We study the approximation of zero for sum of accretive operators using a modified Mann type forward-backward splitting algorithm and obtain strong convergence of the sequence generated by our scheme to the zero of sum of accretive operators in uniformly convex real Banach spaces which are also uniformly smooth. Our result is new and complements many recent and important results in this direction in the literature.

Yekini Shehu

Iterative Approximations for Zeros of Sum of Accretive Operators in Banach Spaces

We introduce a new subclass of analytic functions in the unit disk <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2455397pt" id="M1" height="8.63352pt" version="1.1" viewBox="-0.0498162 -8.38798 9.70353 8.63352" width="9.70353pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g121-83" d="M716 650H478V622C549 617 570 605 575 566C579 535 582 493 582 401V326C582 161 537 40 393 40C269 40 205 133 205 294V521C205 611 213 617 291 622V650H33V622C111 615 120 611 120 521V272C120 154 148 80 208 34C252 0 309 -14 368 -14C434 -14 501 7 548 55C605 116 623 201 623 319V401C623 493 626 536 629 563C633 605 656 618 716 622V650Z"/></g></svg>, defined by using the generalized hypergeometric functions, which extends some previous well-known classes defined by different authors. Inclusion results, radius problems, and some connections with the Bernardi-Libera-Livingston integral operator are discussed.

Badr S. Alkahtani

Saima Mustafa

Teodor Bulboac&#x103;

