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12 pages, 392 KiB

Open AccessArticle

Ulam–Hyers Stability of Linear Differential Equation with General Transform

by Sandra Pinelas, Arunachalam Selvam and Sriramulu Sabarinathan

Symmetry 2023, 15(11), 2023; https://doi.org/10.3390/sym15112023 - 5 Nov 2023

Cited by 14 | Viewed by 1829

The main aim of this study is to implement the general integral transform technique to determine Ulam-type stability and Ulam–Hyers–Mittag–Leffer stability. We are given suitable examples to validate and support the theoretical results. As an application, the general integral transform is used to find Ulam stability of differential equations arising in Thevenin equivalent electrical circuit system. The results are graphically represented, which provides a clear and thorough explanation of the suggested method. Full article

(This article belongs to the Special Issue Theory and Applications of Special Functions II)

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17 pages, 336 KiB

Open AccessArticle

C*-Algebra-Valued Partial Modular Metric Spaces and Some Fixed Point Results

by Santanu Narzary, Dipankar Das, Yumnam Mahendra Singh, Mohammad Saeed Khan and Salvatore Sessa

Symmetry 2023, 15(6), 1135; https://doi.org/10.3390/sym15061135 - 23 May 2023

Cited by 3 | Viewed by 2125

Abstract

In the present paper, we introduce the notion of

C^{*}

-algebra-valued partial modular metric space satisfying the symmetry property that generalizes partial modular metric space,

C^{*}

-algebra-valued partial metric space, and

C^{*}

-algebra-valued modular metric space and discuss it with [...] Read more.

In the present paper, we introduce the notion of

C^{*}

-algebra-valued partial modular metric space satisfying the symmetry property that generalizes partial modular metric space,

C^{*}

-algebra-valued partial metric space, and

C^{*}

-algebra-valued modular metric space and discuss it with examples. Some fixed point results using

(ϕ, MF)

-contraction mapping are discussed in such space. In addition, we study the stability of obtained results in the spirit of Ulam and Hyers. As an application, we also provide the existence and uniqueness of the solution for a system of Fredholm integral equations. Full article

(This article belongs to the Section Mathematics)

40 pages, 783 KiB

Open AccessArticle

Nonlocal Impulsive Fractional Integral Boundary Value Problem for (ρ_k,ϕ_k)-Hilfer Fractional Integro-Differential Equations

by Marisa Kaewsuwan, Rachanee Phuwapathanapun, Weerawat Sudsutad, Jehad Alzabut, Chatthai Thaiprayoon and Jutarat Kongson

Mathematics 2022, 10(20), 3874; https://doi.org/10.3390/math10203874 - 18 Oct 2022

Cited by 6 | Viewed by 1770

Abstract

In this paper, we establish the existence and stability results for the

(ρ_{k}, ϕ_{k})

-Hilfer fractional integro-differential equations under instantaneous impulse with non-local multi-point fractional integral boundary conditions. We achieve the formulation of the solution to the [...] Read more.

In this paper, we establish the existence and stability results for the

(ρ_{k}, ϕ_{k})

-Hilfer fractional integro-differential equations under instantaneous impulse with non-local multi-point fractional integral boundary conditions. We achieve the formulation of the solution to the

(ρ_{k}, ϕ_{k})

-Hilfer fractional differential equation with constant coefficients in term of the Mittag–Leffler kernel. The uniqueness result is proved by applying Banach’s fixed point theory with the Mittag–Leffler properties, and the existence result is derived by using a fixed point theorem due to O’Regan. Furthermore, Ulam–Hyers stability and Ulam–Hyers–Rassias stability results are demonstrated via the non-linear functional analysis method. In addition, numerical examples are designed to demonstrate the application of the main results. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)

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