In this article, we study the approximate fixed point sequence of an evolution family. A family <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="51.597pt" style="vertical-align:-3.42947pt" id="M1" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 51.597 12.7178"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-70" d="M593 650H142L136 622C219 616 224 611 209 531L130 120C114 39 108 34 23 28L17 0H501C518 32 557 130 568 158L540 169C519 132 495 93 469 70C439 43 409 35 332 35C272 35 241 37 225 50C211 61 211 89 222 149L253 320H350C441 320 443 312 441 239H471L510 434H481C455 366 448 358 360 358H260L303 572C311 613 312 615 348 615H431C509 615 522 603 532 581C544 555 546 531 545 491L574 493C583 555 592 631 593 650Z"/></g><g transform="matrix(.013,0,0,-0.013,11.64,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.013,0,0,-0.013,22.903,0)"><path id="g113-124" d="M293 -169V-141C218 -131 209 -90 209 -44C209 19 222 85 222 152C222 207 198 255 139 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d="M556 393C556 426 537 448 514 448C496 448 478 435 471 420C468 414 462 401 466 394C473 382 480 368 480 346C480 268 392 143 338 67H336C329 163 319 253 309 330S286 448 254 448C215 448 160 383 127 337L143 311C167 344 200 373 208 373S222 365 229 320C246 214 264 66 268 -20C219 -83 131 -171 17 -239L25 -261L137 -235C248 -110 273 -79 335 8C384 77 481 215 520 287C544 332 556 363 556 393Z"/></g><g transform="matrix(.013,0,0,-0.013,61.306,0)"><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(.013,0,0,-0.013,65.804,0)"><path id="g113-60" d="M114 412C81 412 58 388 58 355C58 321 81 297 113 297S169 321 169 355C169 388 145 412 114 412ZM95 130C70 130 46 114 46 88C46 72 54 65 59 64C93 56 121 33 121 -3C121 -41 93 -68 45 -88L56 -118C117 -99 186 -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="18.53pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="70.89718380000001 -9.28833 18.53 12.7178"><g transform="matrix(.013,0,0,-0.013,70.947,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.013,0,0,-0.013,81.846,0)"><path id="g117-94" d="M531 285V335L57 550V494L473 310L57 127V71L531 285ZM531 -40V10H57V-40H531Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="18.792pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="93.0591838 -9.28833 18.792 12.7178"><g transform="matrix(.013,0,0,-0.013,93.109,0)"><use xlink:href="#g113-122"/></g><g transform="matrix(.013,0,0,-0.013,104.27,0)"><use xlink:href="#g117-94"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="11.217pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="115.4831838 -9.28833 11.217 12.7178"><g transform="matrix(.013,0,0,-0.013,115.533,0)"><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,-0.013,121.773,0)"><path id="g113-126" d="M283 255V284C195 290 192 337 192 375C192 436 200 508 200 580C200 696 116 709 54 709V681C127 671 139 634 139 588C138 524 125 452 125 387C125 333 149 286 209 272V268C148 253 125 206 125 151C126 87 139 16 139 -47C139 -92 127 -131 54 -141V-169C115 -169 200 -152 200 -41C200 32 192 104 192 164C192 202 195 249 283 255Z"/></g></svg></span> of a bounded nonlinear operator acting on a metric space <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="M2" 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)"><use xlink:href="#g113-41"/></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)"><use xlink:href="#g113-45"/></g></svg><span class="ipunc"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="11.931pt" style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="22.3981838 -9.28833 11.931 11.5564"><g transform="matrix(.013,0,0,-0.013,22.448,0)"><path id="g113-101" d="M530 686C535 705 530 712 521 712C504 712 448 684 359 674L358 648H393C437 648 439 646 429 593L400 435C372 447 345 448 332 448C286 448 194 414 144 373C68 311 23 203 23 111C23 26 57 -12 91 -12C120 -12 147 3 188 29C227 54 290 102 341 170H343L322 71C308 6 320 -12 341 -12C373 -12 442 27 501 96L485 120C455 91 422 67 408 67C401 67 401 76 404 91C440 294 479 473 530 686ZM387 375L355 241C326 187 200 53 142 53C126 53 109 73 109 130C109 217 154 337 218 381C240 396 265 404 297 