Subclasses of Analytic Functions Defined by Generalized Hypergeometric Functions

We consider gradient estimates for two types of nonlinear parabolic equations under the Ricci flow: one is the equation <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.38377pt" id="M1" height="12.6721pt" version="1.1" viewBox="-0.0498162 -9.28833 122.176 12.6721" width="122.176pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-118" d="M515 96L502 119C471 88 431 62 423 62C416 62 411 70 417 101C440 223 469 341 497 448H486L412 422L380 277C330 188 210 57 152 57C137 57 126 69 139 124L195 366C210 431 205 448 182 448C155 448 89 413 23 350L36 326C73 354 103 376 112 376C118 376 118 365 113 340L61 118C54 90 52 68 52 51C52 0 75 -12 98 -12S151 -3 181 17C242 58 305 119 362 193H364L345 104C323 3 339 -12 359 -12C390 -12 464 35 515 96Z"/></g><g transform="matrix(.0091,0,0,-0.0091,6.929,3.132)"><path id="g50-117" d="M329 433H203L239 587L230 596L147 534L123 433H57L30 395L34 388H115L61 129C37 16 59 -12 85 -12C147 -12 222 58 260 98L241 125C212 95 160 62 144 62C132 62 127 71 138 126L192 386L305 394L329 433Z"/></g><g transform="matrix(.013,0,0,-0.013,14.228,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.013,0,0,-0.013,25.491,0)"><path id="g113-133" d="M600 0V24L339 665L311 657L43 24V0H600ZM497 50H107L303 539L497 50Z"/></g><g transform="matrix(.013,0,0,-0.013,33.818,0)"><use xlink:href="#g113-118"/></g><g transform="matrix(.013,0,0,-0.013,43.718,0)"><path id="g117-36" d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"/></g><g transform="matrix(.013,0,0,-0.013,54.254,0)"><path id="g113-98" d="M483 97L471 123C436 91 401 65 392 65C388 65 384 74 390 106C414 239 444 378 457 429L455 433C444 433 429 436 416 439C392 444 368 448 344 448C281 448 204 415 152 376C71 315 23 205 23 103C23 21 57 -12 85 -12C114 -12 149 6 185 34C231 70 285 119 329 183H331L309 81C292 0 308 -12 326 -12C350 -12 421 24 483 97ZM374 387C370 363 356 291 345 261C315 193 181 50 139 50C124 50 110 71 110 118C110 224 153 331 218 379C238 394 271 402 301 402C329 402 359 394 374 387Z"/></g><g transform="matrix(.013,0,0,-0.013,60.835,0)"><use xlink:href="#g113-118"/></g><g transform="matrix(.013,0,0,-0.013,70.008,0)"><path id="g121-106" d="M238 0V26C174 32 166 38 166 104V712C132 700 70 683 18 677V653C81 647 87 645 87 577V104C87 38 78 32 15 26V0H238Z"/></g><g transform="matrix(.013,0,0,-0.013,73.297,0)"><path id="g121-109" d="M257 449C165 449 37 374 37 209C37 98 119 -12 256 -12C355 -12 473 65 473 226C473 349 381 449 257 449ZM244 416C333 416 380 320 380 204C380 67 329 21 267 21C184 21 130 115 130 241C130 354 184 416 244 416Z"/></g><g transform="matrix(.013,0,0,-0.013,79.927,0)"><path id="g121-101" d="M463 437C426 431 375 425 327 422C297 440 264 449 231 449H230C153 449 51 396 51 283C51 215 94 168 139 149C123 129 91 103 51 88C50 78 53 60 62 46C75 25 100 2 136 -9C112 -28 73 -59 53 -79C38 -94 29 -113 29 -135C30 -192 91 -257 203 -257C336 -257 452 -160 452 -59C452 39 371 59 309 59C275 59 240 58 203 58C158 58 140 77 140 96C140 110 157 129 170 138C186 135 205 133 221 133C306 133 396 185 396 293C396 328 384 360 366 381L423 378C439 387 459 413 468 429L463 437ZM219 418C277 418 314 362 314 284C314 205 275 166 231 165C176 165 137 221 137 299C137 376 177 418 219 418ZM241 -11C285 -11 314 -14 339 -24C367 -36 384 -61 384 -95C384 -157 335 -206 240 -206C166 -206 108 -165 108 -110C108 -82 128 -54 154 -32C172 -17 195 -11 241 -11Z"/></g><g transform="matrix(.013,0,0,-0.013,88.25,0)"><use xlink:href="#g113-118"/></g><g transform="matrix(.013,0,0,-0.013,98.15,0)"><use xlink:href="#g117-36"/></g><g transform="matrix(.013,0,0,-0.013,108.686,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g><g transform="matrix(.013,0,0,-0.013,114.788,0)"><use xlink:href="#g113-118"/></g></svg> with <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.57657pt" id="M2" height="10.8649pt" version="1.1" viewBox="-0.0498162 -9.28833 17.9764 10.8649" width="17.9764pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-98"/></g><g transform="matrix(.013,0,0,-0.013,6.58,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.013,0,0,-0.013,11.723,0)"><use xlink:href="#g113-99"/></g></svg> being two real constants; the other is <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.29111pt" id="M3" height="13.2475pt" version="1.1" viewBox="-0.0498162 -9.95639 74.5379 13.2475" width="74.5379pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-118"/></g><g transform="matrix(.0091,0,0,-0.0091,6.929,3.132)"><use xlink:href="#g50-117"/></g><g transform="matrix(.013,0,0,-0.013,14.228,0)"><use xlink:href="#g117-34"/></g><g transform="matrix(.013,0,0,-0.013,25.491,0)"><use xlink:href="#g113-133"/></g><g transform="matrix(.013,0,0,-0.013,33.818,0)"><use xlink:href="#g113-118"/></g><g transform="matrix(.013,0,0,-0.013,43.718,0)"><use xlink:href="#g117-36"/></g><g transform="matrix(.013,0,0,-0.013,54.254,0)"><path id="g113-233" d="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z"/></g><g transform="matrix(.013,0,0,-0.013,61.428,0)"><use xlink:href="#g113-118"/></g><g transform="matrix(.0091,0,0,-0.0091,68.423,-5.741)"><path id="g50-223" d="M558 106L533 129C501 84 476 67 464 67C446 67 435 116 414 239C457 297 508 367 550 440L539 451L484 438C458 390 429 338 404 300H402C375 407 351 451 285 451C168 451 24 312 24 153C24 56 66 -12 132 -12C206 -12 284 64 345 156H347C367 24 386 -12 422 -12C457 -12 503 7 558 106ZM338 206C272 99 215 56 173 56C140 56 116 96 116 170C116 303 194 405 255 405C304 405 324 299 338 206Z"/></g></svg> with <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.57657pt" id="M4" height="10.8649pt" version="1.1" viewBox="-0.0498162 -9.28833 19.8684 10.8649" width="19.8684pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-233"/></g><g transform="matrix(.013,0,0,-0.013,7.174,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.013,0,0,-0.013,12.317,0)"><path id="g113-223" d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"/></g></svg> being two real constants. By a suitable scaling for the above two equations, we obtain Hamilton-Souplet-Zhang-type gradient estimates.

