In this research article, by introducing a mapping <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M1" height="9.39034pt" version="1.1" viewBox="-0.0498162 -5.96091 7.69399 9.39034" width="7.69399pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-253" d="M559 271C559 395 493 448 433 448C370 448 304 394 275 297C259 245 235 121 219 37C157 54 113 110 113 201C113 310 180 386 276 429L265 457C131 410 23 326 23 186C23 83 86 5 211 -10C180 -159 167 -220 158 -252L165 -261C198 -255 236 -246 251 -218L287 -5C421 9 559 119 559 271ZM484 273C484 144 400 40 293 33L352 331C364 389 382 406 405 406C431 406 484 370 484 273Z"/></g></svg> defined on <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.2681pt" id="M2" height="13.8595pt" version="1.1" viewBox="-0.0498162 -11.5914 38.7901 13.8595" width="38.7901pt"><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,-0.013,4.485,0)"><path id="g113-49" d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"/></g><g transform="matrix(.013,0,0,-0.013,10.725,0)"><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,-0.013,15.868,0)"><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><g transform="matrix(.013,0,0,-0.013,29.128,0)"><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,-0.0091,33.626,-5.741)"><path id="g50-53" d="M462 177V227H365V632H320C217 496 116 350 21 208V177H284V109C284 43 280 38 190 31V0H451V31C369 38 365 43 365 108V177H462ZM284 227H88C155 336 218 430 282 519H284V227Z"/></g></svg>,</span> with some axioms, we define two generalized contractions called <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.9783pt" id="M3" height="16.0921pt" version="1.1" viewBox="-0.0498162 -11.1138 19.7999 16.0921" width="19.7999pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-71" d="M584 650H137L131 622C214 614 217 612 200 521L125 127C109 41 101 35 23 28L17 0H288L294 28C201 35 193 42 209 128L242 309H348C440 309 442 300 443 226H471L510 422H482C452 354 449 348 357 348H251L295 575C302 609 304 615 338 615H426C502 615 517 604 526 581C534 560 536 524 537 492L565 494C574 554 583 631 584 650Z"/></g><g transform="matrix(.0091,0,0,-0.0091,7.895,-6.899)"><path id="g50-253" d="M569 272C569 396 502 451 440 451C377 451 308 398 280 296C262 229 239 121 223 41C163 58 116 110 116 203C116 313 187 386 283 428L269 460C133 413 24 332 24 189C24 85 87 3 215 -10C187 -153 174 -212 164 -249L173 -257C208 -253 245 -243 260 -214L295 -5C427 6 569 118 569 272ZM493 270C493 139 401 42 302 37L357 327C369 388 390 405 412 405C438 405 493 369 493 270Z"/></g><g transform="matrix(.0091,0,0,-0.0091,6.786,4.928)"><path id="g50-73" d="M870 650H600L593 618C681 609 684 605 669 516L641 367H261L286 516C302 603 313 610 391 618L398 650H132L123 618C214 610 218 606 200 516L127 130C113 46 107 40 25 32L18 0H290L298 32C204 40 200 42 214 130L250 321H631L596 131C580 47 573 39 484 32L475 0H753L762 32C673 39 668 46 685 130L757 516C772 606 778 610 861 618L870 650Z"/></g><g transform="matrix(.0065,0,0,-0.0065,14.496,1.732)"><path id="g180-36" d="M578 242V297H351V519H286V297H59V242H286V-6H351V242H578Z"/></g></svg>-</span>contractions and <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.971001pt" id="M4" height="14.7677pt" version="1.1" viewBox="-0.0498162 -10.7967 26.4864 14.7677" width="26.4864pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-253"/></g><g transform="matrix(.013,0,0,-0.013,7.437,0)"><path id="g113-73" d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z"/></g><g transform="matrix(.0091,0,0,-0.0091,18.36,-5.741)"><path id="g54-36" d="M556 236V289H337V504H275V289H56V236H275V-4H337V236H556Z"/></g><g transform="matrix(.0091,0,0,-0.0091,17.512,3.784)"><path id="g198-17" d="M856 669C824 657 796 643 772 628C709 667 631 687 551 687C314 687 158 548 158 412C158 332 216 282 300 282C445 282 516 426 527 561H507C478 406 414 312 302 312C245 312 199 352 199 408C199 535 345 657 550 657C617 657 687 644 742 609C644 538 583 430 495 271C383 71 317 16 193 16C122 16 73 60 73 95H75C82 84 97 83 105 83C133 83 148 106 148 131C148 160 125 179 97 179C64 179 38 151 38 108C38 42 104 -15 211 -15C368 -15 496 71 610 302C672 426 704 522 770 588C803 559 825 518 825 465C825 382 787 316 731 316C710 316 697 328 697 345C697 358 700 380 733 399L723 417C688 398 665 367 665 338C665 303 688 272 737 272C821 272 886 359 886 447C886 516 850 572 797 612C816 627 836 639 863 650L856 669Z"/></g></svg>-</span>contractions. We investigate their mutual relation and establish an existence theorem addressing <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.9783pt" id="M5" height="16.0921pt" version="1.1" viewBox="-0.0498162 -11.1138 19.7999 16.0921" width="19.7999pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-71"/></g><g transform="matrix(.0091,0,0,-0.0091,7.895,-6.899)"><use xlink:href="#g50-253"/></g><g transform="matrix(.0091,0,0,-0.0091,6.786,4.928)"><use xlink:href="#g50-73"/></g><g transform="matrix(.0065,0,0,-0.0065,14.496,1.732)"><use xlink:href="#g180-36"/></g></svg>-</span>contractions with some applications.

