We introduce and study variational-like inequalities for generalized pseudomonotone set-valued mappings in Banach spaces. By using KKM technique, we obtain the existence of solutions for variational-like inequalities for generalized pseudomonotone set-valued mappings in reflexive Banach spaces. The results presented in this paper are generalizations and improvements of the several well-known results in the literature.

Syed Shakaib Irfan

Mohammad Firdosh Khan

Variational-Like Inequalities for Weakly Relaxed <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-3.42943pt" height="9.39034pt" version="1.1" viewBox="-0.0498162 -5.96091 18.6027 9.39034" width="18.6027pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-229" d="M471 401C471 442 455 448 440 448C421 448 389 439 366 424C314 390 233 331 165 242H163L187 345C193 372 197 393 197 409C197 435 189 448 174 448C146 448 80 408 23 349L40 327C64 351 97 373 107 373C115 373 118 366 111 337L29 -4L36 -12C56 -4 83 3 110 9C123 69 136 126 150 172C227 283 336 381 377 381C392 381 396 370 379 298L326 74C290 -77 273 -172 273 -197C273 -219 316 -261 346 -261L353 -246C348 -227 352 -174 374 -66L457 310C467 352 471 381 471 401Z" /></g><g transform="matrix(.013,0,0,-0.013,6.423,0)"><path id="g190-46" d="M300 253H71L57 198H286L300 253Z" /></g><g transform="matrix(.013,0,0,-0.013,11.069,0)"><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> Pseudomonotone Set-Valued Mappings in Banach Space

International Journal of Analysis

Retracted: Quasilinear Inner Product Spaces and Hilbert Quasilinear Spaces

Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.29107pt" id="M2" height="12.5794pt" version="1.1" viewBox="-0.0498162 -9.28833 28.1489 12.5794" width="28.1489pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-100"/></g><g transform="matrix(.0091,0,0,-0.0091,4.173,3.132)"><use xlink:href="#g50-79"/></g><g transform="matrix(.013,0,0,-0.013,12.453,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,16.951,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.013,0,0,-0.013,23.478,0)"><use xlink:href="#g113-42"/></g></svg> denote the number of bipartitions <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42947pt" id="M3" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 28.7285 12.7178" width="28.7285pt"><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="g113-233" 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,-0.013,11.672,0)"><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,-0.013,16.815,0)"><path id="g113-234" d="M538 96L524 119C492 88 454 63 446 63C439 63 434 70 440 101C463 223 491 341 518 448H508L433 422L401 276C355 192 240 56 188 56C163 56 154 89 163 133C184 233 207 338 235 443L230 448L152 424L58 17C40 -60 23 -143 23 -185C23 -241 42 -261 63 -261C92 -261 117 -241 125 -227L124 -221C109 -209 89 -165 89 -103C89 -55 96 -13 105 26H107C121 -3 134 -12 151 -12C172 -12 194 -5 221 16C279 61 330 124 384 194H386C381 159 377 141 368 105C343 3 363 -12 383 -12C416 -12 486 35 538 96Z"/></g><g transform="matrix(.013,0,0,-0.013,24.031,0)"><use xlink:href="#g113-42"/></g></svg> of a positive integer <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M4" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 6.6501 6.1673" width="6.6501pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-111"/></g></svg> subject to the restriction that each part of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M5" height="9.39034pt" version="1.1" viewBox="-0.0498162 -5.96091 7.34167 9.39034" width="7.34167pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-234"/></g></svg> is divisible by <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M6" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 11.0475 8.8423" width="11.0475pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-79" d="M822 650H589L583 622C660 617 677 607 674 561C672 534 664 481 647 390L600 137H596L273 650H126L120 622C176 620 194 615 207 594C221 571 225 557 214 504L161 257C141 166 129 112 121 85C108 42 83 30 29 28L23 0H260L266 28C193 33 173 42 176 89C178 122 186 172 202 255L256 527H259L583 -8H612L690 390C708 481 720 535 728 558C744 603 756 619 816 622L822 650Z"/></g></svg>. In this paper, we prove some congruence properties of the function <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.29107pt" id="M7" height="12.5794pt" version="1.1" viewBox="-0.0498162 -9.28833 28.1489 12.5794" width="28.1489pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-100"/></g><g transform="matrix(.0091,0,0,-0.0091,4.173,3.132)"><use xlink:href="#g50-79"/></g><g transform="matrix(.013,0,0,-0.013,12.453,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,16.951,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.013,0,0,-0.013,23.478,0)"><use xlink:href="#g113-42"/></g></svg> for <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M8" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 32.2343 8.8423" width="32.2343pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-79"/></g><g transform="matrix(.013,0,0,-0.013,14.542,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.013,0,0,-0.013,25.805,0)"><path id="g113-56" d="M447 623H65C61 580 56 530 47 475H76C100 541 106 550 172 550H388C308 376 196 170 91 -1L98 -12L172 -2C268 204 360 408 455 611L447 623Z"/></g></svg>, 11, and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M9" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 12.4307 8.8423" width="12.4307pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-54" d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z"/></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path id="g113-127" d="M442 569C442 628 407 660 362 660C328 660 292 648 255 616C200 568 165 508 123 334L106 264C80 242 52 222 23 202L30 176C54 191 76 205 100 222L87 166C73 103 79 43 95 19C108 0 139 -12 169 -12C242 -12 318 38 369 106L354 129C322 93 260 49 215 49C173 49 147 81 170 187L189 274C293 344 442 452 442 569ZM380 565C380 478 267 377 196 322L216 406C254 556 282 623 332 623C364 623 380 596 380 565Z"/></g></svg>, for any integer <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.11981pt" id="M10" height="9.75571pt" version="1.1" viewBox="-0.0498162 -8.6359 27.354 9.75571" width="27.354pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-127"/></g><g transform="matrix(.013,0,0,-0.013,9.679,0)"><path id="g117-94" d="M531 285V335L57 550V494L473 310L57 127V71L531 285ZM531 -40V10H57V-40H531Z"/></g><g transform="matrix(.013,0,0,-0.013,20.942,0)"><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></svg>, by employing Ramanujan’s theta-function identities.