Guangyue Huang

Bingqing Ma

Hamilton-Souplet-Zhang&#x2019;s Gradient Estimates for Two Types of Nonlinear Parabolic Equations under the Ricci Flow

Continuity (both lower and upper semicontinuities) results of the Pareto/efficient solution mapping for a parametric vector variational inequality with a polyhedral constraint set are established via scalarization approaches, within the framework of strict pseudomonotonicity assumptions. As a direct application, the continuity of the solution mapping to a parametric weak Minty vector variational inequality is also discussed. Furthermore, error bounds for the weak vector variational inequality in terms of two known regularized gap functions are also obtained, under strong pseudomonotonicity assumptions.

Xin Zuo

Hong-Zhi Wei

Chun-Rong Chen

Continuity Results and Error Bounds on Pseudomonotone Vector Variational Inequalities via Scalarization

We study the stability of approximative <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M1" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 6.40217 6.1673" width="6.40217pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-241" d="M471 456L444 459C426 433 414 430 388 430C324 430 270 434 216 434C103 434 51 374 23 338L43 317C96 366 146 380 221 375L154 109C149 86 147 68 147 52C147 4 168 -12 197 -12C240 -12 291 25 334 71L320 96C295 75 268 58 252 58C238 58 227 79 238 138C251 211 272 296 292 372C310 372 332 368 350 368C391 368 421 369 434 371C444 388 455 413 471 456Z"/></g></svg>-compactness, where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M2" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 6.40217 6.1673" width="6.40217pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-241"/></g></svg> is the norm or the weak topology. Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M3" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 9.05107 8.68572" width="9.05107pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-140" d="M671 0V28C608 36 593 47 562 130C495 309 432 486 368 665L336 656C270 481 202 311 134 139C98 47 84 39 20 28V0H241V28C167 38 156 47 178 115C217 236 276 383 327 530H329C375 398 437 223 484 92C499 47 489 37 424 28V0H671Z"/></g></svg> be an index set and for every <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.57657pt" id="M4" height="10.8649pt" version="1.1" viewBox="-0.0498162 -9.28833 33.3825 10.8649" width="33.3825pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-233" d="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z"/></g><g transform="matrix(.013,0,0,-0.013,10.806,0)"><path id="g117-173" d="M448 1V51H364C248 51 153 129 140 230H448V280H140C153 381 248 459 364 459H448V509H365C208 509 80 395 80 255S208 1 365 1H448Z"/></g><g transform="matrix(.013,0,0,-0.013,21.302,0)"><use xlink:href="#g113-140"/></g><g transform="matrix(.013,0,0,-0.013,30.212,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g></svg> let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.291101pt" id="M5" height="11.927pt" version="1.1" viewBox="-0.0498162 -8.6359 12.951 11.927" width="12.951pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-90" d="M640 650H413L406 622L438 617C475 611 483 605 463 571C422 503 346 400 296 333C264 403 215 529 199 575C189 604 191 614 228 618L262 622L271 650H30L23 622C90 615 97 609 119 553L212 321C222 295 223 285 218 259L193 127C176 39 167 33 82 28L76 0H350L356 28C266 36 257 40 275 127L303 266C308 290 314 302 328 322C396 418 454 493 498 546C551 610 565 616 631 622L640 650Z"/></g><g transform="matrix(.0091,0,0,-0.0091,7.111,3.132)"><path id="g50-233" d="M547 100L523 123C489 76 461 62 452 62C442 62 433 71 427 109C405 243 387 408 378 495C360 666 322 710 263 710C230 710 182 688 161 667L170 639C184 646 203 651 218 651C247 651 272 634 288 562C297 521 300 485 304 436C230 267 111 104 24 11L33 -12L116 6C163 73 263 255 311 362L347 84C355 26 373 -12 406 -12C440 -12 489 12 547 100Z"/></g></svg> be a subspace of a Banach space <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.291101pt" id="M6" height="11.927pt" version="1.1" viewBox="-0.0498162 -8.6359 15.7304 11.927" width="15.7304pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-89" d="M748 650H522L515 622L546 617C580 611 587 604 565 575C518 513 469 451 419 393C376 474 349 534 330 580C318 609 325 612 361 618L383 622L392 650H151L144 622C214 616 224 612 257 543L360 327C270 218 187 124 159 95C106 40 92 34 26 28L17 0H252L259 28L236 31C189 37 188 47 209 78C249 136 308 210 377 294L478 79C494 44 487 37 449 32L418 28L409 0H673L680 28C596 34 591 39 554 116L436 361C526 469 574 521 604 553C659 612 669 614 739 622L748 650Z"/></g><g transform="matrix(.0091,0,0,-0.0091,9.88,3.132)"><use xlink:href="#g50-233"/></g></svg>. For <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42938pt" id="M7" height="11.7782pt" version="1.1" viewBox="-0.0498162 -8.34882 57.2573 11.7782" width="57.2573pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g><g transform="matrix(.013,0,0,-0.013,9.872,0)"><path id="g117-93" d="M531 71V127L115 310L531 494V550L57 335V285L531 71ZM531 -40V10H57V-40H531Z"/></g><g transform="matrix(.013,0,0,-0.013,21.135,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g><g transform="matrix(.013,0,0,-0.013,32.477,0)"><path id="g117-91" d="M512 -3V55L134 254V256L512 456V514L75 281V230L512 -3Z"/></g><g transform="matrix(.013,0,0,-0.013,43.74,0)"><path id="g113-30" d="M983 225C983 348 891 448 766 448C646 448 579 377 521 276C458 383 382 448 256 448C165 448 37 373 37 208C37 98 119 -12 256 -12C375 -12 440 60 498 161C561 53 639 -12 766 -12C865 -12 983 64 983 225ZM478 196C428 106 365 21 266 21C183 21 129 114 129 240C129 353 183 415 244 415C352 415 420 301 478 196ZM889 204C889 66 838 21 776 21C666 21 598 135 540 241C590 331 655 415 754 415C842 415 889 320 889 204Z"/></g></svg>, let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-6.9069pt" id="M8" height="15.5428pt" version="1.1" viewBox="-0.0498162 -8.6359 55.4535 15.5428" width="55.4535pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.013,0,0,-0.013,13.577,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.013,0,0,-0.013,24.84,0)"><path id="g117-68" d="M573 255C573 407 448 530 294 530S15 407 15 255S140 -20 294 -20S573 103 573 255ZM520 280H323V484C426 471 508 386 520 280ZM265 280H68C80 386 162 471 265 484V280ZM265 26C162 39 80 124 68 230H265V26ZM520 230C508 124 426 39 323 26V230H520Z"/></g><g transform="matrix(.0091,0,0,-0.0091,32.5,3.132)"><path id="g50-109" d="M238 675C244 702 241 710 231 710C220 710 155 682 74 673L70 643H101C147 643 151 635 142 594L39 111C18 11 36 -12 55 -12C89 -12 170 38 223 103L207 129C176 98 130 68 118 68C113 68 108 72 114 102L238 675Z"/></g><g transform="matrix(.0065,0,0,-0.0065,34.738,5.233)"><path id="g176-113" d="M575 299C575 406 528 454 412 454C385 454 342 447 313 440L330 514L318 525C293 510 258 487 242 466L227 419C194 403 158 384 123 361L126 327C155 342 183 355 220 364L107 -148C96 -198 76 -210 19 -215L14 -253L259 -241V-206L240 -205C184 -202 183 -188 192 -141L220 -1C242 -10 269 -12 284 -12C352 5 434 48 484 102C542 165 575 238 575 299ZM479 290C479 170 380 41 305 41C281 41 249 60 237 72L300 395C329 399 353 401 374 401C426 401 479 377 479 290Z"/></g><g transform="matrix(.013,0,0,-0.013,39.657,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.0091,0,0,-0.0091,49.538,3.132)"><use xlink:href="#g50-233"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-6.9069pt" id="M9" height="15.5428pt" version="1.1" viewBox="-0.0498162 -8.6359 51.1996 15.5428" width="51.1996pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-90"/></g><g transform="matrix(.013,0,0,-0.013,12.113,0)"><use xlink:href="#g117-34"/></g><g transform="matrix(.013,0,0,-0.013,23.376,0)"><use xlink:href="#g117-68"/></g><g transform="matrix(.0091,0,0,-0.0091,31.036,3.132)"><use xlink:href="#g50-109"/></g><g transform="matrix(.0065,0,0,-0.0065,33.274,5.233)"><use xlink:href="#g176-113"/></g><g