Sang Og Kim

Muhammad Nazam

Existence Theorems on Advanced Contractions with Applications

In this paper, we prove a sufficient condition that every nonempty closed convex bounded pair <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="20.269pt" style="vertical-align:-2.26807pt" id="M1" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 20.269 11.5564"><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="g113-78" d="M962 650H795L470 145L347 650H176L170 622C268 613 275 606 244 503L190 322C150 188 132 126 118 91C102 50 80 33 18 28L12 0H245L251 28C175 35 170 48 174 93C177 128 191 180 220 284L292 542H294C331 392 383 150 409 4H432L774 555H776L714 137C700 40 694 34 612 28L606 0H868L874 28C793 34 784 37 797 137L849 533C859 612 863 616 956 622L962 650Z"/></g><g transform="matrix(.013,0,0,-0.013,17.305,0)"><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><span class="ipunc"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="15.636pt" style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="22.3981838 -9.28833 15.636 11.5564"><g transform="matrix(.013,0,0,-0.013,22.448,0)"><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><g transform="matrix(.013,0,0,-0.013,33.358,0)"><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></span> in a reflexive Banach space <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M2" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 7.94191 8.68572" width="7.94191pt"><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></svg> satisfying Opial’s condition has proximal normal structure. We analyze the relatively nonexpansive self-mapping <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M3" height="9.01194pt" version="1.1" viewBox="-0.0498162 -8.96212 8.41168 9.01194" width="8.41168pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-85" d="M620 675H597C578 656 570 650 541 650H144C112 650 104 653 94 675H72C59 618 42 552 23 493L53 491C71 534 88 564 105 585C124 608 144 615 238 615H290L197 121C182 40 174 34 88 28L82 0H361L367 28C275 34 266 38 281 121L374 615H441C522 615 543 608 553 583C562 560 566 531 565 493L597 494C603 551 612 629 620 675Z"/></g></svg> on <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="23.344pt" style="vertical-align:-0.2064009pt" id="M4" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 23.344 8.8423"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-78"/></g><g transform="matrix(.013,0,0,-0.013,15.713,0)"><path id="g117-59" d="M535 194V548H477V194C477 102 391 35 294 35S111 102 111 194V548H53V194C53 72 166 -15 294 -15S535 72 535 194Z"/></g></svg><span class="ibiop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="11.122pt" style="vertical-align:-0.2064009pt" height="8.8423pt" version="1.1" viewBox="26.1991838 -8.6359 11.122 8.8423"><g transform="matrix(.013,0,0,-0.013,26.249,0)"><use xlink:href="#g113-79"/></g></svg></span> satisfying <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="41.351pt" style="vertical-align:-2.26807pt" id="M5" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 41.351 11.5564"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-85"/></g><g transform="matrix(.013,0,0,-0.013,8.285,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,12.783,0)"><use xlink:href="#g113-78"/></g><g transform="matrix(.013,0,0,-0.013,25.59,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.013,0,0,-0.013,33.72,0)"><path id="g117-75" d="M531 95V145H339C215 145 115 224 115 322S215 500 339 500H531V550H338C183 550 57 448 57 322S183 95 338 95H531ZM531 -40V10H86V-40H531Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="13.088pt" style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="44.933183799999995 -9.28833 13.088 11.5564"><g transform="matrix(.013,0,0,-0.013,44.983,0)"><use xlink:href="#g113-78"/></g></svg></span> and <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="39.454pt" style="vertical-align:-2.26807pt" id="M6" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 39.454 11.5564"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-85"/></g><g transform="matrix(.013,0,0,-0.013,8.285,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,12.783,0)"><use xlink:href="#g113-79"/></g><g transform="matrix(.013,0,0,-0.013,23.693,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.013,0,0,-0.013,31.823,0)"><use xlink:href="#g117-75"/></g></svg><span