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1
Approximate controllability results in α-norm for some partial functional integrodifferential equations with nonlocal initial conditions in Banach spaces
Ndambomve Patrice, Kpoumie Moussa El-Khalil, Ezzinbi Khalil
Journal of Applied Analysis
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2023
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Vol. 29, nr 1
127--142
EN
In this work, we discuss the approximate controllability of some nonlinear partial functional integrodifferential equations with nonlocal initial condition in Hilbert spaces.We assume that the corresponding linear part is approximately controllable. The results are obtained by using fractional power theory and α-norm, the measure of noncompactness and theMönch fixed-point theorem, and the theory of analytic resolvent operators for integral equations. As a result, we obtain a generalization of the work of Mahmudov [N. I. Mahmudov, Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Anal. 68 (2008), no. 3, 536-546], without assuming the compactness of the resolvent operator. Our results extend and complement many other important results in the literature. Finally, a concrete example is given to illustrate the application of the main results.
2
Content available On the hybrid Caputo-proportional fractional differential inclusions in Banach spaces
Abbas Mohamed I.
Journal of Mathematics and Applications
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2022
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Vol. 45
5--20
EN
The current article concerns an existence criteria of solutions of nonlinear fractional differential inclusions in the sense of the hybrid Caputo-proportional fractional derivatives in Banach space. The investigation of the main result relies on the set-valued issue of Mönch fixed point theorem incorporated with the Kuratowski measure of non-compactness.
3
Analyzing the existence of solution of a fractional order integral equation: A fixed point approach
Saha Dipankar, Sen Mausumi, Roy Santanu
Journal of Applied Analysis
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2022
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Vol. 28, nr 2
199--210
EN
This paper signifies the beauty of fixed point theory in the context of attaining the sufficient condition for the existence of the solution of a non-linear functional-integral equation in an unbounded interval. Here a non-linear integral equation (NLIE) involving a fractional operator is taken in the form of an operator equation. Utilizing the perception of measure of noncompactness (MNC) along with certain relevant assumptions, it is proved that the operator equation satisfies the Darbo condition for the product of operators in a Banach algebra. Finally, the result obtained is verified by the assistance of the numerical example.
4
Controllability of multi-term time-fractional differential systems with state-dependent delay
Chaudhary Renu, Singh Vikram, Pandey D. N.
Journal of Applied Analysis
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2020
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Vol. 26, nr 2
241--255
EN
In this paper, controllability results for a class of multi-term time-fractional differential systems with state-dependent delay have been studied. The concept of fractional calculus, measure of noncompactness and Mönch fixed-point theorem has been implemented to obtain a new set of controllability results. Finally, an application is given to illustrate the obtained results.
5
Content available The existence of monotonic solutions of a class of quadratic integral equations of Volterra type
Karakurt Osman, Temizer Ӧmer Faruk
Journal of Mathematics and Applications
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2019
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Vol. 42
117--133
EN
Using the technique associated with measure of noncompactness we prove the existence of monotonic solutions of a class of quadratic integral equation of Volterra type in the Banach space of real functions defined and continuous on a bounded and closed interval.
6
Content available Nonincreasing solutions for quadratic integral equations of Convolution type
El-Sayed W. G., Abd El-Mowla A. A. H.
Journal of Mathematics and Applications
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2019
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Vol. 42
95--107
EN
We study a nonlinear quadratic integral equation of Convolution type in the Banach space of real functions defined and continuous on a bounded and closed interval. By using a suitable measure of noncompactness, we show that the integral equation has monotonic solutions.
7
Content available remote Generalizations of Darbo’s fixed point theorem for new condensing operators with application to a functional integral equation
Rehman Habib ur, Gopal Dhananjay, Kumam Poom
Demonstratio Mathematica
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2019
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Vol. 52, nr 1
166--182
EN
In this paper, we provide some generalizations of the Darbo’s fixed point theorem associated with the measure of noncompactness and present some results on the existence of the coupled fixed point theorems for a special class of operators in a Banach space. To acquire this result, we define α-ψ and β-ψ con-densing operators and using them we propose new fixed point results. Our results generalize and extendsome comparable results from the literature. Additionally, as an application, we apply the obtained fixedpoint theorems to study the nonlinear functional integral equations.
8
Content available remote Essential norm estimates for multilinear singular and fractional integrals
Meskhi Alexander
Commentationes Mathematicae
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2019
|
Vol. 59, No. 1/2
81--93
EN
We derive lower two-weight estimates for the essential norm (measure of noncompactness) for multilinear Hilbert and Riesz transforms, and Riesz potential operators in Banach function lattices. As a corollary we have appropriate results in weighted Lebesgue spaces. From these statements we conclude that there is no (m + 1)-tuple of weights (v, w1, wm) for which these operators are compact from Lw1p1 ×…×Lwmpm to Lvq.
9
Content available remote On the existence of solutions of a perturbed functional integral equation in the space of Lebesgue integrable functions on R+
Al Sayed W., Darwish M. A.
Journal of Mathematics and Applications
|
2018
|
Vol. 41
19--27
EN
In this paper, we investigate and study the existence of solutions for perturbed functional integral equations of convolution type using Darbo's fixed point theorem, which is associated with the measure of noncompactness in the space of Lebesgue integrable functions on R+. Finally, we offer an example to demonstrate that our abstract result is applicable.