transform="matrix(.013,0,0,-0.013,38.193,0)"><use xlink:href="#g113-90"/></g><g transform="matrix(.0091,0,0,-0.0091,45.304,3.132)"><use xlink:href="#g50-233"/></g></svg>. We prove that <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M10" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 8.6074 8.68572" width="8.6074pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-90"/></g></svg> (resp., <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.1815pt" id="M11" height="11.8174pt" version="1.1" viewBox="-0.0498162 -8.6359 14.4307 11.8174" width="14.4307pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-67" d="M578 512C578 619 494 650 387 650H140L134 622C219 615 223 609 210 542L127 119C112 40 104 34 23 28L17 0H235C317 0 387 7 444 33C515 65 564 122 564 195C564 289 495 335 412 352V354C505 373 578 422 578 512ZM486 510C486 422 423 367 314 367H257L294 565C299 591 305 602 315 608C324 614 339 617 362 617C421 617 486 595 486 510ZM466 200C466 100 393 35 296 35C222 35 198 51 212 127L250 333H303C388 333 466 303 466 200Z"/></g><g transform="matrix(.0091,0,0,-0.0091,7.748,3.132)"><path id="g50-90" d="M679 650H432L424 619L456 615C499 610 507 601 486 568C442 499 366 400 315 336C281 403 229 526 211 570C199 600 202 610 239 614C239 614 281 618 281 619L288 650H33L24 619C98 612 106 606 129 552L226 325C238 297 238 282 234 259L211 136C193 42 184 38 97 32L86 0H375L382 32C285 39 277 45 296 136L324 267C329 290 337 304 350 322C420 416 479 489 523 539C582 607 595 613 668 619L679 650Z"/></g></svg>) is approximatively <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M12" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 6.40217 6.1673" width="6.40217pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-241"/></g></svg>-compact in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M13" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 10.0819 8.68572" width="10.0819pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-89"/></g></svg> if and only if, for every <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.6370001pt" id="M14" height="9.92533pt" version="1.1" viewBox="-0.0498162 -9.28833 30.4074 9.92533" width="30.4074pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-233"/></g><g transform="matrix(.013,0,0,-0.013,10.806,0)"><use xlink:href="#g117-173"/></g><g transform="matrix(.013,0,0,-0.013,21.302,0)"><use xlink:href="#g113-140"/></g></svg>, <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.291101pt" id="M15" height="11.927pt" version="1.1" viewBox="-0.0498162 -8.6359 12.951 11.927" width="12.951pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-90"/></g><g transform="matrix(.0091,0,0,-0.0091,7.111,3.132)"><use xlink:href="#g50-233"/></g></svg> (resp., <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.341001pt" id="M16" height="13.9769pt" version="1.1" viewBox="-0.0498162 -8.6359 17.8126 13.9769" width="17.8126pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-67"/></g><g transform="matrix(.0091,0,0,-0.0091,7.748,3.132)"><use xlink:href="#g50-90"/></g><g transform="matrix(.0065,0,0,-0.0065,12.826,5.233)"><path id="g176-233" d="M564 102L538 129C503 76 473 65 466 65C456 65 445 75 439 113L392 494C376 654 335 708 271 708C239 708 190 685 169 665L179 633C192 639 211 644 229 644C256 644 282 628 297 563C306 524 310 487 315 438C237 265 111 102 25 13L34 -12L123 5C175 81 275 258 321 359L356 85C364 28 384 -12 420 -12C455 -12 503 11 564 102Z"/></g></svg>) is approximatively <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M17" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 6.40217 6.1673" width="6.40217pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-241"/></g></svg>-compact in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.291101pt" id="M18" height="11.927pt" version="1.1" viewBox="-0.0498162 -8.6359 15.7304 11.927" width="15.7304pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.0091,0,0,-0.0091,9.88,3.132)"><use xlink:href="#g50-233"/></g></svg>.

