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Journal articles on the topic 'Frechet derivative, Banach spaces'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Mirotin, A. R. "Perturbation determinants on Banach spaces and operator differentiability for Hirsch functional calculus." Filomat 34, no. 4 (2020): 1105–15. http://dx.doi.org/10.2298/fil2004105m.

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We consider a perturbation determinant for pairs of nonpositive (in a sense of Komatsu) operators on Banach space with nuclear difference and prove the formula for the logarithmic derivative of this determinant. To this end the Frechet differentiability of operator monotonic (negative complete Bernstein) functions of negative and nonpositive operators on Banach spaces is investigated. The results may be regarded as a contribution to the Hirsch functional calculus.

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2

Mirotin, A. R. "Perturbation determinants on Banach spaces and operator differentiability for Hirsch functional calculus." Filomat 34, no. 4 (2020): 1105–15. http://dx.doi.org/10.2298/fil2004105m.

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We consider a perturbation determinant for pairs of nonpositive (in a sense of Komatsu) operators on Banach space with nuclear difference and prove the formula for the logarithmic derivative of this determinant. To this end the Frechet differentiability of operator monotonic (negative complete Bernstein) functions of negative and nonpositive operators on Banach spaces is investigated. The results may be regarded as a contribution to the Hirsch functional calculus.

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3

PRASHANTH, M., and D. K. GUPTA. "A CONTINUATION METHOD AND ITS CONVERGENCE FOR SOLVING NONLINEAR EQUATIONS IN BANACH SPACES." International Journal of Computational Methods 10, no. 04 (April 23, 2013): 1350021. http://dx.doi.org/10.1142/s0219876213500217.

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A continuation method is a parameter based iterative method establishing a continuous connection between two given functions/operators and used for solving nonlinear equations in Banach spaces. The semilocal convergence of a continuation method combining Chebyshev's method and Convex acceleration of Newton's method for solving nonlinear equations in Banach spaces is established in [J. A. Ezquerro, J. M. Gutiérrez and M. A. Hernández [1997] J. Appl. Math. Comput.85: 181–199] using majorizing sequences under the assumption that the second Frechet derivative satisfies the Lipschitz continuity condition. The aim of this paper is to use recurrence relations instead of majorizing sequences to establish the convergence analysis of such a method. This leads to a simpler approach with improved results. An existence–uniqueness theorem is given. Also, a closed form of error bounds is derived in terms of a real parameter α ∈ [0, 1]. Four numerical examples are worked out to demonstrate the efficacy of our convergence analysis. On comparing the existence and uniqueness region and error bounds for the solution obtained by our analysis with those obtained by using majorizing sequences, it is found that our analysis gives better results in three examples, whereas in one example it gives the same results. Further, we have observed that for particular values of the α, our analysis reduces to those for Chebyshev's method (α = 0) and Convex acceleration of Newton's method (α = 1) respectively with improved results.

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4

Talakua, Mozart W., and Stenly J. Nanuru. "TEOREMA REPRESENTASI RIESZ–FRECHET PADA RUANG HILBERT." BAREKENG: Jurnal Ilmu Matematika dan Terapan 5, no. 2 (December 1, 2011): 1–8. http://dx.doi.org/10.30598/barekengvol5iss2pp1-8.

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Hilbert space is a very important idea of the Davids Hilbert invention. In 1907, Riesz and Fréchet developed one of the theorem in Hilbert space called the Riesz-Fréchet representationtheorem. This research contains some supporting definitions Banach space, pre-Hilbert spaces, Hilbert spaces, the duality of Banach and Riesz-Fréchet representation theorem. On Riesz-Fréchet representation theorem will be shown that a continuous linear functional that exist in the Hilbert space is an inner product, in other words, there is no continuous linear functional on a Hilbert space except the inner product.

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5

Osinuga, I. A., S. A. Ayinde, J. A. Oguntuase, and G. A. Adebayo. "On Fermat-Torricelli Problem in Frechet Spaces." Journal of Nepal Mathematical Society 3, no. 2 (December 30, 2020): 16–26. http://dx.doi.org/10.3126/jnms.v3i2.33956.

