We study global behavior of the following max-type difference equation <svg style="vertical-align:-3.3907pt;width:188.03751px;" id="M2" height="16.4" version="1.1" viewBox="0 0 188.03751 16.4" width="188.03751" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.112)"><use xlink:href="#x1D465" /></g><g transform="matrix(.012,-0,0,-.012,9.563,16.2)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.012,-0,0,-.012,15.536,16.2)"><use xlink:href="#x2B" /></g><g transform="matrix(.012,-0,0,-.012,22.52,16.2)"><use xlink:href="#x31" /></g><g transform="matrix(.017,-0,0,-.017,33.575,12.112)"><use xlink:href="#x3D" /></g><g transform="matrix(.017,-0,0,-.017,48.262,12.112)"><use xlink:href="#x6D" /></g><g transform="matrix(.017,-0,0,-.017,62.184,12.112)"><use xlink:href="#x61" /></g><g transform="matrix(.017,-0,0,-.017,69.646,12.112)"><use xlink:href="#x78" /></g><g transform="matrix(.017,-0,0,-.017,83.347,12.112)"><use xlink:href="#x7B" /></g><g transform="matrix(.017,-0,0,-.017,89.246,12.112)"><use xlink:href="#x31" /></g><g transform="matrix(.017,-0,0,-.017,97.405,12.112)"><use xlink:href="#x2F" /></g><g transform="matrix(.017,-0,0,-.017,104.409,12.112)"><use xlink:href="#x1D465" /></g><g transform="matrix(.012,-0,0,-.012,113.912,16.2)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.017,-0,0,-.017,120.512,12.112)"><use xlink:href="#x2C" /></g><g transform="matrix(.017,-0,0,-.017,127.21,12.112)"><use xlink:href="#x1D434" /></g><g transform="matrix(.012,-0,0,-.012,139.688,16.2)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.017,-0,0,-.017,146.275,12.112)"><use xlink:href="#x2F" /></g><g transform="matrix(.017,-0,0,-.017,153.278,12.112)"><use xlink:href="#x1D465" /></g><g transform="matrix(.012,-0,0,-.012,162.788,16.2)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.012,-0,0,-.012,168.761,16.2)"><use xlink:href="#x2212" /></g><g transform="matrix(.012,-0,0,-.012,175.745,16.2)"><use xlink:href="#x31" /></g><g transform="matrix(.017,-0,0,-.017,182.075,12.112)"><use xlink:href="#x7D" /></g></svg>, <svg style="vertical-align:-1.59705pt;width:81.012497px;" id="M3" height="12.9125" version="1.1" viewBox="0 0 81.012497 12.9125" width="81.012497" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.017,-0,0,-.017,13.305,10.862)"><use xlink:href="#x3D" /></g><g transform="matrix(.017,-0,0,-.017,28.008,10.862)"><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" /></g><g transform="matrix(.017,-0,0,-.017,36.168,10.862)"><use xlink:href="#x2C" /></g><g transform="matrix(.017,-0,0,-.017,42.882,10.862)"><use xlink:href="#x31" /></g><g transform="matrix(.017,-0,0,-.017,51.042,10.862)"><use xlink:href="#x2C" /></g><g transform="matrix(.017,-0,0,-.017,57.739,10.862)"><path id="x2026" d="M160 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16zM485 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16zM806 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5&#xA;t16 -40.5q0 -26 -15.5 -42t-41.5 -16z" /></g><g transform="matrix(.017,-0,0,-.017,77.067,10.862)"><use xlink:href="#x2C" /></g></svg> where <svg style="vertical-align:-3.98311pt;width:50.275002px;" id="M4" height="18.575001" version="1.1" viewBox="0 0 50.275002 18.575001" width="50.275002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,13.55)"><use xlink:href="#x7B" /></g><g transform="matrix(.017,-0,0,-.017,5.961,13.55)"><use xlink:href="#x1D434" /></g><g transform="matrix(.012,-0,0,-.012,18.425,17.638)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.017,-0,0,-.017,25.012,13.55)"><use xlink:href="#x7D" /></g><g transform="matrix(.012,-0,0,-.012,30.912,5.388)"><path id="x221E" d="M983 225q0 -112 -67 -174.5t-150 -62.5q-91 0 -154.5 43.5t-113.5 129.5q-49 -85 -104 -129t-138 -44q-98 0 -158.5 66t-60.5 154q0 59 21 106.5t54.5 75.5t70.5 43t73 15q90 0 152.5 -43.5t112.5 -128.5q48 84 104.5 128t140.5 44q93 0 155 -65t62 -158zM478 196&#xA;q-27 49 -47 80t-50 67t-64 54t-73 18q-48 0 -81.5 -47t-33.5 -128q0 -96 37.5 -157.5t99.5 -61.5q68 0 117.5 47t94.5 128zM889 204q0 91 -35.5 151t-99.5 60q-68 0 -119 -47t-95 -127q27 -49 47 -80.5t50 -67.5t65 -54t74 -18q113 0 113 183z" /></g><g transform="matrix(.012,-0,0,-.012,30.912,18.375)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.012,-0,0,-.012,36.886,18.375)"><use xlink:href="#x3D" /></g><g transform="matrix(.012,-0,0,-.012,43.87,18.375)"><use xlink:href="#x30" /></g></svg> is a sequence of positive real numbers with <svg style="vertical-align:-3.3907pt;width:172.875px;" id="M5" height="16.450001" version="1.1" viewBox="0 0 172.875 16.450001" width="172.875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x30" /></g><g transform="matrix(.017,-0,0,-.017,12.931,12.162)"><path id="x2264" d="M531 