The concepts of bounded boundary rotation and parabolic domain are used to define certain new classes of analytic functions such as <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="36.72pt" style="vertical-align:-5.47417pt" id="M2" height="14.7625pt" version="1.1" viewBox="-0.0498162 -9.28833 36.72 14.7625"><g transform="matrix(.013,0,0,-0.013,0,0)"><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(.0091,0,0,-0.0091,7.332,3.132)"><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(.0091,0,0,-0.0091,11.764,3.132)"><path id="g50-45" d="M98 134C72 134 46 117 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class="ipunc"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="10.363pt" style="vertical-align:-6.201pt" height="17.3148pt" version="1.1" viewBox="40.838183799999996 -11.1138 10.363 17.3148"><g transform="matrix(.013,0,0,-0.013,40.888,0)"><use xlink:href="#g113-104"/></g><g transform="matrix(.013,0,0,-0.013,48.287,0)"><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="11.356pt" style="vertical-align:-6.201pt" height="17.3148pt" version="1.1" viewBox="53.3801838 -11.1138 11.356 17.3148"><g transform="matrix(.013,0,0,-0.013,53.43,0)"><path id="g113-98" d="M483 97L471 123C436 91 401 65 392 65C388 65 384 74 390 106C414 239 444 378 457 429L455 433C444 433 429 436 416 439C392 444 368 448 344 448C281 448 204 415 152 376C71 315 23 205 23 103C23 21 57 -12 85 -12C114 -12 149 6 185 34C231 70 285 119 329 183H331L309 81C292 0 308 -12 326 -12C350 -12 421 24 483 97ZM374 387C370 363 356 291 345 261C315 193 181 50 139 50C124 50 110 71 110 118C110 224 153 331 218 379C238 394 271 402 301 402C329 402 359 394 374 387Z"/></g><g transform="matrix(.013,0,0,-0.013,60.01,0)"><use xlink:href="#g113-42"/></g></svg>.</span></span> The difference of coefficients, necessary conditions, distortion bounds, radius problem, and several other interesting properties of these newly defined classes are also studied.

Khalida Inayat Noor

Alina Alb Lupas

Shujaat Ali Shah

Alanod M. Sibih

S. Abdel-Khalek

Study of Generalized <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 6.50656 9.39034" width="6.50656pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-114" d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z" /></g></svg>-Close-to-Convex Functions Related to Parabolic Domain

In this paper, the existence result of at least two positive solutions is obtained for a nonlinear Riemann-Liouville fractional differential equation subject to nonlocal boundary conditions, where fractional derivatives and Riemann-Stieltjes integrals are involved. The nonlinearity possesses singularities on both its time and space variables. The discussion is based on the fixed point index theory on cones.

Lufeng Gu

Qiuyan Zhong

Zhuyan Shao

On Multiple Positive Solutions for Singular Fractional Boundary Value Problems with Riemann-Stieltjes Integrals

In this paper, we use the Faber polynomial expansion to obtain bounds for the general coefficients <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.29107pt" id="M1" height="12.5794pt" version="1.1" viewBox="-0.0498162 -9.28833 18.1535 12.5794" width="18.1535pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-9" d="M162 -163V703H101V-163H162Z"/></g><g transform="matrix(.013,0,0,-0.013,3.419,0)"><path id="g113-98" d="M483 97L471 123C436 91 401 65 392 65C388 65 384 74 390 106C414 239 444 378 457 429L455 433C444 433 429 436 416 439C392 444 368 448 344 448C281 448 204 415 152 376C71 315 23 205 23 103C23 21 57 -12 85 -12C114 -12 149 6 185 34C231 70 285 119 329 183H331L309 81C292 0 308 -12 326 -12C350 -12 421 24 483 97ZM374 387C370 363 356 291 345 261C315 193 181 50 139 50C124 50 110 71 110 118C110 224 153 331 218 379C238 394 271 402 301 402C329 402 359 394 374 387Z"/></g><g transform="matrix(.0091,0,0,-0.0091,9.412,3.132)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/></g><g transform="matrix(.013,0,0,-0.013,14.587,0)"><use xlink:href="#g113-9"/></g></svg> of bi-univalent functions in the family of analytic functions in the open unit disk. Estimation of the bound value for the initial coefficients of the functions in these classes is also established.