404S372 390 387 375Z"/></g><g transform="matrix(.013,0,0,-0.013,29.664,0)"><use xlink:href="#g113-42"/></g></svg></span> is said to be an evolution family if <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="24.183pt" style="vertical-align:-2.26807pt" id="M3" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 24.183 11.5564"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-86"/></g><g transform="matrix(.013,0,0,-0.013,9.454,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,13.952,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.013,0,0,-0.013,21.219,0)"><use xlink:href="#g113-45"/></g></svg><span class="ipunc"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="32.505pt" style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="26.3121838 -9.28833 32.505 11.5564"><g transform="matrix(.013,0,0,-0.013,26.362,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.013,0,0,-0.013,33.63,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.013,0,0,-0.013,41.76,0)"><use xlink:href="#g117-34"/></g><g transform="matrix(.013,0,0,-0.013,53.022,0)"><path id="g113-74" d="M405 650H141L135 622C222 616 230 610 215 535L133 116C118 41 113 33 29 28L23 0H289L295 28C209 33 205 40 219 116L298 535C312 609 317 616 399 622L405 650Z"/></g></svg></span> and <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" 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xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,52.342,0)"><use xlink:href="#g113-122"/></g><g transform="matrix(.013,0,0,-0.013,59.871,0)"><use xlink:href="#g113-45"/></g></svg><span class="ipunc"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="50.052pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="64.9641838 -9.28833 50.052 12.7178"><g transform="matrix(.013,0,0,-0.013,65.014,0)"><path id="g113-123" d="M475 445L452 468C423 436 401 426 385 426S352 427 305 436C268 443 216 448 185 448C156 448 128 425 102 382C89 360 77 338 64 304L89 291C117 340 143 380 190 380C214 380 268 376 295 373C320 370 344 371 362 374L70 101C32 65 23 36 23 18C23 5 27 -3 31 -8C36 -8 39 -6 42 -1C52 15 67 31 80 40S110 49 143 33C195 8 257 -20 290 -20C339 -20 382 11 439 129L416 148C366 74 334 55 305 55C278 55 236 68 203 81C169 94 141 100 122 96C205 170 348 308 411 376L475 445Z"/></g><g transform="matrix(.013,0,0,-0.013,71.489,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.013,0,0,-0.013,79.619,0)"><use xlink:href="#g117-34"/></g><g transform="matrix(.013,0,0,-0.013,90.882,0)"><use xlink:href="#g113-86"/></g><g transform="matrix(.013,0,0,-0.013,100.336,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,104.834,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.013,0,0,-0.013,112.102,0)"><use xlink:href="#g113-45"/></g></svg><span class="ipunc"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="11.485pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="117.19518380000001 -9.28833 11.485 12.7178"><g transform="matrix(.013,0,0,-0.013,117.245,0)"><use xlink:href="#g113-123"/></g><g transform="matrix(.013,0,0,-0.013,123.72,0)"><use xlink:href="#g113-42"/></g></svg></span> for all <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42938pt" id="M5" height="11.7782pt" version="1.1" viewBox="-0.0498162 -8.34882 72.4807 11.7782" width="72.4807pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.013,0,0,-0.013,10.899,0)"><use xlink:href="#g117-94"/></g><g transform="matrix(.013,0,0,-0.013,22.162,0)"><use xlink:href="#g113-122"/></g><g transform="matrix(.013,0,0,-0.013,33.323,0)"><use xlink:href="#g117-94"/></g><g transform="matrix(.013,0,0,-0.013,44.586,0)"><use xlink:href="#g113-123"/></g><g transform="matrix(.013,0,0,-0.013,54.693,0)"><use xlink:href="#g117-94"/></g><g transform="matrix(.013,0,0,-0.013,65.956,0)"><use xlink:href="#g113-49"/></g></svg>.</span> We prove that the common approximate fixed point sequence is equal to the intersection of the approximate fixed point sequence of two operators from the family. Furthermore, we apply the Ishikawa iteration process to construct an approximate fixed point sequence of an evolution family of nonlinear mapping.

