homoclinic Solution ◽  
Equations Of Mixed Type ◽  
Functional Differential ◽  
Equation Of Mixed Type

We study a homoclinic bifurcation in a general functional differential equation of mixed type. More precisely, we investigate the case when the asymptotic steady state of a homoclinic solution undergoes a Hopf bifurcation. Bifurcations of this kind are diffcult to analyse due to the lack of Fredholm properties. In particular, a straightforward application of a Lyapunov–Schmidt reduction is not possible.As one of the main results we prove the existence of centre-stable and centre-unstable manifolds of steady states near homoclinic orbits. With their help, we can analyse the bifurcation scenario similar to the case for ordinary differential equations and can show the existence of solutions which bifurcate near the homoclinic orbit, are decaying in one direction and oscillatory in the other direction. These solutions can be visualized as an interaction of the homoclinic orbit and small periodic solutions that exist on account of the Hopf bifurcation, for exactly one asymptotic direction t→8 or t→−∞.

scholarly journals Terminal value problems of impulsive integro-differential equations in Banach spaces

1997 ◽  
Vol 10 (1) ◽  
pp. 71-78 ◽  
Author(s):  
Dajun Guo
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Banach Spaces ◽  
Mixed Type ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
First Order ◽  
Monotone Iterative ◽  
Equations Of Mixed Type ◽  
Nonnegative Solutions

This paper uses cone theory and the monotone iterative technique to investigate the existence of minimal nonnegative solutions of terminal value problems for first order nonlinear impulsive integro-differential equations of mixed type in a Banach space.

scholarly journals Asymptotic behaviour of solutions of nonlinear delay difference equations in Banach spaces

2005 ◽  
Vol 2005 (17) ◽  
pp. 2769-2774
Author(s):  
Anna Kisiolek ◽  
Ireneusz Kubiaczyk
Keyword(s):  
Banach Space ◽  
Asymptotic Behaviour ◽  
Banach Spaces ◽  
Difference Equations ◽  
Measure Of Noncompactness ◽  
Nonlinear Difference Equations ◽  
Measure Of Weak Noncompactness ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonlinear Delay

We consider the second-order nonlinear difference equations of the formΔ(rn−1Δxn−1)+pnf(xn−k)=hn. We show that there exists a solution(xn), which possesses the asymptotic behaviour‖xn−a∑j=0n−1(1/rj)+b‖=o(1),a,b∈ℝ. In this paper, we extend the results of Agarwal (1992), Dawidowski et al. (2001), Drozdowicz and Popenda (1987), M. Migda (2001), and M. Migda and J. Migda (1988). We suppose thatfhas values in Banach space and satisfies some conditions with respect to the measure of noncompactness and measure of weak noncompactness.

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 Global Asymptotic Stability for Nonlinear Functional Integral Equation of Mixed Type

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Zhinan Xia
Keyword(s):  
Integral Equation ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Mixed Type ◽  
Measure Of Noncompactness ◽  
Functional Integral ◽  
Nonlinear Functional ◽  
The Fixed Point Theorem ◽  
Equation Of Mixed Type ◽  
Stability Of The Solution

The existence results of global asymptotic stability of the solution are proved for functional integral equation of mixed type. The measure of noncompactness and the fixed-point theorem of Darbo are the main tools in carrying out our proof. Furthermore, some examples are given to show the efficiency and usefulness of the main findings.

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.

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