Existence and Stability Analysis for Fractional Differential Equations with Mixed Nonlocal Conditions

Asawasamrit, Suphawat; Nithiarayaphaks, Woraphak; Ntouyas, Sotiris K.; Tariboon, Jessada

doi:10.3390/math7020117

Cited by 11 publications

(4 citation statements)

References 27 publications

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“…Since the Ascoli‐Arzelà theorem is a powerful tool to study the compactness of a family of linear operators for evolution equations, while it just is considered on finite interval, see some applications in fractional initial value problems, 16,17 boundary value problems, 18–20 impulsive problems, 21,22 and so forth. In this paper, we will use the Schauder fixed point theorem to discuss the existence of solutions to problem (), where we need to obtain the compactness of a family of linear operator, for this purpose, the classical Ascoli‐Arzelà theorem shall do with some appropriate extensions, and we find, when doing not assume that function

g\left(t,\cdotp \right)

satisfies the Lipschitz condition, the existence of mild solutions on

\left(0,\infty \right)

will exist for problem ().…”

Section: Introductionmentioning

confidence: 99%

Cauchy problems for Hilfer fractional evolution equations on an infinite interval

Zhou

2022

Math Methods in App Sciences

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It is well known that the classical Ascoli-Arzelà theorem is powerful technique to give a necessary and sufficient condition for investigating the relative compactness of a family of abstract continuous functions, while it is limited to finite compact interval. In this paper, we shall generalize the Ascoli-Arzelà theorem on an infinite interval. As its application, we investigate an initial value problem for fractional evolution equations on infinite interval in the sense of Hilfer type, which is a generalization of both Riemann-Liuoville and Caputo fractional derivatives. Our methods are based on the Hausdorff theorem, classical/generalized Ascoli-Arzelà theorem, Schauder fixed point theorem, Wright function, and Kuratowski measure of noncompactness. We obtain the existence of mild solutions on an infinite interval when the semigroup is compact as well as noncompact.

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g\left(t,\cdotp \right)

satisfies the Lipschitz condition, the existence of mild solutions on

\left(0,\infty \right)

will exist for problem ().…”

Section: Introductionmentioning

confidence: 99%

Cauchy problems for Hilfer fractional evolution equations on an infinite interval

Zhou

2022

Math Methods in App Sciences

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“…Many researchers use this theorem to prove the existence and uniqueness of solutions in various nonlinear problems, i.e., differential equations, integral equations, optimization problems, etc. (see [1,11,17] and references therein).…”

Section: Introductionmentioning

confidence: 99%

On approximating fixed points of weak enriched contraction mappings via Kirk's iterative algorithm in Banach spaces

NITHIARAYAPHAKS¹,

SINTUNAVARAT²

2022

CJM

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Recently Berinde and Păcurar [Approximating fixed points of enriched contractions in Banach spaces. {\em J. Fixed Point Theory Appl.} {\bf 22} (2020), no. 2., 1--10], first introduced the idea of enriched contraction mappings and proved the existence of a fixed point of an enriched contraction mapping using the well-known fact that any fixed point of {the averaged mapping $T_\lambda$, where $\lambda\in (0,1]$, is also a fixed point of the initial mapping $T$}. In this work, we introduce the idea of weak enriched contraction mappings, and a new generalization of an averaged mapping called double averaged mapping. The first attempt is to prove the existence and uniqueness of the fixed point of a double averaged mapping associated with a weak enriched contraction mapping. Based on this result on Banach spaces, we give some sufficient conditions for the equality of all fixed points of a double averaged mapping and the set {of all fixed points of a weak enriched contraction mapping.} Moreover, our results show that an appropriate Kirk's iterative algorithm can be used to approximate a fixed point of a weak enriched contraction mapping. An illustrative example for showing the efficiency of our results is given.

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“…In addition, one of the most powerful techniques for stability analysis is Ulam's stability, which includes Ulam-Hyers (U H) stability, generalized Ulam-Hyers (GU H) stability, Ulam-Hyers-Rassias (U HR) stability, and generalized Ulam-Hyers-Rassias (GU HR) stability. It is useful because the properties of Ulam's stability guarantee the existence of solutions, and when the problem under consideration is Ulam's stability, it ensures that a close exact solution exists; see [32][33][34][35][36][37][38][39][40][41][42] and references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a Nonlinear ψ-Hilfer Fractional Integro-Differential Equation Describing Cantilever Beam Model with Nonlinear Boundary Conditions

et al. 2021

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In this paper, we establish sufficient conditions to approve the existence and uniqueness of solutions of a nonlinear implicit ψ-Hilfer fractional boundary value problem of the cantilever beam model with nonlinear boundary conditions. By using Banach’s fixed point theorem, the uniqueness result is proved. Meanwhile, the existence result is obtained by applying the fixed point theorem of Schaefer. Apart from this, we utilize the arguments related to the nonlinear functional analysis technique to analyze a variety of Ulam’s stability of the proposed problem. Finally, three numerical examples are presented to indicate the effectiveness of our results.

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Existence and Stability Analysis for Fractional Differential Equations with Mixed Nonlocal Conditions

Cited by 11 publications

References 27 publications

Cauchy problems for Hilfer fractional evolution equations on an infinite interval

Cauchy problems for Hilfer fractional evolution equations on an infinite interval

On approximating fixed points of weak enriched contraction mappings via Kirk's iterative algorithm in Banach spaces

Analysis of a Nonlinear ψ-Hilfer Fractional Integro-Differential Equation Describing Cantilever Beam Model with Nonlinear Boundary Conditions

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