71l-474 214v50l474 215v-56l-416 -184l416 -183v-56zM531 -40h-474v50h474v-50z" /></g><g transform="matrix(.017,-0,0,-.017,30.473,12.162)"><path id="x69" d="M135 536q-20 0 -35 15.5t-15 35.5q0 22 15 37t36 15t35.5 -15t14.5 -37q0 -21 -15 -36t-36 -15zM252 0h-220v26q48 5 59 17t11 63v206q0 47 -9 58t-54 18v24q78 12 142 39v-345q0 -51 11.5 -63t59.5 -17v-26z" /></g><g transform="matrix(.017,-0,0,-.017,35.029,12.162)"><path id="x6E" d="M524 0h-209v26q44 5 55 18t11 65v167q0 114 -91 114q-50 0 -109 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10.5t-46.5 23.5q-5 11 -11 51t-6 67l27 5q14 -53 46.5 -88.5t75.5 -35.5q29 0 50 19.5t21 50.5t-19.5 51.5t-59.5 39.5&#xA;q-28 12 -46 22.5t-38.5 27t-30.5 38.5t-10 49q0 54 42.5 92t109.5 38q48 0 88 -18q6 -15 13 -50.5t9 -55.5z" /></g><g transform="matrix(.017,-0,0,-.017,102.577,12.162)"><path id="x75" d="M518 50v-26q-32 -4 -77 -16l-37 -11t-33 -9l-6 6v71q-56 -42 -74 -52q-43 -25 -77 -25q-55 0 -89.5 36.5t-34.5 112.5v196q0 38 -6.5 48.5t-26.5 15.5l-28 6v23q19 1 72 7q59 9 71 11q-3 -31 -3 -145v-135q0 -113 88 -113q59 0 108 48v234q0 38 -8 49t-34 15l-36 6v23&#xA;q44 2 90 8q37 4 67 10v-334q0 -37 8.5 -47t38.5 -12z" /></g><g transform="matrix(.017,-0,0,-.017,111.603,12.162)"><path id="x70" d="M169 380l92 53q26 16 49 16q80 0 128.5 -56t48.5 -133q0 -108 -72.5 -180.5t-176.5 -91.5q-38 1 -69 18v-131q0 -57 13 -70t71 -18v-27h-236v27q49 5 61 17t12 61v452q0 48 -9.5 58.5t-56.5 16.5v24q66 11 145 43v-79zM169 346v-270q37 -39 94 -39q63 0 101.5 50&#xA;t38.5 134q0 78 -38.5 123t-95.5 45q-45 0 -100 -43z" /></g><g transform="matrix(.017,-0,0,-.017,126.171,12.162)"><use xlink:href="#x1D434" /></g><g transform="matrix(.012,-0,0,-.012,138.638,16.25)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.017,-0,0,-.017,149.962,12.162)"><path id="x3C" d="M512 -3l-437 233v51l437 233v-58l-378 -200v-2l378 -199v-58z" /></g><g transform="matrix(.017,-0,0,-.017,164.649,12.162)"><use xlink:href="#x31" /></g></svg>. The special case when <svg style="vertical-align:-3.98311pt;width:50.275002px;" id="M6" height="18.575001" version="1.1" viewBox="0 0 50.275002 18.575001" width="50.275002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,13.55)"><use xlink:href="#x7B" /></g><g transform="matrix(.017,-0,0,-.017,5.961,13.55)"><use xlink:href="#x1D434" /></g><g transform="matrix(.012,-0,0,-.012,18.425,17.638)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.017,-0,0,-.017,25.012,13.55)"><use xlink:href="#x7D" /></g><g transform="matrix(.012,-0,0,-.012,30.912,5.388)"><use xlink:href="#x221E" /></g><g transform="matrix(.012,-0,0,-.012,30.912,18.375)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.012,-0,0,-.012,36.886,18.375)"><use xlink:href="#x3D" /></g><g transform="matrix(.012,-0,0,-.012,43.87,18.375)"><use xlink:href="#x30" /></g></svg> is a periodic sequence with period <svg style="vertical-align:-0.1638pt;width:8.6750002px;" id="M7" height="12.4375" version="1.1" viewBox="0 0 8.6750002 12.4375" width="8.6750002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x1D458" d="M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1&#xA;l5 26q38 4 74 13.5t57 17t25 7.5q12 0 4 -32l-104 -443h2q35 38 97 93q39 35 65.5 56t62 41.5t58.5 20.5q19 0 30.5 -10t11.5 -22z" /></g></svg> and <svg style="vertical-align:-3.3907pt;width:72.387497px;" id="M8" height="16.4125" version="1.1" viewBox="0 0 72.387497 16.4125" width="72.387497" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D434" /></g><g transform="matrix(.012,-0,0,-.012,12.525,16.212)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.017,-0,0,-.017,23.838,12.138)"><path id="x2208" d="M448 1h-83q-118 0 -201.5 74.5t-83.5 179.5t83.5 179.5t201.5 74.5h83v-50h-84q-87 0 -150.5 -51.5t-73.5 -127.5h308v-50h-308q10 -76 73.5 -127.5t150.5 -51.5h84v-50z" /></g><g transform="matrix(.017,-0,0,-.017,37.521,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,43.403,12.138)"><use xlink:href="#x30" /></g><g transform="matrix(.017,-0,0,-.017,51.562,12.138)"><use xlink:href="#x2C" /></g><g transform="matrix(.017,-0,0,-.017,58.277,12.138)"><use xlink:href="#x31" /></g><g transform="matrix(.017,-0,0,-.017,66.436,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g></svg> for every <svg style="vertical-align:-0.546pt;width:36.237499px;" id="M9" height="11.6" version="1.1" viewBox="0 0 36.237499 11.6" width="36.237499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.017,-0,0,-.017,13.305,10.862)"><path id="x2265" d="M531 285l-474 -214v56l416 183l-416 184v56l474 -215v-50zM531 -40h-474v50h474v-50z" /></g><g transform="matrix(.017,-0,0,-.017,28.008,10.862)"><use xlink:href="#x30" /></g></svg> has been completely investigated by Y. Chen. Here we extend his results to the general case.