class="irelop"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="11.175pt" style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="43.036183799999996 -9.28833 11.175 11.5564"><g transform="matrix(.013,0,0,-0.013,43.086,0)"><use xlink:href="#g113-79"/></g></svg>,</span></span> to show that Ishikawa’s and Halpern’s iteration converges to the best proximity point. Also, we prove that under relatively isometry self-mapping <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M7" height="9.01194pt" version="1.1" viewBox="-0.0498162 -8.96212 8.41168 9.01194" width="8.41168pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-85"/></g></svg> on <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="23.344pt" style="vertical-align:-0.2064009pt" id="M8" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 23.344 8.8423"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-78"/></g><g transform="matrix(.013,0,0,-0.013,15.713,0)"><use xlink:href="#g117-59"/></g></svg><span class="ibiop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="11.122pt" style="vertical-align:-0.2064009pt" height="8.8423pt" version="1.1" viewBox="26.1991838 -8.6359 11.122 8.8423"><g transform="matrix(.013,0,0,-0.013,26.249,0)"><use xlink:href="#g113-79"/></g></svg></span> satisfying <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="39.454pt" style="vertical-align:-2.26807pt" id="M9" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 39.454 11.5564"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-85"/></g><g transform="matrix(.013,0,0,-0.013,8.285,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,12.783,0)"><use xlink:href="#g113-79"/></g><g transform="matrix(.013,0,0,-0.013,23.693,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.013,0,0,-0.013,31.823,0)"><use xlink:href="#g117-75"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="11.175pt" style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="43.036183799999996 -9.28833 11.175 11.5564"><g transform="matrix(.013,0,0,-0.013,43.086,0)"><use xlink:href="#g113-79"/></g></svg></span> and <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="41.351pt" style="vertical-align:-2.26807pt" id="M10" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 41.351 11.5564"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-85"/></g><g transform="matrix(.013,0,0,-0.013,8.285,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,12.783,0)"><use xlink:href="#g113-78"/></g><g transform="matrix(.013,0,0,-0.013,25.59,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.013,0,0,-0.013,33.72,0)"><use xlink:href="#g117-75"/></g></svg><span class="irelop"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="13.088pt" style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="44.933183799999995 -9.28833 13.088 11.5564"><g transform="matrix(.013,0,0,-0.013,44.983,0)"><use xlink:href="#g113-78"/></g></svg>,</span></span> Ishikawa’s iteration converges to the best proximity point in the collection of all Chebyshev centers of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M11" 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)"><use xlink:href="#g113-79"/></g></svg> relative to <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M12" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 12.9526 8.68572" width="12.9526pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-78"/></g></svg>.</span> Some illustrative examples are provided to support our results.

Laishram Shanjit

Yumnam Rohen

Sumit Chandok

M. Bina Devi

Some Results on Iterative Proximal Convergence and Chebyshev Center

In this paper, the problem of constructing the Lie point symmetries group of the nonlinear partial differential equation appeared in mathematical physics known as the generalized KdV-Like equation is discussed. By using the Lie symmetry method for the generalized KdV-Like equation, the point symmetry operators are constructed and are used to reduce the equation to another fractional ordinary differential equation based on Erd&#x00E9;lyi-Kober differential operator. The symmetries of this equation are also used to construct the conservation Laws by applying the new conservation theorem introduced by Ibragimov. Furthermore, another type of solutions is given by means of power series method and the convergence of the solutions is provided; also, some graphics of solutions are plotted in 3D.

