scholarly journals Best proximity point results and application to a system of integro-differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Anupam Das ◽  
Hemant Kumar Nashine ◽  
Rabha W. Ibrahim ◽  
Manuel De la Sen
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Measure Of Noncompactness ◽  
Best Proximity Point ◽  
Condensing Operator ◽  
The Stability ◽  
Best Proximity Pair

AbstractIn this work, we solve the system of integro-differential equations (in terms of Caputo–Fabrizio calculus) using the concepts of the best proximity pair (point) and measure of noncompactness. We first introduce the concept of cyclic (noncyclic) Θ-condensing operator for a pair of sets using the measure of noncompactness and then establish results on the best proximity pair (point) on Banach spaces and strictly Banach spaces. In addition, we have illustrated the considered system of integro-differential equations by three examples and discussed the stability, efficiency, and accuracy of solutions.

scholarly journals Cyclic (noncyclic) phi-condensing operator and its application to a system of differential equations

2019 ◽  
Vol 24 (6) ◽  
Author(s):  
Moosa Moosa Gabeleh ◽  
Philemon Moshokoa ◽  
Calogero Vetro
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Measure Of Noncompactness ◽  
Optimal Solution ◽  
Global Optimal Solution ◽  
Condensing Operator ◽  
Global Optimal ◽  
Condensing Operators ◽  
Strictly Convex Banach Spaces ◽  
Best Proximity Pair

We establish a best proximity pair theorem for noncyclic φ-condensing operators in strictly convex Banach spaces by using a measure of noncompactness. We also obtain a counterpart result for cyclic φ-condensing operators in Banach spaces to guarantee the existence of best proximity points, and so, an extension of Darbo’s fixed point theorem will be concluded. As an application of our results, we study the existence of a global optimal solution for a system of ordinary differential equations.

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.

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 k-Generalized Ψ-Hilfer differential equations in Banach spaces

2021 ◽  
Author(s):  
Salim Abdelkrim ◽  
Mouffak Benchohra ◽  
Jamal Lazreg ◽  
Gaston NGuerekata
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Measure Of Noncompactness ◽  
Initial Value ◽  
Hilfer Fractional Derivative ◽  
The Subject ◽  
Fractional Differential ◽  
Stability Results ◽  
Rassias Stability

In this paper, we prove some existence and Ulam-Hyers-Rassias stability results for a class of initial value problem for implicit nonlinear fractional differential equations and generalized Ψ-Hilfer fractional derivative in Banach spaces. The results are based on fixed point theorems of Darbo and Monch associated with the technique of measure of noncompactness. Illustrative examples are the subject of the last section.

On the Existence of Solutions of Ordinary Differential Equations in Banach Spaces

2015 ◽  
Vol 65 (3) ◽  
Author(s):  
Aldona Dutkiewicz
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Measure Of Noncompactness ◽  
Existence Of Solutions ◽  
Kuratowski Measure Of Noncompactness ◽  
Japan Acad

AbstractIn this paper we prove an existence theorem for ordinary differential equations in Banach spaces. The main assumptions in our results, formulated in terms of the Kuratowski measure of noncompactness, are motivated by the paper [CONSTANTIN, A.: On Nagumo’s theorem, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), 41-44].

scholarly journals An existence theorem for ordinary differential equations in Banach spaces

1984 ◽  
Vol 30 (3) ◽  
pp. 449-456 ◽  
Author(s):  
Bogdan Rzepecki
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Regularity Condition ◽  
Measure Of Noncompactness ◽  
Bounded Solution ◽  
Carathéodory Conditions

We prove the existence of bounded solution of the differential equation y′ = A(t)y + f(t, y) in a Banach space. The method used here is based on the concept of “admissibility” due to Massera and Schäffer when f satisfies the Caratheodory conditions and some regularity condition expressed in terms of the measure of noncompactness α.

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.

scholarly journals ON THE CONTROLLABILITY OF IMPULSIVE FUNCTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH SPACES

2019 ◽  
Vol 6 (1) ◽  
pp. 17-22
Author(s):  
Valliammal N ◽  
Ravichandran C
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Banach Spaces ◽  
Point Theorem ◽  
Measure Of Noncompactness ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Mönch Fixed Point Theorem

In this paper, we established the some sufficient conditions for controllability of impulsive functional integrodifferential equations with nonlocal conditions by using the measure of noncompactness and Monch fixed point theorem.

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