Taixiang Sun

Bin Qin

Hongjian Xi

Caihong Han

Global Behavior of the Max-Type Difference Equation <svg style="vertical-align:-3.3907pt" height="16.4" version="1.1" viewBox="0 0 185.2 16.4" width="185.2" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
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This paper investigates the relations between the particular eigensolutions of a limiting functional differential equation of any order, which is the nominal (unperturbed) linear autonomous differential equations, and the associate ones of the corresponding perturbed functional differential equation. Both differential equations involve point and distributed delayed dynamics including Volterra class dynamics. The proofs are based on a Perron-type theorem for functional equations so that the comparison is governed by the real part of a dominant zero of the characteristic equation of the nominal differential equation. The obtained results are also applied to investigate the global stability of the perturbed equation based on that of its corresponding limiting equation.

M. De la Sen

Asymptotic Comparison of  the  Solutions of Linear Time-Delay Systems with Point and  Distributed Lags with Those of Their Limiting Equations

In the theorems of Galambos-Bojani&#x107;-Seneta&#x27;s type, the asymptotic behavior of the functions <svg style="vertical-align:-4.5044pt;width:68.0625px;" id="M1" height="16.299999" version="1.1" viewBox="0 0 68.0625 16.299999" width="68.0625"  xmlns="http://www.w3.org/2000/svg">
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				<tspan style="font-size: 8.75px; " x="0" y="0">[</tspan>
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			<text transform="matrix(1,0,0,-1,-55.62,63.5)">
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</svg>, for <svg style="vertical-align:-0.51414pt;width:62.262501px;" id="M2" height="9.3125" version="1.1" viewBox="0 0 62.262501 9.3125" width="62.262501"  xmlns="http://www.w3.org/2000/svg">
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			<text transform="matrix(1,0,0,-1,-71.95,65.11)">
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				<tspan style="font-size: 12.50px; " x="29.557093" y="0">+</tspan>
				<tspan style="font-size: 12.50px; " x="38.121647" y="0">∞</tspan>
			</text>

		</g>
	</g>
</svg>, is investigated by the asymptotic behavior of the given sequence of positive
numbers <svg style="vertical-align:-3.20526pt;width:23.075001px;" id="M3" height="14.6875" version="1.1" viewBox="0 0 23.075001 14.6875" width="23.075001"  xmlns="http://www.w3.org/2000/svg">
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			<text transform="matrix(1,0,0,-1,-62.6,60.37)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan>
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			<text transform="matrix(1,0,0,-1,-57.75,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
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</svg>, as <svg style="vertical-align:-0.51414pt;width:61.4375px;" id="M4" height="9.3125" version="1.1" viewBox="0 0 61.4375 9.3125" width="61.4375"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,9.3125)">
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			<text transform="matrix(1,0,0,-1,-71.95,65.11)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑛</tspan>
				<tspan style="font-size: 12.50px; " x="11.765323" y="0">→</tspan>
				<tspan style="font-size: 12.50px; " x="28.894432" y="0">+</tspan>
				<tspan style="font-size: 12.50px; " x="37.458988" y="0">∞</tspan>
			</text>

		</g>
	</g>
</svg> and vice versa. The main result of this paper is one theorem of such a type for sequences
of positive numbers <svg style="vertical-align:-3.20526pt;width:23.075001px;" id="M5" height="14.6875" version="1.1" viewBox="0 0 23.075001 14.6875" width="23.075001"  xmlns="http://www.w3.org/2000/svg">
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				<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
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	</g>
</svg> which satisfy an asymptotic condition of the Karamata type <svg style="vertical-align:-4.64723pt;width:53.549999px;" id="M6" height="16.5875" version="1.1" viewBox="0 0 53.549999 16.5875" width="53.549999"  xmlns="http://www.w3.org/2000/svg">
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				<tspan style="font-size: 12.50px; " x="6.9516678" y="0">m</tspan>
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	</g>
</svg>&#x2009;<svg style="vertical-align:-4.5044pt;width:69.974998px;" id="M7" height="16.299999" version="1.1" viewBox="0 0 69.974998 16.299999" width="69.974998"  xmlns="http://www.w3.org/2000/svg">
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			</text>

			<text transform="matrix(1,0,0,-1,-66.76,60.37)">
				<tspan style="font-size: 8.75px; " x="0" y="0">[</tspan>
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				<tspan style="font-size: 8.75px; " x="7.6492481" y="0">𝑛</tspan>
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			<text transform="matrix(1,0,0,-1,-51.35,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">/</tspan>
				<tspan style="font-size: 12.50px; " x="3.4758339" y="0">𝑐</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-42.68,60.37)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan>
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				<tspan style="font-size: 12.50px; " x="0" y="0">&gt;</tspan>
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	</g>
</svg>, for <svg style="vertical-align:-0.30096pt;width:35.787498px;" id="M8" height="11.0625" version="1.1" viewBox="0 0 35.787498 11.0625" width="35.787498"  xmlns="http://www.w3.org/2000/svg">
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	</g>
</svg>.

