scholarly journals Existence, Comparison, and Convergence Results for a Class of Elliptic Hemivariational Inequalities

2021 ◽  
Author(s):  
Claudia M. Gariboldi ◽  
Stanisław Migórski ◽  
Anna Ochal ◽  
Domingo A. Tarzia
Keyword(s):  
Heat Conduction Problem ◽  
Elliptic Boundary ◽  
Sufficient Conditions ◽  
Hemivariational Inequalities ◽  
Pseudomonotone Operators ◽  
Generalized Gradient ◽  
Convergence Results ◽  
Locally Lipschitz ◽  
Steady State Heat ◽  
The Heat Transfer Coefficient

AbstractIn this paper we study a class of elliptic boundary hemivariational inequalities which originates in the steady-state heat conduction problem with nonmonotone multivalued subdifferential boundary condition on a portion of the boundary described by the Clarke generalized gradient of a locally Lipschitz function. First, we prove a new existence result for the inequality employing the theory of pseudomonotone operators. Next, we give a result on comparison of solutions, and provide sufficient conditions that guarantee the asymptotic behavior of solution, when the heat transfer coefficient tends to infinity. Further, we show a result on the continuous dependence of solution on the internal energy and heat flux. Finally, some examples of convex and nonconvex potentials illustrate our hypotheses.

Equivalence of Some Multi-Valued Elliptic Variational Inequalities and Variational-Hemivariational Inequalities

2011 ◽  
Vol 11 (2) ◽  
Author(s):  
Siegfried Carl
Keyword(s):  
Variational Inequalities ◽  
Lipschitz Functions ◽  
Hemivariational Inequalities ◽  
Locally Lipschitz Functions ◽  
Generalized Gradient ◽  
Elliptic Variational Inequalities ◽  
Comparison Results ◽  
Locally Lipschitz ◽  
Additional Regularity ◽  
Definition Of

AbstractFirst, we prove existence and comparison results for multi-valued elliptic variational inequalities involving Clarke’s generalized gradient of some locally Lipschitz functions as multi-valued term. Only by applying the definition of Clarke’s gradient it is well known that any solution of such a multi-valued elliptic variational inequality is also a solution of a corresponding variational-hemivariational inequality. The reverse is known to be true if the locally Lipschitz functions are regular in the sense of Clarke. Without imposing this kind of regularity the equivalence of the two problems under consideration is not clear at all. The main goal of this paper is to show that the equivalence still holds true without any additional regularity, which will fill a gap in the literature. Existence and comparison results for both multi-valued variational inequalities and variational-hemivariational inequalities are the main tools in the proof of the equivalence of these problems.

scholarly journals On nonsmooth multiobjective fractional programming problems involving (p, r)− ρ −(η ,θ)- invex functions

2013 ◽  
Vol 23 (3) ◽  
pp. 367-386 ◽  
Author(s):  
Anurag Jayswal ◽  
Ashish Prasad ◽  
I.M. Stancu-Minasian
Keyword(s):  
Fractional Programming ◽  
Sufficient Conditions ◽  
Generalized Gradient ◽  
Multiobjective Fractional Programming ◽  
Dual Problems ◽  
Invex Functions ◽  
Locally Lipschitz ◽  
Clarke Generalized Gradient ◽  
Sufficient Conditions For Optimality ◽  
Definition Of

A class of multiobjective fractional programming problems (MFP) is considered where the involved functions are locally Lipschitz. In order to deduce our main results, we introduce the definition of (p,r)?? ?(?,?)-invex class about the Clarke generalized gradient. Under the above invexity assumption, sufficient conditions for optimality are given. Finally, three types of dual problems corresponding to (MFP) are formulated, and appropriate dual theorems are proved.

Approximate controllability for stochastic fractional hemivariational inequalities of degenerate type

2020 ◽  
Vol 23 (5) ◽  
pp. 1506-1531
Author(s):  
Yatian Pei ◽  
Yong-Kui Chang
Keyword(s):  
Approximate Controllability ◽  
Directional Derivative ◽  
Inverse Operator ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Hemivariational Inequalities ◽  
Sobolev Type ◽  
Resolvent Family ◽  
Locally Lipschitz ◽  
Degenerate Type

Abstract This paper is mainly concerned with stochastic fractional hemivariational inequalities of degenerate (or Sobolev) type in Caputo and Riemann-Liouville derivatives with order (1, 2), respectively. Based upon some properties of fractional resolvent family and generalized directional derivative of a locally Lipschitz function, some sufficient conditions are established for the existence and approximate controllability of the aforementioned systems. Particularly, the uniform boundedness for some nonlinear terms, the existence and compactness of certain inverse operator are not necessarily needed in obtained approximate controllability results.

scholarly journals No Infimum Gap and Normality in Optimal Impulsive Control Under State Constraints