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We study the Fermat-Torricelli problem (FTP) for Frechet space X, where X is considered as an inverse limit of projective system of Banach spaces. The FTP is defined by using fixed countable collection of continuous seminorms that defines the topology of X as gauges. For a finite set A in X consisting of n distinct and fixed points, the set of minimizers for the sum of distances from the points in A to a variable point is considered. In particular, for the case of collinear points in X, we prove the existence of the set of minimizers for FTP in X and for the case of non collinear points, existence and uniqueness of the set of minimizers are shown for reflexive space X as a result of strict convexity of the space.

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6

Zheng, Xi Yin. "Very Differentiable and Generic Frechet Differentiable Convex Functions on Banach Spaces." Journal of Mathematical Analysis and Applications 235, no. 1 (July 1999): 168–79. http://dx.doi.org/10.1006/jmaa.1999.6388.

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7

Jokhadze, O. "Spatial Problem of Darboux Type for One Model Equation of Third Order." gmj 3, no. 6 (December 1996): 547–64. http://dx.doi.org/10.1515/gmj.1996.547.

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Abstract For a hyperbolic type model equation of third order a Darboux type problem is investigated in a dihedral angle. It is shown that there exists a real number ρ0 such that for α > ρ0 the problem under consideration is uniquely solvable in the Frechet space. In the case where the coefficients are constants, Bochner's method is developed in multidimensional domains, and used to prove the uniquely solvability of the problem both in Frechet and in Banach spaces.

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8

Dorokhov, Alexander, and Michael Karpov. "On the existence of fixed points in completely continuous operators in F -space." Tambov University Reports. Series: Natural and Technical Sciences, no. 125 (2019): 26–32. http://dx.doi.org/10.20310/1810-0198-2019-24-125-26-32.

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This work is dedicated to the development of the theory of fixed points of completely continuous operators. We prove existence of new theorems of fixed points of completely continuous operators in F -space (Frechet space). This class of spaces except Banach includes such important space as a countably normed space and Lp(0 < p < 1), lp(0 < p < 1).

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9

Helal, Mohamed. "Fractional order differential inclusions on an unbounded domain with infinite delay." MATHEMATICA 62 (85), no. 2 (November 15, 2020): 167–78. http://dx.doi.org/10.24193/mathcluj.2020.2.06.

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We provide sufficient conditions for the existence of solutions to initial value problems, for partial hyperbolic differential inclusions of fractional order involving Caputo fractional derivative with infinite delay by applying the nonlinear alternative of Frigon type for multivalued admissible contraction in Frechet spaces.

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10

Carrión, H., P. Galindo, and M. L. Lourenço. "Biholomorphic Mappings on Banach Spaces." Proceedings of the Edinburgh Mathematical Society 62, no. 4 (February 27, 2019): 913–24. http://dx.doi.org/10.1017/s0013091518000883.

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AbstractWe present an infinite-dimensional version of Cartan's theorem concerning the existence of a holomorphic inverse of a given holomorphic self-map of a bounded convex open subset of a dual Banach space. No separability is assumed, contrary to previous analogous results. The main assumption is that the derivative operator is power bounded, and which we, in turn, show to be diagonalizable in some cases, like the separable Hilbert space.

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11

Bozkurt, Hacer, Sümeyye Çakan, and Yılmaz Yılmaz. "Quasilinear Inner Product Spaces and Hilbert Quasilinear Spaces." International Journal of Analysis 2014 (March 11, 2014): 1–7. http://dx.doi.org/10.1155/2014/258389.

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Aseev launched a new branch of functional analysis by introducing the theory of quasilinear spaces in the framework of the topics of norm, bounded quasilinear operators and functionals (Aseev (1986)). Furthermore, some quasilinear counterparts of classical nonlinear analysis that lead to such result as Frechet derivative and its applications were examined deal with. This pioneering work causes a lot of results in such applications such as (Rojas-Medar et al. (2005), Talo and Başar (2010), and Nikol'skiĭ (1993)). His work has motivated us to introduce the concept of quasilinear inner product spaces. Thanks to this new notion, we obtain some new theorems and definitions which are quasilinear counterparts of fundamental definitions and theorems in linear functional analysis. We claim that some new results related to this concept provide an important contribution to the improvement of quasilinear functional analysis.

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12

Fry, R. "Approximation by functions with bounded derivative on Banach spaces." Bulletin of the Australian Mathematical Society 69, no. 1 (February 2004): 125–31. http://dx.doi.org/10.1017/s0004972700034316.

