{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T04:06:59Z","timestamp":1759032419371,"version":"3.37.3"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"2","funder":[{"DOI":"10.13039\/501100008530","name":"FEDER","doi-asserted-by":"crossref","award":["COMPETE, Programa Operacional Factores de Competitividade"],"award-info":[{"award-number":["COMPETE, Programa Operacional Factores de Competitividade"]}],"id":[{"id":"10.13039\/501100008530","id-type":"DOI","asserted-by":"crossref"}]},{"name":"University of Aveiro, Center for Research and Development in Mathematics and Applications"},{"name":"Portuguese Foundation for Science and Technology","award":["PEst-C\/MAT\/UI4106\/2011, FCOMP-01-0124-FEDER-022690"],"award-info":[{"award-number":["PEst-C\/MAT\/UI4106\/2011, FCOMP-01-0124-FEDER-022690"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2015,5,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper we consider a fractional difference inclusion of the form\n<jats:inline-formula id=\"eq1_w2aab2b8b8b1b7b1aab1c13b1b1Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/anly-2012-1281_48b519ad2de2e66eeae02ed41ae2e8c3.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:msub>\n                                    <m:mspace width=\"0.166667em\"\/>\n                                    <m:mrow>\n                                       <m:mi>\u03b1<\/m:mi>\n                                       <m:mo>-<\/m:mo>\n                                       <m:mn>1<\/m:mn>\n                                    <\/m:mrow>\n                                 <\/m:msub>\n                                 <m:msup>\n                                    <m:mi>\u0394<\/m:mi>\n                                    <m:mi>\u03b1<\/m:mi>\n                                 <\/m:msup>\n                                 <m:mi>y<\/m:mi>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>t<\/m:mi>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                              <m:mo>\u2208<\/m:mo>\n                              <m:mi>\u03bb<\/m:mi>\n                              <m:mi>y<\/m:mi>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>t<\/m:mi>\n                                 <m:mo>+<\/m:mo>\n                                 <m:mi>\u03b1<\/m:mi>\n                                 <m:mo>-<\/m:mo>\n                                 <m:mn>1<\/m:mn>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                              <m:mo>+<\/m:mo>\n                              <m:mi>f<\/m:mi>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>t<\/m:mi>\n                                 <m:mo>+<\/m:mo>\n                                 <m:mi>\u03b1<\/m:mi>\n                                 <m:mo>-<\/m:mo>\n                                 <m:mn>1<\/m:mn>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mi>y<\/m:mi>\n                                 <m:mrow>\n                                    <m:mo>(<\/m:mo>\n                                    <m:mi>t<\/m:mi>\n                                    <m:mo>+<\/m:mo>\n                                    <m:mi>\u03b1<\/m:mi>\n                                    <m:mo>-<\/m:mo>\n                                    <m:mn>1<\/m:mn>\n                                    <m:mo>)<\/m:mo>\n                                 <\/m:mrow>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$({\\,}_{\\alpha -1}\\Delta ^{\\alpha }y)(t) \\in \\lambda y(t+\\alpha -1)+f(t+\\alpha -1,y(t+\\alpha -1))$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>,\n<jats:inline-formula id=\"eq2_w2aab2b8b8b1b7b1aab1c13b1b3Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/anly-2012-1281_9c15e69b9e410697530f3649befb62cd.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>y<\/m:mi>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>\u03b1<\/m:mi>\n                                 <m:mo>-<\/m:mo>\n                                 <m:mn>1<\/m:mn>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                              <m:mo>=<\/m:mo>\n                              <m:mi>y<\/m:mi>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>\u03b1<\/m:mi>\n                                 <m:mo>-<\/m:mo>\n                                 <m:mn>1<\/m:mn>\n                                 <m:mo>+<\/m:mo>\n                                 <m:mi>T<\/m:mi>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                              <m:mo>+<\/m:mo>\n                              <m:msubsup>\n                                 <m:mo>\u2211<\/m:mo>\n                                 <m:mrow>\n                                    <m:mi>j<\/m:mi>\n                                    <m:mo>=<\/m:mo>\n                                    <m:mn>1<\/m:mn>\n                                 <\/m:mrow>\n                                 <m:mi>N<\/m:mi>\n                              <\/m:msubsup>\n                              <m:msub>\n                                 <m:mi>F<\/m:mi>\n                                 <m:mi>j<\/m:mi>\n                              <\/m:msub>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>y<\/m:mi>\n                                 <m:mrow>\n                                    <m:mo>(<\/m:mo>\n                                    <m:msub>\n                                       <m:mi>t<\/m:mi>\n                                       <m:mi>j<\/m:mi>\n                                    <\/m:msub>\n                                    <m:mo>)<\/m:mo>\n                                 <\/m:mrow>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$y(\\alpha -1)=y(\\alpha -1+T)+\\sum _{j=1}^{N}F_j(y(t_j))$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>,\nwhere <jats:inline-formula id=\"eq3_w2aab2b8b8b1b7b1aab1c13b1b5Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/anly-2012-1281_cb6394a824b0f672b915a5be665b27f8.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mn>0<\/m:mn>\n                              <m:mo>&lt;<\/m:mo>\n                              <m:mi>\u03b1<\/m:mi>\n                              <m:mo>&lt;<\/m:mo>\n                              <m:mn>1<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${0&amp;lt;\\alpha &amp;lt;1}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>. Consequently, we treat the situation in which the boundary condition is both nonlocal and nonlinear. Thus, the boundary conditions under study can be very general. We provide conditions under which this problem has at least one positive solution and then discuss the application of the results. Finally, since we allow <jats:inline-formula id=\"eq4_w2aab2b8b8b1b7b1aab1c13b1b7Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/anly-2012-1281_0b8c488a2cd48c41433ecc41b7e7ae64.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:msub>\n                                 <m:mi>F<\/m:mi>\n                                 <m:mi>j<\/m:mi>\n                              <\/m:msub>\n                              <m:mo>\u2261<\/m:mo>\n                              <m:mn>0<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${F_j\\equiv 0}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, for each <jats:italic>j<\/jats:italic>, our results are also new for the case of periodic boundary conditions.<\/jats:p>","DOI":"10.1515\/anly-2012-1281","type":"journal-article","created":{"date-parts":[[2015,3,19]],"date-time":"2015-03-19T17:04:55Z","timestamp":1426784695000},"page":"73-83","source":"Crossref","is-referenced-by-count":7,"title":["On positive solutions to fractional difference inclusions"],"prefix":"10.1515","volume":"35","author":[{"given":"Rui A. C.","family":"Ferreira","sequence":"first","affiliation":[{"name":"Department of Mathematics, Lusophone University of Humanities and Technologies, 1749-024 Lisbon, Portugal"}]},{"given":"Christopher S.","family":"Goodrich","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Creighton Preparatory School, NE 68114 Omaha, USA"}]}],"member":"374","published-online":{"date-parts":[[2015,3,19]]},"container-title":["Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/anly-2012-1281\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/anly-2012-1281\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T21:42:50Z","timestamp":1680298970000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/anly-2012-1281\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,3,19]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2015,3,19]]},"published-print":{"date-parts":[[2015,5,1]]}},"alternative-id":["10.1515\/anly-2012-1281"],"URL":"https:\/\/doi.org\/10.1515\/anly-2012-1281","relation":{},"ISSN":["0174-4747","2196-6753"],"issn-type":[{"type":"print","value":"0174-4747"},{"type":"electronic","value":"2196-6753"}],"subject":[],"published":{"date-parts":[[2015,3,19]]}}}