Adel A. Attiya

Abeer M. Albalahi

Taher S. Hassan

Coefficient Estimates for Certain Families of Analytic Functions Associated with Faber Polynomial

The fractional-stochastic Radhakrishnan-Kundu-Lakshmanan equation (FSRKLE) is considered here. To attain new hyperbolic, elliptic, rational, and trigonometric stochastic-fractional solutions, we use two various methods such as the sine-cosine and the Jacobi elliptic function methods. The solutions acquired are important in understanding some interesting physical phenomena due to the significance of the Radhakrishnan-Kundu-Lakshmanan equation in designing the propagation of solitons through an optical fiber. Furthermore, we graph some of the obtained solutions in 3D to display the influence of fractional derivative and multiplicative noise on these solutions. Finally, we show that when the order of fractional derivative decreases, the surface shrinks, while the multiplicative noise stabilizes the solutions of FSRKLE a round zero.

Farah M. Al-Askar

Impact of Fractional Derivative and Brownian Motion on the Solutions of the Radhakrishnan-Kundu-Lakshmanan Equation

In this paper, we consider a class of fractional <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="M2" height="10.2124pt" version="1.1" viewBox="-0.0498162 -6.78297 7.83752 10.2124" width="7.83752pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-113"/></g></svg>-</span>Laplacian problems with critical and negative exponents. By decomposition of the Nehari manifold, the existence and multiplicity of nontrivial solutions for the above problems are established with respect to a sufficiently small parameter.

Qinghao Zhu

Jianming Qi

Critical Fractional <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-3.42943pt" height="10.2124pt" version="1.1" viewBox="-0.0498162 -6.78297 7.83752 10.2124" width="7.83752pt"><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></svg>-Laplacian System with Negative Exponents

The aim in this paper is to establish a new duality property of Morrey spaces and to discover the complex interpolation space between Morrey spaces and Lebesgue spaces. For that purpose, a new space <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-6.12661pt" id="M1" height="16.083pt" version="1.1" viewBox="-0.0498162 -9.95639 45.6769 16.083" width="45.6769pt"><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><g transform="matrix(.0091,0,0,-0.0091,10.91,-5.741)"><path id="g50-116" d="M357 391C357 416 325 451 270 451C241 451 179 431 146 400C109 365 97 334 97 307C97 248 143 213 196 180C250 146 261 123 261 101C261 75 237 40 190 40C153 40 110 72 86 109C81 116 68 119 55 112C37 102 24 87 24 66C24 30 85 -12 137 -12C229 -12 331 62 331 140C331 189 304 218 236 260C197 284 164 309 164 346C164 381 194 400 220 400C259 400 282 380 304 350C310 342 321 339 330 344C347 353 357 372 357 391Z"/></g><g transform="matrix(.0091,0,0,-0.0091,10.062,3.784)"><path id="g50-113" d="M573 302C573 402 527 451 414 451C386 451 343 446 313 437L330 513L320 522C295 508 261 484 243 463L230 415C194 400 159 383 126 359L131 330C159 344 187 357 222 368L109 -147C96 -204 80 -214 18 -223L13 -255L256 -244L259 -212L236 -210C184 -205 180 -195 191 -141L219 -1C240 -10 268 -12 284 -12C352 4 431 48 484 104C543 166 573 240 573 302ZM481 290C481 165 381 37 305 37C280 37 249 56 235 71L302 395C328 399 353 402 372 402C427 402 481 378 481 290Z"/></g><g transform="matrix(.0091,0,0,-0.0091,15.496,3.784)"><path id="g50-45" d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"/></g><g transform="matrix(.0091,0,0,-0.0091,17.652,3.784)"><path id="g50-114" d="M481 430L460 436C440 442 393 451 366 451C348 451 325 448 313 446C267 437 195 409 147 367C79 308 24 211 24 105C24 37 56 -12 93 -12C121 -12 156 1 199 32C246 66 313 122 349 168H351L286 -147C274 -205 261 -215 210 -224L199 -226L193 -255L438 -243L442 -214L417 -212C364 -208 359 -200 370 -140L433 204C455 324 469 389 481 430ZM391 385C382 336 366 260 358 238C321 178 209 59 143 59C129 59 115 83 115 131C115 207 152 317 220 373C243 392 282 403 314 403C348 403 374 395 391 385Z"/></g><g transform="matrix(.013,0,0,-0.013,22.736,0)"><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,-0.013,27.234,0)"><path id="g187-83" d="M660 23C638 31 621 43 599 66C580 87 558 119 523 175C493 221 456 287 441 317C508 344 566 392 566 480C566 542 539 586 499 613C458 640 408 650 320 650H39V615C112 608 121 598 121 518V134C121 51 110 42 35 35V0H337V35C261 42 252 51 252 134V280H271C303 280 321 267 337 237C366 184 393 130 419 89C460 26 498 -2 624 -8C637 -8 648 -9 654 -9L660 23ZM252 320V563C252 584 254 594 261 600C267 606 281 610 306 610C362 610 424 572 424 465C424 414 410 374 376 346C354 327 320 320 278 320H252Z"/></g><g transform="matrix(.0091,0,0,-0.0091,35.833,-5.741)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/></g><g transform="matrix(.013,0,0,-0.013,41.008,0)"><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> is introduced by using methods of tent spaces. The space generalizes Morrey spaces and Lebesgue spaces. Furthermore, this scale of spaces is amenable to the complex interpolation space.