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Let X be a separable Banach space which admits a C1-smooth norm, and let G ⊂ X be an open subset. Then any real-valued, bounded and uniformly continuous map on G can be uniformly approximated on G by C1-smooth functions with bounded derivative.

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13

Rashid, MH, and A. Sarder. "Convergence of the Newton-Type Method for Generalized Equations." GANIT: Journal of Bangladesh Mathematical Society 35 (June 28, 2016): 27–40. http://dx.doi.org/10.3329/ganit.v35i0.28565.

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Let X and Y be real or complex Banach spaces. Suppose that f: X->Y is a Frechet differentiable function and F: X => 2Yis a set-valued mapping with closed graph. In the present paper, we study the Newton-type method for solving generalized equation 0 ? f(x) + F(x). We prove the existence of the sequence generated by the Newton-type method and establish local convergence of the sequence generated by this method for generalized equation.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 27-40

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14

McLaughlin, David P., and Jon D. Vanderwerff. "Higher order Gateaux smooth bump functions on Banach spaces." Bulletin of the Australian Mathematical Society 51, no. 2 (April 1995): 291–300. http://dx.doi.org/10.1017/s000497270001412x.

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For Г uncountable and p ≥ 1 odd, it is shown ℓp(г) admits no continuous p-times Gateaux differentiable bump function. A space is shown to admit a norm with Hölder derivative on its sphere if it admits a bounded bump function with uniformly directionally Hölder derivative. Some results on smooth approximation are obtained for spaces that admit bounded uniformly Gateaux differentiable bump functions.

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15

Argyros, Ioannis K., and Santhosh George. "On a sixth-order Jarratt-type method in Banach spaces." Asian-European Journal of Mathematics 08, no. 04 (November 17, 2015): 1550065. http://dx.doi.org/10.1142/s1793557115500655.

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We present a local convergence analysis of a sixth-order Jarratt-type method in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies such as [X. Wang, J. Kou and C. Gu, Semilocal convergence of a sixth-order Jarratt method in Banach spaces, Numer. Algorithms 57 (2011) 441–456.] require hypotheses up to the third Fréchet-derivative. Numerical examples are also provided in this study.

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16

Huang, Zhengda. "Newton method under weak Lipschitz continuous derivative in Banach spaces." Applied Mathematics and Computation 140, no. 1 (July 2003): 115–26. http://dx.doi.org/10.1016/s0096-3003(02)00215-1.

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17

Ndoutoume, James Louis, and Michel Théra. "Generalised second-order derivatives of convex functions in reflexive Banach spaces." Bulletin of the Australian Mathematical Society 51, no. 1 (February 1995): 55–72. http://dx.doi.org/10.1017/s0004972700013897.

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Generalised second-order derivatives introduced by Rockafellar in the finite dimensional setting are extended to convex functions defined on reflexive Banach spaces. Our approach is based on the characterisation of convex generalised quadratic forms defined in reflexive Banach spaces, from the graph of the associated subdifferentials. The main result which is obtained is the exhibition of a particular generalised Hessian when the function admits a generalised second derivative. Some properties of the generalised second derivative are pointed out along with further justifications of the concept.

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18

Argyros, Ioannis K., and Santhosh George. "MODIFICATION OF THE KANTOROVICH-TYPE CONDITIONS FOR NEWTON'S METHOD INVOLVING TWICE FRECHET DIFFERENTIABLE OPERATORS." Asian-European Journal of Mathematics 06, no. 03 (September 2013): 1350026. http://dx.doi.org/10.1142/s1793557113500265.

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We expand the applicability of Newton's method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. The nonlinear operator involved is twice Fréchet differentiable. We introduce more precise majorizing sequences than in earlier studied (see [Concerning the convergence and application of Newton's method under hypotheses on the first and second Fréchet derivative, Comm. Appl. Nonlinear Anal.11 (2004) 103–119; A new semilocal convergence theorem for Newton's method, J. Comp. Appl. Math.79 (1997) 131–145; A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math.47 (1993) 211–217]). This way, our convergence criteria can be weaker; the error estimates tighter and the information on the location of the solution more precise. Numerical examples are presented to show that our results apply in cases not covered before such as [Concerning the convergence and application of Newton's method under hypotheses on the first and second Fréchet derivative, Comm. Appl. Nonlinear Anal.11 (2004) 103–119; A new semilocal convergence theorem for Newton's method, J. Comp. Appl. Math.79 (1997) 131–145; A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math.47 (1993) 211–217].

