scholarly journals On solvability of the impulsive Cauchy problem for integro-differential inclusions with non-densely defined operators

2021 ◽  
Vol 379 (2191) ◽  
pp. 20190384
Author(s):  
Irene Benedetti ◽  
Valeri Obukhovskii ◽  
Valentina Taddei
Keyword(s):  
Cauchy Problem ◽  
Fixed Point ◽  
Compact Convex ◽  
Linear Part ◽  
Main Tool ◽  
Densely Defined Operator ◽  
Fan Theorem ◽  
Locally Convex ◽  
Ky Fan ◽  
Densely Defined

We prove the existence of at least one integrated solution to an impulsive Cauchy problem for an integro-differential inclusion in a Banach space with a non-densely defined operator. Since we look for integrated solution we do not need to assume that A is a Hille Yosida operator. We exploit a technique based on the measure of weak non-compactness which allows us to avoid any hypotheses of compactness both on the semigroup generated by the linear part and on the nonlinear term. As the main tool in the proof of our existence result, we are using the Glicksberg–Ky Fan theorem on a fixed point for a multivalued map on a compact convex subset of a locally convex topological vector space. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

CONTROLLABILITY OF NON-DENSELY DEFINED ABSTRACT MIXED VOLTERRA–FREDHOLM NEUTRAL FUNCTIONAL INTEGRODIFFERENTIAL EQUATIONS

2013 ◽  
Vol 06 (02) ◽  
pp. 1350025 ◽  
Author(s):  
Kishor D. Kucche ◽  
M. B. Dhakne
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Point Theorem ◽  
Semigroup Theory ◽  
Linear Part ◽  
Integrodifferential Equations ◽  
Resolvent Estimate ◽  
Densely Defined ◽  
Controllability Result

In this paper we establish the controllability result for class of mixed Volterra–Fredholm neutral functional integrodifferential equations in Banach spaces where the linear part is non-densely defined and satisfies the resolvent estimate of the Hille–Yosida condition. The results are obtained using the integrated semigroup theory and the Sadovskii's fixed point theorem.

scholarly journals Invariant subspace theorems for amenable groups

1989 ◽  
Vol 32 (3) ◽  
pp. 415-430 ◽  
Author(s):  
A. T. Lau ◽  
A. L. T. Paterson ◽  
J. C. S. Wong
Keyword(s):  
Invariant Subspace ◽  
Closed Subspace ◽  
Compact Convex ◽  
Amenable Group ◽  
Dimensional Subspace ◽  
Linear Operators ◽  
Finite Codimension ◽  
Subspace Theorem ◽  
Locally Convex ◽  
Ky Fan

In [5], Ky Fan proved the following remarkable amenability “invariant subspace” theorem:Let G be an amenable group of continuous, invertible linear operators acting on a locally convex space E. Let H be a closed subspace of finite codimension n in E and X⊂E be such that:(i) H and X are G-invariant;(ii) (e + H) ∩X is compact convex for all e ∈ E;(iii) X contains an n-dimensional subspace V of E. Then there exists an n-dimensional subspace of E contained in X and invariant under G.

scholarly journals A fixed point theorem and an application for the Cauchy problem in the scale of Banach spaces

Filomat ◽  
2020 ◽  
Vol 34 (13) ◽  
pp. 4387-4398 ◽  
Author(s):  
Vo Tri ◽  
Erdal Karapinar
Keyword(s):  
Cauchy Problem ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Convex Space ◽  
Locally Convex Space ◽  
Normed Spaces ◽  
Scale Of Banach Spaces ◽  
The Cauchy Problem ◽  
Locally Convex

The main aim of this paper is to prove the existence of the fixed point of the sum of two operators in setting of the cone-normed spaces with the values of cone-norm belonging to an ordered locally convex space. We apply this result to prove the existence of global solution of the Cauchy problem with perturbation of the form (x?(t) = f[t,x(t)] + g[t,x(t)], t ? [0,?), x(0) = x0? F1, in a scale of Banach spaces {(Fs,||.||) : s ? (0, 1]}.

The Embedding of Compact Convex Sets in Locally Convex Spaces

1978 ◽  
Vol 30 (03) ◽  
pp. 449-454 ◽  
Author(s):  
James W. Roberts
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Locally Convex Spaces ◽  
Hausdorff Topological Vector Space ◽  
Compact Convex Set ◽  
Affine Functions ◽  
Locally Convex

In studying compact convex sets it is usually assumed that the compact convex set X is contained in a Hausdorff topological vector space L where the topology on X is the relative topology. Usually one assumes that L is locally convex. The reason for this is that most of the major theorems such as the Krein-Milman, Choquet-Bishop-de Leeuw, and most of the fixed point theorems require that there be enough continuous affine functions on X to separate points.

A Set-Valued Generalization of Fan's Best Approximation Theorem

1992 ◽  
Vol 44 (4) ◽  
pp. 784-796 ◽  
Author(s):  
Xie Ping Ding ◽  
Kok-Keong Tan
Keyword(s):  
Fixed Point ◽  
Convex Subset ◽  
Best Approximation ◽  
Fixed Point Theorems ◽  
Approximation Theorem ◽  
Compact Convex ◽  
Convex Approximation ◽  
Weakly Compact ◽  
Hausdorff Topological Vector Space ◽  
Locally Convex

AbstractLet (E, T) be a Hausdorff topological vector space whose topological dual separates points of E, X be a non-empty weakly compact convex subset of E and W be the relative weak topology on X. If F: (X, W) → 2(E,T) is continuous (respectively, upper semi-continuous if £ is locally convex), approximation and fixed point theorems are obtained which generalize the corresponding results of Fan, Park, Reich and Sehgal-Singh-Smithson (respectively, Ha, Reich, Park, Browder and Fan) in several aspects.

scholarly journals On some optimization problems in not necessarily locally convex space

2008 ◽  
Vol 18 (2) ◽  
pp. 167-172
Author(s):  
Ljiljana Gajic
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Existence Theorem ◽  
Convex Space ◽  
Optimization Problems ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Equilibrium Existence ◽  
Cooperative Equilibrium ◽  
Locally Convex

In this note, by using O. Hadzic's generalization of a fixed point theorem of Himmelberg, we prove a non - cooperative equilibrium existence theorem in non - compact settings and a generalization of an existence theorem for non - compact infinite optimization problems, all in not necessarily locally convex spaces.

scholarly journals Hopf bifurcation for an SIR model with age structure

2021 ◽  
Author(s):  
HUI CAO ◽  
Dongxue Yan ◽  
Xiaxia Xu
Keyword(s):  
Cauchy Problem ◽  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Age Structure ◽  
Equilibrium Points ◽  
Sir Model ◽  
Numerical Examples ◽  
System Parameters ◽  
Stability And Instability ◽  
Densely Defined

This paper deals with an SIR model with age structure of infected individuals. We formulate the model as an abstract non-densely defined Cauchy problem and derive the conditions for the existence of all the feasible equilibrium points of the system. The criteria for both stability and instability involving system parameters are obtained. Bifurcation analysis indicates that the system with age structure exhibits Hopf bifurcation which is the main result of this paper. Finally, some numerical examples are provided to illustrate our obtained results.

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