Zhenghua Luo

Longfa Sun

Wen Zhang

A Remark on the Stability of Approximative Compactness

We completely characterize the pluriharmonic symbols for (semi)commuting dual Toeplitz operators on the orthogonal complement of the pluriharmonic Dirichlet space in Sobolev space of the unit ball. We show that, for <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42947pt" id="M1" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 8.47692 12.7178" width="8.47692pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-103" d="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M2" height="9.39034pt" version="1.1" viewBox="-0.0498162 -5.96091 7.52435 9.39034" width="7.52435pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-104" d="M546 430L539 434C529 434 505 438 495 440C473 444 450 448 430 448C352 448 265 412 213 366C145 306 96 203 96 103C96 22 135 -12 160 -12C190 -12 238 14 262 32C310 68 368 120 411 184H413C403 117 396 75 384 21C353 -118 325 -158 291 -184C270 -200 241 -205 208 -205C133 -205 90 -164 74 -110C70 -98 58 -100 49 -107C34 -119 23 -140 23 -155C23 -190 74 -261 166 -261C219 -261 280 -233 314 -208C383 -157 446 -79 470 81C491 223 529 388 546 430ZM456 386C452 357 433 283 420 252C402 216 366 174 325 129C288 88 239 56 212 56C192 56 182 77 182 120C182 165 199 242 226 292C256 348 281 377 311 389C327 395 353 402 375 402C408 402 436 394 456 386Z"/></g></svg> pluriharmonic functions, <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.529pt" id="M3" height="14.1649pt" version="1.1" viewBox="-0.0498162 -8.6359 63.7143 14.1649" width="63.7143pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-84" d="M449 634C442 637 425 643 405 650C376 660 341 666 307 666C181 666 98 590 98 485C98 400 170 343 215 310L246 288C307 243 343 204 343 147C343 67 291 18 219 18C104 18 61 124 51 202L23 199C28 124 27 71 27 47C47 22 122 -16 204 -16C324 -16 428 60 428 174C428 256 379 309 307 360L276 382C223 419 179 455 179 516C179 576 221 632 293 632C379 632 410 564 418 487L448 490C446 536 446 592 449 634Z"/></g><g transform="matrix(.0091,0,0,-0.0091,6.071,3.132)"><path id="g50-103" d="M620 667C620 684 593 710 552 710C515 710 459 687 409 634C360 582 334 506 320 433H247L215 400L220 388H310L259 80C224 -129 199 -161 185 -174C175 -183 160 -192 141 -192C123 -192 90 -181 75 -165C69 -158 64 -156 54 -164C39 -178 24 -195 24 -208C24 -231 61 -257 94 -257C124 -257 162 -243 209 -194C267 -134 303 -66 343 107C369 220 378 279 393 385L503 397L523 433H404C432 619 464 660 500 660C522 660 542 646 566 621C573 614 583 615 593 621C603 628 620 648 620 667Z"/></g><g transform="matrix(.013,0,0,-0.013,12.452,0)"><use xlink:href="#g113-84"/></g><g transform="matrix(.0091,0,0,-0.0091,18.522,3.132)"><path id="g50-104" d="M554 433L549 437C539 437 513 441 503 443C481 447 456 451 436 451C354 451 268 415 216 368C150 308 99 205 99 106C99 24 136 -12 165 -12C196 -12 243 14 268 33C318 70 374 118 415 181H419L395 39C374 -84 332 -149 295 -175C269 -193 245 -197 212 -197C137 -197 93 -156 76 -102C71 -87 61 -87 51 -96C36 -110 24 -130 24 -147C24 -183 76 -257 171 -257C226 -257 285 -227 323 -200C387 -155 455 -76 480 94C499 225 538 391 554 433ZM460 386C457 361 438 287 426 254C408 218 373 176 332 133C294 93 246 61 217 61C198 61 186 80 186 122C186 167 206 250 231 294C262 349 288 375 317 387C334 394 358 400 381 400C413 400 439 393 460 386Z"/></g><g transform="matrix(.013,0,0,-0.013,27.932,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.013,0,0,-0.013,39.195,0)"><use xlink:href="#g113-84"/></g><g transform="matrix(.0091,0,0,-0.0091,45.266,3.132)"><use xlink:href="#g50-104"/></g><g transform="matrix(.013,0,0,-0.013,51.044,0)"><use xlink:href="#g113-84"/></g><g transform="matrix(.0091,0,0,-0.0091,57.114,3.132)"><use xlink:href="#g50-103"/></g></svg> on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.2911pt" id="M4" height="14.4063pt" version="1.1" viewBox="-0.0498162 -11.1152 32.8722 14.4063" width="32.8722pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path id="g198-5" d="M810 587C770 574 761 568 725 543C654 630 540 687 408 687C217 687 69 577 69 430C69 319 160 247 276 247C369 247 511 319 511 495C511 510 512 523 509 538H490C490 354 393 277 274 277C174 277 110 344 110 422C110 559 219 655 415 655C524 655 636 602 695 519C628 463 572 392 516 303C433 170 377 97 313 57C252 89 199 109 143 109C79 109 36 87 36 54C36 15 82 -9 165 -9C222 -9 276 0 326 18C373 -1 423 -15 482 -15C645 -15 804 133 804 324C804 393 783 458 747 514C776 546 781 551 818 567L810 587ZM756 336C756 135 623 16 478 16C439 16 402 24 367 36C386 45 401 56 420 68C505 127 569 206 616 302C647 362 679 427 719 480C744 435 756 387 756 336ZM276 39C243 26 207 21 164 21C106 21 77 32 77 54C77 75 125 79 144 79C191 79 227 66 276 39Z"/></g><g