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19

Albanese, A. A., G. Metafune, and V. B. Moscatelli. "Representations of the spaces Cm(Ω) ∩ Hk, p (Ω)." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 3 (October 1996): 489–98. http://dx.doi.org/10.1017/s0305004100075034.

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The present work has its motivation in the papers [2] and [6] on distinguished Fréchet function spaces. Recall that a Fréchet space E is distinguished if it is the projective limit of a sequence of Banach spaces En such that the strong dual E′β is the inductive limit of the sequence of the duals E′n. Clearly, the property of being distinguished is inherited by complemented subspaces and in [6] Taskinen proved that the Fréchet function space C(R) ∩ L1(R) (intersection topology) is not distinguished, by showing that it contains a complemented subspace of Moscatelli type (see Section 1) that is not distinguished. Because of the criterion in [1], it is easy to decide when a Frechet space of Moscatelli type is distinguished. Using this, in [2], Bonet and Taskinen obtained that the spaces open in RN) are distinguished, by proving that they are isomorphic to complemented subspaces of distinguished Fréchet spaces of Moscatelli type.

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20

Torrea, José L., and Chao Zhang. "Fractional vector-valued Littlewood–Paley–Stein theory for semigroups." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 144, no. 3 (May 16, 2014): 637–67. http://dx.doi.org/10.1017/s0308210511001302.

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We consider the fractional derivative of a general Poisson semigroup. With this fractional derivative, we define the generalized fractional Littlewood–Paley g-function for semigroups acting on Lp-spaces of functions with values in Banach spaces. We give a characterization of the classes of Banach spaces for which the fractional Littlewood–Paley g-function is bounded on Lp-spaces. We show that the class of Banach spaces is independent of the order of derivation and coincides with the classical (Lusin-type/-cotype) case. We also show that the same kind of results exist for the case of the fractional area function and the fractional gλ*-function on ℝn. Finally, we consider the relationship of the almost sure finiteness of the fractional Littlewood–Paley g-function, the area function and the gλ*-function with the Lusin-cotype property of the underlying Banach space. As a byproduct of the techniques developed, one can find some results of independent interest for vector-valued Calderón–Zygmund operators. For example, one can find the following characterization: a Banach space is the unconditional martingale difference if and only if, for some (or, equivalently, for every) p ∈ [1, ∞), dy exists for almost every x ∈ ℝ and every .

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21

Giles, J. R., and Scott Sciffer. "Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces." Bulletin of the Australian Mathematical Society 47, no. 2 (April 1993): 205–12. http://dx.doi.org/10.1017/s0004972700012430.

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For a locally Lipschitz function on a separable Banach space the set of points of Gâteaux differentiability is dense but not necessarily residual. However, the set of points where the upper Dini derivative and the Clarke derivative agree is residual. It follows immediately that the set of points of intermediate differentiability is also residual and the set of points where the function is Gâteaux but not strictly differentiable is of the first category.

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22

Arnold, Loris. "Derivative bounded functional calculus of power bounded operators on Banach spaces." Acta Scientiarum Mathematicarum 87, no. 12 (2021): 265–94. http://dx.doi.org/10.14232/actasm-020-040-y.

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23

Seddiki, Fakhruddeen, Mohammad Al-horani, and Roshdi Rashid Khalil. "Finite Rank Solution for Conformable Degenerate First-Order Abstract Cauchy Problem in Hilbert Spaces." European Journal of Pure and Applied Mathematics 14, no. 2 (May 18, 2021): 493–505. http://dx.doi.org/10.29020/nybg.ejpam.v14i2.3950.

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In this paper, we find a solution of finite rank form of fractional Abstract Cauchy Problem. The fractional derivative used is the Conformable derivative. The main idea of the proofs are based on theory of tensor product of Banach spaces.

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24

Azagra, D., M. Fabian, and M. Jiménez-Sevilla. "Exact Filling of Figures with the Derivatives of Smooth Mappings Between Banach Spaces." Canadian Mathematical Bulletin 48, no. 4 (December 1, 2005): 481–99. http://dx.doi.org/10.4153/cmb-2005-045-9.