Nipen Saikia

Chayanika Boruah

Some Congruence Properties of a Restricted Bipartition Function <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-3.29107pt" height="12.5794pt" version="1.1" viewBox="-0.0498162 -9.28833 28.1489 12.5794" width="28.1489pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-100" d="M383 397C383 424 344 448 299 448C244 448 172 409 132 375C66 319 23 227 23 146C23 42 74 -12 146 -12C208 -12 298 30 359 103L343 124C315 95 248 48 192 48C145 48 111 85 111 163C111 228 129 294 151 330C171 363 201 401 241 401C275 401 302 384 325 356C332 347 339 344 348 348C373 360 383 381 383 397Z" /></g><g transform="matrix(.0091,0,0,-0.0091,4.173,3.132)"><path id="g50-79" d="M870 650H622L613 619C694 615 708 606 708 559C708 531 702 477 686 396L636 136H633L281 650H127L120 619C180 616 198 611 212 592C227 572 232 558 220 503L166 254C148 169 139 131 126 89C112 43 85 35 33 32L24 0H280L286 32C208 37 189 44 189 93C189 124 198 179 212 250L267 526H271L622 -8H655L732 396C747 477 762 532 770 556C785 600 798 616 862 619L870 650Z" /></g><g transform="matrix(.013,0,0,-0.013,12.453,0)"><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,-0.013,16.951,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z" /></g><g transform="matrix(.013,0,0,-0.013,23.478,0)"><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></svg>

