Optimization (10294945)73(5)pp. 1589-1609
In this article, we introduce the notions of l-transfer lower continuous and q-level intersectionally closed for set-valued mappings with respect to the lower set less relation. Then, we obtain some existence results for strict weak l-efficient solutions of such set-valued mappings. Moreover, we prove some existence results for nonconvex set optimization problems via asymptotic analysis tools, in the setting of the Banach spaces equipped with a Hausdorff topology σ coarser than the norm topology. © 2023 Informa UK Limited, trading as Taylor & Francis Group.
Fakhar, M.,
Khodakhah m.r., M.,
Soubeyran a., A.,
Zafarani j., J. OPTIMIZATION (02331934)72(1)pp. 215-239
This paper has two parts. The mathematical part provides generalized versions of the robust Ekeland variational principle in terms of set-valued EVP with variable preferences, uncertain parameters and changing weights given to vectorial perturbation functions. The behavioural part that motivates our findings models the formation and stability of a partnership in a changing, uncertain and complex environment in the context of the variational rationality approach of stop, continue and go human dynamics. Our generalizations allow us to consider two very important psychological effects relative to ego depletion and goal gradient hypothesis.
Fakhar, M.,
Khodakhah m.r., M.,
Soubeyran a., A.,
Zafarani j., J. Journal of Applied and Numerical Optimization (25625535)4(1)pp. 37-51
In this paper, in the context of quasi-metric spaces, we obtain two set-valued versions of the Ekeland variational-type principle by means of lower and upper set less relations, for the case where the perturbations need not satisfy the triangle inequality. An application in terms of migration problems and quality of life is given. ©2022 Journal of Applied and Numerical Optimization
Fakhar, M.,
Khodakhah m.r., M.,
Mazyaki, A.,
Soubeyran a., A.,
Zafarani j., J. Journal of Global Optimization (9255001)82(1)pp. 161-177
This paper has two aspects. Mathematically, in the context of global optimization, it provides the existence of an optimum of a perturbed optimization problem that generalizes the celebrated Ekeland variational principle and equivalent formulations (Caristi, Takahashi), whenever the perturbations need not satisfy the triangle inequality. Behaviorally, it is a continuation of the recent variational rationality approach of stay (stop) and change (go) human dynamics. It gives sufficient conditions for the existence of traps in a changing environment. In this way it emphasizes even more the striking correspondence between variational analysis in mathematics and variational rationality in psychology and behavioral sciences. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Optimization Letters (18624472)15(3)pp. 923-931
Different aspects of the KKM Lemma besides its vast applications in nonlinear analysis have been investigated over the years. Some researchers tried to present the KKM results such as generalized L-KKM type theorems in the absence of usual convexity. Here, first we present a counterexample to show that the main results in some generalized KKM type theorems and their consequences are not valid. Afterward, the uniform mapconvex space is introduced and a nonempty intersection result in this space is proved. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
Journal of Mathematical Analysis and Applications (10960813)502(1)
In this paper, we first obtain a characterization of transfer weakly lower continuous functions. Then, by introducing the class of nearly quasi-closed set-valued mappings, we obtain some characterizations of set-valued mappings whose displacement functions are transfer weakly lower continuous. We also present some fixed point theorems for nearly quasi-closed set-valued mappings which are either nearly almost convex or almost affine. Finally, we construct an almost affine mapping T:[0,1)→R, which is not α-almost convex for any continuous and strictly increasing function α:[0,+∞)→[0,+∞) with α(0)=0. This example gives an affirmative response to the Question 3 of Jachymski (2015) [8]. © 2021 Elsevier Inc.
Analysis Mathematica (1333852)46(1)pp. 1-12
In this article we obtain a characterization of the class of p-convergent operators between two Banach spaces in terms of p-(V) subsets of the dual space. Also, for 1 ≤ p < q ≤ ∞, by introducing the concepts of Pelczyński's properties (V)p,q and (V*)p,q, we obtain a condition that ensures that q-convergent operators are p-convergent operators. Some characterizations of the p-Schur property of Banach spaces and their dual spaces are deduced. © 2020, Akadémiai Kiadó, Budapest.
