Let <svg     style="vertical-align:-0.0pt;width:13.5875px;" id="M1" height="11.175" version="1.1" viewBox="0 0 13.5875 11.175" width="13.5875"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D44B" d="M775 650l-6 -28q-60 -6 -81.5 -16t-61.5 -54l-175 -191l125 -243q30 -58 48.5 -71t82.5 -19l-5 -28h-275l7 28l35 4q31 4 37 12t-6 34l-108 216q-140 -165 -177 -219q-16 -22 -10.5 -30.5t41.5 -13.5l22 -3l-7 -28h-244l8 28q52 4 75 15.5t67 52.5q48 46 206 231&#xA;l-110 215q-26 51 -44 63t-72 17l6 28h250l-6 -28l-27 -4q-30 -5 -35 -10t3 -27q17 -43 95 -190q70 78 154 185q15 21 10 29.5t-33 12.5l-30 4l5 28h236z" /></g>

		
	
</svg> and <svg     style="vertical-align:-0.0pt;width:11.6625px;" id="M2" height="11.175" version="1.1" viewBox="0 0 11.6625 11.175" width="11.6625"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D44C" d="M667 650l-9 -28q-53 -5 -76 -17t-64 -59q-51 -61 -175 -225q-21 -29 -27 -55l-27 -136q-13 -65 -0.5 -80t83.5 -22l-7 -28h-280l8 28q64 4 81 19t30 83l25 128q6 35 -7 65l-98 231q-17 41 -32.5 52t-68.5 16l8 28h252l-6 -28l-40 -4q-27 -3 -33 -12.5t2 -31.5&#xA;q8 -26 43 -107.5t61 -134.5q114 145 174 240q14 24 8 33t-37 13l-34 4l8 28h238z" /></g>

		
	
</svg> be Banach spaces, and let <svg     style="vertical-align:-0.1638pt;width:84.037498px;" id="M3" height="11.6625" version="1.1" viewBox="0 0 84.037498 11.6625" width="84.037498"  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="#x1D44B"/></g><g transform="matrix(.017,-0,0,-.017,48.645,11.4)"><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,-.017,72.426,11.4)"><use xlink:href="#x1D44C"/></g>

		
	
</svg> be a bounded linear operator. In this paper, we first define and characterize the quasi-linear operator (resp., out) generalized inverse <svg     style="vertical-align:-5.3793pt;width:29.5625px;" id="M4" height="23.450001" version="1.1" viewBox="0 0 29.5625 23.450001" width="29.5625"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,16.7)"><use xlink:href="#x1D434"/></g>

			<g transform="matrix(.012,-0,0,-.012,12.525,8.538)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.012,-0,0,-.012,16.642,8.538)"><path id="x210E" d="M499 88q-41 -46 -83.5 -73t-68.5 -27q-41 0 -19 98l52 225q10 43 8 59.5t-16 16.5q-37 0 -112 -65t-117 -135q-14 -42 -38 -179q-36 -7 -75 -20l-7 8l127 594q8 40 2.5 49t-35.5 9h-35l5 25q36 3 71.5 13t56.5 18t25 8q11 0 3 -38l-88 -417h2q77 102 195 167q44 24 79 24&#xA;q61 0 30 -130l-52 -220q-8 -33 4 -33q8 0 31 14.5t43 33.5z" /></g><g transform="matrix(.012,-0,0,-.012,22.782,8.538)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g><g transform="matrix(.012,-0,0,-.012,12.525,22)"><path id="x1D447" d="M649 676l-22 -187l-33 -2q3 56 -12 94q-8 20 -31.5 26.5t-86.5 6.5h-74l-90 -491q-4 -23 -6 -36.5t1 -25t7 -16.5t18 -9t27 -5t41 -3l-6 -28h-286l4 28q68 5 84 18.5t28 76.5l94 491h-55q-74 0 -100.5 -6t-41.5 -23q-24 -29 -54 -98l-32 1q32 98 53 188h22q7 -18 15 -22&#xA;t37 -4h417q23 0 33.5 5t25.5 21h23z" /></g><g transform="matrix(.012,-0,0,-.012,20.45,22)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.012,-0,0,-.012,23.162,22)"><path id="x1D446" d="M457 488l-30 -3q-17 148 -131 148q-53 0 -84.5 -34.5t-31.5 -82.5q0 -42 25.5 -72t74.5 -62l33 -22q63 -42 95 -85t32 -102q0 -84 -67 -137t-163 -53q-58 0 -113 22t-70 43l-4 152l27 4q4 -32 15 -62.5t31 -59.5t53.5 -47t76.5 -18q56 0 92 35t36 96q0 39 -25 70t-78 68&#xA;l-31 22q-32 23 -53.5 41.5t-45 57t-23.5 77.5q0 82 58 132.5t156 50.5q46 0 101 -17l18.5 -6t17 -6t8.5 -3q-4 -55 0 -147z" /></g>

