Mathematical Researches

fa عملگرهای یکنوای تعمیم یافته و رویکرد قطبیِ مجموعه های یکنوای تعمیم‌یافته Generalized monotone operators and polarity approach to generalized monotone sets ریاضی Mat علمی پژوهشی بنیادی S ابتدا  نامساوی فنشل- موراِ  به توابع σ-محدب تعمیم داده میشود و سپس با استفاده از تابع فیتزپاتریک تعمیم یافته، تظریفی برای نامساوی فنشل- موراِ تعمیم یافته ارائه میشود. در ادامه، قطبی مجموعه های σ -یکنوا معرفی شده و نتایج مرتبط با آن مورد مطالعه قرار میگیرد. Introduction Many Suppose that <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image002.gif" style="width:19px; height:17px" >is a Banach Space with topological dual space <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image004.gif" style="width:27px; height:21px" >We will denote by <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image006.gif" style="width:119px; height:24px" > the duality pairing between X and  <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image008.gif" style="width:23px; height:20px" >. For <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image010.gif" style="width:49px; height:17px" >, we denote by <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image012.gif" style="width:12px; height:19px" ><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image014.gif" style="width:36px; height:19px" >the boundary points of Ω and by <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image016.gif" style="width:36px; height:19px" > the interior of Ω. Also we will denote by <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image018.gif" style="width:24px; height:24px" > the real nonnegative numbers. Let <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image020.gif" style="width:15px; height:17px" >be a set-valued map from <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image002.gif" style="width:19px; height:17px" > to <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image008.gif" style="width:23px; height:20px" >. The domain and graph of <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image020.gif" style="width:15px; height:17px" > are, respectively, defined by <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image024.gif" style="width:319px; height:61px" > We recall that a set valued operator <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image020.gif" style="width:15px; height:17px" >is monotone if  <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image027.gif" style="width:124px; height:24px" > for all <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image029.gif" style="width:57px; height:21px" > and  <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image031.gif" style="width:137px; height:27px" > For two multivalued operators <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image020.gif" style="width:15px; height:17px" > and <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image034.gif" style="width:15px; height:19px" >we write <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image036.gif" style="width:43px; height:20px" > if<img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image034.gif" style="width:15px; height:19px" >is an extension of <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image020.gif" style="width:15px; height:17px" >, i.e., <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image040.gif" style="width:73px; height:21px" >. A monotone operator is called maximal monotone if it has no monotone extension other than itself. In 1988, The Fitzpatrick function of a monotone operator was introduced by Fitzpatrick. The Fitzpatrick function makes a bridge between the results of convex functions and results on maximal monotone operators. For a monotone operator <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image020.gif" style="width:15px; height:17px" >, its Fitzpatrick function is defined by                            <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image043.gif" style="width:313px; height:61px" > It is a convex and norm to weak lower semicontinuous and function. Let <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image045.gif" style="width:123px; height:27px" >be an extended real-valued function. Its effective domain is defined by <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image047.gif" style="width:196px; height:27px" > The function is called proper if <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image049.gif" style="width:85px; height:21px" >. Let <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image051.gif" style="width:16px; height:21px" > be a proper function. The subdifferential (in the sense of Convex Analysis) of <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image051.gif" style="width:16px; height:21px" > at <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image054.gif" style="width:83px; height:27px" > is defined by                       <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image056.gif" style="width:353px; height:29px" > Given a proper function <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image051.gif" style="width:16px; height:21px" > and a map <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image059.gif" style="width:120px; height:27px" >, then <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image051.gif" style="width:16px; height:21px" > is called <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image062.gif" style="width:16px; height:15px" >-convex if          <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image064.gif" style="width:459px; height:27px" > For all <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image029.gif" style="width:57px; height:21px" >and for all <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image067.gif" style="width:57px; height:21px" > Given an operator <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image020.gif" style="width:15px; height:17px" > and a map <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image070.gif" style="width:103px; height:27px" >. Then <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image020.gif" style="width:15px; height:17px" >is called <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image062.gif" style="width:16px; height:15px" >-monotone if for all <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image029.gif" style="width:57px; height:21px" > and <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image075.gif" style="width:133px; height:27px" > we have                              <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image077.gif" style="width:299px; height:27px" > Also <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image020.gif" style="width:15px; height:17px" >is called maximal <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image062.gif" style="width:16px; height:15px" >-monotone if it has no <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image062.gif" style="width:16px; height:15px" >-monotone extension other than itself. We recall for a proper function <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image051.gif" style="width:16px; height:21px" > the <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image062.gif" style="width:16px; height:15px" >-subdifferential of <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image051.gif" style="width:16px; height:21px" > at <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image054.gif" style="width:83px; height:27px" > is defined by         <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image086.gif" style="width:463px; height:29px" > and <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image088.gif" style="width:84px; height:27px" > if <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image090.gif" style="width:83px; height:27px" >.  Main results   The definition we use for the Fitzpatrick function is the same as for monotone operators. Assume that <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image051.gif" style="width:16px; height:21px" >is a proper <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image062.gif" style="width:16px; height:15px" >-convex function its conjugate is defined by                                <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image094.gif" style="width:281px; height:56px" > First we have the following refinement of the Fenchel-Moreau inequality:                     <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image096.gif" style="width:373px; height:59px" > where <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image098.gif" style="width:16px; height:24px" >is the indicator function and <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image100.gif" style="width:189px; height:27px" >. Also we have the following refinement, when <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image051.gif" style="width:16px; height:21px" >is a proper, <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image062.gif" style="width:16px; height:15px" >-convex and lower semicontinuous function and <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image102.gif" style="width:31px; height:24px" > is a maximal <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image062.gif" style="width:16px; height:15px" >-monotone operator:                                  <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip11clip_image105.gif" style="width:269px; height:61px" > Moreover, we approach generalized monotonicity from the point of view of the classical concept of polarity. Besides, we introduce and study the notion of generalized monotone polar of a set A. Moreover, we find some equivalent relations between polarity and maximal generalized monotonicity.   عملگرهای یکنوا و σ -یکنوا, نامساوی فنشل- موراِ, تابع فیتزپاتریک, قطبی مجموعه های یکنوا و σ -یکنوا Monotone and sigma-monotone operators, Fitzpatrik function, Fenchel inequality, Monotone polar 151 163 http://mmr.khu.ac.ir/browse.php?a_code=A-10-496-1&slc_lang=fa&sid=1 Mohammad Alizadeh محمد حسین علی زاده m.alizadeh@iasbs.ac.ir 10031947532846005796 10031947532846005796 Yes IASBS دانشگاه تحصیلات تکمیلی علوم پایه زنجان