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Journal articles on the topic 'Multivalued function'

Author: Grafiati

Published: 4 June 2021

Last updated: 12 February 2022

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1

Naimpally, Som. "Multivalued function spaces and Atsuji spaces." Applied General Topology 4, no. 2 (2003): 201. http://dx.doi.org/10.4995/agt.2003.2025.

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<p>In this paper we present two themes. The first one describes a transparent treatment of some of the recent results in graph topologies on multi-valued functions. The study includes Vietoris topology, Fell topology, Fell uniform topology on compacta and uniform topology on compacta. The second theme concerns when continuity is equivalent to proximal continuity or uniform continuity.</p>

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2

HANCER, HATICE ASLAN, MURAT OLGUN, and ISHAK ALTUN. "Some new types multivalued F-contractions on quasi metric spaces and their fixed points." Carpathian Journal of Mathematics 35, no. 1 (2019): 41–50. http://dx.doi.org/10.37193/cjm.2019.01.05.

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In this paper we present two new results for the existence of fixed points of multivalued mappings with closed values on quasi metric space. First we introduce the multivalued Fd-contraction on quasi metric space (X, d) and give a fixed point result related to this concept. Then taking into account the Q-function on a quasi metric space, we establish a Q-function version of this concept as multivalued Fq-contraction and hence we present a fixed point result to see the effect of Q-function to existence of fixed point of multivalued mappings on quasi metric space.

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3

Donaldson, Simon. "Deformations Of Multivalued Harmonic Functions." Quarterly Journal of Mathematics 72, no. 1-2 (2021): 199–235. http://dx.doi.org/10.1093/qmath/haab018.

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Abstract We consider harmonic sections of a bundle over the complement of a codimension 2 submanifold in a Riemannian manifold, which can be thought of as multivalued harmonic functions. We prove a result to the effect that these are stable under small deformations of the data. The proof is an application of a version of the Nash-Moser implicit function theorem.

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4

Leśniak, K. "Infinite Iterated Function Systems: A Multivalued Approach." Bulletin of the Polish Academy of Sciences Mathematics 52, no. 1 (2004): 1–8. http://dx.doi.org/10.4064/ba52-1-1.

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5

Huang, Zongchao, Zhaogong Zhang, and Guanwen Yu. "Multivalued function recognition based on spectral clustering." Journal of Physics: Conference Series 1453 (January 2020): 012145. http://dx.doi.org/10.1088/1742-6596/1453/1/012145.

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6

Ban, Hyunju, and William D. Kalies. "A Computational Approach to Conley’s Decomposition Theorem." Journal of Computational and Nonlinear Dynamics 1, no. 4 (2006): 312–19. http://dx.doi.org/10.1115/1.2338651.

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Background. The discrete dynamics generated by a continuous map can be represented combinatorially by an appropriate multivalued map on a discretization of the phase space such as a cubical grid or triangulation. Method of approach. We describe explicit algorithms for computing dynamical structures for the combinatorial multivalued maps. Results. We provide computational complexity bounds and numerical examples. Specifically we focus on the computation attractor-repeller pairs and Lyapunov functions for Morse decompositions. Conclusions. The computed discrete Lyapunov functions are weak Lyapunov functions and well-approximate a continuous Lyapunov function for the underlying map.

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7

Okeke, Godwin Amechi, та Safeer Hussain Khan. "Approximation of fixed point of multivalued ρ-quasi-contractive mappings in modular function spaces". Arab Journal of Mathematical Sciences 26, № 1/2 (2019): 75–93. http://dx.doi.org/10.1016/j.ajmsc.2019.02.001.

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The purpose of this paper is to extend the recent results of Okeke et al. (2018) to the class of multivalued ρ-quasi-contractive mappings in modular function spaces. We approximate fixed points of this class of nonlinear multivalued mappings in modular function spaces. Moreover, we extend the concepts of T-stability, almost T-stability and summably almost T-stability to modular function spaces and give some results.