Si Fuan

Rizwan Ullah

Gul Rahmat

Muhammad Numan

Saad Ihsan Butt

Xun Ge

Approximate Fixed Point Sequences of an Evolution Family on a Metric Space

In this paper, we study a class of 3-species multidelay Lotka&#x2013;Volterra ratio-dependent predator-prey model with feedback controls and prey diffusion. By using the theory of delay differential inequalities and developing some new analysis methods as well as constructing a suitable Lyapunov function, some sufficient conditions are obtained to guarantee the permanence of the system and the global attractivity of the positive solution for the predator-prey system. Furthermore, the corresponding periodic system is discussed, and some conditions are established about the existence, uniqueness, and stability of the positive periodic solution for the periodic system by using the fixed-point theory and theoretical analysis. In additional, some numerical solutions of the equations describing the system are given to verify the obtained criteria. Finally, the corresponding stochastic time-delay predator-prey model with multiplicative noise sources is solved numerically, and some interesting new dynamics behavior is obtained.

Lili Jia

Analysis for a Delayed Three-Species Predator-Prey Model with Feedback Controls and Prey Diffusion

Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M1" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 7.31558 8.68572" width="7.31558pt"><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></svg> be the free nilpotent Lie algebra of finite rank over a field of characteristic zero. We define the concepts of marginal ideals and marginal automorphisms of <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="M2" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 7.31558 8.68572" width="7.31558pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-77"/></g></svg>,</span> and we give some results on marginal automorphisms.

&#xd6;zge &#xd6;ztekin

On Marginal Automorphisms of Free Nilpotent Lie Algebras

Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M1" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 9.02496 8.8423" width="9.02496pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-72" d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z"/></g></svg>be a graph with <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 6.6501 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> vertices and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M3" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 22.7653 11.5564" width="22.7653pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-132" d="M485 501C480 550 475 615 475 650H42V622C119 616 127 606 127 525V126C127 44 119 34 41 28V0H311V28C220 34 212 43 212 126V584C212 612 216 615 244 615H318C382 615 404 608 420 587C435 568 445 541 455 497L485 501Z"/></g><g transform="matrix(.013,0,0,-0.013,6.595,0)"><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,-0.013,11.093,0)"><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(.013,0,0,-0.013,18.087,0)"><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> be the degree of its <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="M4" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 7.11985 6.1673" 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363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"/></g><g transform="matrix(.013,0,0,-0.013,18.36,0)"><use xlink:href="#g113-42"/></g></svg> is the degree 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="M6" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 7.11985 6.1673" width="7.11985pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-118"/></g></svg>).</span> In this article, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper <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="M7" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 8.02022 8.68572" width="8.02022pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-71" d="M584 650H137L131 622C214 614 217 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Xing-Long Wang

Jia-Bao Liu

Akbar Jahanbani

Muhammad Kamran Siddiqui

Nader Jafari Rad

Roslan Hasni

On Generalized Topological Indices of Silicon-Carbon

A tiling of the Euclidean plane, by regular polygons, is called 2-uniform tiling if it has two orbits of vertices under the action of its symmetry group. There are 20 distinct 2-uniform tilings of the plane. Plane being the universal cover of torus and Klein bottle, it is natural to ask about the exploration of maps on these two surfaces corresponding to the 2-uniform tilings. We call such maps as doubly semiequivelar maps. In the present study, we compute and classify (up to isomorphism) doubly semiequivelar maps on torus and Klein bottle. This classification of semiequivelar maps is useful in classifying a category of symmetrical maps which have two orbits of vertices, named as 2-uniform maps.

Anand K. Tiwari

Amit Tripathi

Yogendra Singh

Punam Gupta

Doubly Semiequivelar Maps on Torus and Klein Bottle

Let <i>M</i> be an almost cosymplectic 3-<i>h</i>-<i>a</i>-manifold. In this paper, we prove that the Ricci operator of <i>M</i> is transversely Killing if and only if <i>M</i> is locally isometric to a product space of an open interval and a surface of constant Gauss curvature, or a unimodular Lie group equipped with a left invariant almost cosymplectic structure. Some corollaries of this result and some examples illustrating main results are given.

Quanxiang Pan

Yajie Wang

Remarks on Almost Cosymplectic 3-Manifolds with RICCI Operators

We herein present the detailed results for the existence and uniqueness of mild solution for multifractional order impulsive integrodifferential control equations with a nonlocal condition involving several types of semigroups of bounded linear operators, which were established on probability density functions related with the fractional differential equation. Additionally, we present the necessary and sufficient conditions to investigate Schauder’s fixed point theorem with Holder’s inequality <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 7.13289 9.39034" width="7.13289pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-239"/></g></svg>–mean continuity and infinite delay parameter to guarantee the uniqueness of a fixed point.

