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- W1982949491 abstract "We study the existence and multiplicity of solutions for the following fractional boundary value problem: <svg style=vertical-align:-4.52803pt;width:581.28748px; id=M1 height=20.700001 version=1.1 viewBox=0 0 581.28748 20.700001 width=581.28748 xmlns=http://www.w3.org/2000/svg> <g transform=matrix(1.25,0,0,-1.25,0,20.7)> <g transform=translate(72,-55.44)> <text transform=matrix(1,0,0,-1,-71.95,60)> <tspan style=font-size: 12.50px; x=0 y=0>(</tspan> <tspan style=font-size: 12.50px; x=4.1634989 y=0>𝑑</tspan> <tspan style=font-size: 12.50px; x=10.815095 y=0>/</tspan> <tspan style=font-size: 12.50px; x=14.290929 y=0>𝑑</tspan> <tspan style=font-size: 12.50px; x=20.942526 y=0>𝑡</tspan> <tspan style=font-size: 12.50px; x=24.855965 y=0>)</tspan> <tspan style=font-size: 12.50px; x=29.019463 y=0>(</tspan> <tspan style=font-size: 12.50px; x=33.182961 y=0>(</tspan> <tspan style=font-size: 12.50px; x=37.346462 y=0>1</tspan> <tspan style=font-size: 12.50px; x=43.597961 y=0>/</tspan> <tspan style=font-size: 12.50px; x=47.073795 y=0>2</tspan> <tspan style=font-size: 12.50px; x=53.325294 y=0>)</tspan> </text> <text transform=matrix(1,0,0,-1,-14.46,56.88)> <tspan style=font-size: 8.75px; x=0 y=0>0</tspan> </text> <text transform=matrix(1,0,0,-1,-9.59,60)> <tspan style=font-size: 12.50px; x=0 y=0>𝐷</tspan> </text> <text transform=matrix(1,0,0,-1,0.48,66.1)> <tspan style=font-size: 8.75px; x=-0.19 y=9.8599997>𝑡</tspan> <tspan style=font-size: 8.75px; x=0 y=0>−</tspan> <tspan style=font-size: 8.75px; x=5.99512 y=0>𝛽</tspan> </text> <text transform=matrix(1,0,0,-1,11.86,60)> <tspan style=font-size: 12.50px; x=0 y=0>(</tspan> <tspan style=font-size: 12.50px; x=4.1634989 y=0>𝑢</tspan> <tspan style=font-size: 12.50px; x=10.089921 y=0>′</tspan> <tspan style=font-size: 12.50px; x=13.703288 y=0>(</tspan> <tspan style=font-size: 12.50px; x=17.866787 y=0>𝑡</tspan> <tspan style=font-size: 12.50px; x=21.780226 y=0>)</tspan> <tspan style=font-size: 12.50px; x=25.943726 y=0>)</tspan> <tspan style=font-size: 12.50px; x=32.870388 y=0>+</tspan> <tspan style=font-size: 12.50px; x=44.22311 y=0>(</tspan> <tspan style=font-size: 12.50px; x=48.386608 y=0>1</tspan> <tspan style=font-size: 12.50px; x=54.638111 y=0>/</tspan> <tspan style=font-size: 12.50px; x=58.113945 y=0>2</tspan> <tspan style=font-size: 12.50px; x=64.36544 y=0>)</tspan> </text> <text transform=matrix(1,0,0,-1,80.39,56.88)> <tspan style=font-size: 8.75px; x=0 y=0>𝑡</tspan> </text> <text transform=matrix(1,0,0,-1,83.63,60)> <tspan style=font-size: 12.50px; x=0 y=0>𝐷</tspan> </text> <text transform=matrix(1,0,0,-1,93.69,66.1)> <tspan style=font-size: 8.75px; x=-0.18000001 y=10.61>𝑇</tspan> <tspan style=font-size: 8.75px; x=0 y=0>−</tspan> <tspan style=font-size: 8.75px; x=5.99512 y=0>𝛽</tspan> </text> <text transform=matrix(1,0,0,-1,105.07,60)> <tspan style=font-size: 12.50px; x=0 y=0>(</tspan> <tspan style=font-size: 12.50px; x=4.1634989 y=0>𝑢</tspan> <tspan style=font-size: 12.50px; x=10.089921 y=0>′</tspan> <tspan style=font-size: 12.50px; x=13.703288 y=0>(</tspan> <tspan style=font-size: 12.50px; x=17.866787 y=0>𝑡</tspan> <tspan style=font-size: 12.50px; x=21.780226 y=0>)</tspan> <tspan style=font-size: 