		
	
</svg> (resp., <svg     style="vertical-align:-5.3793pt;width:36px;" id="M5" height="23.450001" version="1.1" viewBox="0 0 36 23.450001" width="36"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,16.7)"><use xlink:href="#x1D434"/></g>

			<g transform="matrix(.012,-0,0,-.012,12.525,8.538)"><use xlink:href="#x28"/></g><g transform="matrix(.012,-0,0,-.012,16.642,8.538)"><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><g transform="matrix(.012,-0,0,-.012,22.353,8.538)"><use xlink:href="#x2C"/></g><g transform="matrix(.012,-0,0,-.012,25.066,8.538)"><use xlink:href="#x210E"/></g><g transform="matrix(.012,-0,0,-.012,31.206,8.538)"><use xlink:href="#x29"/></g><g transform="matrix(.012,-0,0,-.012,12.525,22)"><use xlink:href="#x1D447"/></g><g transform="matrix(.012,-0,0,-.012,20.45,22)"><use xlink:href="#x2C"/></g><g transform="matrix(.012,-0,0,-.012,23.162,22)"><use xlink:href="#x1D446"/></g>

		
	
</svg>) for the operator <svg     style="vertical-align:-0.0pt;width:12.5875px;" id="M6" height="11.4625" version="1.1" viewBox="0 0 12.5875 11.4625" width="12.5875"  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)"><use xlink:href="#x1D434"/></g>

		
	
</svg>, where <svg     style="vertical-align:-0.0pt;width:44.325001px;" id="M7" height="11.6125" version="1.1" viewBox="0 0 44.325001 11.6125" width="44.325001"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.55)"><use xlink:href="#x1D447"/></g><g transform="matrix(.017,-0,0,-.017,16.092,11.55)"><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,-.017,30.796,11.55)"><use xlink:href="#x1D44B"/></g>

		
	
</svg> and <svg     style="vertical-align:-0.23206pt;width:39.25px;" id="M8" height="11.75" version="1.1" viewBox="0 0 39.25 11.75" width="39.25"  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)"><use xlink:href="#x1D446"/></g><g transform="matrix(.017,-0,0,-.017,12.931,11.4)"><use xlink:href="#x2282"/></g><g transform="matrix(.017,-0,0,-.017,27.634,11.4)"><use xlink:href="#x1D44C"/></g>

		
	
</svg> are homogeneous subsets. Then, we further investigate the perturbation problems of the generalized inverses <svg     style="vertical-align:-5.3793pt;width:29.5625px;" id="M9" height="23.450001" version="1.1" viewBox="0 0 29.5625 23.450001" width="29.5625"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,16.7)"><use xlink:href="#x1D434"/></g>

			<g transform="matrix(.012,-0,0,-.012,12.525,8.538)"><use xlink:href="#x28"/></g><g transform="matrix(.012,-0,0,-.012,16.642,8.538)"><use xlink:href="#x210E"/></g><g transform="matrix(.012,-0,0,-.012,22.782,8.538)"><use xlink:href="#x29"/></g><g transform="matrix(.012,-0,0,-.012,12.525,22)"><use xlink:href="#x1D447"/></g><g transform="matrix(.012,-0,0,-.012,20.45,22)"><use xlink:href="#x2C"/></g><g transform="matrix(.012,-0,0,-.012,23.162,22)"><use xlink:href="#x1D446"/></g>