2021 ◽  
Author(s):  
Giovanni Fusco ◽  
Monica Motta
Keyword(s):  
Original Problem ◽  
State Constraints ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Continuous Data ◽  
Lipschitz Continuous ◽  
Extended Sense ◽  
Unbounded Controls ◽  
Locally Lipschitz ◽  
Optimal Impulsive Control

AbstractIn this paper we consider an impulsive extension of an optimal control problem with unbounded controls, subject to endpoint and state constraints. We show that the existence of an extended-sense minimizer that is a normal extremal for a constrained Maximum Principle ensures that there is no gap between the infima of the original problem and of its extension. Furthermore, we translate such relation into verifiable sufficient conditions for normality in the form of constraint and endpoint qualifications. Links between existence of an infimum gap and normality in impulsive control have previously been explored for problems without state constraints. This paper establishes such links in the presence of state constraints and of an additional ordinary control, for locally Lipschitz continuous data.

Steady-State Thermal Stresses in Wedge-Shaped Cutting Tools

1975 ◽  
Vol 97 (3) ◽  
pp. 1060-1066
Author(s):  
P. F. Thomason
Keyword(s):  
Thermal Stress ◽  
Steady State ◽  
Metal Cutting ◽  
Thermal Stresses ◽  
Heat Conduction Problem ◽  
Equivalent Stress ◽  
Cutting Tools ◽  
Thermoelastic Stress ◽  
Plastic State ◽  
Steady State Heat

Closed form expressions for the steady-state thermal stresses in a π/2 wedge, subject to constant-temperature heat sources on the rake and flank contact segments, are obtained from a conformal mapping solution to the steady-state heat conduction problem. It is shown, following a theorem of Muskhelishvili, that the only nonzero thermal stress in the plane-strain wedge is that acting normal to the wedge plane. The thermal stress solutions are superimposed on a previously published isothermal cutting-load solution, to give the complete thermoelastic stress distribution at the wedge surfaces. The thermoelastic stresses are then used to determine the distribution of the equivalent stress, and this gives an indication of the regions on a cutting tool which are likely to be in the plastic state. The results are discussed in relation to the problems of flank wear and rakeface crater wear in metal cutting tools.

scholarly journals ON EIGENVALUE PROBLEMS FOR ELLIPTIC HEMIVARIATIONAL INEQUALITIES

2008 ◽  
Vol 51 (2) ◽  
pp. 407-419 ◽  
Author(s):  
Zhenhai Liu ◽  
Guifang Liu
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problems ◽  
Hemivariational Inequalities ◽  
Generalized Gradient ◽  
At Resonance ◽  
Quasilinear Elliptic ◽  
First Eigenfunction

AbstractThis paper is devoted to the Dirichlet problem for quasilinear elliptic hemivariational inequalities at resonance as well as at non-resonance. Using Clarke's notion of the generalized gradient and the property of the first eigenfunction, we also build a Landesman–Lazer theory in the non-smooth framework of quasilinear elliptic hemivariational inequalities.

scholarly journals Fixed Point Theorems for Generalized αs-ψ-Contractions with Applications

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Zhenhua Ma ◽  
Muhammad Nazam ◽  
Sami Ullah Khan ◽  
Xiangling Li
Keyword(s):  
Fixed Point ◽  
Boundary Value Problems ◽  
Common Fixed Point ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Elliptic Boundary ◽  
Sufficient Conditions ◽  
Elliptic Boundary Value Problems ◽  
Elliptic Boundary Value ◽  
Unique Common Fixed Point

We study the sufficient conditions for the existence of a unique common fixed point of generalized αs-ψ-Geraghty contractions in an αs-complete partial b-metric space. We give an example in support of our findings. Our work generalizes many existing results in the literature. As an application of our findings we demonstrate the existence of the solution of the system of elliptic boundary value problems.

scholarly journals Selected Topics in Homogenization of Transport Processes in Historical Masonry Structures

2012 ◽  
Vol 6 (1) ◽  
pp. 148-159
Author(s):  
Jan Sykora ◽  
Jan Zeman ◽  
Michal Ŝejnoha
Keyword(s):  
Initial Conditions ◽  
Heat Conduction Problem ◽  
Transport Processes ◽  
Random Pattern ◽  
Binary Sample ◽  
Homogenization Approach ◽  
Heat And Moisture Transport ◽  
Historical Masonry ◽  
Steady State Heat ◽  
Heat And Moisture

The paper reviews several topics associated with the homogenization of transport processed in historical ma-sonry structures. Since these often experience an irregular or random pattern, we open the subject by summarizing essen-tial steps in the formulation of a suitable computational model in the form of Statistically Equivalent Periodic Unit Cell (SEPUC). Accepting SEPUC as a reliable representative volume element is supported by application of the Fast Fourier Transform to both the SEPUC and large binary sample of real masonry in search for effective thermal conductivities lim-ited here to a steady state heat conduction problem. Fully coupled non-stationary heat and moisture transport is addressed next in the framework of two-scale first-order homogenization approach with emphases on the application of boundary and initial conditions on the meso-scale.

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