{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,15]],"date-time":"2025-11-15T17:15:10Z","timestamp":1763226910051,"version":"3.37.3"},"reference-count":54,"publisher":"Wiley","license":[{"start":{"date-parts":[[2020,10,19]],"date-time":"2020-10-19T00:00:00Z","timestamp":1603065600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61673169"],"award-info":[{"award-number":["61673169"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2020,10,19]]},"abstract":"<jats:p>Discrete fractional calculus <jats:inline-formula>\n                     <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\">\n                        <a:mfenced open=\"(\" close=\")\" separators=\"|\">\n                           <a:mrow>\n                              <a:mi>\u2131<\/a:mi>\n                              <a:mi mathvariant=\"script\">C<\/a:mi>\n                           <\/a:mrow>\n                        <\/a:mfenced>\n                     <\/a:math>\n                  <\/jats:inline-formula> is proposed to depict neural systems with memory impacts. This research article aims to investigate the consequences in the frame of the discrete proportional fractional operator. <jats:inline-formula>\n                     <g:math xmlns:g=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\">\n                        <g:mi>\u210f<\/g:mi>\n                     <\/g:math>\n                  <\/jats:inline-formula>-discrete exponential functions are assumed in the kernel of the novel generalized fractional sum defined on the time scale <jats:inline-formula>\n                     <i:math xmlns:i=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\">\n                        <i:mi>\u210f<\/i:mi>\n                        <i:mi>\u2124<\/i:mi>\n                     <\/i:math>\n                  <\/jats:inline-formula>. The nabla <jats:inline-formula>\n                     <k:math xmlns:k=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\">\n                        <k:mi>\u210f<\/k:mi>\n                     <\/k:math>\n                  <\/jats:inline-formula>-fractional sums are accounted in particular. The governing high discretization of problems is an advanced version of the existing forms that can be transformed into linear and nonlinear difference equations using appropriately adjusted transformations invoking property of observing the new chaotic behaviors of the logistic map. Based on the theory of discrete fractional calculus, explicit bounds for a class of positive functions <jats:inline-formula>\n                     <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\">\n                        <m:mi>n<\/m:mi>\n                        <m:mfenced open=\"(\" close=\")\" separators=\"|\">\n                           <m:mrow>\n                              <m:mi>n<\/m:mi>\n                              <m:mo>\u2208<\/m:mo>\n                              <m:mi>\u2115<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:mfenced>\n                     <\/m:math>\n                  <\/jats:inline-formula> concerned are established. These variants can be utilized as a convenient apparatus in the qualitative analysis of solutions of discrete fractional difference equations. With respect to applications, we can apply the introduced outcomes to explore boundedness, uniqueness, and continuous reliance on the initial value problem for the solutions of certain underlying worth problems of fractional difference equations.<\/jats:p>","DOI":"10.1155\/2020\/8845867","type":"journal-article","created":{"date-parts":[[2020,10,19]],"date-time":"2020-10-19T22:35:16Z","timestamp":1603146916000},"page":"1-13","source":"Crossref","is-referenced-by-count":15,"title":["On Discrete Fractional Integral Inequalities for a Class of Functions"],"prefix":"10.1155","volume":"2020","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7137-1720","authenticated-orcid":true,"given":"Saima","family":"Rashid","sequence":"first","affiliation":[{"name":"Department of Mathematics, Government College University, Faisalabad, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5438-5407","authenticated-orcid":true,"given":"Hijaz","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan"}]},{"given":"Aasma","family":"Khalid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College Women University, Faisalabad, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0944-2134","authenticated-orcid":true,"given":"Yu-Ming","family":"Chu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Huzhou University, Huzhou 313000, China"},{"name":"Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha 410114, China"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2020.109860"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2020.109787"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2017.08.048"},{"key":"4","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-25562-0","volume-title":"Discrete Fractional Calculus","author":"C. 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