Dragan Djur&#x10D;i&#x107;

Aleksandar Torga&#x161;ev

A Theorem of Galambos-Bojani&#x107;-Seneta Type

This paper is devoted to studying growth of solutions of linear differential
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-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" /></g><g transform="matrix(.017,-0,0,-.017,278.6,17.65)"><use xlink:href="#x28" /></g><g transform="matrix(.017,-0,0,-.017,284.482,17.65)"><use xlink:href="#x1D467" /></g><g transform="matrix(.017,-0,0,-.017,292.947,17.65)"><use xlink:href="#x29" /></g><g transform="matrix(.017,-0,0,-.017,298.829,17.65)"><use xlink:href="#x1D453" /></g><g transform="matrix(.017,-0,0,-.017,314.467,17.65)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,329.154,17.65)"><path id="x1D43B" d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28&#xA;h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z" /></g><g transform="matrix(.017,-0,0,-.017,344.062,17.65)"><use xlink:href="#x28" /></g><g transform="matrix(.017,-0,0,-.017,349.944,17.65)"><use xlink:href="#x1D467" /></g><g transform="matrix(.017,-0,0,-.017,358.409,17.65)"><use xlink:href="#x29" /></g></svg> where <svg style="vertical-align:-5.76988pt;width:134.58749px;" id="M2" height="19.4125" version="1.1" viewBox="0 0 134.58749 19.4125" width="134.58749" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D434" /></g><g transform="matrix(.012,-0,0,-.012,12.525,16.25)"><path id="x1D457" d="M394 607q0 -25 -15.5 -43t-39.5 -18q-19 0 -32 13t-13 35t16 41.5t40 19.5q20 0 32 -13.5t12 -34.5zM360 349l-57 -283q-38 -188 -122 -263q-72 -64 -114 -64q-16 0 -30 10.5t-14 22.5q0 24 21 45q15 13 26 6q30 -20 54 -20q41 0 65 95q37 147 86 415q6 34 6.5 42.5&#xA;t-5.5 11.5q-18 0 -79 -54l-14 25q48 48 96.5 79t71.5 31q29 0 9 -99z" /></g><g transform="matrix(.017,-0,0,-.017,21.013,12.162)"><use xlink:href="#x28" /></g><g transform="matrix(.017,-0,0,-.017,26.894,12.162)"><use xlink:href="#x1D457" /></g><g transform="matrix(.017,-0,0,-.017,38.793,12.162)"><use xlink:href="#x3D" /></g><g transform="matrix(.017,-0,0,-.017,53.497,12.162)"><use xlink:href="#x30" /></g><g transform="matrix(.017,-0,0,-.017,61.657,12.162)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,68.371,12.162)"><path id="x2026" d="M160 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16zM485 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16zM806 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5&#xA;t16 -40.5q0 -26 -15.5 -42t-41.5 -16z" /></g><g transform="matrix(.017,-0,0,-.017,87.682,12.162)"><use xlink:href="#x2C" /></g><g transform="matrix(.017,-0,0,-.017,94.396,12.162)"><use xlink:href="#x1D458" /></g><g transform="matrix(.017,-0,0,-.017,106.72,12.162)"><use xlink:href="#x2212" /></g><g transform="matrix(.017,-0,0,-.017,120.472,12.162)"><use xlink:href="#x31" /></g><g transform="matrix(.017,-0,0,-.017,128.632,12.162)"><use xlink:href="#x29" /></g></svg> and <svg style="vertical-align:-0.0pt;width:15.0375px;" id="M3" height="11.175" version="1.1" viewBox="0 0 15.0375 11.175" width="15.0375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D43B" /></g></svg> are entire functions of finite order.

Jun Wang

Ilpo Laine

Growth of Solutions of Nonhomogeneous Linear Differential Equations

Here we introduce the <svg style="vertical-align:-0.10033pt;width:7.8874998px;" id="M1" height="7.1374998" version="1.1" viewBox="0 0 7.8874998 7.1374998" width="7.8874998"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,7.1375)">
		<g transform="translate(72,-66.29)">
			<text transform="matrix(1,0,0,-1,-71.95,66.44)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑛</tspan>
			</text>

		</g>
	</g>
</svg>th
 weighted space on the upper half-plane <svg style="vertical-align:-3.49493pt;width:163.7625px;" id="M2" height="15.1125" version="1.1" viewBox="0 0 163.7625 15.1125" width="163.7625"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,15.1125)">
		<g transform="translate(72,-59.91)">
			<text transform="matrix(1,0,0,-1,-71.95,63.45)">
				<tspan style="font-size: 12.50px; " x="0" y="0">Π</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-62.92,60.32)">
				<tspan style="font-size: 8.75px; " x="0" y="0">+</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-52.96,63.45)">
				<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
				<tspan style="font-size: 12.50px; " x="12.040389" y="0">{</tspan>
				<tspan style="font-size: 12.50px; " x="18.041828" y="0">𝑧</tspan>
				<tspan style="font-size: 12.50px; " x="27.744158" y="0">∈</tspan>
				<tspan style="font-size: 12.50px; " x="39.784546" y="0">ℂ</tspan>
				<tspan style="font-size: 12.50px; " x="52.074993" y="0">∶</tspan>
				<tspan style="font-size: 12.50px; " x="61.939861" y="0">I</tspan>
				<tspan style="font-size: 12.50px; " x="66.103363" y="0">m</tspan>
				<tspan style="font-size: 12.50px; " x="77.906197" y="0">𝑧</tspan>
				<tspan style="font-size: 12.50px; " x="87.621025" y="0">&gt;</tspan>
				<tspan style="font-size: 12.50px; " x="99.661415" y="0">0</tspan>
				<tspan style="font-size: 12.50px; " x="105.91291" y="0">}</tspan>
			</text>

		</g>
	</g>
</svg> in the complex plane <svg style="vertical-align:-0.17555pt;width:11.1375px;" id="M3" height="10.9125" version="1.1" viewBox="0 0 11.1375 10.9125" width="11.1375"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
		<g transform="translate(72,-63.27)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">ℂ</tspan>
			</text>