Sameer Qasim Hasan

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This paper deals with the existence of mild solutions for the following Cauchy problem: <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="134.314pt" style="vertical-align:-3.4295pt" id="M1" height="13.6599pt" version="1.1" viewBox="-0.0498162 -10.2304 134.314 13.6599"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g190-101" d="M517 51L485 54C448 58 441 63 441 115V712C404 700 337 684 285 678V653C357 648 362 645 362 580V437C339 446 309 449 295 449C159 449 38 340 38 201C38 61 143 -12 223 -12C234 -12 261 -6 301 17L362 53V-12C420 9 495 22 517 26V51ZM362 85C338 67 301 51 266 51C201 51 128 109 128 228C128 373 212 411 259 411C296 411 338 395 362 360V85Z"/></g><g transform="matrix(.0091,0,0,-0.0091,6.89,-6.016)"><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 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transform="matrix(.013,0,0,-0.013,287.91,0)"><use xlink:href="#g113-45"/></g></svg><span class="ipunc"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="11.72pt" style="vertical-align:-3.4295pt" height="13.6599pt" version="1.1" viewBox="293.0031838 -10.2304 11.72 13.6599"><g transform="matrix(.013,0,0,-0.013,293.053,0)"><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><g transform="matrix(.013,0,0,-0.013,299.335,0)"><path id="g113-94" d="M226 -163V703H56V676C162 667 165 662 165 584V-43C165 -122 162 -126 56 -136V-163H226Z"/></g></svg>,</span></span> where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.2682pt" id="M2" height="12.4986pt" version="1.1" viewBox="-0.0498162 -10.2304 47.338 12.4986" width="47.338pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g190-101"/></g><g transform="matrix(.0091,0,0,-0.0091,6.89,-6.016)"><use xlink:href="#g50-223"/></g><g transform="matrix(.013,0,0,-0.013,12.704,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,17.202,0)"><path id="g113-47" d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"/></g><g transform="matrix(.013,0,0,-0.013,20.166,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.013,0,0,-0.013,24.664,0)"><use xlink:href="#g113-48"/></g><g transform="matrix(.013,0,0,-0.013,30.025,0)"><use xlink:href="#g190-101"/></g><g transform="matrix(.013,0,0,-0.013,36.915,0)"><use xlink:href="#g113-117"/></g><g transform="matrix(.0091,0,0,-0.0091,41.348,-5.741)"><use xlink:href="#g50-223"/></g></svg> is the so-called conformable fractional derivative. The linear part <i>A</i> is the infinitesimal generator of a uniformly continuous semigroup <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.95787pt" id="M3" height="13.2462pt" version="1.1" viewBox="-0.0498162 -9.28833 44.6177 13.2462" width="44.6177pt"><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="g113-85" 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><g transform="matrix(.013,0,0,-0.013,12.783,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,17.281,0)"><use xlink:href="#g113-117"/></g><g transform="matrix(.013,0,0,-0.013,21.714,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.013,0,0,-0.013,26.212,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.0091,0,0,-0.0091,30.71,3.132)"><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(.0091,0,0,-0.0091,33.868,3.132)"><path id="g54-94" d="M552 293V344L59 566V507L489 319L59 131V72L552 293ZM552 -41V12H59V-41H552Z"/></g><g transform="matrix(.0091,0,0,-0.0091,39.428,3.132)"><use xlink:href="#g50-49"/></g></svg> on a Banach space <i>X</i>, <i>f</i> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M4" 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 given functions. The main result is proved by using the Darbo–Sadovskii fixed point theorem without assuming the compactness of the family <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.63817pt" id="M5" height="12.9265pt" version="1.1" viewBox="-0.0498162 -9.28833 44.6177 12.9265" width="44.6177pt"><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="#g113-85"/></g><g transform="matrix(.013,0,0,-0.013,12.783,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,17.281,0)"><use xlink:href="#g113-117"/></g><g transform="matrix(.013,0,0,-0.013,21.714,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.013,0,0,-0.013,26.212,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.0091,0,0,-0.0091,30.71,3.132)"><use xlink:href="#g50-117"/></g><g transform="matrix(.0091,0,0,-0.0091,33.868,3.132)"><path id="g54-92" d="M530 234V290L81 530V468L467 263V261L81 57V-6L530 234Z"/></g><g transform="matrix(.0091,0,0,-0.0091,39.428,3.132)"><use xlink:href="#g50-49"/></g></svg> and the Lipshitz condition on the nonlocal part <span class="nowrap"><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>.</span>