Takahiro Ono

Tent Space Approach of Morrey Spaces and Their Application to Duality and Complex Interpolation

In the present study, the optimality approach is applied to find the exact solution of the Landau-Ginzburg-Higgs Equation (LGHE) using new transformations. This method is a direct algebraic method for obtaining exact solutions of nonlinear differential equations. We find suitable solutions of the LGHE in terms of elliptic Jacobi functions by applying transformations of basic functions. Exact solutions of the equations are obtained with the help of symbolic software (Maple) which allows the computation of equations with parameter constants. It is exposed that PIM is influential, suitable, and shortest and offers an exact solution of LGHE.

Khalil Ahmad

Khudija Bibi

Muhammad Shoaib Arif

Kamaleldin Abodayeh

New Exact Solutions of Landau-Ginzburg-Higgs Equation Using Power Index Method

Let <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="20.155pt" style="vertical-align:-2.26807pt" id="M2" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 20.155 11.5564"><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><g transform="matrix(.013,0,0,-0.013,12.524,0)"><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="14.99pt" style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="23.7371838 -9.28833 14.99 11.5564"><g transform="matrix(.013,0,0,-0.013,23.787,0)"><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,-0.013,28.285,0)"><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,-0.013,35.813,0)"><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="12.769pt" style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="40.9061838 -9.28833 12.769 11.5564"><g transform="matrix(.013,0,0,-0.013,40.956,0)"><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,-0.013,48.964,0)"><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> be a connected finite graph and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.4742pt" id="M3" height="14.1101pt" version="1.1" viewBox="-0.0498162 -8.6359 16.0462 14.1101" width="16.0462pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-133" d="M600 0V24L339 665L311 657L43 24V0H600ZM497 50H107L303 539L497 50Z"/></g><g transform="matrix(.0091,0,0,-0.0091,9.958,3.132)"><path id="g50-113" d="M573 302C573 402 527 451 414 451C386 451 343 446 313 437L330 513L320 522C295 508 261 484 243 463L230 415C194 400 159 383 126 359L131 330C159 344 187 357 222 368L109 -147C96 -204 80 -214 18 -223L13 -255L256 -244L259 -212L236 -210C184 -205 180 -195 191 -141L219 -1C240 -10 268 -12 284 -12C352 4 431 48 484 104C543 166 573 240 573 302ZM481 290C481 165 381 37 305 37C280 37 249 56 235 71L302 395C328 399 353 402 372 402C427 402 481 378 481 290Z"/></g></svg> be the <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="M4" height="10.2124pt" version="1.1" viewBox="-0.0498162 -6.78297 7.83752 10.2124" width="7.83752pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-113"/></g></svg>-</span>Laplacian on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M5" 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)"><use xlink:href="#g113-72"/></g></svg> with <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="18.973pt" style="vertical-align:-3.42938pt" id="M6" height="11.7782pt" version="1.1" viewBox="-0.0498162 -8.34882 18.973 11.7782"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-113"/></g><g transform="matrix(.013,0,0,-0.013,11.342,0)"><path id="g117-92" d="M512 230V281L75 514V456L453 256V254L75 55V-3L512 230Z"/></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="6.419pt" style="vertical-align:-3.42938pt" height="11.7782pt" version="1.1" viewBox="22.555183800000002 -8.34882 6.419 11.7782"><g transform="matrix(.013,0,0,-0.013,22.605,0)"><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>.