In this work, we investigate the existence, uniqueness, and stability of fractional differential equation with multi-point integral boundary conditions involving the Caputo fractional derivative. By utilizing the Laplace transform technique, the existence of solution is accomplished. By applying the Bielecki-norm and the classical fixed point theorem, the Ulam stability results of the studied system are presented. An illustrative example is provided at the last part to validate all our obtained theoretical results.
In this article, we make analysis of the implicit fractional differential equations involving integral boundary conditions associated with Stieltjes integral and its corresponding coupled system. We use some sufficient conditions to achieve the existence and uniqueness results for the given problems by applying the Banach contraction principle, Schaefer’s fixed point theorem, and Leray–Schauder result of the cone type. Moreover, we present different kinds of stability such as Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam–Rassias stability, and generalized Hyers–Ulam–Rassias stability by using the classical technique of functional analysis. At the end, the results are verified with the help of examples.
In this article, we make analysis of the implicit q‐fractional differential equations involving integral boundary conditions associated with Stieltjes integral and its corresponding coupled system. We are using some sufficient conditions to achieve the existence and uniqueness results for the given problems by applying the Banach contraction principle, Schaefer's fixed point theorem, and Leray–Schauder result of cone type. Moreover, we present different kinds of stability such as Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam–Rassias stability, and generalized Hyers–Ulam–Rassias stability using the classical technique of functional analysis. At the end, the results are verified with the help of examples.
Ethylenediaminetetraacetic acid or EDTA was used complementarily with nisin to give a synergistic antibacterial effect against Gram positive and negative bacteria. Nisin and EDTA were encapsulated in nonionic surfactant vesicles or niosomes. Sorbitan monooleate and polyethylene glycol were precursors in preparation of niosomes. Size reduction of niosomes was conducted via extrusion through polycarbonate membrane with pore size of 200 nm. Diameters of prepared blank niosomes and nisin-EDTA-encapsulated niosomes were approximately 130 nm and 270 nm, respectively. Bilayer structure of niosomes was observed from negative staining-transmission electron microscopic images. Long-termed investigation of antimicrobial activity of nisin-EDTA-encapsulated niosomes and free nisin/EDTA were conducted against Staphylococcus aureus and Escherichia coli. Bacterial counts denoted a slow release of nisin-EDTA-encapsulated niosomes overtime whilst free nisin/EDTA gave a sudden bactericidal activity. Due to that free nisin/EDTA was immediately exploited at the beginning, bacterial counts then tended towards higher during the latter time of antimicrobial activity test.
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