		</g>
	</g>
</svg>. For the case <svg style="vertical-align:-0.10033pt;width:35.099998px;" id="M4" height="10.8125" version="1.1" viewBox="0 0 35.099998 10.8125" width="35.099998"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,10.8125)">
		<g transform="translate(72,-63.35)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑛</tspan>
				<tspan style="font-size: 12.50px; " x="9.6773224" y="0">=</tspan>
				<tspan style="font-size: 12.50px; " x="21.71771" y="0">2</tspan>
			</text>

		</g>
	</g>
</svg>, we
call it the Zygmund-type space,  and denote it by <svg style="vertical-align:-3.49493pt;width:44.5px;" id="M5" height="15.2125" version="1.1" viewBox="0 0 44.5 15.2125" width="44.5"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,15.2125)">
		<g transform="translate(72,-59.83)">
			<text transform="matrix(1,0,0,-1,-71.95,63.36)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝒵</tspan>
				<tspan style="font-size: 12.50px; " x="11.652796" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="15.816295" y="0">Π</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-47.11,60.23)">
				<tspan style="font-size: 8.75px; " x="0" y="0">+</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-40.61,63.36)">
				<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg>. The main result of the
paper gives some necessary and sufficient conditions for the boundedness of
the composition operator <svg style="vertical-align:-4.74141pt;width:112.4875px;" id="M6" height="16.6" version="1.1" viewBox="0 0 112.4875 16.6" width="112.4875"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,16.6)">
		<g transform="translate(72,-58.72)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝐶</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-63.56,60.37)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝜑</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-57.23,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑓</tspan>
				<tspan style="font-size: 12.50px; " x="6.9391651" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="11.102664" y="0">𝑧</tspan>
				<tspan style="font-size: 12.50px; " x="17.341661" y="0">)</tspan>
				<tspan style="font-size: 12.50px; " x="24.968491" y="0">=</tspan>
				<tspan style="font-size: 12.50px; " x="37.008881" y="0">𝑓</tspan>
				<tspan style="font-size: 12.50px; " x="43.948044" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="48.111546" y="0">𝜑</tspan>
				<tspan style="font-size: 12.50px; " x="56.438541" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="60.602039" y="0">𝑧</tspan>
				<tspan style="font-size: 12.50px; " x="66.841042" y="0">)</tspan>
				<tspan style="font-size: 12.50px; " x="71.004539" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg> from the Hardy space <svg style="vertical-align:-3.49493pt;width:50.962502px;" id="M7" height="15.5625" version="1.1" viewBox="0 0 50.962502 15.5625" width="50.962502"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,15.5625)">
		<g transform="translate(72,-59.55)">
			<text transform="matrix(1,0,0,-1,-71.95,63.09)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝐻</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-59.91,68.09)">
				<tspan style="font-size: 8.75px; " x="0" y="0">𝑝</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-55.13,63.09)">
				<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">Π</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-41.94,59.96)">
				<tspan style="font-size: 8.75px; " x="0" y="0">+</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-35.45,63.09)">
				<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg> on the upper half-plane, to the Zygmund-type space, where <svg style="vertical-align:-2.29482pt;width:10.5375px;" id="M8" height="9.875" version="1.1" viewBox="0 0 10.5375 9.875" width="10.5375"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,9.875)">
		<g transform="translate(72,-64.1)">
			<text transform="matrix(1,0,0,-1,-71.95,66.44)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝜑</tspan>
			</text>

		</g>
	</g>
</svg> is an analytic
self-map of the upper half-plane.

Stevo Stevi&#x0107;

Composition Operators from the Hardy Space to the Zygmund-Type Space on the Upper Half-Plane