		
	
</svg> and <svg     style="vertical-align:-5.3793pt;width:36px;" id="M10" height="23.450001" version="1.1" viewBox="0 0 36 23.450001" width="36"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,16.7)"><use xlink:href="#x1D434"/></g>

			<g transform="matrix(.012,-0,0,-.012,12.525,8.538)"><use xlink:href="#x28"/></g><g transform="matrix(.012,-0,0,-.012,16.642,8.538)"><use xlink:href="#x32"/></g><g transform="matrix(.012,-0,0,-.012,22.353,8.538)"><use xlink:href="#x2C"/></g><g transform="matrix(.012,-0,0,-.012,25.066,8.538)"><use xlink:href="#x210E"/></g><g transform="matrix(.012,-0,0,-.012,31.206,8.538)"><use xlink:href="#x29"/></g><g transform="matrix(.012,-0,0,-.012,12.525,22)"><use xlink:href="#x1D447"/></g><g transform="matrix(.012,-0,0,-.012,20.45,22)"><use xlink:href="#x2C"/></g><g transform="matrix(.012,-0,0,-.012,23.162,22)"><use xlink:href="#x1D446"/></g>

		
	
</svg>. The results obtained in this paper extend some well-known results for linear operator generalized inverses with prescribed range and kernel.

Jianbing Cao

Yifeng Xue

The Quasi-Linear Operator Outer Generalized Inverse with Prescribed Range and Kernel in Banach Spaces

We introduce the sequence space <svg     style="vertical-align:-3.50804pt;width:142.39999px;" id="M2" height="17.987499" version="1.1" viewBox="0 0 142.39999 17.987499" width="142.39999"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,13.55)"><use xlink:href="#x1D438"/></g>

			<g transform="matrix(.012,-0,0,-.012,10.8,5.388)"><use xlink:href="#x1D45E"/></g><g transform="matrix(.012,-0,0,-.012,10.8,17.638)"><use xlink:href="#x1D45B"/></g>

			<g transform="matrix(.017,-0,0,-.017,17.4,13.55)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,23.282,13.55)"><use xlink:href="#x2133"/></g><g transform="matrix(.017,-0,0,-.017,41.504,13.55)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,48.202,13.55)"><use xlink:href="#x1D45D"/></g><g transform="matrix(.017,-0,0,-.017,58.282,13.55)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,64.997,13.55)"><use xlink:href="#x1D460"/></g><g transform="matrix(.017,-0,0,-.017,71.371,13.55)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,78.086,13.55)"><use xlink:href="#x7C"/></g><g transform="matrix(.017,-0,0,-.017,82.556,13.55)"><use xlink:href="#x7C"/></g><g transform="matrix(.017,-0,0,-.017,87.027,13.55)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,90.903,13.55)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,97.6,13.55)"><use xlink:href="#x2026"/></g><g transform="matrix(.017,-0,0,-.017,116.928,13.55)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,123.642,13.55)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,127.518,13.55)"><use xlink:href="#x7C"/></g><g transform="matrix(.017,-0,0,-.017,131.989,13.55)"><use xlink:href="#x7C"/></g><g transform="matrix(.017,-0,0,-.017,136.459,13.55)"><use xlink:href="#x29"/></g>

		
	
</svg> defined by a Musielak-Orlicz function <svg     style="vertical-align:-3.3907pt;width:73.5px;" id="M3" height="16.4125" version="1.1" viewBox="0 0 73.5 16.4125" width="73.5"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x2133"/></g><g transform="matrix(.017,-0,0,-.017,22.994,12.138)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,37.698,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,43.579,12.138)"><path id="x1D440" d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28&#xA;q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" /></g>

			<g transform="matrix(.012,-0,0,-.012,60.95,16.212)"><path id="x1D458" d="M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1&#xA;l5 26q38 4 74 13.5t57 17t25 7.5q12 0 4 -32l-104 -443h2q35 38 97 93q39 35 65.5 56t62 41.5t58.5 20.5q19 0 30.5 -10t11.5 -22z" /></g>

			<g transform="matrix(.017,-0,0,-.017,67.55,12.138)"><use xlink:href="#x29"/></g>

		
	
</svg>. We also study some topological properties and prove some inclusion relations involving this space.

