Type: Article
Publication Date: 2013-01-01
Citations: 9
DOI: https://doi.org/10.1155/2013/813635
We studied the approximate split equality problem (ASEP) in the framework of infinite-dimensional Hilbert spaces. Let <svg style="vertical-align:-3.27605pt;width:21.362499px;" id="M1" height="15.25" version="1.1" viewBox="0 0 21.362499 15.25" width="21.362499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path id="x1D43B" d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28
h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z" /></g> <g transform="matrix(.012,-0,0,-.012,14.975,15.187)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g> </svg>, <svg style="vertical-align:-3.27605pt;width:21.362499px;" id="M2" height="15.25" version="1.1" viewBox="0 0 21.362499 15.25" width="21.362499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D43B"/></g> <g transform="matrix(.012,-0,0,-.012,14.975,15.187)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g> </svg>, and  <svg style="vertical-align:-3.3907pt;width:21.362499px;" id="M3" height="15.4" version="1.1" viewBox="0 0 21.362499 15.4" width="21.362499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D43B"/></g> <g transform="matrix(.012,-0,0,-.012,14.975,15.187)"><path id="x33" d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5
q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" /></g> </svg> be infinite-dimensional real Hilbert spaces, let <svg style="vertical-align:-3.27605pt;width:52.775002px;" id="M4" height="15.5375" version="1.1" viewBox="0 0 52.775002 15.5375" width="52.775002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D436" d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5
t325.5 101.5q114 0 204 -38z" /></g><g transform="matrix(.017,-0,0,-.017,16.755,11.4)"><path id="x2282" d="M512 1h-130q-127 0 -217 74.5t-90 179.5t90 179.5t217 74.5h130v-50h-131q-103 0 -175.5 -59.5t-72.5 -144.5t72.5 -144.5t175.5 -59.5h131v-50z" /></g><g transform="matrix(.017,-0,0,-.017,31.459,11.4)"><use xlink:href="#x1D43B"/></g> <g transform="matrix(.012,-0,0,-.012,46.375,15.475)"><use xlink:href="#x31"/></g> </svg> and  <svg style="vertical-align:-3.27605pt;width:53.837502px;" id="M5" height="15.5375" version="1.1" viewBox="0 0 53.837502 15.5375" width="53.837502" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D444" d="M745 361q0 -134 -83.5 -233t-214.5 -130l16 -11q97 -67 250 -132l-8 -23q-76 3 -131 16q-81 19 -242 125l-20 13q-129 8 -209 91t-80 208q0 160 116 271t289 111q136 0 226.5 -83t90.5 -223zM645 356q0 127 -57.5 201.5t-169.5 74.5q-126 0 -210.5 -104.5t-84.5 -248.5
q0 -97 46 -166.5t129 -87.5l84 15l29 -19q104 21 169 121.5t65 213.5z" /></g><g transform="matrix(.017,-0,0,-.017,17.826,11.4)"><use xlink:href="#x2282"/></g><g transform="matrix(.017,-0,0,-.017,32.53,11.4)"><use xlink:href="#x1D43B"/></g> <g transform="matrix(.012,-0,0,-.012,47.45,15.475)"><use xlink:href="#x32"/></g> </svg> be two nonempty closed convex sets, and let <svg style="vertical-align:-3.3907pt;width:101.5125px;" id="M6" height="15.6875" version="1.1" viewBox="0 0 101.5125 15.6875" width="101.5125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D434" d="M716 28l-8 -28h-254l8 28q55 5 66 17.5t5 47.5l-30 145h-232l-74 -138q-21 -43 -11 -54.5t72 -17.5l-6 -28h-235l8 28q53 7 73.5 21t53.5 70l319 540l33 8q6 -43 33 -176l80 -381q10 -49 26.5 -63t72.5 -19zM495 281l-49 264h-2l-149 -264h200z" /></g><g transform="matrix(.017,-0,0,-.017,17.231,11.4)"><path id="x2236" d="M223 355q0 -25 -15.5 -41.5t-39.5 -16.5t-40 16.5t-16 41.5t16 41t40 16t39.5 -16t15.5 -41zM223 46q0 -26 -16 -42t-40 -16t-39.5 16t-15.5 42q0 24 16 40.5t39 16.5q24 0 40 -16.5t16 -40.5z" /></g><g transform="matrix(.017,-0,0,-.017,27.634,11.4)"><use xlink:href="#x1D43B"/></g> <g transform="matrix(.012,-0,0,-.012,42.55,15.475)"><use xlink:href="#x31"/></g> <g transform="matrix(.017,-0,0,-.017,56.438,11.4)"><path id="x2192" d="M901 255q-71 -62 -185 -187l-22 15l102 147h-727v50h727l-102 147l22 15q114 -125 185 -187z" /></g><g transform="matrix(.017,-0,0,-.017,80.202,11.4)"><use xlink:href="#x1D43B"/></g> <g transform="matrix(.012,-0,0,-.012,95.125,15.475)"><use xlink:href="#x33"/></g> </svg> and  <svg style="vertical-align:-3.3907pt;width:99.550003px;" id="M7" height="15.4" version="1.1" viewBox="0 0 99.550003 15.4" width="99.550003" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path id="x1D435" d="M594 511q0 -122 -171 -157l1 -2q158 -33 158 -159q0 -52 -34.5 -95t-90.5 -65q-76 -33 -217 -33h-223l8 28q63 5 79.5 19t26.5 72l83 426q9 48 -2.5 60t-77.5 17l6 28h259q195 0 195 -139zM499 509q0 59 -37 83t-91 24q-36 0 -51 -9q-17 -9 -22 -44l-35 -195h62
q82 0 128 37t46 104zM481 199q0 71 -48 102.5t-121 31.5h-56l-37 -201q-11 -58 7.5 -77t80.5 -19q76 0 125 44.5t49 118.5z" /></g><g transform="matrix(.017,-0,0,-.017,15.259,11.112)"><use xlink:href="#x2236"/></g><g transform="matrix(.017,-0,0,-.017,25.663,11.112)"><use xlink:href="#x1D43B"/></g> <g transform="matrix(.012,-0,0,-.012,40.575,15.187)"><use xlink:href="#x32"/></g> <g transform="matrix(.017,-0,0,-.017,54.475,11.112)"><use xlink:href="#x2192"/></g><g transform="matrix(.017,-0,0,-.017,78.239,11.112)"><use xlink:href="#x1D43B"/></g> <g transform="matrix(.012,-0,0,-.012,93.15,15.187)"><use xlink:href="#x33"/></g> </svg> be two bounded linear operators. The ASEP in infinite-dimensional Hilbert spaces is to minimize the function <svg style="vertical-align:-4.0404pt;width:183.5px;" id="M8" height="20.875" version="1.1" viewBox="0 0 183.5 20.875" width="183.5" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path id="x1D453" d="M619 670q0 -13 -9 -26t-18 -19q-13 -10 -25 2q-36 38 -66 38q-31 0 -54.5 -50t-45.5 -185h120l-20 -31l-107 -12q-23 -138 -57 -293q-27 -122 -55 -184.5t-75 -109.5q-60 -61 -114 -61q-25 0 -47.5 15t-22.5 31q0 17 31 44q11 8 20 -1q10 -11 31 -19t35 -8q26 0 47 19
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