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8

Ali, Muhammad Usman, Quanita Kiran та Naseer Shahzad. "Fixed Point Theorems for Multivalued Mappings Involvingα-Function". Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/409467.

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We obtain some fixed point theorems with error estimates for multivalued mappings satisfying a newα-ψ-contractive type condition. Our theorems generalize many existing fixed point theorems, including some fixed point theorems proved forα-ψ-contractive type conditions.

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9

Ukhobotov, V. I., and V. N. Ushakov. "On one control problem with disturbance and vectograms depending linearly on given sets." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 30, no. 3 (2020): 429–43. http://dx.doi.org/10.35634/vm200306.

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A control problem with a given end time is considered, in which the control vectograms and disturbance depend linearly on the given convex compact sets. A multivalued mapping of the phase space of the control problem to the linear normed space E is given. The goal of constructing a control is that at the end of the control process the fixed vector of the space E belongs to the image of the multivalued mapping for any admissible realization of the disturbance. A stable bridge is defined in terms of multivalued functions. The presented procedure constructs, according to a given multivalued function which is a stable bridge, a control that solves the problem. Explicit formulas are obtained that determine a stable bridge in the considered control problem. Conditions are found under which the constructed stable bridge is maximal. Some problems of group pursuit can be reduced to the considered control problem with disturbance. The article provides such an example.

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10

Ransford, Thomas. "On the range of an analytic multivalued function." Pacific Journal of Mathematics 123, no. 2 (1986): 421–39. http://dx.doi.org/10.2140/pjm.1986.123.421.

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11

Chen, Jin-Ping, and Shie-Jue Lee. "Modified Multivalued Neuron With Periodic Tolerant Activation Function." IEEE Transactions on Neural Networks and Learning Systems 25, no. 9 (2014): 1645–58. http://dx.doi.org/10.1109/tnnls.2013.2276012.

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12

Okeke, Godwin Amechi, Sheila Amina Bishop та Safeer Hussain Khan. "Iterative Approximation of Fixed Point of Multivalued ρ-Quasi-Nonexpansive Mappings in Modular Function Spaces with Applications". Journal of Function Spaces 2018 (2018): 1–9. http://dx.doi.org/10.1155/2018/1785702.

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Recently, Khan and Abbas initiated the study of approximating fixed points of multivalued nonlinear mappings in modular function spaces. It is our purpose in this study to continue this recent trend in the study of fixed point theory of multivalued nonlinear mappings in modular function spaces. We prove some interesting theorems for ρ-quasi-nonexpansive mappings using the Picard-Krasnoselskii hybrid iterative process. We apply our results to solving certain initial value problem.

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13

Došenovic, Tatjana, Dušan Rakic, Biljana Caric, and Stojan Radenovic. "Multivalued generalizations of fixed point results in fuzzy metric spaces." Nonlinear Analysis: Modelling and Control 21, no. 2 (2016): 211–22. http://dx.doi.org/10.15388/na.2016.2.5.

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This paper attempts to prove fixed and coincidence point results in fuzzy metric space using multivalued mappings. Altering distance function and multivalued strong {bn}-fuzzy contraction are used in order to do that. Presented theorems are generalization of some well known single valued results. Two examples are given to support the theoretical results.

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14

Merrikh-Bayat, Farshad, and Masoud Karimi-Ghartemani. "On the Essential Instabilities Caused by Fractional-Order Transfer Functions." Mathematical Problems in Engineering 2008 (2008): 1–13. http://dx.doi.org/10.1155/2008/419046.

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The exact stability condition for certain class of fractional-order (multivalued) transfer functions is presented. Unlike the conventional case that the stability is directly studied by investigating the poles of the transfer function, in the systems under consideration, the branch points must also come into account as another kind of singularities. It is shown that a multivalued transfer function can behave unstably because of the numerator term while it has no unstable poles. So, in this case, not only the characteristic equation but the numerator term is of significant importance. In this manner, a family of unstable fractional-order transfer functions is introduced which exhibit essential instabilities, that is, those which cannot be removed by feedback. Two illustrative examples are presented; the transfer function of which has no unstable poles but the instability occurred because of the unstable branch points of the numerator term. The effect of unstable branch points is studied and simulations are presented.