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AbstractWe establish sufficient conditions on the shape of a set A included in the space (X, Y ) of the n-linear symmetric mappings between Banach spaces X and Y , to ensure the existence of a Cn-smooth mapping f: X → Y, with bounded support, and such that f(n)(X) = A, provided that X admits a Cn-smooth bump with bounded n-th derivative and dens X = dens ℒn(X, Y ). For instance, when X is infinite-dimensional, every bounded connected and open set U containing the origin is the range of the n-th derivative of such amapping. The same holds true for the closure of U, provided that every point in the boundary of U is the end point of a path within U. In the finite-dimensional case, more restrictive conditions are required. We also study the Fréchet smooth case for mappings from ℝn to a separable infinite-dimensional Banach space and the Gâteaux smooth case for mappings defined on a separable infinite-dimensional Banach space and with values in a separable Banach space.

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25

ARGYROS, IOANNIS K. "A CONVERGENCE ANALYSIS FOR NEWTON-LIKE METHODS IN BANACH SPACE UNDER WEAK HYPOTHESES AND APPLICATIONS." Tamkang Journal of Mathematics 30, no. 4 (December 1, 1999): 253–61. http://dx.doi.org/10.5556/j.tkjm.30.1999.4231.

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In this study we use Newton-like methods to approximate solutions of nonlinear equations in a Banach space setting. Most convergence results for Newton-like methods involve some type of a Lipschitz continuity condition on the Frechet-derivative of the operator involved However there are many interesting real life problems already in the literature where the operator can only satisfy a Holder continuity condition. That is why here we chose the Frcchet-derivativc of the operator involved to be only Holder continuous, which allows us to consider a wider range of problems than before. Special choices of our parameters reduce our results to earlier ones. An error analysis is also provided for our method. At the end of our study, we provide applications to show that our results apply where earlier results do not. In paricular we solve a two point boundary value problem appearing in physics in connection with the problem of bending of beams.

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26

Fedorov, Vladimir E., Marina V. Plekhanova, and Elizaveta M. Izhberdeeva. "Initial Value Problems of Linear Equations with the Dzhrbashyan–Nersesyan Derivative in Banach Spaces." Symmetry 13, no. 6 (June 11, 2021): 1058. http://dx.doi.org/10.3390/sym13061058.

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Among the many different definitions of the fractional derivative, the Riemann–Liouville and Gerasimov–Caputo derivatives are most commonly used. In this paper, we consider the equations with the Dzhrbashyan–Nersesyan fractional derivative, which generalizes the Riemann–Liouville and the Gerasimov–Caputo derivatives; it is transformed into such derivatives for two sets of parameters that are, in a certain sense, symmetric. The issues of the unique solvability of initial value problems for some classes of linear inhomogeneous equations of general form with the fractional Dzhrbashyan–Nersesyan derivative in Banach spaces are investigated. An inhomogeneous equation containing a bounded operator at the fractional derivative is considered, and the solution is presented using the Mittag–Leffler functions. The result obtained made it possible to study the initial value problems for a linear inhomogeneous equation with a degenerate operator at the fractional Dzhrbashyan–Nersesyan derivative in the case of relative p-boundedness of the operator pair from the equation. Abstract results were used to study a class of initial boundary value problems for equations with the time-fractional Dzhrbashyan–Nersesyan derivative and with polynomials in a self-adjoint elliptic differential operator with respect to spatial variables.

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27

Parhi, S. K., and D. K. Gupta. "A Stirling-like method with Hölder continuous first derivative in Banach spaces." Applied Mathematics and Computation 217, no. 23 (August 2011): 9567–74. http://dx.doi.org/10.1016/j.amc.2011.04.032.

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28

Wang, JinRong, Zhenbin Fan, and Yong Zhou. "Nonlocal Controllability of Semilinear Dynamic Systems with Fractional Derivative in Banach Spaces." Journal of Optimization Theory and Applications 154, no. 1 (February 18, 2012): 292–302. http://dx.doi.org/10.1007/s10957-012-9999-3.

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29

Fedorov, V. E., D. M. Gordievskikh, and M. V. Plekhanova. "Equations in Banach spaces with a degenerate operator under a fractional derivative." Differential Equations 51, no. 10 (October 2015): 1360–68. http://dx.doi.org/10.1134/s0012266115100110.