Numerical Functional Analysis and Optimization (15322467)40(5)pp. 603-619
In this article, in the setting of metric spaces we introduce the notions of noncyclic and cyclic Fisher quasi-contraction mappings. We establish the existence of an optimal pair of fixed points for a noncyclic Fisher quasi-contraction mapping and iterative algorithms are furnished to determine such optimal pair of fixed points. For a cyclic Fisher quasi-contraction mapping, we also study the existence of best proximity points. Presented results extend and improve some recent results in the literature. © 2019, © 2019 Taylor & Francis Group, LLC.
Carpathian Journal of Mathematics (15842851)35(3)pp. 407-416
In this article, by using the notions of contingent derivative, contingent epiderivative and generalized contingent epiderivative, we obtain some characterizations of the Lagrange multiplier rule at points which are not necessarily local minima. © 2019, SINUS Association. All rights reserved.
Fakhar, M.,
Mahyarinia m.r., M.R.,
Zafarani j., J. Optimization (2331934)68(9)pp. 1653-1683
We introduce a new concept of generalized convexity of ‘degree n’ for a multiobjective optimization problem and is compared it to the previous notions of generalized convex functions. Some examples to justify the importance of the term ‘degree n’ are provided. Namely, the conclusions of our results may fail if this term is dropped. By applying our new definition to nonsmooth robust multiobjective optimization problems, we establish the nonsmooth robust optimality conditions and robust duality theory for robust ϵ-quasi-(weakly) efficient solutions. A robust ϵ-Mond-Weir type duality of degree n for an uncertain multi-objective optimization problem under our generalized convexity assumption is presented. Furthermore, we introduce an ϵ-approximate scalar saddle-point and an ϵ-approximate weak vector saddle-point of degree n for the robust multi-objective optimization problem. The relationships between these two concepts with robust ϵ-approximate (KKT) condition and robust ϵ-weakly efficient solutions are also given. © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group.
Journal of Fixed Point Theory and Applications (16617738)21(3)
In this article, we introduce relaxed μ-quasimonotone set-valued vector field on Hadamard manifolds and prove the existence of solutions of the Stampacchia variational inequality for such mappings. We also present the notion of relaxed μ-quasiconvexity and show that the Upper Dini and Clarke–Rockafellar subdifferentials of a relaxed μ-quasiconvex function is relaxed μ-quasimonotone. Under relaxed μ-quasiconvexity in the nondifferentiable sense, we establish the connection between the Stampacchia variational inequality problem and a nonsmooth constrained optimization problem. © 2019, Springer Nature Switzerland AG.
Journal of Global Optimization (09255001)75(1)pp. 131-141
In this paper, we characterize the nonemptiness of the set of weak minimal elements for a nonempty subset of a linear space. Moreover, we obtain some existence results for a nonconvex set-valued optimization problem under weaker topological conditions. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
Journal of Mathematical Analysis and Applications (10960813)467(2)pp. 1168-1173
In this paper, we define the class of (α,β)-nonexpansive mappings which is properly larger than the class of α-nonexpansive mappings and prove that every (α,β)-nonexpansive mapping T:C→C has an approximate fixed point sequence, where C is a nonempty bounded subset of a Banach space X, α>0 and β≥0. This, in particular, gives an affirmative answer to the open question posed by Ariza-Ruiz and et al. concerning the existence of an approximate fixed point sequence for α-nonexpansive mappings, Ariza-Ruiz et al. (2016) [4]. © 2018 Elsevier Inc.
Journal of Nonlinear and Convex Analysis (13454773)19(7)pp. 1189-1198
In this paper, we study the existence of fixed point for asymptotic compact absorbing contraction map and generalized a-set contraction maps. Moreover, we present structure of fixed point set results for this maps. Also, the asymptotic version of the Meir-Keeler, Boyd-Wong, Nadler contractions for the KKM maps on metric space are given. © 2018 Yokohama Publications.
Fixed Point Theory (15835022)19(1)pp. 211-218
In this paper the concept of set-valued cyclic Meir–Keeler contraction map is introduced. The existence of best proximity point for such maps on a metric space with the UC property is presented. © 2018, House of the Book of Science. All rights reserved.
