This paper considers the optimal stopping problem for continuous-time Markov processes. We describe the methodology and solve the optimal stopping problem for a broad class of reward functions. Moreover, we illustrate the outcomes by some typical Markov processes including diffusion and L&#xe9;vy processes with jumps. For each of the processes, the explicit formula for value function and optimal stopping time is derived. Furthermore, we relate the derived optimal rules to some other optimal problems.

Lu Ye

Value Function and Optimal Rule on the Optimal Stopping Problem for Continuous-Time Markov Processes

Considering a function <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="35.086pt" style="vertical-align:-3.429399pt" id="M1" height="15.0208pt" version="1.1" viewBox="-0.0498162 -11.5914 35.086 15.0208"><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><g transform="matrix(.013,0,0,-0.013,8.352,0)"><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,-0.013,12.85,0)"><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,-0.013,19.325,0)"><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,-0.013,27.455,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="51.936pt" style="vertical-align:-3.429399pt" height="15.0208pt" version="1.1" viewBox="38.6681838 -11.5914 51.936 15.0208"><g transform="matrix(.013,0,0,-0.013,38.718,0)"><use xlink:href="#g113-123"/></g><g transform="matrix(.013,0,0,-0.013,45.193,0)"><path id="g113-48" d="M368 703H309L44 -163H104L368 703Z"/></g><g transform="matrix(.013,0,0,-0.013,50.554,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,55.052,0)"><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,-0.013,64.198,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.013,0,0,-0.013,74.734,0)"><use xlink:href="#g113-123"/></g><g transform="matrix(.0091,0,0,-0.0091,81.21,-5.741)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g><g transform="matrix(.013,0,0,-0.013,86.156,0)"><use xlink:href="#g113-42"/></g></svg></span> which is analytic and starlike in the open unit disc <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M2" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 9.58606 8.8423" width="9.58606pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-86" d="M715 650H476L470 622C550 619 562 610 562 570C562 536 553 481 535 384L523 321C485 121 408 37 283 37C178 37 99 109 134 297L178 532C193 612 199 616 278 622L284 650H29L23 622C105 616 109 609 94 532L48 291C23 160 51 83 98 40C143 -1 202 -16 260 -16C321 -16 387 3 439 49C509 111 542 204 563 312L577 384C595 479 608 533 619 569C629 603 637 620 709 622L715 650Z"/></g></svg> and a function <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="35.086pt" style="vertical-align:-3.42947pt" id="M3" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 35.086 12.7178"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-103"/></g><g transform="matrix(.013,0,0,-0.013,8.352,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,12.85,0)"><use xlink:href="#g113-123"/></g><g transform="matrix(.013,0,0,-0.013,19.325,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.013,0,0,-0.013,27.455,0)"><use xlink:href="#g117-34"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="46.99pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="38.6681838 -9.28833 46.99 12.7178"><g transform="matrix(.013,0,0,-0.013,38.718,0)"><use xlink:href="#g113-123"/></g><g transform="matrix(.013,0,0,-0.013,45.193,0)"><use xlink:href="#g113-48"/></g><g transform="matrix(.013,0,0,-0.013,50.554,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,55.052,0)"><use xlink:href="#g113-50"/></g><g transform="matrix(.013,0,0,-0.013,64.198,0)"><use xlink:href="#g117-33"/></g><g transform="matrix(.013,0,0,-0.013,74.734,0)"><use xlink:href="#g113-123"/></g><g transform="matrix(.013,0,0,-0.013,81.21,0)"><use xlink:href="#g113-42"/></g></svg></span> which is analytic and convex in <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="11.505pt" style="vertical-align:-1.576501pt" id="M4" height="10.2124pt" version="1.1" viewBox="-0.0498162 -8.6359 11.505 10.2124"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-86"/></g><g transform="matrix(.013,0,0,-0.013,8.541,0)"><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></span> we introduce two new classes <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.9436pt" id="M5" height="14.0461pt" version="1.1" viewBox="-0.0498162 -10.1025 32.485 14.0461" width="32.485pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g198-20" d="M397 