Let <svg style="vertical-align:-2.3205pt;width:37.287498px;" id="M3" height="15.0875" version="1.1" viewBox="0 0 37.287498 15.0875" width="37.287498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B" /></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28" /></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><path id="x1D435" d="M594 511q0 -122 -171 -157l1 -2q158 -33 158 -159q0 -52 -34.5 -95t-90.5 -65q-76 -33 -217 -33h-223l8 28q63 5 79.5 19t26.5 72l83 426q9 48 -2.5 60t-77.5 17l6 28h259q195 0 195 -139zM499 509q0 59 -37 83t-91 24q-36 0 -51 -9q-17 -9 -22 -44l-35 -195h62&#xA;q82 0 128 37t46 104zM481 199q0 71 -48 102.5t-121 31.5h-56l-37 -201q-11 -58 7.5 -77t80.5 -19q76 0 125 44.5t49 118.5z" /></g><g transform="matrix(.017,-0,0,-.017,31.34,12.138)"><use xlink:href="#x29" /></g></svg> denote the space of all holomorphic functions on the
unit ball <svg style="vertical-align:-0.0pt;width:10.6125px;" id="M4" height="11.175" version="1.1" viewBox="0 0 10.6125 11.175" width="10.6125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D435" /></g></svg>. Let <svg style="vertical-align:-2.3205pt;width:64.849998px;" id="M5" height="15.0875" version="1.1" viewBox="0 0 64.849998 15.0875" width="64.849998" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x1D462" d="M515 96q-37 -44 -84.5 -76t-71.5 -32q-12 0 -18 7t-6 34t10 75l19 89h-2q-87 -113 -181 -176q-43 -29 -83 -29q-46 0 -46 63q0 31 9 67l52 222q7 36 -1 36q-10 0 -76 -50l-13 24q47 45 92 71.5t67 26.5q18 0 21 -16.5t-8 -65.5l-56 -242q-16 -67 13 -67q41 0 114 74&#xA;t114 146l32 145l74 26h11q-52 -199 -80 -347q-8 -39 6 -39q7 0 32 17.5t47 39.5z" /></g><g transform="matrix(.017,-0,0,-.017,13.916,12.138)"><path id="x2208" d="M448 1h-83q-118 0 -201.5 74.5t-83.5 179.5t83.5 179.5t201.5 74.5h83v-50h-84q-87 0 -150.5 -51.5t-73.5 -127.5h308v-50h-308q10 -76 73.5 -127.5t150.5 -51.5h84v-50z" /></g><g transform="matrix(.017,-0,0,-.017,27.617,12.138)"><use xlink:href="#x1D43B" /></g><g transform="matrix(.017,-0,0,-.017,42.525,12.138)"><use xlink:href="#x28" /></g><g transform="matrix(.017,-0,0,-.017,48.407,12.138)"><use xlink:href="#x1D435" /></g><g transform="matrix(.017,-0,0,-.017,58.895,12.138)"><use xlink:href="#x29" /></g></svg> and <svg style="vertical-align:-3.56265pt;width:10.0125px;" id="M6" height="12.325" version="1.1" viewBox="0 0 10.0125 12.325" width="10.0125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.825)"><path id="x1D711" d="M559 271q0 -109 -83 -187t-189 -89l-36 -213q-16 -30 -86 -43l-7 9q10 36 53 242q-91 11 -139.5 63.5t-48.5 132.5q0 186 242 271l11 -28q-76 -34 -119.5 -91.5t-43.5 -136.5q0 -66 29 -108.5t77 -55.5q39 205 56 260q21 71 65 111t93 40q50 0 88 -45.5t38 -131.5z&#xA;M484 273q0 65 -26.5 99t-52.5 34q-37 0 -53 -75l-59 -298q80 5 135.5 74.5t55.5 165.5z" /></g></svg> be a holomorphic
self-map of <svg style="vertical-align:-0.0pt;width:10.6125px;" id="M7" height="11.175" version="1.1" viewBox="0 0 10.6125 11.175" width="10.6125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D435" /></g></svg>. In this paper, we investigate the boundedness and
compactness of the weighted composition operator <svg style="vertical-align:-5.76988pt;width:28.799999px;" id="M8" height="18.65" version="1.1" viewBox="0 0 28.799999 18.65" width="28.799999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D462" /></g><g transform="matrix(.017,-0,0,-.017,9.208,11.4)"><path id="x1D436" d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5&#xA;t325.5 101.5q114 0 204 -38z" /></g><g transform="matrix(.012,-0,0,-.012,21.188,15.475)"><use xlink:href="#x1D711" /></g></svg> from
the general function space <svg style="vertical-align:-3.50804pt;width:60.674999px;" id="M9" height="16.5625" version="1.1" viewBox="0 0 60.674999 16.5625" width="60.674999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D439" /></g><g transform="matrix(.017,-0,0,-.017,10.636,12.138)"><use xlink:href="#x28" /></g><g transform="matrix(.017,-0,0,-.017,16.517,12.138)"><use xlink:href="#x1D45D" /></g><g transform="matrix(.017,-0,0,-.017,26.598,12.138)"><use xlink:href="#x2C" /></g><g transform="matrix(.017,-0,0,-.017,33.295,12.138)"><use xlink:href="#x1D45E" /></g><g transform="matrix(.017,-0,0,-.017,41.641,12.138)"><use xlink:href="#x2C" /></g><g transform="matrix(.017,-0,0,-.017,48.356,12.138)"><use xlink:href="#x1D460" /></g><g transform="matrix(.017,-0,0,-.017,54.73,12.138)"><use xlink:href="#x29" /></g></svg> to the weighted-type space
<svg style="vertical-align:-5.76988pt;width:27.7875px;" id="M10" height="20.799999" version="1.1" viewBox="0 0 27.7875 20.799999" width="27.7875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,13.55)"><use xlink:href="#x1D43B" /></g><g transform="matrix(.012,-0,0,-.012,14.975,5.388)"><use xlink:href="#x221E" /></g><g transform="matrix(.012,-0,0,-.012,14.975,17.638)"><use xlink:href="#x1D707" /></g></svg> in the unit ball.