Kuldip Raj

Sunil K. Sharma

Some Properties of the Sequence Space <svg style="vertical-align:-3.50804pt" height="17.987499" version="1.1" viewBox="0 0 142.39999 17.987499" width="142.39999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,13.55)"><path id="x1D438" d="M609 650l-19 -162l-30 -2q2 69 -14 94q-9 18 -28.5 26t-74.5 8h-88q-30 0 -37 -6.5t-12 -36.5l-41 -212h103q48 0 69 5.5t32 19t26 50.5h29l-40 -198h-30q2 58 -11.5 70.5t-85.5 12.5h-99l-30 -167q-17 -87 3 -101q18 -15 111 -15q59 0 89 7t54 27q8 7 15.5 16t16 21.5&#xA;l14 20.5t15 23.5t13.5 21.5l29 -10q-52 -129 -71 -163h-500l6 28q66 4 83 17.5t28 73.5l77 409q11 61 -0.5 75.5t-78.5 18.5l10 28h467z" /></g><g transform="matrix(.012,-0,0,-.012,10.8,5.388)"><path id="x1D45E" d="M474 429q-19 -69 -47 -222l-65 -347q-8 -49 0 -60t49 -16l26 -3l-4 -26l-246 -12l4 25l17 3q37 6 50 20.5t23 60.5l67 321h-2q-65 -79 -150 -138q-69 -47 -104 -47q-28 0 -48.5 32t-20.5 81q0 78 35 148t90 117q67 57 161 77q23 5 58 5q43 0 90 -15zM387 387&#xA;q-30 16 -75 16q-58 0 -92 -27q-49 -39 -78.5 -112t-29.5 -136q0 -34 8.5 -52.5t21.5 -18.5q40 0 109.5 63.5t103.5 115.5q16 53 32 151z" /></g><g transform="matrix(.012,-0,0,-.012,10.8,17.638)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g><g transform="matrix(.017,-0,0,-.017,17.4,13.55)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,23.282,13.55)"><path id="x2133" d="M1048 672l8 -13q-142 -225 -231 -393q-88 -167 -88 -213q0 -35 33 -35q54 0 170 132l19 -16q-120 -146 -203 -146q-38 0 -62 22t-24 60q0 148 308 563l-1 2q-65 -25 -145 -120q-79 -95 -185 -284q-82 -144 -113 -181q-46 -54 -100 -54q-25 0 -40.5 19t-15.5 51&#xA;q0 87 143 307q126 197 192 270l-1 1q-54 -23 -119 -91q-65 -69 -120 -156l-52 -84l-47 -76l-40 -64q-21 -33 -39 -58q-47 -63 -70 -83q-61 -51 -107 -47q-34 0 -57 22.5t-23 55.5q0 27 17.5 46t41.5 19q21 0 36.5 -15.5t15.5 -36.5q0 -18 -12.5 -31t-30.5 -13q-8 0 -15 4h-3&#xA;q3 -21 31 -21q50 0 98 51q47 50 127 184q146 243 238 333q94 91 180 91q3 0 17 -2l15 -12q-100 -141 -192 -297q-89 -151 -130 -245q-17 -46 -17 -61q0 -18 13 -18q17 0 34.5 20.5t51.5 74.5l60 100l68 117q30 51 72 110q42 58 83 103q41 44 89 74t94 36q12 0 28 -2z" /></g><g transform="matrix(.017,-0,0,-.017,41.504,13.55)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,48.202,13.55)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 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/></g><g transform="matrix(.017,-0,0,-.017,82.556,13.55)"><use xlink:href="#x7C"/></g><g transform="matrix(.017,-0,0,-.017,87.027,13.55)"><path id="x22C5" d="M170 255q0 -25 -16 -41t-40 -16t-40 16t-16 41t16 41.5t40 16.5t40 -16.5t16 -41.5z" /></g><g transform="matrix(.017,-0,0,-.017,90.903,13.55)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,97.6,13.55)"><path id="x2026" d="M160 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16zM485 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16zM806 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5&#xA;t16 -40.5q0 -26 -15.5 -42t-41.5 -16z" /></g><g transform="matrix(.017,-0,0,-.017,116.928,13.55)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,123.642,13.55)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,127.518,13.55)"><use xlink:href="#x7C"/></g><g transform="matrix(.017,-0,0,-.017,131.989,13.55)"><use xlink:href="#x7C"/></g><g transform="matrix(.017,-0,0,-.017,136.459,13.55)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g></svg>