</span></span> We consider a perturbed <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="M7" height="10.2124pt" version="1.1" viewBox="-0.0498162 -6.78297 7.83752 10.2124" width="7.83752pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-113"/></g></svg>-</span>th Yamabe equation <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="99.22pt" style="vertical-align:-5.474199pt" id="M8" height="17.0656pt" version="1.1" viewBox="-0.0498162 -11.5914 99.22 17.0656"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.013,0,0,-0.013,7.631,0)"><use xlink:href="#g113-133"/></g><g transform="matrix(.0091,0,0,-0.0091,17.589,3.132)"><use xlink:href="#g50-113"/></g><g transform="matrix(.013,0,0,-0.013,23.54,0)"><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,-0.013,33.44,0)"><use xlink:href="#g117-33"/></g><g transform="matrix(.013,0,0,-0.013,43.976,0)"><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,-0.013,51.15,0)"><path id="g113-9" d="M162 -163V703H101V-163H162Z"/></g><g transform="matrix(.013,0,0,-0.013,54.569,0)"><use xlink:href="#g113-118"/></g><g transform="matrix(.013,0,0,-0.013,61.564,0)"><use xlink:href="#g113-9"/></g><g transform="matrix(.0091,0,0,-0.0091,64.982,-5.741)"><use xlink:href="#g50-113"/></g><g transform="matrix(.0091,0,0,-0.0091,70.416,-5.741)"><path id="g54-33" d="M556 236V289H56V236H556Z"/></g><g transform="matrix(.0091,0,0,-0.0091,75.976,-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,-0.013,80.963,0)"><use xlink:href="#g113-118"/></g><g transform="matrix(.013,0,0,-0.013,91.589,0)"><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="53.914pt" style="vertical-align:-5.474199pt" height="17.0656pt" version="1.1" viewBox="102.80218380000001 -11.5914 53.914 17.0656"><g transform="matrix(.013,0,0,-0.013,102.852,0)"><path id="g113-105" d="M499 88L487 113C459 86 422 65 413 65C403 65 403 74 409 98L461 318C487 426 460 448 431 448C408 448 383 441 352 424C305 398 223 344 157 257H155L243 674C249 702 249 712 240 712C227 712 170 680 87 673L82 648H117C155 648 162 644 150 590L23 -4L30 -12C55 -4 81 3 105 8C113 53 131 150 143 187C199 281 323 387 372 387C390 387 393 369 380 311L328 86C313 19 319 -12 347 -12C379 -12 442 24 499 88Z"/></g><g transform="matrix(.013,0,0,-0.013,109.56,0)"><use xlink:href="#g113-9"/></g><g transform="matrix(.013,0,0,-0.013,112.979,0)"><use xlink:href="#g113-118"/></g><g transform="matrix(.013,0,0,-0.013,119.973,0)"><use xlink:href="#g113-9"/></g><g transform="matrix(.0091,0,0,-0.0091,123.392,-5.741)"><path id="g50-223" d="M558 106L533 129C501 84 476 67 464 67C446 67 435 116 414 239C457 297 508 367 550 440L539 451L484 438C458 390 429 338 404 300H402C375 407 351 451 285 451C168 451 24 312 24 153C24 56 66 -12 132 -12C206 -12 284 64 345 156H347C367 24 386 -12 422 -12C457 -12 503 7 558 106ZM338 206C272 99 215 56 173 56C140 56 116 96 116 170C116 303 194 405 255 405C304 405 324 299 338 206Z"/></g><g transform="matrix(.0091,0,0,-0.0091,128.689,-5.741)"><use xlink:href="#g54-33"/></g><g transform="matrix(.0091,0,0,-0.0091,134.249,-5.741)"><use xlink:href="#g50-51"/></g><g transform="matrix(.013,0,0,-0.013,139.236,0)"><use xlink:href="#g113-118"/></g><g transform="matrix(.013,0,0,-0.013,149.135,0)"><path id="g117-36" d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"/></g></svg><span class="ibiop"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="12.778pt" style="vertical-align:-5.474199pt" height="17.0656pt" version="1.1" viewBox="159.6221838 -11.5914 12.778 17.0656"><g transform="matrix(.013,0,0,-0.013,159.672,0)"><path