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15

Choudhury, Binayak S., and Chaitali Bandyopadhyay. "Stability of Fixed Point Sets of a Class of Multivalued Nonlinear Contractions." Journal of Mathematics 2015 (2015): 1–4. http://dx.doi.org/10.1155/2015/302012.

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We consider a problem of stability of fixed point sets for a sequence of multivalued mappings defined on a metric space converging to a limit function where the convergence is with respect to the Pompeiu-Hausdorff distance. The members of the sequence are assumed to be multivalued almost contractions. We show that the fixed point sets of this sequence of mappings are stable.

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16

Kutbi, MarwanA, and Abdul Latif. "Fixed Points of Multivalued Maps in Modular Function Spaces." Fixed Point Theory and Applications 2009, no. 1 (2009): 786357. http://dx.doi.org/10.1155/2009/786357.

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17

Alvarez, Teresa, Fatma Fakhfakh, and Maher Mnif. "Coperturbation function and lower semi-Browder multivalued linear operators." Linear and Multilinear Algebra 61, no. 4 (2013): 494–516. http://dx.doi.org/10.1080/03081087.2012.691972.

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18

Benavoli, Alessio. "Belief function and multivalued mapping robustness in statistical estimation." International Journal of Approximate Reasoning 55, no. 1 (2014): 311–29. http://dx.doi.org/10.1016/j.ijar.2013.04.014.

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19

Zigmunt, V. "Remarks on the superposition measurability of a multivalued function." Mathematical Notes of the Academy of Sciences of the USSR 48, no. 3 (1990): 923–24. http://dx.doi.org/10.1007/bf01157435.

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20

LI, ZHIQIANG, and DAIZHAN CHENG. "ALGEBRAIC APPROACH TO DYNAMICS OF MULTIVALUED NETWORKS." International Journal of Bifurcation and Chaos 20, no. 03 (2010): 561–82. http://dx.doi.org/10.1142/s0218127410025892.

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Using semi-tensor product of matrices, a matrix expression for multivalued logic is proposed, where a logical variable is expressed as a vector, and a logical function is expressed as a multilinear mapping. Under this framework, the dynamics of a multivalued logical network is converted into a standard discrete-time linear system. Analyzing the network transition matrix, easily computable formulas are obtained to show (a) the number of equilibriums; (b) the numbers of cycles of different lengths; (c) transient period, the minimum time for all points to enter the set of attractors, respectively. A method to reconstruct the logical network from its network transition matrix is also presented. This approach can also be used to convert the dynamics of a multivalued control network into a discrete-time bilinear system. Then, the structure and the controllability of multivalued logical control networks are revealed.

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21

Altun, Ishak, and Gonca Durmaz. "The influence of theta-function to the class of MWP operators." Mathematica Moravica 24, no. 2 (2020): 21–31. http://dx.doi.org/10.5937/matmor2002021a.

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In this work, taking into account the theta-function, we present a general class of multivalued weakly Picard operators on complete metric space. We also provide an example showing that it includes some earlier classes as properly.

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22

Debnath, Pradip, and Hari Mohan Srivastava. "New Extensions of Kannan’s and Reich’s Fixed Point Theorems for Multivalued Maps Using Wardowski’s Technique with Application to Integral Equations." Symmetry 12, no. 7 (2020): 1090. http://dx.doi.org/10.3390/sym12071090.