We prove some common fixed point theorems for two pairs of weakly compatible mappings in 2-metric spaces via an implicit relation. As an application to our main result, we derive Bryant's type generalized fixed point theorem for four finite families of self-mappings which can be utilized to derive common fixed point theorems involving any finite number of mappings. Our results improve and extend a host of previously known results. Moreover, we study the existence of solutions of a nonlinear integral equation.

Sunny Chauhan

Mohammad Imdad

Calogero Vetro

Unified Metrical Common Fixed Point Theorems in 2-Metric Spaces via an Implicit Relation

Motivated by Suzuki (2008), we prove a Suzuki-type fixed point theorem employing Chatterjea contraction on partial metric spaces.

Ali Erduran

Suzuki-Type Generalization of Chatterjea Contraction Mappings on
Complete Partial Metric Spaces

We prove some common fixed point theorems for two pairs of weakly compatible mappings satisfying a rational type contractive condition in the framework of complex valued metric spaces. The proved results generalize and extend some of the known results in the literature.

Sumit Chandok

Deepak Kumar

Some Common Fixed Point Results for Rational Type Contraction Mappings in Complex Valued Metric Spaces

We provide several characterizations of the bounded and the compact weighted composition operators from the Bloch space <svg     style="vertical-align:-0.20474pt;width:16.275px;" id="M1" height="12.0625" version="1.1" viewBox="0 0 16.275 12.0625" width="16.275"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.738)"><path id="x212C" d="M895 660l7 -19q-44 -20 -72 -40q72 -59 72 -138q0 -51 -45 -94q-43 -43 -108 -51v-2q43 -18 66.5 -53.5t23.5 -76.5q0 -45 -26 -89.5t-70 -74.5q-49 -33 -112 -33q-42 0 -68 23t-26 55q0 51 36.5 86t101.5 46l4 -23q-43 -5 -74 -33t-31 -61q0 -20 14.5 -34t35.5 -14&#xA;q36 0 61 15q36 21 57 63t21 91q0 70 -46 95q-24 -6 -49 -6q-38 0 -47 19q-158 -326 -414 -326q-76 0 -125 35q-48 36 -48 87q0 31 17 51.5t42 20.5q22 0 36.5 -13.5t14.5 -34.5t-11.5 -34.5t-31.5 -13.5q-23 0 -30 12h-2q0 -28 35 -54q34 -25 85 -25q94 0 161 53&#xA;q67 54 151 202q70 124 134 203q64 77 141 127q-82 56 -210 56q-91 0 -178 -32q-85 -32 -139 -92q-55 -61 -55 -137q0 -48 30 -77t81 -29q84 0 144 75q61 76 65 196h20q0 -125 -64 -213t-170 -88q-65 0 -106 38t-41 105q0 64 39 120.5t103 91.5q134 72 269 72q144 0 244 -67&#xA;q40 22 87 40zM841 485q0 54 -40 95q-43 -36 -79 -89t-83 -143q7 4 23 4q25 0 55 -14q60 4 92 49q32 44 32 98z" /></g>

		
	
</svg> and the analytic Besov spaces <svg     style="vertical-align:-5.73167pt;width:18.2875px;" id="M2" height="18.3125" version="1.1" viewBox="0 0 18.2875 18.3125" width="18.2875"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><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&#xA;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(.012,-0,0,-.012,10.55,15.188)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g>

		
	