Mohamed Bouaouid

Mohamed Hannabou

Khalid Hilal

Nonlocal Conformable-Fractional Differential Equations with a Measure of Noncompactness in Banach Spaces

This article considers modified formulas for the standard conjugate gradient (CG) technique that is planned by Li and Fukushima. A new scalar parameter <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.943601pt" id="M1" height="15.6674pt" version="1.1" viewBox="-0.0498162 -11.7238 23.8662 15.6674" width="23.8662pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-230" d="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z"/></g><g transform="matrix(.0091,0,0,-0.0091,6.475,-5.741)"><path id="g190-79" d="M719 650H483V622C553 619 576 606 581 560C585 532 588 487 588 394V148H585L167 650H20V622C67 618 88 610 109 585C127 562 129 557 129 484V264C129 173 126 124 123 93C118 44 92 31 35 28V0H272V28C204 32 181 44 176 95C173 124 170 172 170 264V515H172L598 -9H629V394C629 487 631 532 635 563C639 606 663 619 719 622V650Z"/></g><g transform="matrix(.0091,0,0,-0.0091,13.008,-5.741)"><path id="g190-102" d="M380 106C343 72 306 56 265 56C195 56 116 112 115 248C235 252 361 262 377 265C396 269 400 277 400 297C400 374 333 449 250 449H249C198 449 144 421 103 376S37 269 37 201C37 88 109 -12 232 -12C263 -12 332 6 395 84L380 106ZM225 412C281 412 315 364 314 312C314 297 308 292 290 292C232 290 176 289 120 289C135 370 180 412 225 412Z"/></g><g transform="matrix(.0091,0,0,-0.0091,16.949,-5.741)"><path id="g190-120" d="M696 437H516V411C580 402 587 397 574 345C555 271 527 183 505 114H503C461 222 420 328 383 432H357C315 316 278 214 241 118H239C214 195 189 274 165 350C151 396 156 402 216 411V437H9V411C57 405 65 398 83 346C125 226 162 108 199 -11H227C267 100 308 199 346 300C387 195 428 97 468 -11H497C542 124 587 257 617 337C637 392 646 404 696 411V437Z"/></g><g transform="matrix(.0091,0,0,-0.0091,6.149,3.784)"><path id="g50-108" d="M485 418C485 434 469 451 441 451C389 451 312 387 255 335C221 304 194 278 159 242H157L259 678C264 698 264 710 255 710C242 710 183 681 96 673L90 643L128 642C165 641 171 640 162 602L24 -5L31 -12C51 -4 81 4 110 9L145 181C154 195 180 220 205 242C233 170 263 106 291 53C318 1 335 -12 360 -12C383 -12 425 0 483 82L463 105C437 79 411 57 400 57C388 57 378 68 355 110C329 156 288 244 268 298C299 333 355 376 405 376C426 376 440 373 448 368C453 365 465 368 468 373C477 386 485 405 485 418Z"/></g></svg> for this CG technique of unconstrained optimization is planned. The descent condition and global convergent property are established below using strong Wolfe conditions. Our numerical experiments show that the new proposed algorithms are more stable and economic as compared to some well-known standard CG methods.

Eman T. Hamed

Rana Z. Al-Kawaz

Abbas Y. Al-Bayati

New Investigation for the Liu-Story Scaled Conjugate Gradient Method for Nonlinear Optimization