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The metric function generalizes the concept of distance between two points and hence includes the symmetric property. The aim of this article is to introduce a new and proper extension of Kannan’s fixed point theorem to the case of multivalued maps using Wardowski’s F-contraction. We show that our result is applicable to a class of mappings where neither the multivalued version of Kannan’s theorem nor that of Wardowski’s can be applied to determine the existence of fixed points. Application of our result to the solution of integral equations has been provided. A multivalued Reich type generalized version of the result is also established.

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23

Ibrahim, A. G., and F. Aladsani. "Convex Sweeping Processes with Noncompact Perturbations and with Delay in Banach Spaces." Abstract and Applied Analysis 2016 (2016): 1–13. http://dx.doi.org/10.1155/2016/3853205.

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We prove two results concerning the existence of solutions for functional differential inclusions that are governed by sweeping processes, with noncompact valued perturbations in Banach spaces. Indeed, we have two goals. The first is to give a technique that allows considering sweeping processes with noncompact valued perturbations and associated with a multivalued function depending on time. The second is to give a technique to overcome the arising problem from the nonlinearity of the normalized mappings, when we deal with sweeping processes with noncompact valued perturbations and associated with a multivalued function depending on time and state.

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24

Shuai, W. C., K. L. Xiang, and W. Y. Zhang. "Constrained Weak Nash-Type Equilibrium Problems." Abstract and Applied Analysis 2014 (2014): 1–4. http://dx.doi.org/10.1155/2014/307903.

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A constrained weak Nash-type equilibrium problem with multivalued payoff functions is introduced. By virtue of a nonlinear scalarization function, some existence results are established. The results extend the corresponding one of Fu (2003). In particular, if the payoff functions are singlevalued, our existence theorem extends the main results of Fu (2003) by relaxing the assumption of convexity.

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25

Ali, Muhammad Usman, and Calogero Vetro. "Fixed point theorems for multivalued maps via new auxiliary function." Nonlinear Analysis: Modelling and Control 2017, no. 1 (2016): 84–98. http://dx.doi.org/10.15388/na.2017.1.6.

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26

Rouhani, B. Djafari, and B. Ahmadi Kakavandi. "Infinite Time-Dependent Network Equilibria with a Multivalued Cost Function." Journal of Optimization Theory and Applications 131, no. 3 (2006): 405–15. http://dx.doi.org/10.1007/s10957-006-9159-8.

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27

Akewe, Hudson, та Hallowed Olaoluwa. "Multistep-type construction of fixed point for multivalued ρ-quasi-contractive-like maps in modular function spaces". Arab Journal of Mathematical Sciences 27, № 2 (2021): 189–213. http://dx.doi.org/10.1108/ajms-07-2020-0026.

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PurposeIn this paper, the explicit multistep, explicit multistep-SP and implicit multistep iterative sequences are introduced in the context of modular function spaces and proven to converge to the fixed point of a multivalued map T such that PρT, an associate multivalued map, is a ρ-contractive-like mapping.Design/methodology/approachThe concepts of relative ρ-stability and weak ρ-stability are introduced, and conditions in which these multistep iterations are relatively ρ-stable, weakly ρ-stable and ρ-stable are established for the newly introduced strong ρ-quasi-contractive-like class of maps.FindingsNoor type, Ishikawa type and Mann type iterative sequences are deduced as corollaries in this study.Originality/valueThe results obtained in this work are complementary to those proved in normed and metric spaces in the literature.

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28

Hicks, Troy, and B. E. Rhoades. "Fixed points and continuity for multivalued mappings." International Journal of Mathematics and Mathematical Sciences 15, no. 1 (1992): 15–30. http://dx.doi.org/10.1155/s0161171292000024.

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Many of the contractive definitions do not require continuity of the map. However, in a previous paper the second author has shown in [1] that, in most cases, the function is continuous at a fixed point. In this paper we show that the same behavior is exhibited for many multivalued mappings.

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29

WU, JUN, and YICHENG LIU. "NEW COINCIDENCE POINT THEOREMS IN CONTINUOUS FUNCTION SPACES AND APPLICATIONS." Bulletin of the Australian Mathematical Society 80, no. 1 (2009): 26–37. http://dx.doi.org/10.1017/s0004972708001123.