Xiangling Zhu

Weighted Composition Operators from <svg style="vertical-align:-3.50804pt" height="16.5625" version="1.1" viewBox="0 0 60.674999 16.5625" width="60.674999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x1D439" d="M599 650l-18 -160l-30 -2q-2 70 -11 94q-5 16 -26 24t-75 8h-94q-27 0 -33.5 -6.5t-12.5 -32.5l-43 -226h108q52 0 72.5 5t31 18t28.5 52h28l-40 -200h-28q-1 42 -6 56.5t-24.5 21t-70.5 6.5h-108l-32 -177q-13 -65 1 -80.5t88 -22.5l-8 -28h-279l6 28q61 5 77.5 20.5&#xA;t29.5 81.5l72 387l7.5 40.5t1 26.5t-4 18t-15.5 10t-24 6t-37 4l8 28h461z" /></g><g transform="matrix(.017,-0,0,-.017,10.636,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,16.517,12.138)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g><g transform="matrix(.017,-0,0,-.017,26.598,12.138)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,33.295,12.138)"><path id="x1D45E" d="M474 429q-19 -69 -47 -222l-65 -347q-8 -49 0 -60t49 -16l26 -3l-4 -26l-246 -12l4 25l17 3q37 6 50 20.5t23 60.5l67 321h-2q-65 -79 -150 -138q-69 -47 -104 -47q-28 0 -48.5 32t-20.5 81q0 78 35 148t90 117q67 57 161 77q23 5 58 5q43 0 90 -15zM387 387&#xA;q-30 16 -75 16q-58 0 -92 -27q-49 -39 -78.5 -112t-29.5 -136q0 -34 8.5 -52.5t21.5 -18.5q40 0 109.5 63.5t103.5 115.5q16 53 32 151z" /></g><g transform="matrix(.017,-0,0,-.017,41.641,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,48.356,12.138)"><path id="x1D460" d="M352 391q0 -31 -27 -44q-14 -7 -24 6q-39 48 -84 48q-23 0 -39.5 -15t-16.5 -40q0 -43 73 -90q49 -32 70 -58t21 -57q0 -58 -62 -105.5t-129 -47.5q-40 0 -75.5 25t-35.5 52q0 28 32 46q7 4 15 3t11 -6q19 -31 48.5 -50.5t54.5 -19.5q34 0 54 19.5t20 42.5q0 43 -65 81&#xA;q-97 56 -97 123q0 50 51 96q19 17 58 32.5t62 15.5q37 0 61 -18t24 -39z" /></g><g transform="matrix(.017,-0,0,-.017,54.73,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g></svg> Spaces to <svg style="vertical-align:-5.76988pt" height="20.799999" version="1.1" viewBox="0 0 27.7875 20.799999" width="27.7875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,13.55)"><path id="x1D43B" d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28&#xA;h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z" /></g><g transform="matrix(.012,-0,0,-.012,14.975,5.388)"><path id="x221E" d="M983 225q0 -112 -67 -174.5t-150 -62.5q-91 0 -154.5 43.5t-113.5 129.5q-49 -85 -104 -129t-138 -44q-98 0 -158.5 66t-60.5 154q0 59 21 106.5t54.5 75.5t70.5 43t73 15q90 0 152.5 -43.5t112.5 -128.5q48 84 104.5 128t140.5 44q93 0 155 -65t62 -158zM478 196&#xA;q-27 49 -47 80t-50 67t-64 54t-73 18q-48 0 -81.5 -47t-33.5 -128q0 -96 37.5 -157.5t99.5 -61.5q68 0 117.5 47t94.5 128zM889 204q0 91 -35.5 151t-99.5 60q-68 0 -119 -47t-95 -127q27 -49 47 -80.5t50 -67.5t65 -54t74 -18q113 0 113 183z" /></g><g transform="matrix(.012,-0,0,-.012,14.975,17.638)"><path id="x1D707" d="M538 96q-38 -45 -84 -76.5t-71 -31.5q-44 0 -15 117q12 49 18 89h-2q-99 -128 -163 -178q-36 -28 -70 -28q-26 0 -44 38h-2q-16 -69 -16 -129q0 -43 11 -75.5t24 -42.5l1 -6q-7 -12 -24.5 -23t-37.5 -11q-40 0 -40 76q0 52 35 202l94 407l78 24l5 -5q-32 -122 -72 -310&#xA;q-7 -33 0 -55t25 -22q36 0 105.5 75t107.5 145l32 146l75 26h10q-50 -197 -78 -347q-7 -38 6 -38q7 0 31 17t47 39z" /></g></svg> Spaces

By using the ordinary fermionic <svg style="vertical-align:-3.50804pt;width:10.2px;" id="M1" height="13.3" version="1.1" viewBox="0 0 10.2 13.3" width="10.2" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,8.863)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g></svg>-adic invariant integral on <svg style="vertical-align:-5.73167pt;width:20.5px;" id="M2" height="18.737499" version="1.1" viewBox="0 0 20.5 18.737499" width="20.5" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.537)"><path id="x2124" d="M734 170q-25 -130 -36 -170h-661l-8 15l420 599h-173q-77 0 -102 -7t-39 -29q-19 -29 -35 -87l-29 1l12 183h24q9 -16 19 -20.5t34 -4.5h543l7 -14l-416 -598l166 -2q94 0 131.5 7.5t58.5 29.5q28 29 55 103zM639 614h-89l-397 -567l88 -7z" /></g><g transform="matrix(.012,-0,0,-.012,12.763,15.612)"><use xlink:href="#x1D45D" /></g></svg>, we derive
some interesting identities related to the Frobenius-Euler polynomials.

Taekyun Kim

Byungje Lee

Some Identities of the Frobenius-Euler Polynomials

The paper is concerned with a two-dimensional Landau-Lifshitz
equation which was first raised by A. DeSimone and F. Otto, and so fourth, when studying thin film micromagnetics. We get the existence of a local weak solution by approximating it with a higher-order equation. Penalty approximation and semigroup theory are employed to deal with the higher-order equation.