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30

Prashanth, M., and D. K. Gupta. "Convergence of a continuation method under Lipschitz continuous derivative in Banach spaces." Journal of Applied Mathematics and Computing 39, no. 1-2 (November 9, 2011): 253–70. http://dx.doi.org/10.1007/s12190-011-0522-z.

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31

Guida, Karim, Khalid Hilal, Lahcen Ibnelazyz, and Ming Mei. "Existence of Mild Solutions for a Class of Impulsive Hilfer Fractional Coupled Systems." Advances in Mathematical Physics 2020 (September 28, 2020): 1–12. http://dx.doi.org/10.1155/2020/8406509.

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The aim of this paper is to give existence results for a class of coupled systems of fractional integrodifferential equations with Hilfer fractional derivative in Banach spaces. We first give some definitions, namely the Hilfer fractional derivative and the Hausdorff’s measure of noncompactness and the Sadovskii’s fixed point theorem.

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32

Pavlíček, Libor. "Monotonically Controlled Mappings." Canadian Journal of Mathematics 63, no. 2 (April 1, 2011): 460–80. http://dx.doi.org/10.4153/cjm-2011-004-0.

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Abstract We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó–Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.

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33

Erovenko, V. A. "SOME RESULTS ON ESSENTIAL SPECTRA OF DIFFERENTIAL OPERATORS IN BANACH SPACES." Mathematical Modelling and Analysis 8, no. 3 (September 30, 2003): 203–16. http://dx.doi.org/10.3846/13926292.2003.9637224.

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In this paper we investigate spectral and semi‐Predholm properties of maximum and minimum Puchsian differential operators on Lebesgue spaces on a semi‐axis. These results are applied for determination of various essential spectra and spectrum of ordinary differential operators with polynomial coefficients, which order does not exceed the order of the corresponding derivative.

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34

Saxena, Akanksha, Ioannis K. Argyros, Jai P. Jaiswal, Christopher Argyros, and Kamal R. Pardasani. "On the Local Convergence of Two-Step Newton Type Method in Banach Spaces under Generalized Lipschitz Conditions." Mathematics 9, no. 6 (March 21, 2021): 669. http://dx.doi.org/10.3390/math9060669.

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The motive of this paper is to discuss the local convergence of a two-step Newton-type method of convergence rate three for solving nonlinear equations in Banach spaces. It is assumed that the first order derivative of nonlinear operator satisfies the generalized Lipschitz i.e., L-average condition. Also, some results on convergence of the same method in Banach spaces are established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak L-average particularly it is assumed that L is positive integrable function but not necessarily non-decreasing. Our new idea gives a tighter convergence analysis without new conditions. The proposed technique is useful in expanding the applicability of iterative methods. Useful examples justify the theoretical conclusions.

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35

Ojha, Bhuwan Prasad. "Different Concepts of Derivatives." Journal of Advanced College of Engineering and Management 3 (January 10, 2018): 11. http://dx.doi.org/10.3126/jacem.v3i0.18809.

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<p>In this paper, different concept of derivatives with some properties has been introduced. In differential calculus, the partial derivative, directional derivative and total derivative are studied. Their generalization for Banach spaces are the Gateaux differential and Freshet derivative.</p><p><strong>Journal of Advanced College of Engineering and Management,</strong> Vol.3, 2017, Page: 11-14</p>

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36

Keten, Ayşegül, Mehmet Yavuz, and Dumitru Baleanu. "Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces." Fractal and Fractional 3, no. 2 (May 16, 2019): 27. http://dx.doi.org/10.3390/fractalfract3020027.

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We investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo–Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and hence it removed the computational complexities arising from the singular kernel functions inherit in the conventional fractional derivatives. The method used in this study is based on the Banach contraction mapping principle. Moreover, we gave a numerical example which shows the applicability of the obtained results.

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37

Derbazi, Choukri, Zidane Baitiche, Mouffak Benchohra, and G. N’Guérékata. "Existence, Uniqueness, and Mittag–Leffler–Ulam Stability Results for Cauchy Problem Involving ψ -Caputo Derivative in Banach and Fréchet Spaces." International Journal of Differential Equations 2020 (October 13, 2020): 1–16. http://dx.doi.org/10.1155/2020/6383916.