transform="matrix(.0091,0,0,-0.0091,16.003,3.132)"><path id="g50-105" d="M509 90L493 118C465 91 427 70 418 70S409 77 415 103L469 327C494 429 466 451 436 451C412 451 387 444 356 427C307 400 222 342 160 264H158L246 667C253 701 252 710 243 710C228 710 169 680 87 673L83 644H117C155 644 160 640 149 588L24 -4L32 -12C57 -4 86 4 108 9L143 187C198 275 324 386 375 386C394 386 399 367 385 311L331 87C315 22 322 -12 351 -12C387 -12 452 24 509 90Z"/></g><g transform="matrix(.013,0,0,-0.013,21.315,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g><g transform="matrix(.0091,0,0,-0.0091,25.83,-5.996)"><path id="g54-183" d="M645 0V53H381V555H319V53H55V0H645Z"/></g></svg> if and only if <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42947pt" id="M5" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 8.47692 12.7178" width="8.47692pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-103"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M6" height="9.39034pt" version="1.1" viewBox="-0.0498162 -5.96091 7.52435 9.39034" width="7.52435pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-104"/></g></svg> satisfy one of the following conditions: <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M7" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 15.3928 11.5564" width="15.3928pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g><g transform="matrix(.013,0,0,-0.013,10.738,0)"><use xlink:href="#g113-42"/></g></svg> both <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42947pt" id="M8" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 8.47692 12.7178" width="8.47692pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-103"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M9" height="9.39034pt" version="1.1" viewBox="-0.0498162 -5.96091 7.52435 9.39034" width="7.52435pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-104"/></g></svg> are holomorphic; <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M10" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 15.3928 11.5564" width="15.3928pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g><g transform="matrix(.013,0,0,-0.013,10.738,0)"><use xlink:href="#g113-42"/></g></svg> both <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.4294pt" id="M11" height="15.9799pt" version="1.1" viewBox="-0.0498162 -12.5505 8.47692 15.9799" width="8.47692pt"><rect height="0.65243" width="8.37729" x="0" y="-11.8482"/><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-103"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42944pt" id="M12" height="12.6525pt" version="1.1" viewBox="-0.0498162 -9.22306 7.52435 12.6525" width="7.52435pt"><rect height="0.65243" width="7.42472" x="0" y="-8.52081"/><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-104"/></g></svg> are holomorphic; <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M13" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 15.3928 11.5564" width="15.3928pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path id="g113-52" d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z"/></g><g transform="matrix(.013,0,0,-0.013,10.738,0)"><use xlink:href="#g113-42"/></g></svg> there are constants <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M14" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 7.51131 6.1673" width="7.51131pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-223" d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42947pt" id="M15" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 7.68094 12.7178" width="7.68094pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-224" d="M558 587C558 666 497 712 432 712C379 712 330 691 284 650C212 586 178 508 148 348L71 -65C49 -185 35 -229 23 -235L27 -261C49 -259 101 -251 124 -224C131 -216 138 -178 159 -24C171 66 197 200 227 356C264 550 295 668 393 668C443 668 479 632 479 575C479 516 446 458 383 418C361 404 344 398 318 397L296 350C395 338 460 281 460 192C460 110 411 40 300 40C258 40 215 55 192 69L181 51C191 18 222 -16 266 -16C308 -16 351 1 397 26C471 67 545 142 545 221C545 315 486 365 401 395C469 437 558 498 558 587Z"/></g></svg>, both not being zero, such that <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42947pt" id="M16" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 42.4079 12.7178" width="42.4079pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-223"/></g><g transform="matrix(.013,0,0,-0.013,6.08,0)"><use xlink:href="#g113-103"/></g><g transform="matrix(.013,0,0,-0.013,17.338,0)"><path id="g117-36" d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"/></g><g transform="matrix(.013,0,0,-0.013,27.874,0)"><use xlink:href="#g113-224"/></g><g transform="matrix(.013,0,0,-0.013,34.776,0)"><use xlink:href="#g113-104"/></g></svg> is constant.