id="g113-247" d="M387 375C387 402 357 448 257 448C172 448 82 404 82 326C82 289 108 255 156 241V239C85 223 23 181 23 116C23 39 89 -12 182 -12C265 -12 336 31 378 91L361 114C320 73 269 47 216 47C157 47 115 82 115 137C115 191 160 219 218 219C243 219 262 218 272 217L304 259L302 266C295 265 281 264 255 264C195 264 163 294 163 335C163 377 200 416 249 416C293 416 321 389 329 342C331 332 335 329 341 329C355 329 387 352 387 375Z"/></g><g transform="matrix(.013,0,0,-0.013,163.568,0)"><path id="g113-103" d="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z"/></g></svg>,</span></span> where <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="9.672pt" style="vertical-align:-3.42947pt" id="M9" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 9.672 12.7178"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-105"/></g><g transform="matrix(.013,0,0,-0.013,6.708,0)"><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="13.643pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="11.8011838 -9.28833 13.643 12.7178"><g transform="matrix(.013,0,0,-0.013,11.851,0)"><use xlink:href="#g113-103"/></g><g transform="matrix(.013,0,0,-0.013,22.53,0)"><path id="g113-59" d="M116 412C83 412 59 388 59 355C59 321 83 297 115 297S170 321 170 355C170 388 147 412 116 412ZM114 -12C147 -12 171 11 171 46C171 78 147 103 115 103C84 103 60 78 60 46C60 11 84 -12 114 -12Z"/></g></svg><span class="ipunc"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="31.034pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="29.076183800000003 -9.28833 31.034 12.7178"><g transform="matrix(.013,0,0,-0.013,29.126,0)"><use xlink:href="#g113-87"/></g><g transform="matrix(.013,0,0,-0.013,41.982,0)"><path id="g117-149" d="M567 230V280H69V230H567Z"/></g><g transform="matrix(.013,0,0,-0.013,47.758,0)"><path id="g117-148" d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H796L694 83L716 68C790 149 861 220 901 255Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="9.947pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="63.8191838 -9.28833 9.947 12.7178"><g transform="matrix(.013,0,0,-0.013,63.869,0)"><path id="g197-29" d="M712 0L497 285C582 312 628 378 628 466C628 609 518 662 390 662H70V0H236V271H310L505 0H712ZM491 604C551 577 584 532 584 464C584 401 550 345 491 321V604ZM447 320C428 317 411 315 392 315H236V618H397C412 618 432 613 447 610V320ZM626 44H528L363 271C392 271 422 270 451 275L626 44ZM192 44H114V618H192V44Z"/></g></svg></span> are functions with <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="9.672pt" style="vertical-align:-3.42947pt" id="M10" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 9.672 12.7178"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-105"/></g><g transform="matrix(.013,0,0,-0.013,6.708,0)"><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="19.615pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="11.8011838 -9.28833 19.615 12.7178"><g transform="matrix(.013,0,0,-0.013,11.851,0)"><use xlink:href="#g113-103"/></g><g transform="matrix(.013,0,0,-0.013,23.835,0)"><use xlink:href="#g117-92"/></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="6.453pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="35.0481838 -9.28833 6.453 12.7178"><g transform="matrix(.013,0,0,-0.013,35.098,0)"><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></svg>;</span></span> <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="17.503pt" style="vertical-align:-3.42938pt" id="M11" height="11.7782pt" version="1.1" viewBox="-0.0498162 -8.34882 17.503 11.7782"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-50"/></g><g transform="matrix(.013,0,0,-0.013,9.872,0)"><path id="g117-91" d="M512 -3V55L134 254V256L512 