Maria Ihsane El Bahi

Khalid Hilal

Lie Symmetry Analysis, Exact Solutions, and Conservation Laws for the Generalized Time-Fractional KdV-Like Equation

This article characterizes the isometries between spaces of all differentiable functions from a compact interval of the real line into a strictly convex Banach space.

Alireza Ranjbar-Motlagh

Isometries between Spaces of Vector-Valued Differentiable Functions

In this paper, the variable-order fractional Laplacian equations with variable exponents and the Kirchhoff-type problem driven by <span class="nowrap"><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 19.8424 12.7178" width="19.8424pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><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,-0.013,7.71,0)"><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,-0.013,12.208,0)"><path id="g113-46" d="M170 255C170 288 146 313 114 313S58 288 58 255C58 221 82 198 114 198S170 221 170 255Z"/></g><g transform="matrix(.013,0,0,-0.013,15.172,0)"><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>-</span>fractional Laplace with variable exponents were studied. By using variational method, the authors obtain the existence and uniqueness results.

Yating Guo

Guoju Ye

Existence and Uniqueness of Weak Solutions to Variable-Order Fractional Laplacian Equations with Variable Exponents

Fractional derivative is nonlocal, which is more suitable to simulate physical phenomena and provides more accurate models of physical systems such as earthquake vibration and polymers. The present study suggested a new numerical approach for the fractional diffusion-wave equation (FDWE). The fractional-order derivative is in the Riemann-Liouville (R-L) sense. Discussed the theoretical analysis of stability, consistency, and convergence. The numerical examples demonstrate that the method is more workable and excellently holds the theoretical analysis, showing the scheme&#x2019;s feasibility.

Umair Ali

Muhammad Asim Khan

Mostafa M. A. Khater

A. A. Mousa

Raghda A. M. Attia

A New Numerical Approach for Solving 1D Fractional Diffusion-Wave Equation

In this paper, we study the properties of <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M3" 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>-</span>polynomial <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.503pt" id="M4" height="12.2089pt" version="1.1" viewBox="-0.0498162 -9.7059 6.01071 12.2089" width="6.01071pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-228"/></g></svg>-</span>preinvex functions and establish some integral inequalities of Hermite-Hadamard type via this class of convex functions. Moreover, we discuss some special cases which provide a significant complement to the integral estimations of preinvex functions. Applications of the obtained results to the inequalities for special means are also considered.

Shanhe Wu

Muhammad Uzair Awan

Muhammad Ubaid Ullah

Sadia Talib

Artion Kashuri

Some Integral Inequalities for <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-0.2063904pt" 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)"><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></svg>-Polynomial <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-2.503pt" height="12.2089pt" version="1.1" viewBox="-0.0498162 -9.7059 6.01071 12.2089" width="6.01071pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-228" d="M430 689C430 695 425 700 413 700C377 700 332 667 290 635C232 640 213 677 213 696C213 713 220 724 226 732L209 744C166 731 135 700 135 674C135 636 184 601 255 599C144 503 23 345 23 181C23 82 68 44 135 20C156 12 177 7 202 2C238 -5 252 -25 252 -53C252 -98 211 -144 192 -164L214 -187C257 -152 328 -92 328 -19C328 31 298 58 251 70C230 75 208 81 189 88C142 105 108 134 108 208C108 346 202 512 303 595C352 599 430 626 430 689Z" /></g></svg>-Preinvex Functions

In this article, we consider and study a system of generalized set-valued variational inequalities involving relaxed cocoercive mappings in Hilbert spaces. Using the projection method and Banach contraction principle, we prove the existence of a solution for the considered problem. Further, we propose an iterative algorithm and discuss its convergence. Moreover, we establish equivalence between the system of variational inequalities and altering points problem. Some parallel iterative algorithms are proposed, and the strong convergence of the sequences generated by these iterative algorithms is discussed. Finally, a numerical example is constructed to illustrate the convergence analysis of the proposed parallel iterative algorithms.

M. Akram

Aysha Khan

M. Dilshad

Convergence of Some Iterative Algorithms for System of Generalized Set-Valued Variational Inequalities

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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 2021

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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