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Our aim in this paper is to investigate the existence, uniqueness, and Mittag–Leffler–Ulam stability results for a Cauchy problem involving ψ -Caputo fractional derivative with positive constant coefficient in Banach and Fréchet Spaces. The techniques used are a variety of tools for functional analysis. More specifically, we apply Weissinger’s fixed point theorem and Banach contraction principle with respect to the Chebyshev and Bielecki norms to obtain the uniqueness of solution on bounded and unbounded domains in a Banach space. However, a new fixed point theorem with respect to Meir–Keeler condensing operators combined with the technique of Hausdorff measure of noncompactness is used to investigate the existence of a solution in Banach spaces. After that, by means of new generalizations of Grönwall’s inequality, the Mittag–Leffler–Ulam stability of the proposed problem is studied on a compact interval. Meanwhile, an extension of the well-known Darbo’s fixed point theorem in Fréchet spaces associated with the concept of measures of noncompactness is applied to obtain the existence results for the problem at hand. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility of the main theorems.

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Mordukhovich, Boris S., and Bingwu Wang. "Restrictive metric regularity and generalized differential calculus in Banach spaces." International Journal of Mathematics and Mathematical Sciences 2004, no. 50 (2004): 2653–80. http://dx.doi.org/10.1155/s0161171204405183.

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We consider nonlinear mappingsf:X→Ybetween Banach spaces and study the notion ofrestrictive metric regularityoffaround some pointx¯, that is, metric regularity offfromXinto the metric spaceE=f(X). Some sufficient as well as necessary and sufficient conditions for restrictive metric regularity are obtained, which particularly include an extension of the classical Lyusternik-Graves theorem in the case whenfis strictly differentiable atx¯but its strict derivative∇f(x¯)is not surjective. We develop applications of the results obtained and some other techniques in variational analysis to generalized differential calculus involving normal cones to nonsmooth and nonconvex sets, coderivatives of set-valued mappings, as well as first-order and second-order subdifferentials of extended real-valued functions.

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Ibrahim, Rabha W. "On Generalized Hyers-Ulam Stability of Admissible Functions." Abstract and Applied Analysis 2012 (2012): 1–10. http://dx.doi.org/10.1155/2012/749084.

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We consider the Hyers-Ulam stability for the following fractional differential equations in sense of Srivastava-Owa fractional operators (derivative and integral) defined in the unit disk:Dzβf(z)=G(f(z),Dzαf(z),zf'(z);z),0<α<1<β≤2, in a complex Banach space. Furthermore, a generalization of the admissible functions in complex Banach spaces is imposed, and applications are illustrated.

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MUÑOZ, GUSTAVO A., and YANNIS SARANTOPOULOS. "Bernstein and Markov-type inequalities for polynomials on real Banach spaces." Mathematical Proceedings of the Cambridge Philosophical Society 133, no. 3 (November 2002): 515–30. http://dx.doi.org/10.1017/s0305004102006217.

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In this work we generalize Markov's inequality for any derivative of a polynomial on a real Hilbert space and provide estimates for the second and third derivatives of a polynomial on a real Banach space. Our result on a real Hilbert space answers a question raised by L. A. Harris in his commentary on problem 74 in the Scottish Book [20]. We also provide generalizations of previously obtained inequalities of the Bernstein and Markov-type for polynomials with curved majorants on a real Hilbert space.

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Miller, T. L., and V. G. Miller. "An operator satisfying Dunford's condition (C) but without bishop's property (β)." Glasgow Mathematical Journal 40, no. 3 (September 1998): 427–30. http://dx.doi.org/10.1017/s0017089500032754.

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For X a complex Banach space and U an open subset of the complex plane С, let O (U, X) denote the space of analytic X- valued functions defined on U. This is a Frechet space when endowed with the topology of uniform convergence on compact subsets, and the space X may be viewed as simply the constants in O(U, X). Every bounded operator T on X induces a continuous mapping TU on O(U, X) given by (Tuf)(λ) = (λ – T)f(λ) for every f e O(U, X) and λ e U. Corresponding to each closed F ⊂ С there is also an associated analytic subspace XT(F) = X ∩ ran(7c//F). For an arbitrary T e L(X), the spaces XT(F) are T-invariant, generally non-closed linear manifolds in X.

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Ling, Yonghui, Xiubin Xu, and Shaohua Yu. "Convergence Behavior for Newton-Steffensen’s Method under -Condition of Second Derivative." Abstract and Applied Analysis 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/682167.

