The goal of this paper is to study the boundedness, the persistence, and the asymptotic behavior of the positive solutions of the system of two difference equations of exponential form: <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.1023pt" id="M1" height="15.3522pt" version="1.1" viewBox="-0.0498162 -11.2499 415.641 15.3522" width="415.641pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-121" d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 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transform="matrix(.0135,0,0,-0.0135,412.283,0)"><use xlink:href="#g113-45"/></g></svg> where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.6409pt" id="M2" height="11.3188pt" version="1.1" viewBox="-0.0498162 -9.6779 49.355 11.3188" width="49.355pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-98"/></g><g transform="matrix(.0135,0,0,-0.0135,6.858,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.0135,0,0,-0.0135,14.489,0)"><use xlink:href="#g113-99"/></g><g transform="matrix(.0135,0,0,-0.0135,20.847,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.0135,0,0,-0.0135,28.478,0)"><use xlink:href="#g113-100"/></g><g transform="matrix(.0135,0,0,-0.0135,33.983,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.0135,0,0,-0.0135,41.614,0)"><use xlink:href="#g113-101"/></g></svg>, and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129898pt" id="M3" height="9.89089pt" version="1.1" viewBox="-0.0498162 -9.6779 7.11668 9.89089" width="7.11668pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-105"/></g></svg> are positive constants and the initial values <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.57194pt" id="M4" height="9.7821pt" version="1.1" viewBox="-0.0498162 -6.21016 32.8059 9.7821" width="32.8059pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.0095,0,0,-0.0095,7.506,3.264)"><path id="g50-49" d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z"/></g><g transform="matrix(.0135,0,0,-0.0135,12.64,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.0135,0,0,-0.0135,20.271,0)"><use xlink:href="#g113-122"/></g><g transform="matrix(.0095,0,0,-0.0095,27.505,3.264)"><use xlink:href="#g50-49"/></g></svg> are positive real values. Also, we determine the rate of convergence of a solution that converges to the equilibrium <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.571899pt" id="M5" height="13.2498pt" version="1.1" viewBox="-0.0498162 -9.6779 54.2869 13.2498" width="54.2869pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><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(.0135,0,0,-0.0135,12.131,0)"><use xlink:href="#g117-34"/></g><g transform="matrix(.0135,0,0,-0.0135,23.869,0)"><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><rect height="0.679945" width="7.60182" x="28.5564" y="-8.88012"/><g transform="matrix(.0135,0,0,-0.0135,28.556,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.0135,0,0,-0.0135,36.158,0)"><use xlink:href="#g113-45"/></g><rect height="0.679945" width="7.87381" x="41.5182" y="-8.88012"/><g transform="matrix(.0135,0,0,-0.0135,41.518,0)"><use xlink:href="#g113-122"/></g><g transform="matrix(.0135,0,0,-0.0135,49.392,0)"><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 this system.

Vu Van Khuong

Tran Hong Thai

Asymptotic Behavior of the Solutions of System of Difference Equations of Exponential Form

A numerical method for finding the solution of Duffing-harmonic oscillator is proposed. The approach is based on hybrid functions approximation. The properties of hybrid functions that consist of block-pulse and Chebyshev cardinal functions are discussed. The associated operational matrices of integration and product are then utilized to reduce the solution of a strongly nonlinear oscillator to the solution of a system of algebraic equations. The method is easy to implement and computationally very attractive. The results are compared with the exact solution and results from several recently published methods, and the comparisons showed proper accuracy of this method.

Mohammad Heydari

Ghasem Brid Loghmani

Seyed Mohammad. Hosseini

Seyed Mehdi Karbassi

Application of Hybrid Functions for Solving Duffing-Harmonic Oscillator

We consider an almost periodic discrete Schoener&#x2019;s competition model with delays. By means of an almost periodic functional hull theory and constructing a suitable Lyapunov function, sufficient conditions are obtained for the existence of a unique strictly positive almost periodic solution which is globally attractive. An example together with numerical simulation indicates the feasibility of the main result.

Hui Zhang

Yingqi Li

Bin Jing

Xiaofeng Fang

Jing Wang

Almost Periodic Solution of a Discrete Schoener&#x2019;s Competition Model with Delays

A functional differential equation with deviated argument coupled with impulsive conditions is studied for the existence and uniqueness of the mild solution and exact controllability of the system. The results are obtained by using Banach contraction principle and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.25793pt;width:18.4375px;" id="M1" height="15.6875" version="1.1" viewBox="0 0 18.4375 15.6875" width="18.4375">
	
		
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			<g transform="matrix(.012,-0,0,-.012,12.05,15.475)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z"></path></g>

		
	
</svg> semigroup theory without imposing additional assumptions such as analyticity and compactness conditions on the generated semigroup and the nonlinear term. An example is provided to illustrate the presented theory.

Sanjukta Das

Dwijendra N. Pandey

N. Sukavanam

Exact Controllability of an Impulsive Semilinear System with Deviated Argument in a Banach Space

We proposed a method by utilizing method of steps and Hermite wavelet method, for solving
the fractional delay differential equations. This technique first converts the fractional delay differential equation
to a fractional nondelay differential equation and then applies the Hermite wavelet method on the obtained fractional
nondelay differential equation to find the solution. Several numerical examples are solved to show the
applicability of the proposed method.

Umer Saeed

Mujeeb ur Rehman

Hermite Wavelet Method for Fractional Delay Differential Equations

We study a certain integral modification of well-known Baskakov operators with weight function of beta basis function. We establish rate of convergence for these operators for functions having derivative of bounded variation. Also, we discuss Stancu type generalization of these operators.

Prashantkumar Patel

Vishnu Narayan Mishra

Rate of Convergence of Modified Baskakov-Durrmeyer Type Operators for Functions of Bounded Variation

Consider the difference equation <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.107699pt" id="M1" height="17.5607pt" version="1.1" viewBox="-0.0498162 -13.453 219.901 17.5607" width="219.901pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-121" d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"/></g><g transform="matrix(.0095,0,0,-0.0095,7.506,3.264)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 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-141V-169C115 -169 200 -152 200 -41C200 32 192 104 192 164C192 202 195 249 283 255Z"/></g></svg> are nonnegative real numbers. We investigate the asymptotic behavior of the solutions of the considered equation. We give easy-to-check conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation.

A. Brett

E. J. Janowski

M. R. S. Kulenovi&#x107;

Global Asymptotic Stability for Linear Fractional Difference Equation

We consider the regularity criterion for the 3D MHD equations. It is proved that if the horizontal components of the velocity and magnetic fields satisfy <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.37273pt;width:176.5125px;" id="M1" height="22.75" version="1.1" viewBox="0 0 176.5125 22.75" width="176.5125">
	
		
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</svg> with <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.30096pt;width:62.462502px;" id="M2" height="11.125" version="1.1" viewBox="0 0 62.462502 11.125" width="62.462502">
	
		
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</svg>, then the solution smooth. This improves the result given by Gala (2012).

Zujin Zhang

Tong Tang

Fumin Zhang

A Remark on the Regularity Criterion for the MHD Equations via Two Components in Morrey-Campanato Spaces

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Journal of Difference Equations