Nonlinear Studies (13598678)25(3)pp. 689-700
In this paper we show the stability of the Gelfand-Phillips property of order p under tak- ing injective tensor product, compact operators, and Bochner integrable functions. The concept of L-limited sets of order p; is introduced and some characterizations of limited p-convergent operators are given. Also we define the notion of L-limited property of order p and characterize this property in terms of weak compact operators. Furthermore, we give a new dual characterization of the class of weak* p-convergent operators through L-limited sets of order p: Moreover, some characterizations of the Gelfand-Phillips property of order p in terms of limited p-convergent operators are obtained. In addition by applying our results on the limited p-convergent operators, we obtain some characteriza- tions of the Dunford-Pettis* property of order p. © CSP - Cambridge, UK.
Fakhar, M.,
Mahyarinia m.r., M.R.,
Zafarani j., J. European Journal of Operational Research (3772217)265(1)pp. 39-48
We introduce a new concept of generalized convexity at a given point for a family of real-valued functions and deduce nonsmooth sufficient optimality conditions for robust (weakly) efficient solutions. In addition, we present a robust duality theory and Mond–Weir type duality for an uncertain multiobjective optimization problem. Furthermore, some nonsmooth saddle-point theorems are obtained under our generalized convexity assumption. Finally we show the viability of our new concept of generalized convexity for robust optimization and portfolio optimization. © 2017 Elsevier B.V.
Filomat (3545180)32(6)pp. 2081-2089
In this paper, we study the existence and uniqueness of best proximity points for cyclic Meir– Keeler contraction mappings in metric spaces with the property W-WUC. Also, the existence of best proximity points for set-valued cyclic Meir-Keeler contraction mappings in metric spaces with the property WUC are obtained. © 2018, University of Nis. All rights reserved.
Acosta, M.D.,
Fakhar, M.,
Soleimani-mourchehkhorti, M. Journal of Mathematical Analysis and Applications (0022247X)458(2)pp. 925-936
In this paper, we introduce the notion of the Bishop–Phelps–Bollobás property for numerical radius (BPBp-ν) for a subclass of the space of bounded linear operators. Then, we show that certain subspaces of L(L1(μ)) have the BPBp-ν for every finite measure μ. As a consequence we deduce that the subspaces of finite-rank operators, compact operators and weakly compact operators on L1(μ) have the BPBp-ν. © 2017 Elsevier Inc.
Journal of Fixed Point Theory and Applications (16617738)20(1)
Let (X, d) be a metric space, Y be a nonempty subset of X, and let T: Y→ P(X) be a non-self multivalued mapping. In this paper, by a new technique we study the fixed point theory of multivalued mappings under the assumption of the existence of a bounded sequence (xn)n in Y such that Tnxn⊆ Y, for each n∈ N. Our main result generalizes fixed point theorems due to Matkowski (Diss. Math. 127, 1975), Wȩgrzyk (Diss. Math. (Rozprawy Mat.) 201, 1982), Reich and Zaslavski (Fixed Point Theory 8:303–307, 2007), Petruşel et al. (Set-Valued Var. Anal. 23:223–237, 2015) and provides a solution to the problems posed in Petruşel et al. (Set-Valued Var. Anal. 23:223–237, 2015) and Rus and Şerban (Miskolc Math. Notes 17:1021–1031, 2016). © 2018, Springer International Publishing AG, part of Springer Nature.
Bulletin of the Iranian Mathematical Society (10186301)43(1)pp. 131-135
It is well known that every (real or complex) normed linear space L is isometrically embeddable into C(X) for some compact Hausdorff space X. Here X is the closed unit ball of L* (the set of all continuous scalar-valued linear mappings on L) endowed with the weak* topology, which is compact by the Banach-Alaoglu theorem. We prove that the compact Hausdorff space X can indeed be chosen to be the Stone-Čech compactification of L* \ {0}, where L*\ {0} is endowed with the supremum norm topology. © 2017 Iranian Mathematical Society.