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AbstractIn this paper, some new coincidence point theorems in continuous function spaces are presented. We show the hybrid mapping version and multivalued version of both Lou’s fixed point theorem (Proc. Amer. Math. Soc.127 (1999)) and de Pascale and de Pascale’s fixed point theorem (Proc. Amer. Math. Soc.130 (2002)). Our new results encompass a number of previously known generalizations of the theorems. Two examples are presented.

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30

MIMACHI, Katsuhisa, and Takeshi SASAKI. "MONODROMY REPRESENTATIONS ASSOCIATED WITH THE GAUSS HYPERGEOMETRIC FUNCTION USING INTEGRALS OF A MULTIVALUED FUNCTION." Kyushu Journal of Mathematics 66, no. 1 (2012): 35–60. http://dx.doi.org/10.2206/kyushujm.66.35.

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31

Velinov, Daniel, Marko Kostic, and Stevan Pilipovic. "Degenerate C-distribution cosine functions and degenerate C-ultradistribution cosine functions in locally convex spaces." Filomat 31, no. 11 (2017): 3075–89. http://dx.doi.org/10.2298/fil1711075v.

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The main purpose of this paper is to investigate degenerate C-(ultra)distribution cosine functions in the setting of barreled sequentially complete locally convex spaces. In our approach, the infinitesimal generator of a degenerate C-(ultra)distribution cosine function is a multivalued linear operator and the regularizing operator C is not necessarily injective. We provide a few important theoretical novelties, considering also exponential subclasses of degenerate C-(ultra)distribution cosine functions.

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32

Dinevari, T., and M. Frigon. "Applications of Multivalued Contractions on Graphs to Graph-Directed Iterated Function Systems." Abstract and Applied Analysis 2015 (2015): 1–16. http://dx.doi.org/10.1155/2015/345856.

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We apply a fixed point result for multivalued contractions on complete metric spaces endowed with a graph to graph-directed iterated function systems. More precisely, we construct a suitable metric space endowed with a graphGand a suitableG-contraction such that its fixed points permit us to obtain more information on the attractor of a graph-directed iterated function system.

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33

Maouche, Abdelaziz. "Caractérisations Spectrales du Radical et du Socle d’Une Paire de Jordan-Banach." Canadian Mathematical Bulletin 40, no. 4 (1997): 488–97. http://dx.doi.org/10.4153/cmb-1997-058-9.

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RésuméIf f and g are two analytic functions from a domain D of the complex plane into respectively the Banach spaces V+ and V−, we prove that is an analytic multivalued function. From this derives the subharmonicity of the functions where ρ denotes the spectral radius. We apply these results to obtain nice caracterizations of the radical and the socle of a Banach Jordan pair, and finally we get an algebraic structural theorem.

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34

Samreen, Maria, Khansa Waheed та Quanita Kiran. "Multivalued φ contractions and fixed point theorems". Filomat 32, № 4 (2018): 1209–20. http://dx.doi.org/10.2298/fil1804209s.

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In this paper we establish some fixed point theorems for multivalued mappings satisfying contractive condition involving gauge function when the underlying primary structure is b-metric space. Our proposed iterative scheme converges to the fixed point with higher order. Moreover, we also calculate priori and posteriori estimates for the fixed point. Our main results generalize/extend many perexisting results in literature. Consequently, to substantiate the validity of our result we obtain an existence result for the solution of integral inclusion.

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35

Ali, Amjad, Hüseyin Işık, Hassen Aydi, Eskandar Ameer, Jung Rye Lee та Muhammad Arshad. "On multivalued Suzuki-type θ-contractions and related applications". Open Mathematics 18, № 1 (2020): 386–99. http://dx.doi.org/10.1515/math-2020-0139.

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Abstract In this study, we develop the concept of multivalued Suzuki-type θ-contractions via a gauge function and established two new related fixed point theorems on metric spaces. We also discuss an example to validate our results.

