scholarly journals Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Ahmet Yantir ◽  
Ireneusz Kubiaczyk ◽  
Aneta Sikorska-Nowak
Keyword(s):  
Fixed Point ◽  
Integral Operator ◽  
Banach Spaces ◽  
Existence Result ◽  
Measure Of Noncompactness ◽  
Regularity Conditions ◽  
Kuratowski Measure Of Noncompactness ◽  
Mönch’S Fixed Point Theorem ◽  
Sturm Liouville ◽  
Carathéodory Solutions

AbstractIn this paper, we present the existence result for Carathéodory type solutions for the nonlinear Sturm- Liouville boundary value problem (SLBVP) in Banach spaces on an arbitrary time scale. For this purpose, we introduce an equivalent integral operator to the SLBVP by means of Green’s function on an appropriate set. By imposing the regularity conditions expressed in terms of Kuratowski measure of noncompactness, we prove the existence of the fixed points of the equivalent integral operator. Mönch’s fixed point theorem is used to prove the main result. Finally, we also remark that it is straightforward to guarantee the existence of Carathéodory solutions for the SLBVP if Kuratowski measure of noncompactness is replaced by any axiomatic measure of noncompactness.

scholarly journals Kuratowski MNC method on a generalized fractional Caputo Sturm–Liouville–Langevin q-difference problem with generalized Ulam–Hyers stability

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Abdelatif Boutiara ◽  
Maamar Benbachir ◽  
Sina Etemad ◽  
Shahram Rezapour
Keyword(s):  
Fixed Point ◽  
Measure Of Noncompactness ◽  
Theoretical Method ◽  
Difference Problem ◽  
Kuratowski Measure Of Noncompactness ◽  
Banach Contraction ◽  
Mönch’S Fixed Point Theorem ◽  
Sturm Liouville ◽  
The Banach Contraction Principle ◽  
Theoretical Results

AbstractIn this work, we consider a generalized quantum fractional Sturm–Liouville–Langevin difference problem with terminal boundary conditions. The relevant results rely on Mönch’s fixed point theorem along with a theoretical method by terms of Kuratowski measure of noncompactness (MNC) and the Banach contraction principle (BCP). Furthermore, two dynamical notions of Ulam–Hyers (UH) and generalized Ulam–Hyers (GUH) stability are addressed for solutions of the supposed Sturm–Liouville–Langevin quantum boundary value problem (q-FBVP). Two examples are presented to show the validity and also the effectiveness of theoretical results. In the last part of the paper, we conclude our exposition with some final remarks and observations.

Caputo-Hadamard fractional differential equations in banach spaces

2018 ◽  
Vol 21 (4) ◽  
pp. 1027-1045 ◽  
Author(s):  
Saïd Abbas ◽  
Mouffak Benchohra ◽  
Naima Hamidi ◽  
Johnny Henderson
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Banach Spaces ◽  
Fractional Differential Equations ◽  
Measure Of Noncompactness ◽  
Existence Results ◽  
Hadamard Fractional Differential Equations ◽  
Mönch’S Fixed Point Theorem ◽  
Fractional Differential

Abstract This article deals with some existence results for a class of Caputo–Hadamard fractional differential equations. The results are based on the Mönch’s fixed point theorem associated with the technique of measure of noncompactness. Two illustrative examples are presented.

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

2020 ◽  
Vol 4 (2) ◽  
pp. 104-115
Author(s):  
Khalil Ezzinbi ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Integrodifferential Equation ◽  
Resolvent Operator ◽  
Measure Of Noncompactness ◽  
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.

scholarly journals Investigation of the neutral fractional differential inclusions of Katugampola-type involving both retarded and advanced arguments via Kuratowski MNC technique

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sina Etemad ◽  
Mohammed Said Souid ◽  
Benoumran Telli ◽  
Mohammed K. A. Kaabar ◽  
Shahram Rezapour
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Differential Inclusions ◽  
Measure Of Noncompactness ◽  
Research Work ◽  
Neutral System ◽  
Fractional Differential Inclusions ◽  
Kuratowski Measure Of Noncompactness ◽  
Fractional Differential

AbstractA class of the boundary value problem is investigated in this research work to prove the existence of solutions for the neutral fractional differential inclusions of Katugampola fractional derivative which involves retarded and advanced arguments. New results are obtained in this paper based on the Kuratowski measure of noncompactness for the suggested inclusion neutral system for the first time. On the one hand, this research concerns the set-valued analogue of Mönch fixed point theorem combined with the measure of noncompactness technique in which the right-hand side is convex valued. On the other hand, the nonconvex case is discussed via Covitz and Nadler fixed point theorem. An illustrative example is provided to apply and validate our obtained results.

scholarly journals Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Abdelkrim Salim ◽  
Mouffak Benchohra ◽  
Erdal Karapınar ◽  
Jamal Eddine Lazreg
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Banach Spaces ◽  
Fractional Differential Equations ◽  
Fixed Point Theorems ◽  
Measure Of Noncompactness ◽  
Ulam Stability ◽  
Stability Of Solutions ◽  
Fractional Differential ◽  
Implicit Fractional Differential Equations

Abstract In this manuscript, we examine the existence and the Ulam stability of solutions for a class of boundary value problems for nonlinear implicit fractional differential equations with instantaneous impulses in Banach spaces. The results are based on fixed point theorems of Darbo and Mönch associated with the technique of measure of noncompactness. We provide some examples to indicate the applicability of our results.

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.

scholarly journals Fixed point theory of Mönch type for weakly sequentially upper semicontinuous maps

2000 ◽  
Vol 61 (3) ◽  
pp. 439-449 ◽  
Author(s):  
Donal O'Regan
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Banach Spaces ◽  
Weak Topology ◽  
Existence Result ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Upper Semicontinuous

A variety of fixed point results are presented for weakly sequentially upper semicontinuous maps. In addition an existence result is established for differential equations in Banach spaces relative to the weak topology.

scholarly journals Fixed point theorems for metric spaces with a conical geodesic bicombing

2017 ◽  
Vol 38 (5) ◽  
pp. 1642-1657 ◽  
Author(s):  
GIULIANO BASSO
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Existence Result ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Probability Measures

We derive two fixed point theorems for a class of metric spaces that includes all Banach spaces and all complete Busemann spaces. We obtain our results by the use of a $1$-Lipschitz barycenter construction and an existence result for invariant Radon probability measures. Furthermore, we construct a bounded complete Busemann space that admits an isometry without fixed points.

scholarly journals Solvability for generalized nonlinear two dimensional functional integral equations via measure of noncompactness

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Soniya Singh ◽  
Bhupander Singh ◽  
Kottakkaran Sooppy Nisar ◽  
Abd-Allah Hyder ◽  
M. Zakarya
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Integral Equations ◽  
Point Theorem ◽  
Existence Result ◽  
Measure Of Noncompactness ◽  
Functional Integral ◽  
Two Dimensional ◽  
Functional Integral Equations

AbstractIn this article, we provide the existence result for functional integral equations by using Petryshyn’s fixed point theorem connecting the measure of noncompactness in a Banach space. The results enlarge the corresponding results of several authors. We present fascinating examples of equations.

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