</svg> (with <svg     style="vertical-align:-3.50804pt;width:74.550003px;" id="M3" height="15.2875" version="1.1" viewBox="0 0 74.550003 15.2875" width="74.550003"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,10.862)"><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><g transform="matrix(.017,-0,0,-.017,12.931,10.862)"><path id="x3C" d="M512 -3l-437 233v51l437 233v-58l-378 -200v-2l378 -199v-58z" /></g><g transform="matrix(.017,-0,0,-.017,27.634,10.862)"><use xlink:href="#x1D45D"/></g><g transform="matrix(.017,-0,0,-.017,42.44,10.862)"><use xlink:href="#x3C"/></g><g transform="matrix(.017,-0,0,-.017,57.144,10.862)"><path id="x221E" d="M983 225q0 -112 -67 -174.5t-150 -62.5q-91 0 -154.5 43.5t-113.5 129.5q-49 -85 -104 -129t-138 -44q-98 0 -158.5 66t-60.5 154q0 59 21 106.5t54.5 75.5t70.5 43t73 15q90 0 152.5 -43.5t112.5 -128.5q48 84 104.5 128t140.5 44q93 0 155 -65t62 -158zM478 196&#xA;q-27 49 -47 80t-50 67t-64 54t-73 18q-48 0 -81.5 -47t-33.5 -128q0 -96 37.5 -157.5t99.5 -61.5q68 0 117.5 47t94.5 128zM889 204q0 91 -35.5 151t-99.5 60q-68 0 -119 -47t-95 -127q27 -49 47 -80.5t50 -67.5t65 -54t74 -18q113 0 113 183z" /></g>

		
	
</svg>) into the Zygmund space <svg     style="vertical-align:-0.20474pt;width:15.9625px;" id="M4" height="12.0625" version="1.1" viewBox="0 0 15.9625 12.0625" width="15.9625"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.738)"><path id="x1D4B5" d="M407 519l18 -8q-26 -50 -58 -74t-80 -24q-50 0 -90 32q-38 32 -38 83q0 60 52 109q53 50 145 50q52 0 108 -13q67 -16 94 -26l92 -36q71 46 150 46q55 0 83.5 -16.5t28.5 -43.5q0 -24 -33 -41t-83 -17q-53 0 -133 26q-57 -66 -116 -236q3 -1 11 -1q60 1 93 27l14 -17&#xA;q-56 -59 -118 -59q-15 0 -18 1q-54 -143 -158 -221q107 -36 157 -36q79 0 132 36.5t72 102.5l20 -1q-14 -85 -71.5 -131t-150.5 -46q-34 0 -76 11q-50 13 -121 38q-74 -43 -162 -43q-112 0 -112 60q0 24 33 41t83 17q46 0 142 -31q58 55 120 215q-3 1 -12 1q-52 0 -95 -32&#xA;l-17 16q60 65 126 65q13 0 17 -1q67 163 157 241q-43 17 -88 32q-31 11 -85 21q-56 12 -105 12q-64 0 -100 -40q-35 -40 -35 -80q0 -39 25 -62t62 -23q42 0 68.5 17t51.5 59zM871 598q0 30 -71 30q-62 0 -106 -33q66 -25 112 -25q27 0 46 8.5t19 19.5zM283 51&#xA;q-75 28 -118 28q-27 0 -46 -8.5t-19 -19.5q0 -30 71 -30q65 0 112 30z" /></g>

		
	
</svg>. As a special case, we show that the bounded (resp., compact) composition operators from <svg     style="vertical-align:-0.20474pt;width:16.275px;" id="M5" height="12.0625" version="1.1" viewBox="0 0 16.275 12.0625" width="16.275"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.738)"><use xlink:href="#x212C"/></g>

		
	
</svg>, <svg     style="vertical-align:-5.73167pt;width:18.2875px;" id="M6" height="18.3125" version="1.1" viewBox="0 0 18.2875 18.3125" width="18.2875"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D435"/></g>

			<g transform="matrix(.012,-0,0,-.012,10.55,15.188)"><use xlink:href="#x1D45D"/></g>

		
	
</svg>, and <svg     style="vertical-align:-0.0pt;width:27.7875px;" id="M7" height="13.6125" version="1.1" viewBox="0 0 27.7875 13.6125" width="27.7875"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,13.55)"><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&#xA;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,-.012,14.975,5.388)"><use xlink:href="#x221E"/></g>

		
	
</svg> to <svg     style="vertical-align:-0.20474pt;width:15.9625px;" id="M8" height="12.0625" version="1.1" viewBox="0 0 15.9625 12.0625" width="15.9625"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.738)"><use xlink:href="#x1D4B5"/></g>

		
	
</svg> coincide. In addition, the boundedness and the compactness of the composition operator can be characterized in terms of the boundedness (resp., convergence to 0, under the boundedness assumption of the operator) of the Zygmund norm of the powers of the symbol.