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36

Sun, X. K., та S. J. Li. "Duality and gap function for generalized multivalued ϵ-vector variational inequality". Applicable Analysis 92, № 3 (2013): 482–92. http://dx.doi.org/10.1080/00036811.2011.628940.

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37

ZHANG, Po, Chang-Chun YANG, and You-Ming LI. "3-D Multivalued Seismic wavefield Reconstruction and Computation for Green's Function." Chinese Journal of Geophysics 43, no. 2 (2000): 269–77. http://dx.doi.org/10.1002/cjg2.34.

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38

Khovanskii, A. G. "On the Monodromy of a Multivalued Function along Its Ramification Locus." Functional Analysis and Its Applications 37, no. 2 (2003): 134–41. http://dx.doi.org/10.1023/a:1024409023859.

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39

Tolstonogov, A. A. "Theorem on continuous extension of a multivalued function and its applications." Mathematical Notes of the Academy of Sciences of the USSR 42, no. 4 (1987): 821–27. http://dx.doi.org/10.1007/bf01138318.

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40

MIMACHI, Katsuhisa, and Takeshi SASAKI. "MONODROMY REPRESENTATIONS ASSOCIATED WITH APPELL' S HYPERGEOMETRIC FUNCTION F1 USING INTEGRALS OF A MULTIVALUED FUNCTION." Kyushu Journal of Mathematics 66, no. 1 (2012): 89–114. http://dx.doi.org/10.2206/kyushujm.66.89.

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41

Liu, Jia-Bao, Asma Rashid Butt, Shahzad Nadeem, Shahbaz Ali, and Muhammad Shoaib. "Higher Order of Convergence with Multivalued Contraction Mappings." Journal of Mathematics 2020 (November 11, 2020): 1–8. http://dx.doi.org/10.1155/2020/8867897.

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In this paper, we establish some theorems of fixed point on multivalued mappings satisfying contraction mapping by using gauge function. Furthermore, we use Q - and R -order of convergence. Our main results extend many previous existing results in the literature. Consequently, to substantiate the validity of proposed method, we give its application in integral inclusion.

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42

Bucur. "Fixed Points for Multivalued Weighted Mean Contractions in a Symmetric Generalized Metric Space." Symmetry 12, no. 1 (2020): 134. http://dx.doi.org/10.3390/sym12010134.

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This paper defines two new concepts: the concept of multivalued left-weighted mean contractions in the generalized sense of Nadler in a symmetric generalized metric space and the concept of multivalued right-weighted mean contractions in the generalized sense of Nadler in a symmetric generalized metric space, and demonstrates fixed-point theorems for them. For these, we demonstrated two fixed-point existence theorems and their corollaries, by using the properties of the regular-global-inf function and the properties of symmetric generalized metric spaces, respectively. Moreover, we demonstrated that the theorems can be applied in particular cases of inclusion systems. This article contains not only an example of application in science, but also an example of application in real life, in biology, in order to find an equilibrium solution to a prey–predator-type problem. The results of this paper extend theorems for multivalued left-weighted mean contractions in the generalized sense of Nadler, demonstrating that some of the results given by Rus (2008), Mureșan (2002), and Nadler (1969) in metric spaces can also be proved in symmetric generalized metric spaces.

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43

Damas, Bruno, and José Santos-Victor. "Online Learning of Single- and Multivalued Functions with an Infinite Mixture of Linear Experts." Neural Computation 25, no. 11 (2013): 3044–91. http://dx.doi.org/10.1162/neco_a_00510.

