scholarly journals On Semilinear Integro-Differential Equations with Nonlocal Conditions in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-26
Author(s):  
Tran Dinh Ke ◽  
Valeri Obukhovskii ◽  
Ngai-Ching Wong ◽  
Jen-Chih Yao
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Banach Spaces ◽  
Topological Structure ◽  
Continuous Dependence ◽  
Nonlocal Conditions ◽  
Abstract Cauchy Problem ◽  
Nonlinear Perturbations ◽  
Continuous Dependence Results

We study the abstract Cauchy problem for a class of integrodifferential equations in a Banach space with nonlinear perturbations and nonlocal conditions. By using MNC estimates, the existence and continuous dependence results are proved. Under some additional assumptions, we study the topological structure of the solution set.

scholarly journals An Application Of The Theory Of Scale Of Banach Spaces

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.

scholarly journals Solvability of Impulsive Nonlinear Partial Differential Equations with Nonlocal Conditions

2020 ◽  
pp. 140-152
Author(s):  
Ali Kadhim Jabbar ◽  
Sameer Qasim Hasan
Keyword(s):  
Cauchy Problem ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Theoretical Approach ◽  
Nonlinear Partial Differential Equations ◽  
Nonlocal Conditions ◽  
Abstract Cauchy Problem ◽  
Beam Equation ◽  
Partial Differential

The aim of this paper is to investigate the theoretical approach for solvability of impulsive abstract Cauchy problem for impulsive nonlinear fractional order partial differential equations with nonlocal conditions, where the nonlinear extensible beam equation is a particular application case of this problem.

scholarly journals Kneser's theorem for differential equations in Banach spaces

1986 ◽  
Vol 33 (3) ◽  
pp. 419-434 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Vector Field ◽  
Differential Equations ◽  
Banach Spaces ◽  
Measure Of Noncompactness ◽  
Solution Set ◽  
The Cauchy Problem ◽  
Uniformly Continuous

We consider the Cauchy problem x (t) = f (t,x (t)), x (O) = xO defined in a nonreflexive Banach space and with the vector field f: T × X → X being weakly uniformly continuous. Using a compactness hypothesis that involves the weak measure of noncompactness, we prove that the solution set of the above Cauchy problem is nonempty, connected and compact in .

scholarly journals Normal periodic solutions for the fractional abstract Cauchy problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jennifer Bravo ◽  
Carlos Lizama
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Periodic Solution ◽  
Periodic Boundary ◽  
Abstract Cauchy Problem ◽  
Periodic Boundary Conditions ◽  
Sectorial Operator ◽  
Closed Linear Operator ◽  
Resolvent Set ◽  
Spectral Angle

AbstractWe show that if A is a closed linear operator defined in a Banach space X and there exist $t_{0} \geq 0$ t 0 ≥ 0 and $M>0$ M > 0 such that $\{(im)^{\alpha }\}_{|m|> t_{0}} \subset \rho (A)$ { ( i m ) α } | m | > t 0 ⊂ ρ ( A ) , the resolvent set of A, and $$ \bigl\Vert (im)^{\alpha }\bigl(A+(im)^{\alpha }I \bigr)^{-1} \bigr\Vert \leq M \quad \text{ for all } \vert m \vert > t_{0}, m \in \mathbb{Z}, $$ ∥ ( i m ) α ( A + ( i m ) α I ) − 1 ∥ ≤ M  for all  | m | > t 0 , m ∈ Z , then, for each $\frac{1}{p}<\alpha \leq \frac{2}{p}$ 1 p < α ≤ 2 p and $1< p < 2$ 1 < p < 2 , the abstract Cauchy problem with periodic boundary conditions $$ \textstyle\begin{cases} _{GL}D^{\alpha }_{t} u(t) + Au(t) = f(t), & t \in (0,2\pi ); \\ u(0)=u(2\pi ), \end{cases} $$ { D t α G L u ( t ) + A u ( t ) = f ( t ) , t ∈ ( 0 , 2 π ) ; u ( 0 ) = u ( 2 π ) , where $_{GL}D^{\alpha }$ D α G L denotes the Grünwald–Letnikov derivative, admits a normal 2π-periodic solution for each $f\in L^{p}_{2\pi }(\mathbb{R}, X)$ f ∈ L 2 π p ( R , X ) that satisfies appropriate conditions. In particular, this happens if A is a sectorial operator with spectral angle $\phi _{A} \in (0, \alpha \pi /2)$ ϕ A ∈ ( 0 , α π / 2 ) and $\int _{0}^{2\pi } f(t)\,dt \in \operatorname{Ran}(A)$ ∫ 0 2 π f ( t ) d t ∈ Ran ( A ) .

scholarly journals On the n-parameter abstract Cauchy problem

2004 ◽  
Vol 69 (3) ◽  
pp. 383-394
Author(s):  
M. Janfada ◽  
A. Niknam
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Unique Solution ◽  
Initial Value Problems ◽  
Abstract Cauchy Problem ◽  
Linear Operators ◽  
Uniqueness Of Solution ◽  
Initial Value ◽  
Bounded Linear ◽  
Closed Operators

Let Hi(i = 1, 2, …, n), be closed operators in a Banach space X. The generalised initialvalue problem of the abstract Cauchy problem is studied. We show that the uniqueness of solution u: [0, T1] × [0, T2] × … × [0, Tn] → X of this n-abstract Cauchy problem is closely related to C0-n-parameter semigroups of bounded linear operators on X. Also as another application of C0-n-parameter semigroups, we prove that many n-parameter initial value problems cannot have a unique solution for some initial values.

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

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.

scholarly journals Impulsive integro-differential equations with nonlocal conditions in Banach spaces

2017 ◽  
Vol 171 (3) ◽  
pp. 304-315 ◽  
Author(s):  
Mouhamadou Alpha Diallo ◽  
Khalil Ezzinbi ◽  
Abdoulaye Séne
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Nonlocal Conditions

scholarly journals Weak solutions of differential equations in Banach spaces

1986 ◽  
Vol 33 (3) ◽  
pp. 407-418 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Vector Field ◽  
Banach Spaces ◽  
Existence Result ◽  
Measure Of Noncompactness ◽  
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.

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