Journal of Nonlinear and Convex Analysis (13454773)18(3)pp. 361-368
In this paper, we study P-property and the best proximity point theory. In fact, by using the P-property, we obtain the best proximity counterpart of the many well-known mixed point theorems for multivalued and single valued mappings in the setting of metric spaces and Banach spaces. Presented theorems extend and improve some recent results in the literature. © 2017. Journal of Nonliner and Convex Analysis. All rights reserved.
Journal of Fixed Point Theory and Applications (16617738)19(4)pp. 2349-2360
In this paper, in the setting of complete metric spaces we establish some fixed point theorems for non-self mappings of contractive type satisfying either the Reich–Zaslavski property or the approximate fixed point property. As applications, we obtain some results in endpoint theory. © 2017, Springer International Publishing.
Journal of Nonlinear and Convex Analysis (13454773)17(3)pp. 569-578
Here, we show that continuous set-valued maps which are generalized set contraction on noncompact topological spaces have a maximal invariant (fixed) set. As an application, we prove the existence and uniqueness of endpoints for topological contraction mappings. Also, we present fractal set results for system of continuous set-valued maps on regular topological spaces. As application of our result, we show how some fixed point theorems can be established from these results. © 2016.
Journal of Nonlinear and Convex Analysis (13454773)17(12)pp. 2483-2500
In this paper, we present a KKM theorem in uniform convex spaces. By using this theorem, several generalized vector quasiequilibrium problems under pseudomonotonicity conditions are studied. Moreover, in the presence of monotonicity, we establish sufficient conditions for well-posedness in new classes of mapping by relaxing some topological requirements. © 2016.
SIAM Journal on Optimization (10526234)26(4)pp. 2847-2862
The main goal of this paper is to obtain a generalization of the Weierstrass theorem for transfer weakly lower continuous functions on noncompact topological spaces. To achieve this goal, the notion of a quasi-regular-global-inf (qrgi) function on a topological space is introduced, some equivalent statements are given, and a Weierstrass-Type theorem for such functions is proved. Moreover, the well-posedness of the minimization problem for regular-global-inf (rgi) and qrgi functions is studied. Furthermore, in the setting of reexive Banach spaces the existence of global minimum points of noncoercive qrgi and transfer weakly lower continuous functions are investigated. We also introduce the concept of nearly quasi-convexity of a function, as a generalization of the quasi-convexity notion, and present a result on the minimization problem of these functions. © 2016 Society for Industrial and Applied Mathematics.
Journal of Fixed Point Theory and Applications (16617738)18(3)pp. 601-607
In this paper, we first show that a Banach space X has weak normal structure if and only if X has the weak fixed point property for nonexpansive mappings with respect to (wrt) orbits. Then, we give a counterexample to show that the Goebel–Karlovitz lemma does not hold for minimal invariant sets of nonexpansive mappings wrt orbits, and we present a modified version of the Goebel–Karlovitz lemma. © 2016, Springer International Publishing.
Journal of Fixed Point Theory and Applications (16617738)17(2)pp. 287-300
On a subset of Banach space which is a locally finite union of closed, convex sets, we extend the Lefschetz fixed point theorem for set-valued mappings. As an application of this result we give a partial answer to Nussbaum’s conjecture for set-valued mappings. © 2014, Springer Basel.
Taiwanese Journal of Mathematics (10275487)18(6)pp. 1999-2020
In this paper, we introduce the concept of a weak q-distance and for this distance we derive a set-valued version of Ekeland’s variational principle in the setting of uniform spaces. By using this principle, we prove the existence of solutions to a vector optimization problem with a set-valued map. Moreover, we define the (p, ε)-condition of Takahashi and the (p, ε)-condition of Hamel for a set-valued map. It is shown that these two conditions are equivalent. As an application, we discuss the relationship between an ε-approximate solution and a solution of a vector optimization problem with a set-valued map. Also, a well-posedness result for a vector optimization problem with a set-valued map is given. © 2014, Mathematical Society of the Rep. of China. All rights reserved.
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas (15791505)108(2)pp. 721-732
In this paper, we first introduce a distance space and then give a new fixed point theorem for multi-valued contractions in such spaces. Even in the case of metric spaces, our main theorem unifies and generalizes some recent results in the literature. Some examples are given to show that the fixed point result given here is a genuine generalization. © 2013 Springer-Verlag Italia.
