Search results for: Resolvent

Authors: Jorge Gonzalez-Camus

Abstract:

In this work is proved the existence of at least one minimal and maximal mild solutions to the Cauchy problem, for fractional evolution equation of neutral type, involving a general kernel. An operator A generating a resolvent family and integral resolvent family on a Banach space X and a kernel belonging to a large class appears in the equation, which covers many relevant cases from physics applications, in particular, the important case of time - fractional evolution equations of neutral type. The main tool used in this work was the Kuratowski measure of noncompactness and fixed point theorems, specifically Darbo-type, and an iterative method of lower and upper solutions, based in an order in X induced by a normal cone P. Initially, the equation is a Cauchy problem, involving a fractional derivate in Caputo sense. Then, is formulated the equivalent integral version, and defining a convenient functional, using the theory of resolvent families, and verifying the hypothesis of the fixed point theorem of Darbo type, give us the existence of mild solution for the initial problem. Furthermore, the existence of minimal and maximal mild solutions was proved through in an iterative method of lower and upper solutions, using the Azcoli-Arzela Theorem, and the Gronwall’s inequality. Finally, we recovered the case derivate in Caputo sense.

Keywords: fractional evolution equations, Volterra integral equations, minimal and maximal mild solutions, neutral type equations, non-local in time equations

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Authors: Maryam Moosavi, Hadi Khatibzadeh

Abstract:

The main purpose of this paper is to study the proximal point algorithm for quasi-nonexpansive mappings in Hadamard spaces. △-convergence and strong convergence of cyclic resolvents for a finite family of quasi-nonexpansive mappings one to a fixed point of the mappings are established

Keywords: Fixed point, Hadamard space, Proximal point algorithm, Quasi-nonexpansive sequence of mappings, Resolvent

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Authors: N. Smaoui, B. Chentouf

Abstract:

The boundary stabilization problem of the rotating disk-beam system is a topic of interest in research studies. This system involves a flexible beam attached to the center of a disk, and the control and stabilization of this system have been extensively studied. This research focuses on the case where the center of mass is fixed in an inertial frame, and the rotation of the center is non-uniform. The system is represented by a set of nonlinear coupled partial differential equations and ordinary differential equations. The boundary stabilization problem of this system via a delayed boundary control is considered. We assume that the boundary control is either of a force type control or a moment type control and is subject to the presence of a constant time-delay. The aim of this research is threefold: First, we demonstrate that the rotating disk-beam system is well-posed in an appropriate functional space. Then, we establish the exponential stability property of the system. Finally, we provide numerical simulations that illustrate the theoretical findings. The research utilizes the semigroup theory to establish the well-posedness of the system. The resolvent method is then employed to prove the exponential stability property. Finally, the finite element method is used to demonstrate the theoretical results through numerical simulations. The research findings indicate that the rotating disk-beam system can be stabilized using a boundary control with a time delay. The proof of stability is based on the resolvent method and a variation of constants formula. The numerical simulations further illustrate the theoretical results. The findings have potential implications for the design and implementation of control strategies in similar systems. In conclusion, this research demonstrates that the rotating disk-beam system can be stabilized using a boundary control with time delay. The well-posedness and exponential stability properties are established through theoretical analysis, and these findings are further supported by numerical simulations. The research contributes to the understanding and practical application of control strategies for flexible structures, providing insights into the stability of rotating disk-beam systems.

Keywords: rotating disk-beam, delayed force control, delayed moment control, torque control, exponential stability

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Authors: Francis O. Nwawuru

Abstract:

The convergence analysis of split monotone inclusion problems and fixed point problems of certain nonlinear mappings are investigated in the setting of real Hilbert spaces. Inertial extrapolation term in the spirit of Polyak is incorporated to speed up the rate of convergence. Under standard assumptions, a strong convergence of the proposed algorithm is established without computing the resolvent operator or involving Yosida approximation method. The stepsize involved in the algorithm does not depend on the spectral radius of the linear operator. Furthermore, applications of the proposed algorithm in solving some related optimization problems are also considered. Our result complements and extends numerous results in the literature.

Keywords: fixedpoint, hilbertspace, monotonemapping, resolventoperators

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Authors: Jorge Gonzalez Camus, Carlos Lizama

Abstract:

In this work is proved the existence of at least one globally attractive mild solution to the Cauchy problem, for fractional evolution equation of neutral type, involving the fractional derivate in Caputo sense. An almost sectorial operator on a Banach space X and a kernel belonging to a large class appears in the equation, which covers many relevant cases from physics applications, in particular, the important case of time - fractional evolution equations of neutral type. The main tool used in this work was the Hausdorff measure of noncompactness and fixed point theorems, specifically Darbo-type. Initially, the equation is a Cauchy problem, involving a fractional derivate in Caputo sense. Then, is formulated the equivalent integral version, and defining a convenient functional, using the analytic integral resolvent operator, and verifying the hypothesis of the fixed point theorem of Darbo type, give us the existence of mild solution for the initial problem. Furthermore, each mild solution is globally attractive, a property that is desired in asymptotic behavior for that solution.

Keywords: attractive mild solutions, integral Volterra equations, neutral type equations, non-local in time equations

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Authors: Nitin Rajan, Kashif Shakeel, Shashank Tiwari, Shachan Sagar

Abstract:

Background: - Aegle Marmelos (Bael) leaf extract is taken twice daily to treat ophthalmia, ulcers, and intestinal worms, among other ailments. Poultice made from bael leaf is used in the treatment of eye conditions. The leaf juice has a variety of therapeutic applications, with the most notable being the treatment of diabetes. Fenugreek is used to cure red spots around the eyes, as well as to soften the throat and chest and to give relief from coughing. The use of this plant in the form of infusion, powder, pomade, and decoction has been extremely popular in Iranian traditional medicine. The plant may be used to wash one's vaginal linings. This plant is used as an emollient in the lack of appetite, treatment of pellagra, and gastrointestinal problems, as well as a general tonic. Calendula officinalis leaves are used to treat varicose veins on the outside of the body by infusing them. In Europe, the leaves are diaphoretic and resolvent in nature, while the blooms are employed as an emmenagogue and antispasmodic stimulant in Canada and the United States. The flowers were decocted and served as a posset drink when smallpox and measles were common in England, and the fresh juice was used to treat jaundice. Objective: - This study is done to compare the physicochemical parameter of the alcoholic extract of the leaves of Aegle Marmelos, Calendula Officinalis, and Fenugreek. Materials and Methods: Extraction and Isolation of Aegle Marmelos, Calendula Officinalis, Fenugreek, were done. Preliminary phytochemical study for alkaloids, cardiac glycosides, flavonoids, glycosides, phenols, resins, saponins, steroids, tannins, terpenoids of the extract was done individual by using the standard procedure. Result: - The phytochemical screening of Aegle Marmelos, Calendula Officinalis, and Fenugreek shows the presence of alkaloids, carbohydrates, total phenolics, total flavonoids, tannins, saponins gum. Conclusion: - In this study, we have found that crude aqueous and organic solvent extracts of Aegle Marmelos, Calendula Officinalis, and Fenugreek leaves contain some important bioactive compounds and it justifies their use in the traditional medicines for the treatment of different diseases.

Keywords: Aegle Marmelos, Calendula Officinalis, Fenugreek, physiochemical parameter

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