12.50px; x=25.943726 y=0>)</tspan> <tspan style=font-size: 12.50px; x=30.107224 y=0>)</tspan> <tspan style=font-size: 12.50px; x=37.046391 y=0>+</tspan> <tspan style=font-size: 12.50px; x=48.386608 y=0>∇</tspan> <tspan style=font-size: 12.50px; x=57.526302 y=0>𝐹</tspan> <tspan style=font-size: 12.50px; x=65.253159 y=0>(</tspan> <tspan style=font-size: 12.50px; x=69.416656 y=0>𝑡</tspan> <tspan style=font-size: 12.50px; x=73.330093 y=0>,</tspan> <tspan style=font-size: 12.50px; x=78.543846 y=0>𝑢</tspan> <tspan style=font-size: 12.50px; x=84.470268 y=0>(</tspan> <tspan style=font-size: 12.50px; x=88.633766 y=0>𝑡</tspan> <tspan style=font-size: 12.50px; x=92.547203 y=0>)</tspan> <tspan style=font-size: 12.50px; x=96.710709 y=0>)</tspan> <tspan style=font-size: 12.50px; x=104.33753 y=0>=</tspan> <tspan style=font-size: 12.50px; x=116.37792 y=0>0</tspan> <tspan style=font-size: 12.50px; x=122.62943 y=0>,</tspan> <tspan style=font-size: 12.50px; x=132.00667 y=0>a</tspan> <tspan style=font-size: 12.50px; x=137.558 y=0>.</tspan> <tspan style=font-size: 12.50px; x=140.68376 y=0>e</tspan> <tspan style=font-size: 12.50px; x=146.23509 y=0>.</tspan> <tspan style=font-size: 12.50px; x=153.53683 y=0>𝑡</tspan> <tspan style=font-size: 12.50px; x=160.9136 y=0>∈</tspan> <tspan style=font-size: 12.50px; x=172.95399 y=0>[</tspan> <tspan style=font-size: 12.50px; x=177.11749 y=0>0</tspan> <tspan style=font-size: 12.50px; x=183.369 y=0>,</tspan> <tspan style=font-size: 12.50px; x=188.58275 y=0>𝑇</tspan> <tspan style=font-size: 12.50px; x=195.4594 y=0>]</tspan> <tspan style=font-size: 12.50px; x=199.62289 y=0>,</tspan> <tspan style=font-size: 12.50px; x=209.00015 y=0>𝑢</tspan> <tspan style=font-size: 12.50px; x=214.92657 y=0>(</tspan> <tspan style=font-size: 12.50px; x=219.09007 y=0>0</tspan> <tspan style=font-size: 12.50px; x=225.34157 y=0>)</tspan> <tspan style=font-size: 12.50px; x=232.9809 y=0>=</tspan> <tspan style=font-size: 12.50px; x=245.00879 y=0>𝑢</tspan> <tspan style=font-size: 12.50px; x=250.93521 y=0>(</tspan> <tspan style=font-size: 12.50px; x=255.09871 y=0>𝑇</tspan> <tspan style=font-size: 12.50px; x=261.97537 y=0>)</tspan> <tspan style=font-size: 12.50px; x=269.61469 y=0>=</tspan> <tspan style=font-size: 12.50px; x=281.65509 y=0>0</tspan> </text> </g> </g> </svg>, where <svg style=vertical-align:-2.21957pt;width:36.0625px; id=M2 height=13.45 version=1.1 viewBox=0 0 36.0625 13.45 width=36.0625 xmlns=http://www.w3.org/2000/svg> <g transform=matrix(1.25,0,0,-1.25,0,13.45)> <g transform=translate(72,-61.24)> <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=7.7268538 y=0>(</tspan> <tspan style=font-size: 12.50px; x=11.890353 y=0>𝑡</tspan> <tspan style=font-size: 12.50px; x=15.803792 y=0>,</tspan> <tspan style=font-size: 12.50px; x=21.005039 y=0>⋅</tspan> <tspan style=font-size: 12.50px; x=24.580898 y=0>)</tspan> </text> </g> </g> </svg> are superquadratic, asymptotically quadratic, and subquadratic, respectively. Several examples are presented to illustrate our results." @default.
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- W1982949491 title "Existence and Multiplicity of Solutions for Some Fractional Boundary Value Problem via Critical Point Theory" @default.
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