Flavia Colonna

Songxiao Li

Weighted Composition Operators from the Bloch Space and the Analytic Besov Spaces into the Zygmund Space

We use a method of reducing coupled coincidence point results in (ordered) symmetric <svg style="vertical-align:-0.20473pt;width:12.275px;" id="M2" height="11.6875" version="1.1" viewBox="0 0 12.275 11.6875" width="12.275" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.363)"><use xlink:href="#x47" /></g></svg>-metric spaces to the respective results for mappings with one variable, even obtaining (in some cases) more general theorems. Our results generalize, extend, unify, and complement recent coupled coincidence point theorems in this frame, established by Cho et al. (2012), Aydi et al. (2011), and Choudhury and Maity (2011). Also, by using our method several recent tripled coincidence point results in ordered symmetric <svg style="vertical-align:-0.20473pt;width:12.275px;" id="M3" height="11.6875" version="1.1" viewBox="0 0 12.275 11.6875" width="12.275" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.363)"><use xlink:href="#x47" /></g></svg>-metric spaces can be reduced to the coincidence point results with one variable.

Stojan Radenovi&#x107;

Remarks on Some Recent Coupled Coincidence Point Results in Symmetric <svg style="vertical-align:-0.20473pt" height="11.6875" version="1.1" viewBox="0 0 12.275 11.6875" width="12.275" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.363)"><path id="x47" d="M692 302v-28q-45 -6 -54.5 -17t-9.5 -55v-86q0 -62 7 -95q-7 -2 -32 -7.5l-37 -8l-35.5 -7.5t-40.5 -7t-38.5 -4t-40.5 -2q-165 0 -266 93t-101 243q0 84 32 151t86 108t121.5 63t142.5 22q29 0 62.5 -4t52 -7.5t47.5 -10.5t31 -7l15 -156l-28 -6q-17 87 -66.5 121.5&#xA;t-128.5 34.5q-117 0 -191 -80.5t-74 -216.5q0 -56 17 -110t49.5 -100.5t87.5 -74.5t124 -28q53 0 87 17q17 9 24.5 23t7.5 43v85q0 54 -15.5 67.5t-89.5 18.5v28h254z" /></g></svg>-Metric Spaces

There are quite a few generalizations or applications of the 1984 minimax
inequality of Ky Fan compared with his original 1972 minimax inequality. In a certain
sense, the relationship between the 1984 inequality and several hundreds of known generalizations
of the original 1972 inequality has not been recognized for a long period. Hence,
it would be necessary to seek such relationship. In this paper, we give several generalizations
of the 1984 inequality and some known applications in order to clarify the close
relationship among them. Some new types of minimax inequalities are added.

Sehie Park

Evolution of the Minimax Inequality of Ky Fan

We generalize the result of Pre&#x161;i&#x107; in metric-like spaces by proving some common fixed point theorems for Pre&#x161;i&#x107; type mappings in metric-like spaces. An example is given which shows that the generalization is proper.

Satish Shukla

Brian Fisher

A Generalization of Pre&#x161;i&#x107; Type Mappings in Metric-Like Spaces

We prove some <i>n</i>-tupled coincidence point results whenever <i>n</i> is even. We give here several new definitions like <i>n</i>-tupled fixed point, <i>n</i>-tupled coincidence point, and so forth. The main result is supported with the aid of an illustrative example.

Ahmed H. Soliman

Binayak S. Choudhury

Pradyut Das

On <svg style="vertical-align:-0.1638pt" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g></svg>-Tupled Coincidence Point Results in Metric Spaces

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Journal of Operators