Miskolc Mathematical Notes (17872405)15(1)pp. 87-95
By introducing a new concept called "set-valued asymptotic contraction of final type" in metric spaces, the existence and uniqueness of compact fixed sets for such mappings have been obtained. Furthermore, by adding additional conditions, we prove the existence and uniqueness of endpoints for these maps. © 2014 Miskolc University Press.
Rendiconti del Circolo Matematico di Palermo (0009725X)62(3)pp. 367-377
In this paper, we obtain some fixed point theorems for new set-valued contractions in complete metric spaces. Then by using these results and the scalarization method, we present some fixed point theorems for set-valued contractions in complete cone metric spaces without the normality assumption. We also present some examples to support our results. © 2013 Springer-Verlag Italia.
Journal of Global Optimization (09255001)55(4)pp. 751-770
In this paper, we obtain several new generalized KKM-type theorems under a new coercivity condition and the condition of intersectionally closedness which improves condition of transfer closedness. As applications, we obtain new versions of equilibrium problem, minimax inequality, coincidence theorem, fixed point theorem and an existence theorem for an 1-person game. © 2012 Springer Science+Business Media, LLC.
Mathematical and Computer Modelling (08957177)57(5-6)pp. 1250-1258
In this paper, some extensions of the Ekeland variational principle in metric spaces are given for a generalized pseudodistance. As an application we obtain Caristi's fixed point theorem. Then, by using this result, we establish some fixed point theorems for set-valued contractive mappings. © 2012 Elsevier Ltd.
Fixed Point Theory and Algorithms for Sciences and Engineering (16871812)2013
In this paper, we first introduce two new classes of (ω,δ)- contractions of the first and second kinds and establish some related new fixed point and best proximity point theorems in preordered metric spaces. Our theorems subsume the corresponding recent results of Samet (J. Optim. Theory Appl. (2013), doi:10.1007/s10957-013-0269-9) and extend and generalize many of the well-known results in the literature. An example is also provided to support our main results. ©2013 Amini-Harandi et al.; licensee Springer.
Topology and its Applications (1668641)159(16)pp. 3453-3460
In this paper, we present some asymptotic stationary point results for topological contraction mappings by relaxing the compactness of the space. Moreover, some classes of topological contractions are characterized. © 2012 Elsevier B.V.
Applied Mathematics Letters (18735452)24(4)pp. 428-431
By introducing a new concept called "set-valued asymptotic contraction" in metric spaces, the existence and uniqueness of endpoints for a set-valued asymptotic contraction which has the approximate endpoint property have been established. © 2010 Elsevier Ltd. All rights reserved.
Australian Journal of Basic and Applied Sciences (19918178)71(11)pp. 1782-1787
In this paper, we consider a new class of Vector variational inequalities with multivalued pseudomonotone with respect to (η,F). The results presented in this paper are extension and improvement of the authores.
Nonlinear Analysis, Theory, Methods and Applications (0362546X)72(6)pp. 2891-2895
A new generalized set-valued contraction on topological spaces with respect to a measure of noncompactness is introduced. Two fixed point theorems for the K K M type maps which are either generalized set-contraction or condensing ones are given. Furthermore, applications of these results for existence of coincidence points and maximal elements are deduced. © 2009.
Computers and Mathematics with Applications (08981221)59(11)pp. 3529-3534
Motivated by the scalarization method in vector optimization theory, we take a new approach to fixed point theory on cone metric spaces. By using our method we prove some fixed point theorems and several common fixed point theorems on cone metric spaces in which the cone need not be normal. Our results improve and generalize many well-known results from the literature. © 2010 Elsevier Ltd. All rights reserved.
Fixed Point Theory and Algorithms for Sciences and Engineering (16871812)2010
By using a Dane s ' drop theorem in locally convex spaces we obtain a vectorial form of Ekeland-type variational principle in locally convex spaces. From this theorem, we derive some versions of vectorial Caristi-Kirk's fixed-point theorem, Takahashi's nonconvex minimization theorem, and Oettli-Théra's theorem. Furthermore, we show that these results are equivalent to each other. Also, the existence of solution of vector equilibrium problem is given. Copyright 2010 S. Eshghinezhad and M. Fakhar.