The purpose of this paper is to introduce the concepts of <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="25.397pt" style="vertical-align:-3.42947pt" id="M2" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 25.397 12.7178"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-224"/></g><g transform="matrix(.013,0,0,-0.013,7.554,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,12.052,0)"><use xlink:href="#g113-241"/></g><g transform="matrix(.0091,0,0,-0.0091,17.486,3.132)"><use xlink:href="#g50-50"/></g><g transform="matrix(.013,0,0,-0.013,22.433,0)"><use xlink:href="#g113-45"/></g></svg><span class="ipunc"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="15.09pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="27.526183800000002 -9.28833 15.09 12.7178"><g transform="matrix(.013,0,0,-0.013,27.576,0)"><use xlink:href="#g113-241"/></g><g transform="matrix(.0091,0,0,-0.0091,33.01,3.132)"><use xlink:href="#g50-51"/></g><g transform="matrix(.013,0,0,-0.013,37.956,0)"><use xlink:href="#g113-42"/></g></svg>-</span></span>continuous multifunctions and almost <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="25.397pt" style="vertical-align:-3.42947pt" id="M3" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 25.397 12.7178"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-224"/></g><g transform="matrix(.013,0,0,-0.013,7.554,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,12.052,0)"><use xlink:href="#g113-241"/></g><g transform="matrix(.0091,0,0,-0.0091,17.486,3.132)"><use xlink:href="#g50-50"/></g><g transform="matrix(.013,0,0,-0.013,22.433,0)"><use xlink:href="#g113-45"/></g></svg><span class="ipunc"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="15.09pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="27.526183800000002 -9.28833 15.09 12.7178"><g transform="matrix(.013,0,0,-0.013,27.576,0)"><use xlink:href="#g113-241"/></g><g transform="matrix(.0091,0,0,-0.0091,33.01,3.132)"><use xlink:href="#g50-51"/></g><g transform="matrix(.013,0,0,-0.013,37.956,0)"><use xlink:href="#g113-42"/></g></svg>-</span></span>continuous multifunctions. Moreover, some characterizations of <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="25.397pt" style="vertical-align:-3.42947pt" id="M4" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 25.397 12.7178"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-224"/></g><g transform="matrix(.013,0,0,-0.013,7.554,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,12.052,0)"><use xlink:href="#g113-241"/></g><g transform="matrix(.0091,0,0,-0.0091,17.486,3.132)"><use xlink:href="#g50-50"/></g><g transform="matrix(.013,0,0,-0.013,22.433,0)"><use xlink:href="#g113-45"/></g></svg><span class="ipunc"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="15.09pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="27.526183800000002 -9.28833 15.09 12.7178"><g transform="matrix(.013,0,0,-0.013,27.576,0)"><use xlink:href="#g113-241"/></g><g transform="matrix(.0091,0,0,-0.0091,33.01,3.132)"><use xlink:href="#g50-51"/></g><g transform="matrix(.013,0,0,-0.013,37.956,0)"><use xlink:href="#g113-42"/></g></svg>-</span></span>continuous multifunctions and almost <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="25.397pt" style="vertical-align:-3.42947pt" id="M5" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 25.397 12.7178"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-224"/></g><g transform="matrix(.013,0,0,-0.013,7.554,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,12.052,0)"><use xlink:href="#g113-241"/></g><g transform="matrix(.0091,0,0,-0.0091,17.486,3.132)"><use xlink:href="#g50-50"/></g><g transform="matrix(.013,0,0,-0.013,22.433,0)"><use xlink:href="#g113-45"/></g></svg><span class="ipunc"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="15.09pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="27.526183800000002 -9.28833 15.09 12.7178"><g transform="matrix(.013,0,0,-0.013,27.576,0)"><use xlink:href="#g113-241"/></g><g transform="matrix(.0091,0,0,-0.0091,33.01,3.132)"><use xlink:href="#g50-51"/></g><g transform="matrix(.013,0,0,-0.013,37.956,0)"><use xlink:href="#g113-42"/></g></svg>-</span></span>continuous multifunctions are investigated.

Kaewta Laprom

Chawalit Boonpok

Chokchai Viriyapong

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About this journal

<p><em>Journal of Mathematics</em> is a broad scope journal that publishes original research and review articles on all aspects of both pure and applied mathematics.</p>


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Meet the editorial board

<p>Chief Editor, Professor Jen-Chih Yao, is currently based at Zhejiang Normal University in China. His current research includes dynamic programming, mathematical programming, and operations research.</p>


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We currently have a number of Special Issues open for submission. Special Issues highlight emerging areas of research within a field, or provide a venue for a deeper investigation into an existing research area.

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Journal of Mathematics - Table of contents 2020

title

Journal of Mathematics is a broad scope journal that publishes original research and review articles on all aspects of both pure and applied mathematics.

Journal of Mathematics