We determine the relations between the classes <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.2911pt" id="M1" height="15.2153pt" version="1.1" viewBox="-0.0498162 -11.9242 11.355 15.2153" width="11.355pt"><g transform="matrix(.013,0,0,-0.013,.013,-2.897)"><path id="g119-17" d="M470 557C393 591 314 624 246 666H224C156 624 76 591 0 557L12 532C86 550 163 573 235 596C309 572 384 550 458 532L470 557Z"/></g><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,-0.0091,6.159,3.132)"><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> of almost <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M2" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 7.30254 9.49473" width="7.30254pt"><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></svg>-statistically convergent sequences and the relations between the classes <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.1507pt" id="M3" height="14.0749pt" version="1.1" viewBox="-0.0498162 -11.9242 29.0156 14.0749" width="29.0156pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-92" d="M290 -163V-135C183 -126 181 -122 181 -44V583C181 662 184 666 290 675V703H120V-163H290Z"/></g><g transform="matrix(.013,0,0,-0.013,4.818,-2.897)"><path id="g119-105" d="M658 557C549 593 437 629 340 673H318C222 629 109 593 0 557L9 532C116 551 225 577 329 602C435 576 543 551 649 532L658 557Z"/></g><g transform="matrix(.013,0,0,-0.013,4.485,0)"><path id="g113-87" d="M697 650H468L461 623L492 619C539 613 547 605 518 546C481 471 367 264 278 116H276C239 278 197 500 186 567C180 604 185 613 226 619L252 623L260 650H24L17 623C78 617 92 613 108 533L216 -11H247C365 200 515 462 560 529C616 612 624 615 689 623L697 650Z"/></g><g transform="matrix(.013,0,0,-0.013,12.013,0)"><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,-0.013,17.156,0)"><use xlink:href="#g113-233"/></g><g transform="matrix(.013,0,0,-0.013,24.33,0)"><path id="g113-94" d="M226 -163V703H56V676C162 667 165 662 165 584V-43C165 -122 162 -126 56 -136V-163H226Z"/></g></svg> of strongly almost <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M4" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 29.0417 11.5564" width="29.0417pt"><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)"><use xlink:href="#g113-87"/></g><g transform="matrix(.013,0,0,-0.013,12.026,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.013,0,0,-0.013,17.169,0)"><use xlink:href="#g113-233"/></g><g transform="matrix(.013,0,0,-0.013,24.343,0)"><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></svg>-summable sequences for various sequences <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M5" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 7.30254 9.49473" width="7.30254pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-233"/></g></svg>, <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 7.34167 9.39034" width="7.34167pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-234" d="M538 96L524 119C492 88 454 63 446 63C439 63 434 70 440 101C463 223 491 341 518 448H508L433 422L401 276C355 192 240 56 188 56C163 56 154 89 163 133C184 233 207 338 235 443L230 448L152 424L58 17C40 -60 23 -143 23 -185C23 -241 42 -261 63 -261C92 -261 117 -241 125 -227L124 -221C109 -209 89 -165 89 -103C89 -55 96 -13 105 26H107C121 -3 134 -12 151 -12C172 -12 194 -5 221 16C279 61 330 124 384 194H386C381 159 377 141 368 105C343 3 363 -12 383 -12C416 -12 486 35 538 96Z"/></g></svg> in the class <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M7" 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>. Furthermore we also give the relations between the classes <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.2911pt" id="M8" height="15.2153pt" version="1.1" viewBox="-0.0498162 -11.9242 11.355 15.2153" width="11.355pt"><g transform="matrix(.013,0,0,-0.013,.013,-2.897)"><use xlink:href="#g119-17"/></g><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-84"/></g><g transform="matrix(.0091,0,0,-0.0091,6.159,3.132)"><use xlink:href="#g50-233"/></g></svg> of almost <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M9" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 7.30254 9.49473" width="7.30254pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-233"/></g></svg>-statistically convergent sequences and the classes <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.1507pt" id="M10" height="14.0749pt" version="1.1" viewBox="-0.0498162 -11.9242 29.0156 14.0749" width="29.0156pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-92"/></g><g transform="matrix(.013,0,0,-0.013,4.818,-2.897)"><use xlink:href="#g119-105"/></g><g transform="matrix(.013,0,0,-0.013,4.485,0)"><use xlink:href="#g113-87"/></g><g transform="matrix(.013,0,0,-0.013,12.013,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.013,0,0,-0.013,17.156,0)"><use xlink:href="#g113-233"/></g><g transform="matrix(.013,0,0,-0.013,24.33,0)"><use xlink:href="#g113-94"/></g></svg> of strongly almost <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M11" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 29.0417 11.5564" width="29.0417pt"><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)"><use xlink:href="#g113-87"/></g><g transform="matrix(.013,0,0,-0.013,12.026,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.013,0,0,-0.013,17.169,0)"><use xlink:href="#g113-233"/></g><g transform="matrix(.013,0,0,-0.013,24.343,0)"><use xlink:href="#g113-42"/></g></svg>-summable sequences for various sequences <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42947pt" id="M12" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 42.8037 12.7178" width="42.8037pt"><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)"><use xlink:href="#g113-234"/></g><g transform="matrix(.013,0,0,-0.013,23.165,0)"><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,-0.013,33.661,0)"><use xlink:href="#g113-140"/></g></svg>.

R. &#xc7;olak

&#xc7;. A. Bekta&#x15f;

H. Alt&#x131;nok

S. Ercan

On Inclusion Relations between Some Sequence Spaces

We establish new fixed point theorems in 0-complete ordered partial metric spaces. Also, we give remark on coupled generalized Banach contraction. Some examples illustrate the usability of our results. The theorems presented in this paper are generalizations and improvements of the several well known results in the literature.

Esra Yolacan

Fixed Point Theorems for Geraghty Type Contractive Mappings and Coupled Fixed Point Results in 0-Complete Ordered Partial Metric Spaces

We introduce the notion of dualistic Geraghty&#x2019;s type contractions. We prove some fixed point theorems for ordered mappings satisfying the abovementioned contractions. We discuss an application of our fixed point results to show the existence of solution of integral equations.

Muhammad Nazam

Muhammad Arshad

On a Fixed Point Theorem with Application to Integral Equations

We extend results of Favini, Nashed, and Zhao on singular differential equations using the <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)"><use xlink:href="#g113-104"/></g></svg>-Drazin inverse and the order of a quasinilpotent operator in the sense of Miekka and Nevanlinna. Two classes of singularly perturbed differential equations are studied using the continuity properties of the <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M3" 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>-Drazin inverse obtained by Koliha and Rakočević.

Alrazi Abdeljabbar

Trung Dinh Tran

Singular Differential Equations and <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-3.42943pt" 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>-Drazin Invertible Operators

About this journal

Journal profile

Meet the editorial board

Editor spotlight

Find out more

This journal's articles appear in a wide range of abstracting and indexing databases, and are covered by numerous other services that aid discovery and access. Find out more about where and how the content of this journal is available.

International Journal of Analysis