Journal of Global Optimization (09255001)43(4)pp. 503-511
We obtain a new version of the minimax inequality of Ky Fan. As an application, an existence result for the generalized variational inequality problem with set-valued mappings defined on noncompact sets in Hausdorff topological vector spaces is given. Also, some existence results for the generalized variational inequality problem for quasimonotone and pseudomonotone mappings are obtained. © 2008 Springer Science+Business Media, LLC.
Journal of Optimization Theory and Applications (00223239)136(3)pp. 397-409
We establish an existence result for scalar quasiequilibrium problems without any continuity requirement on noncompact subsets of locally convex topological vector spaces. As a consequence, we obtain a solution of symmetric scalar quasiequilibrium problem. Moreover, using a so-called nonlinear scalarization function, existence theorems for vector quasiequilibrium problems and general symmetric vector quasiequilibrium problems are obtained. © 2007 Springer Science+Business Media, LLC.
Nonlinear Analysis, Theory, Methods and Applications (0362546X)66(1)pp. 14-21
We define KKM mappings and S-KKM mappings similarly to in the case of convex spaces for abstract convex spaces. Some approximate fixed point theorems will be established for the multifunction with the S-KKM property on Φ-spaces. We also obtain a new version of Sadovskii's fixed point theorem in topological spaces. © 2005 Elsevier Ltd. All rights reserved.
Taiwanese Journal of Mathematics (10275487)11(1)pp. 95-105
We give some new generalized. R-KKM theorems in the nonconvexity setting of topological spaces. As an application we answer a question posed by Isac et al. for the lower and upper bounds equilibrium problem in topological spaces.
Journal of Optimization Theory and Applications (223239)126(1)pp. 125-136
Existence results for quasimonotone vector equilibrium problems and quasimonotone vector variational inequalities are obtained starting from an existence result for a scalar equilibrium problem involving two quasimonotone bifunctions. These results are established under weaker conditions than in previous works. © 2005 Springer Science+Business Media, Inc.
Bulletin of the Belgian Mathematical Society - Simon Stevin (13701444)12(2)pp. 235-247
We obtain a generalized continuous selection theorem and a coincidence theorem for generalized convex spaces. Some new Himmelberg type theorems and Eilenberg-Montgomery and Gorniéwicz type fixed point theorems for mappings with KKM property are established in noncompact LG-spaces. Moreover, applications to these fixed point theorems for existence of equilibria are given.
Journal of Optimization Theory and Applications (223239)126(1)pp. 109-124
In this paper, we apply a new version of the Brézis, Nirenberg, and Stampacchia theorem; we use pseudomonotonicity and some coercivity conditions to establish some existence result for a solution of generalized vector equilibrium problems for multivalued bifunctions. The proper quasiconvexity of multivalued bifunctions is introduced and existence theorems for generalized vector equilibrium problems related to multivalued mappings with the KKM property are obtained. The new results extend and modify various existence theorems for similar problems. © 2005 Springer Science+Business Media, Inc.
Nonlinear Analysis, Theory, Methods and Applications (0362546X)60(6)pp. 1045-1052
We introduce the class of KKM-type mappings on metric spaces and establish some fixed point theorems for this class. We also obtain a generalized Fan's matching theorem, a generalized Fan-Browder's type theorem, and a new version of Fan's best approximation theorem. © 2004 Elsevier Ltd. All rights reserved.
Journal of Optimization Theory and Applications (223239)123(2)pp. 349-364
By using quasimonotone and pseudomonotone bifunctions, we derive sufficient conditions which include weak coercivity conditions for existence of equilibrium points. As a consequence, we improve some recent results on the existence of such solutions.
International Journal of Mathematics and Mathematical Sciences (1611712)2003(51)pp. 3267-3276
We give some new versions of KKM theorem for generalized convex spaces. As an application, we answer a question posed by Isac et al. (1999) for the lower and upper bounds equilibrium problem. © 2003 Hindawi Publishing Corporation. All rights reserved.