Jingna Li

A Two-Dimensional Landau-Lifshitz Model in Studying Thin Film Micromagnetics

Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in <svg style="vertical-align:-0.15048pt;width:18.200001px;" id="M2" height="11.7125" version="1.1" viewBox="0 0 18.200001 11.7125" width="18.200001" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,11.7125)"><g transform="translate(72,-62.63)"><text transform="matrix(1,0,0,-1,-71.95,62.83)"><tspan style="font-size: 12.50px; " x="0" y="0">𝐶</tspan></text><text transform="matrix(1,0,0,-1,-62.56,67.83)"><tspan style="font-size: 8.75px; " x="0" y="0">∗</tspan></text></g></g></svg>-algebras and Lie <svg style="vertical-align:-0.15048pt;width:18.200001px;" id="M3" height="11.7125" version="1.1" viewBox="0 0 18.200001 11.7125" width="18.200001" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,11.7125)"><g transform="translate(72,-62.63)"><text transform="matrix(1,0,0,-1,-71.95,62.83)"><tspan style="font-size: 12.50px; " x="0" y="0">𝐶</tspan></text><text transform="matrix(1,0,0,-1,-62.56,67.83)"><tspan style="font-size: 8.75px; " x="0" y="0">∗</tspan></text></g></g></svg>-algebras and also of derivations on
<svg style="vertical-align:-0.15048pt;width:18.200001px;" id="M4" height="11.7125" version="1.1" viewBox="0 0 18.200001 11.7125" width="18.200001" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,11.7125)"><g transform="translate(72,-62.63)"><text transform="matrix(1,0,0,-1,-71.95,62.83)"><tspan style="font-size: 12.50px; " x="0" y="0">𝐶</tspan></text><text transform="matrix(1,0,0,-1,-62.56,67.83)"><tspan style="font-size: 8.75px; " x="0" y="0">∗</tspan></text></g></g></svg>-algebras and Lie <svg style="vertical-align:-0.15048pt;width:18.200001px;" id="M5" height="11.7125" version="1.1" viewBox="0 0 18.200001 11.7125" width="18.200001" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,11.7125)"><g transform="translate(72,-62.63)"><text transform="matrix(1,0,0,-1,-71.95,62.83)"><tspan style="font-size: 12.50px; " x="0" y="0">𝐶</tspan></text><text transform="matrix(1,0,0,-1,-62.56,67.83)"><tspan style="font-size: 8.75px; " x="0" y="0">∗</tspan></text></g></g></svg>-algebras for the Jensen-type functional equation
<svg style="vertical-align:-2.34499pt;width:216.14999px;" id="M6" height="13.6125" version="1.1" viewBox="0 0 216.14999 13.6125" width="216.14999" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,13.6125)"><g transform="translate(72,-61.11)"><text transform="matrix(1,0,0,-1,-71.95,63.5)"><tspan style="font-size: 12.50px; " x="0" y="0">𝑓</tspan><tspan style="font-size: 12.50px; " x="6.9391651" y="0">(</tspan><tspan style="font-size: 12.50px; " x="11.102664" y="0">(</tspan><tspan style="font-size: 12.50px; " x="15.266163" y="0">𝑥</tspan><tspan style="font-size: 12.50px; " x="24.91848" y="0">+</tspan><tspan style="font-size: 12.50px; " x="36.258701" y="0">𝑦</tspan><tspan style="font-size: 12.50px; " x="42.46019" y="0">)</tspan><tspan style="font-size: 12.50px; " x="46.623688" y="0">/</tspan><tspan style="font-size: 12.50px; " x="50.099522" y="0">2</tspan><tspan style="font-size: 12.50px; " x="56.351021" y="0">)</tspan><tspan style="font-size: 12.50px; " x="63.290188" y="0">+</tspan><tspan style="font-size: 12.50px; " x="74.630409" y="0">𝑓</tspan><tspan style="font-size: 12.50px; " x="81.569572" y="0">(</tspan><tspan style="font-size: 12.50px; " x="85.73307" y="0">(</tspan><tspan style="font-size: 12.50px; " x="89.896568" y="0">𝑥</tspan><tspan style="font-size: 12.50px; " x="99.561386" y="0">−</tspan><tspan style="font-size: 12.50px; " x="110.90161" y="0">𝑦</tspan><tspan style="font-size: 12.50px; " x="117.1031" y="0">)</tspan><tspan style="font-size: 12.50px; " x="121.26659" y="0">/</tspan><tspan style="font-size: 12.50px; " x="124.74243" y="0">2</tspan><tspan style="font-size: 12.50px; " x="130.99393" y="0">)</tspan><tspan style="font-size: 12.50px; " x="138.63327" y="0">=</tspan><tspan style="font-size: 12.50px; " x="150.66115" y="0">𝑓</tspan><tspan style="font-size: 12.50px; " x="157.60031" y="0">(</tspan><tspan style="font-size: 12.50px; " x="161.76381" y="0">𝑥</tspan><tspan style="font-size: 12.50px; " x="168.64046" y="0">)</tspan></text></g></g></svg>.

Choonkil Park

John Michael Rassias

Stability of the Jensen-Type Functional Equation in <svg style="vertical-align:-0.23206pt" height="14.0375" version="1.1" viewBox="0 0 19.7125 14.0375" width="19.7125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,13.688)"><path id="x1D436" d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5&#xA;t325.5 101.5q114 0 204 -38z" /></g><g transform="matrix(.012,-0,0,-.012,12.05,5.525)"><path id="x2217" d="M471 153q-22 -15 -61 -13q-25 31 -45.5 49.5t-56.5 41.5q4 -71 28 -134q-17 -33 -42 -46q-24 12 -42 46q24 63 28 134q-36 -23 -56.5 -41.5t-45.5 -49.5q-36 -2 -61 13q0 28 19 59q65 10 130 43q-65 33 -130 43q-19 31 -19 59q22 15 61 13q25 -31 45.5 -49.5t56.5 -41.5&#xA;q-4 71 -28 134q17 33 42 46q24 -12 42 -46q-24 -63 -28 -134q36 23 56.5 41.5t45.5 49.5q36 2 61 -13q0 -28 -19 -59q-65 -10 -130 -43q65 -33 130 -43q19 -31 19 -59z" /></g></svg>-Algebras: A Fixed Point Approach

We construct global smooth approximate solution to a scalar conservation law with arbitrary smooth monotonic initial data. Different kinds of singularities interactions which arise during the evolution of the initial data are
described as well. In order to solve the problem, we use and further develop
the weak asymptotic method, recently introduced technique for investigating nonlinear waves interactions.

V. G. Danilov

D. Mitrovic

Smooth Approximations of Global in Time Solutions to Scalar Conservation Laws

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<p><em>Abstract and Applied Analysis</em> publishes research with an emphasis on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimisation theory, and control theory.</p>


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<p>Chief Editor, Dr Wong, is an associate professor at Nanyang Technological University, Singapore. Her research interests include differential equations, difference equations, integral equations, and numerical mathematics.</p>


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Abstract and Applied Analysis publishes research with an emphasis on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimisation theory, and control theory.

Abstract and Applied Analysis