143C384 252 309 299 234 299C151 299 67 242 67 142C67 29 167 -15 264 -15C419 -15 547 68 547 204C547 336 385 381 385 491C385 582 483 644 560 644C613 644 668 624 668 563C668 499 620 463 559 456V435C627 435 701 491 701 564C701 648 628 680 561 680C454 680 308 615 308 472C308 340 465 297 465 191C465 75 365 15 272 15C192 15 129 67 129 137C129 212 192 256 241 256C296 256 345 244 377 139L397 143Z"/></g><g transform="matrix(.0091,0,0,-0.0091,9.659,-5.741)"><path id="g50-43" d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 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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><g transform="matrix(.013,0,0,-0.013,27.79,0)"><use xlink:href="#g113-42"/></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="M6" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 35.691 12.7178" width="35.691pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g198-12" d="M771 669C722 624 660 598 616 598C593 598 565 610 530 629C460 664 429 678 388 678C284 678 211 602 211 527C211 453 270 401 350 401C418 401 468 443 501 508L483 517C450 464 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xmlns="http://www.w3.org/2000/svg" width="80.308pt" style="vertical-align:-3.42941pt" height="13.3858pt" version="1.1" viewBox="42.8511838 -9.95639 80.308 13.3858"><g transform="matrix(.013,0,0,-0.013,42.901,0)"><use xlink:href="#g113-123"/></g><g transform="matrix(.013,0,0,-0.013,49.376,0)"><use xlink:href="#g113-48"/></g><g transform="matrix(.013,0,0,-0.013,54.737,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,59.235,0)"><use xlink:href="#g113-50"/></g><g transform="matrix(.013,0,0,-0.013,68.381,0)"><use xlink:href="#g117-33"/></g><g transform="matrix(.013,0,0,-0.013,78.917,0)"><use xlink:href="#g113-123"/></g><g transform="matrix(.0091,0,0,-0.0091,85.393,-5.741)"><use xlink:href="#g50-223"/></g><g transform="matrix(.013,0,0,-0.013,91.207,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.013,0,0,-0.013,100.063,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,104.561,0)"><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><g transform="matrix(.013,0,0,-0.013,115.578,0)"><path id="g117-92" d="M512 230V281L75 514V456L453 256V254L75 55V-3L512 230Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="10.738pt" style="vertical-align:-3.42941pt" height="13.3858pt" version="1.1" viewBox="126.7911838 -9.95639 10.738 13.3858"><g transform="matrix(.013,0,0,-0.013,126.841,0)"><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,-0.013,133.081,0)"><use xlink:href="#g113-42"/></g></svg></span>. The object of the present paper is to discuss some interesting properties for functions in the classes <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.9436pt" id="M8" height="14.0461pt" version="1.1" viewBox="-0.0498162 -10.1025 32.485 14.0461" width="32.485pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g198-20"/></g><g transform="matrix(.0091,0,0,-0.0091,9.659,-5.741)"><use xlink:href="#g50-43"/></g><g transform="matrix(.0091,0,0,-0.0091,9.659,3.784)"><use xlink:href="#g50-223"/></g><g transform="matrix(.013,0,0,-0.013,15.738,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,20.236,0)"><use xlink:href="#g113-224"/></g><g transform="matrix(.013,0,0,-0.013,27.79,0)"><use xlink:href="#g113-42"/></g></svg> and <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="38.445pt" style="vertical-align:-3.42947pt" id="M9" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 38.445 12.7178"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g198-12"/></g><g transform="matrix(.0091,0,0,-0.0091,13.117,3.132)"><use xlink:href="#g50-223"/></g><g transform="matrix(.013,0,0,-0.013,18.931,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,23.429,0)"><use xlink:href="#g113-224"/></g><g transform="matrix(.013,0,0,-0.013,30.983,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.013,0,0,-0.013,35.481,0)"><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></svg><span class="ipunc"></span></span>

Maslina Darus

Shigeyoshi Owa

New Subclasses concerning Some Analytic and Univalent Functions

This paper aims to study extensively some results concerning continuous dependence for implicit Kirk-Mann and implicit Kirk-Ishikawa iterations. In order to equipoise the formation of these algorithms, we introduce a general hyperbolic space which is no doubt a free associate of some known hyperbolic spaces. The present results are extension of other results and they can be used in many applications.

