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 Hilfer-type fractional differential switched inclusions with noninstantaneous impulsive and nonlocal conditions

2018 ◽  
Vol 23 (6) ◽  
pp. 921-941 ◽  
Author(s):  
JinRong Wang ◽  
Ahmed Gamal Ibrahim ◽  
Donal O’Regan
Keyword(s):  
Hausdorff Measure ◽  
Measure Of Noncompactness ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Continuous Functions ◽  
Inclusion Problem ◽  
Nonlocal Problems ◽  
Hausdorff Measure Of Noncompactness ◽  
New Class ◽  
Fractional Differential

In this paper, we study a new class of nonlocal problems for noninstantaneous impulsive Hilfer-type fractional differential switched inclusions in Banach spaces. First, we introduce a mild solution formula for this noninstantaneous impulsive inclusion problem. Second, we show the existence of mild solutions using the Hausdorff measure of noncompactness on the space of piecewise weighted continuous functions. Finally, an example is provided to illustrate the theory.

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.

scholarly journals Existence and regularity of mild solutions in some interpolation spaces for functional partial differential equations with nonlocal initial conditions

2018 ◽  
Vol 16 (1) ◽  
pp. 113-126
Author(s):  
Xuping Zhang ◽  
Qiyu Chen ◽  
Yongxiang Li
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Linear Part ◽  
Fractional Power ◽  
Interpolation Spaces ◽  
Partial Differential ◽  
Nonlocal Initial Conditions ◽  
Spatial Derivatives

AbstractThis paper is devoted to study the existence and regularity of mild solutions in some interpolation spaces for a class of functional partial differential equations with nonlocal initial conditions. The linear part is assumed to be a sectorial operator in Banach space X. The fractional power theory and α-norm are used to discuss the problem so that the obtained results can be applied to equations with terms involving spatial derivatives. Moreover, we present an example to illustrate the application of main results.

scholarly journals Solvability of infinite systems of fractional differential equations in the spaces of tempered sequences

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5519-5530 ◽  
Author(s):  
Anupam Das ◽  
Bipan Hazarika ◽  
Ravi Agarwal ◽  
Hemant Nashine
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Hausdorff Measure ◽  
Measure Of Noncompactness ◽  
Existence Of Solution ◽  
Hausdorff Measure Of Noncompactness ◽  
Infinite Systems ◽  
Fractional Differential

In this paper we discuss the existence of solution of infinite systems of fractional differential equations with the help of Hausdorff measure of noncompactness and Meir-Keeler fixed point theorem in the tempered sequence spaces. We provide examples to established the applicability of our results.

Banach spaces of absolutely k-summable series

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mehmet Ali Sarıgöl ◽  
Ravi P. Agarwal
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hausdorff Measure ◽  
Measure Of Noncompactness ◽  
Compact Operators ◽  
Matrix Operator ◽  
Hausdorff Measure Of Noncompactness ◽  
Bk Space ◽  
New Space ◽  
General Banach Space

Abstract In this paper, we present a general Banach space of absolutely k-summable series using a triangle matrix operator and prove that this is a BK-space isometrically isomorphic to the space ℓ k {\ell_{k}} . We also establish the α - {\alpha-} , β - {\beta-} , γ-duals and base of the new space. Finally, we qualify some matrix and compact operators on the new space making use of the Hausdorff measure of noncompactness. Our results include, as particular cases, a number of well-known results.

scholarly journals Infinite System of Differential Equations in Some Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
M. Mursaleen ◽  
Abdullah Alotaibi
Keyword(s):  
Differential Equations ◽  
Hausdorff Measure ◽  
Measure Of Noncompactness ◽  
Infinite System ◽  
Fixed Point Theory ◽  
Point Theory ◽  
System Of Differential Equations ◽  
Hausdorff Measure Of Noncompactness ◽  
Measures Of Noncompactness ◽  
Infinite Dimensional

The first measure of noncompactness was defined by Kuratowski in 1930 and later the Hausdorff measure of noncompactness was introduced in 1957 by Goldenštein et al. These measures of noncompactness have various applications in several areas of analysis, for example, in operator theory, fixed point theory, and in differential and integral equations. In particular, the Hausdorff measure of noncompactness has been extensively used in the characterizations of compact operators between the infinite-dimensional Banach spaces. In this paper, we present a brief survey on the applications of measures of noncompactness to the theory of infinite system of differential equations in some spaces and .

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