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The present paper is concerned with the semilocal as well as the local convergence problems of Newton-Steffensen’s method to solve nonlinear operator equations in Banach spaces. Under the assumption that the second derivative of the operator satisfies -condition, the convergence criterion and convergence ball for Newton-Steffensen’s method are established.

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Parida, P. K., and D. K. Gupta. "Convergence of an iterative method in Banach spaces with Lipschitz continuous first derivative." International Journal of Applied Nonlinear Science 1, no. 4 (2014): 289. http://dx.doi.org/10.1504/ijans.2014.068254.

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Parhi, S. K., and D. K. Gupta. "Semilocal convergence of Stirling's method under Hölder continuous first derivative in Banach spaces." International Journal of Computer Mathematics 87, no. 12 (October 2010): 2752–59. http://dx.doi.org/10.1080/00207160902777922.

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45

Deville, Robert. "On the range of the derivative of a smooth mapping between Banach spaces." Abstract and Applied Analysis 2005, no. 5 (2005): 499–507. http://dx.doi.org/10.1155/aaa.2005.499.

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We survey recent results on the structure of the range of the derivative of a smooth mappingfbetween two Banach spacesXandY. We recall some necessary conditions and some sufficient conditions on a subsetAofℒ(X,Y)for the existence of a Fréchet differentiable mappingffromXintoYso thatf′(X)=A. Wheneverfis only assumed Gâteaux differentiable, new phenomena appear: for instance, there exists a mappingffromℓ1(ℕ)intoℝ2, which is bounded, Lipschitz-continuous, and so that for allx,y∈ℓ1(ℕ), ifx≠y, then‖f′(x)−f′(y)‖>1.

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Wang, JinRong, Linli Lv, and Yong Zhou. "Boundary value problems for fractional differential equations involving Caputo derivative in Banach spaces." Journal of Applied Mathematics and Computing 38, no. 1-2 (February 15, 2011): 209–24. http://dx.doi.org/10.1007/s12190-011-0474-3.

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Shubha, Vorkady S., Santhosh George, and P. Jidesh. "Third-order derivative-free methods in Banach spaces for nonlinear ill-posed equations." Journal of Applied Mathematics and Computing 61, no. 1-2 (February 26, 2019): 137–53. http://dx.doi.org/10.1007/s12190-019-01246-1.

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PARHI, S. K., and D. K. GUPTA. "SEMILOCAL CONVERGENCE OF A STIRLING-LIKE METHOD IN BANACH SPACES." International Journal of Computational Methods 07, no. 02 (June 2010): 215–28. http://dx.doi.org/10.1142/s0219876210002210.

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The aim of this paper is to establish the semilocal convergence of a third order Stirling–like method employed for solving nonlinear equations in Banach spaces by using the first Fréchet derivative, which satisfies the Lipschitz continuity condition. This makes it possible to avoid the evaluation of higher order Fréchet derivatives which are computationally difficult at times or may not even exist. The recurrence relations are used for convergence analysis. A convergence theorem is given for deriving error bounds and the domains of existence and uniqueness of solutions. The R order of the method is also established to be equal to 3. Finally, two numerical examples are worked out, and the results obtained are compared with the existing results. It is observed that our convergence analysis is more effective.

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Fitzpatrick, Simon. "Nearest points to closed sets and directional derivatives of distance functions." Bulletin of the Australian Mathematical Society 39, no. 2 (April 1989): 233–38. http://dx.doi.org/10.1017/s0004972700002707.

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We investigate the circumstances under which the distance function to a closed set in a Banach space having a one-sided directional derivative equal to 1 or −1 implies the existence of nearest points. In reflexive spaces we show that at a dense set of points outside a closed set the distance function has a directional derivative equal to 1.

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Welsh, Stewart C. "Open mappings and solvability of nonlinear equations in Banach space." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 126, no. 2 (1996): 239–46. http://dx.doi.org/10.1017/s030821050002271x.

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We give sufficient conditions under which a funclion f: X → Y is an open mapping, where X and y are Banach spaces. This function is not necessarily continuous, but is assumed to have closed graph. We prove our results without requiring that f be Gateaux differentiable; instead, f is assumed to possess a weak type of Gateaux inverse derivative.

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