On Fractional Semilinear Nonlocal Initial Value Problem with State Dependent Delay
In this paper, we consider a class of fractional order semilinear abstract Cauchy problem with state dependent delay subject to nonlocal initial conditions, and enlarge the existence theory with two different sets of assumptions. Under the first set of assumptions, we establish the existence of Höld...
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| Published in | Differential equations and dynamical systems Vol. 32; no. 3; pp. 685 - 708 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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New Delhi
Springer India
01.07.2024
Springer Nature B.V |
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| ISSN | 0971-3514 0974-6870 |
| DOI | 10.1007/s12591-022-00600-3 |
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| Abstract | In this paper, we consider a class of fractional order semilinear abstract Cauchy problem with state dependent delay subject to nonlocal initial conditions, and enlarge the existence theory with two different sets of assumptions. Under the first set of assumptions, we establish the existence of Hölder classical solution. Since the Hölder exponent appears as an exponent on the metric function in contraction inequality, it is not suitable to use Banach contraction mapping principle. Krasnoselskii’s fixed point theorem becomes effective to overcome this situation. Under the second set of assumptions, we obtain only the existence of mild solution using Schauder’s fixed point theorem. Few examples have been provided to illustrate our results. |
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| AbstractList | In this paper, we consider a class of fractional order semilinear abstract Cauchy problem with state dependent delay subject to nonlocal initial conditions, and enlarge the existence theory with two different sets of assumptions. Under the first set of assumptions, we establish the existence of Hölder classical solution. Since the Hölder exponent appears as an exponent on the metric function in contraction inequality, it is not suitable to use Banach contraction mapping principle. Krasnoselskii’s fixed point theorem becomes effective to overcome this situation. Under the second set of assumptions, we obtain only the existence of mild solution using Schauder’s fixed point theorem. Few examples have been provided to illustrate our results. |
| Author | Alam, Md Mansur Dubey, Shruti |
| Author_xml | – sequence: 1 givenname: Md Mansur surname: Alam fullname: Alam, Md Mansur organization: Department of Mathematics, Indian Institute of Technology Madras – sequence: 2 givenname: Shruti surname: Dubey fullname: Dubey, Shruti email: sdubey@iitm.ac.in organization: Department of Mathematics, Indian Institute of Technology Madras |
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| Keywords | Nonlocal conditions 47D06 Analytic semigroup Primary 34K37 State dependent delay Secondary 49J27 Mild solutions Caputo fractional derivative |
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| Snippet | In this paper, we consider a class of fractional order semilinear abstract Cauchy problem with state dependent delay subject to nonlocal initial conditions,... |
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| SubjectTerms | Boundary value problems Computer Science Engineering Existence theorems Fixed points (mathematics) Initial conditions Mathematics Mathematics and Statistics Original Research |
| Title | On Fractional Semilinear Nonlocal Initial Value Problem with State Dependent Delay |
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