456V514L75 281V230L512 -3Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="18.973pt" style="vertical-align:-3.42938pt" height="11.7782pt" version="1.1" viewBox="21.085183800000003 -8.34882 18.973 11.7782"><g transform="matrix(.013,0,0,-0.013,21.135,0)"><use xlink:href="#g113-113"/></g><g transform="matrix(.013,0,0,-0.013,32.477,0)"><use xlink:href="#g117-91"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="10.594pt" style="vertical-align:-3.42938pt" height="11.7782pt" version="1.1" viewBox="43.6901838 -8.34882 10.594 11.7782"><g transform="matrix(.013,0,0,-0.013,43.74,0)"><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,-0.013,51.125,0)"><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></span> <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M12" 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> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M13" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 5.44961 6.1673" width="5.44961pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-247"/></g></svg> are two positive constants. Using the variational method, we prove that there exists some positive constant <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.1815pt" id="M14" height="9.14241pt" version="1.1" viewBox="-0.0498162 -5.96091 10.2525 9.14241" width="10.2525pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-227" d="M401 397C401 420 368 448 302 448C245 448 169 416 122 377C62 327 23 254 23 169C23 45 83 -12 181 -12C252 -12 323 29 374 85L358 107C305 62 257 43 210 43C147 43 110 98 110 189V214L313 208L321 256L115 250C132 342 190 405 253 405C291 405 323 389 346 360C356 348 364 348 377 357C392 367 401 384 401 397Z"/></g><g transform="matrix(.0091,0,0,-0.0091,5.187,3.132)"><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></svg> such that for all <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="16.009pt" style="vertical-align:-3.18147pt" id="M15" height="12.4698pt" version="1.1" viewBox="-0.0498162 -9.28833 16.009 12.4698"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-227"/></g><g transform="matrix(.013,0,0,-0.013,9.145,0)"><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></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="13.702pt" style="vertical-align:-3.18147pt" height="12.4698pt" version="1.1" viewBox="19.5911838 -9.28833 13.702 12.4698"><g transform="matrix(.013,0,0,-0.013,19.641,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,24.139,0)"><use xlink:href="#g113-49"/></g><g transform="matrix(.013,0,0,-0.013,30.379,0)"><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="14.864pt" style="vertical-align:-3.18147pt" height="12.4698pt" version="1.1" viewBox="35.472183799999996 -9.28833 14.864 12.4698"><g transform="matrix(.013,0,0,-0.013,35.522,0)"><use xlink:href="#g113-227"/></g><g transform="matrix(.0091,0,0,-0.0091,40.709,3.132)"><use xlink:href="#g50-50"/></g><g transform="matrix(.013,0,0,-0.013,45.656,0)"><use xlink:href="#g113-42"/></g></svg>,</span></span> the above equation has two distinct solutions.

Zhongqi Peng

Aimin Zhu

Tingting Zhang

Multiple Solutions of a <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-3.42943pt" height="10.2124pt" version="1.1" viewBox="-0.0498162 -6.78297 7.83752 10.2124" width="7.83752pt"><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></svg>-th Yamabe Equation on Graph

About this journal

<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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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 Function Spaces - Table of contents 2023

title

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

Journal of Function Spaces