K. Rauf

O. T. Wahab

S. M. Alata

Continuous Dependence for Two Implicit Kirk-Type Algorithms in General Hyperbolic Spaces

This paper is devoted to existence and uniqueness of solutions for some stochastic functional differential equations with infinite delay in a fading memory phase space.

Hassane Bouzahir

Brahim Benaid

Chafai Imzegouan

Some Stochastic Functional Differential Equations with Infinite Delay: A Result on Existence and Uniqueness of Solutions in a Concrete Fading Memory Space

Using Leray-Schauder degree or degree for <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 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>-condensing maps we obtain the existence of at least one solution for the boundary value problem of the following type: <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.4294pt" id="M2" height="14.8655pt" version="1.1" viewBox="-0.0498162 -11.4361 105.289 14.8655" width="105.289pt"><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="g113-253" 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><g transform="matrix(.013,0,0,-0.013,12.065,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,16.563,0)"><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 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xlink:href="#g113-49"/></g><g transform="matrix(.013,0,0,-0.013,18.305,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.013,0,0,-0.013,26.435,0)"><use xlink:href="#g117-34"/></g><g transform="matrix(.013,0,0,-0.013,37.698,0)"><use xlink:href="#g113-49"/></g></svg>, <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42947pt" id="M6" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 122.744 12.7178" width="122.744pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-103"/></g><g transform="matrix(.013,0,0,-0.013,10.679,0)"><use xlink:href="#g113-59"/></g><g transform="matrix(.013,0,0,-0.013,17.275,0)"><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,-0.013,21.76,0)"><use xlink:href="#g113-49"/></g><g transform="matrix(.013,0,0,-0.013,28,0)"><use xlink:href="#g113-45"/></g><g 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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> is a positive real number, and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M8" 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> is a real Banach space.

Dionicio Pastor Dallos Santos

Problems with Mixed Boundary Conditions in Banach Spaces

This paper deals with the lateral vibration of a finite double-Rayleigh beam system having arbitrary classical end conditions and traversed by a concentrated moving mass. The system is made up of two identical parallel uniform Rayleigh beams which are continuously joined together by a viscoelastic Winkler type layer. Of particular interest, however, is the effect of the mass of the moving load on the dynamic response of the system. To this end, a solution technique based on the generalized finite integral transform, modified Struble’s method, and differential transform method (DTM) is developed. Numerical examples are given for the purpose of demonstrating the simplicity and efficiency of the technique. The dynamic responses of the system are presented graphically and found to be in good agreement with those previously obtained in the literature for the case of a moving force. The conditions under which the system reaches a state of resonance and the corresponding critical speeds were established. The effects of variations of the ratio <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 19.7651 12.7178" width="19.7651pt"><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="g113-225" d="M478 372C478 418 458 448 431 448C409 448 389 431 389 410C389 404 391 400 394 395C398 388 406 371 406 348C406 253 308 122 251 51H249C254 122 249 257 231 336C212 421 189 448 159 448C126 448 75 412 23 327L48 306C83 354 103 371 115 371C125 371 134 360 144 334C185 224 192 64 183 -19C146 -100 116 -202 110 -244L125 -261C154 -259 208 -234 222 -220C222 -194 225 -84 235 -23C247 -3 273 36 308 79C379 165 478 288 478 372Z"/></g><g transform="matrix(.0091,0,0,-0.0091,10.166,3.132)"><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><g transform="matrix(.013,0,0,-0.013,15.112,0)"><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> of the mass of the moving load to the mass of the beam on the dynamic response are presented. The effects of other parameters on the dynamic response of the system are also examined.

Jacob Abiodun Gbadeyan

Fatai Akangbe Hammed

Influence of a Moving Mass on the Dynamic Behaviour of Viscoelastically Connected Prismatic Double-Rayleigh Beam System Having Arbitrary End Supports

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Chinese Journal of Mathematics