Yinyin Hu

Yufeng Lu

Tao Yu

Properties of Commutativity of Dual Toeplitz Operators on the Orthogonal Complement of Pluriharmonic Dirichlet Space over the Ball

Zhen-Hang Yang

Yu-Ming Chu

A Sharp Lower Bound for Toader-Qi Mean with Applications

The existence of solutions for a coupled system of time-fractional differential equations including continuous functions and the Caputo-Fabrizio fractional derivative is examined. After that we investigated a coupled system of time-fractional differential inclusions including compact- and convex-valued <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04979992pt" id="M1" height="11.6412pt" version="1.1" viewBox="-0.0498162 -11.5914 12.262 11.6412" width="12.262pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-77" d="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z"/></g><g transform="matrix(.0091,0,0,-0.0091,7.185,-5.741)"><path id="g50-50" d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"/></g></svg>-Caratheodory multifunctions and the Caputo-Fabrizio fractional derivative.

Ahmed Alsaedi

Dumitru Baleanu

Sina Etemad

Shahram Rezapour

On Coupled Systems of Time-Fractional Differential Problems by Using a New Fractional Derivative

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<p><em>Journal of Function Spaces</em> publishes research on all aspects of function spaces, functional analysis, and their employment across other mathematical disciplines.</p>


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<p>Chief Editor, Dr Ragusa, is a full professor of mathematical analysis at University of Catania, Italy. Her research interests include partial differential equations and real analysis.</p>


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Journal of Function Spaces - Table of contents 2016

title

Journal of Function Spaces publishes research on all aspects of function spaces, functional analysis, and their employment across other mathematical disciplines.

Journal of Function Spaces