10
Content available On a cubic integral equation of Urysohn type with linear perturbation of second kind
Awad H. K., Darwish M. A., Metwali M. M. A.
Journal of Mathematics and Applications
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2018
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Vol. 41
29--38
EN
In this paper, we concern by a very general cubic integral equation and we prove that this equation has a solution in C[0; 1]. We apply the measure of noncompactness introduced by Banaś and Olszowy and Darbo's fixed point theorem to establish the proof of our main result.
11
Content available Nonlinear fractional differential equations with non-instantaneous impulses in Banach spaces
Benchohra M., Slimane M.
Journal of Mathematics and Applications
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2018
|
Vol. 41
39--51
EN
This paper is devoted to study the existence of solutions for a class of initial value problems for non-instantaneous impulsive fractional differential equations involving the Caputo fractional derivative in a Banach space. The arguments are based upon Monch's fixed point theorem and the technique of measures of noncompactness.
12
Content available remote Second order evolution differential functional equations with infinite delay
Człapiński T., Karpowicz A.
Commentationes Mathematicae
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2018
|
Vol. 58, No. 1/2
93--104
EN
We consider a second order semilinear functional evolution equation with infinite delay in a Banach space. We prove the existence of mild solutions for this equation using the measure of noncompactness technique and the Schauder fixed point theorem.
13
Content available On some qualitative properties of integrable solutions for Cauchy-type problem of fractional order
Metwali M. M. A.
Journal of Mathematics and Applications
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2017
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Vol. 40
121--p134
EN
The paper discusses the existence of solutions for Cauchy-type problem of fractional order in the space of Lebesgue integrable functions on bounded interval. Some qualitative properties of solutions are presented such as monotonicity, uniqueness and continuous dependence on the initial data. The main tools used are measure of weak (strong) noncompactness, Darbo fixed point theorem and fractional calculus.
14
Content available Measures of noncompactness in a Banach algebra and their applications
Dudek S.
Journal of Mathematics and Applications
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2017
|
Vol. 40
69--84
EN
In this paper we study the existence of solutions of a nonlinear quadratic integral equation of fractional order. This equation is considered in the Banach space of real functions defined, continuous and bounded on the real half axis. Additionally, using the technique of measures of noncompactness we obtain some characterization of considered integral equation. We provide also an example illustrating the applicability of our approach.
15
Content available Fractional evolution equation nonlocal problems with noncompact semigroups
Zhang X, Chen P.
Opuscula Mathematica
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2016
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Vol. 36, no. 1
123--137
EN
This paper is concerned with the existence results of mild solutions to the nonlocal problem of fractional semilinear integro-differential evolution equations. New existence theorems are obtained by means of the fixed point theorem for condensing maps. The results extend and improve some related results in this direction.
16
Content available Measure of noncompactness and neutral functional differential equations with state-dependent delay
Benchohra M., Henderson J., Medjadj I.
Journal of Mathematics and Applications
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2016
|
Vol. 39
23--45
EN
Our aim in this work is to study the existence of solutions of first and second order for neutral functional differential equations with state-dependent delay. We use the Mönch’s fixed point theorem for the existence of solutions and the concept of measures of noncompactness.
17
Content available On the solutions of a class of nonlinear functional integral quations in space C [0,a]
Özdemir İ., Çakan Ü.
Journal of Mathematics and Applications
|
2015
|
Vol. 38
105--114
EN
The principal aim of this paper is to give sufficient conditions for solvability of a class of some nonlinear functional integral equations in the space of continuous functions defined on interval [0,a]. The main tool used in our study is associated with the technique of measures of noncompactness. We give also some examples satisfying the conditions of our main theorem but not satisfying the conditions in [8].
18
Content available Solvability of functional quadratic integral equations with perturbation
Metwali M. A. M.
Opuscula Mathematica
|
2013
|
Vol. 33, no. 4
725--739
EN
We study the existence of solutions of the functional quadratic integral equation with a perturbation term in the space of Lebesgue integrable functions on an unbounded interval by using the Krasnoselskii fixed point theory and the measure of weak noncompactness.
19
Content available remote Existence of solutions of the dynamic Cauchy problem in Banach spaces
Cichoń M., Kubiaczyk I., Sikorska-Nowak A., Yantir A.
Demonstratio Mathematica
|
2012
|
Vol. 45, nr 3
561-573
EN
In this paper we obtain the existence of solutions and Carathéodory type solutions of the dynamic Cauchy problem in Banach spaces for functions defined on time scales (…), where f is continuous or f satisfies Carathéodory conditions and some conditions expressed in terms of measures of noncompactness. The Mönch fixed point theorem is used to prove the main result, which extends these obtained for real valued functions.
20
Content available remote An existence theorem for solutions of an integro-differential equation in Banach spaces
Dutkiewicz A.
Demonstratio Mathematica
|
2012
|
Vol. 45, nr 4
895-899
EN
The paper contains an existence theorem for local solutions of an initial value problem for a nonlinear integro-differential equation in Banach spaces. The assumptions and proofs are expressed in terms of measures of noncompactness.
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