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We present a supervised learning algorithm for estimation of generic input-output relations in a real-time, online fashion. The proposed method is based on a generalized expectation-maximization approach to fit an infinite mixture of linear experts (IMLE) to an online stream of data samples. This probabilistic model, while not fully Bayesian, can efficiently choose the number of experts that are allocated to the mixture, this way effectively controlling the complexity of the resulting model. The result is an incremental, online, and localized learning algorithm that performs nonlinear, multivariate regression on multivariate outputs by approximating the target function by a linear relation within each expert input domain and that can allocate new experts as needed. A distinctive feature of the proposed method is the ability to learn multivalued functions: one-to-many mappings that naturally arise in some robotic and computer vision learning domains, using an approach based on a Bayesian generative model for the predictions provided by each of the mixture experts. As a consequence, it is able to directly provide forward and inverse relations from the same learned mixture model. We conduct an extensive set of experiments to evaluate the proposed algorithm performance, and the results show that it can outperform state-of-the-art online function approximation algorithms in single-valued regression, while demonstrating good estimation capabilities in a multivalued function approximation context.

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44

Aizenberg, Igor. "Periodic Activation Function and a Modified Learning Algorithm for the Multivalued Neuron." IEEE Transactions on Neural Networks 21, no. 12 (2010): 1939–49. http://dx.doi.org/10.1109/tnn.2010.2082561.

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45

Kita, Michitake. "On vanishing of the twisted rational de Rham cohomology associated with hypergeometric functions." Nagoya Mathematical Journal 135 (September 1994): 55–85. http://dx.doi.org/10.1017/s0027763000004955.

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Recent development in hypergeometric functions in several variables has made the importance of studying twisted rational de Rham cohomology clear to many specialists. Roughly speaking, a hypergeometric function in our sense is the integral of a product of complex powers of polynomials Pj(u1, . . . . ,un) : ∫ U du1 ∧ · · · ∧ dun, U = Π , integration being taken over some cycle. So we are led naturally to consider the twisted rational de Rham cohomology, which is a direct generalization of the usual de Rham cohomology to multivalued case.

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46

BOUFOUSSI, B., and N. MRHARDY. "MULTIVALUED STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS VIA BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS." Stochastics and Dynamics 08, no. 02 (2008): 271–94. http://dx.doi.org/10.1142/s0219493708002317.

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In this paper, we establish by means of Yosida approximation, the existence and uniqueness of the solution of a backward doubly stochastic differential equation whose coefficient contains the subdifferential of a convex function. We will use this result to prove the existence of stochastic viscosity solution for some multivalued parabolic stochastic partial differential equation.

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47

O'Regan, Donal. "A note on multivalued differential equations on proximate retracts." Journal of Applied Mathematics and Stochastic Analysis 12, no. 2 (1999): 169–78. http://dx.doi.org/10.1155/s1048953399000179.

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This paper discusses viable solutions for differential inclusions in Banach spaces. Existence will be established in two steps. In step 1, a nonlinear alternative of Leray-Schauder type [8] for maps with closed graphs will be used to establish a variety of existence principles for the Cauchy differential inclusion. Step 2 involves using the results in step 1 together with some tricks involving the Bouligand cone (and sometimes the Urysohn function) so that new existence criteria can be established for multivalued differential equations on proximate retracts.

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48

Khan, Safeer Hussain, Mujahid Abbas, and Sartaj Ali. "Fixed point approximation of multivalued rho-quasi-nonexpansive mappings in modular function spaces." Journal of Nonlinear Sciences and Applications 10, no. 06 (2017): 3168–79. http://dx.doi.org/10.22436/jnsa.010.06.30.

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49

Xiexiong, Chen, Zhao Xiaojie, and Wu Xunwei. "Synthesis of multivalued cmos circuits with many variables based on transmission function theory." Journal of Electronics (China) 9, no. 1 (1992): 9–16. http://dx.doi.org/10.1007/bf02778587.

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50

Kovalev, M. M., and A. V. Moshchenskii. "Search for the maximum upper zero of a threshold function of multivalued logic." USSR Computational Mathematics and Mathematical Physics 30, no. 2 (1990): 190–92. http://dx.doi.org/10.1016/0041-5553(90)90096-b.

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