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Existence and uniqueness of solutions of differential equations with respect to non-additive measures
  • Shekhar Negi,
  • Vicenc Torra
Shekhar Negi
Umea University Department of Computing Science

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Vicenc Torra
Umea University Department of Computing Science
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Abstract

By taking Sugeno-derivative into account, firstly, we investigate the existence of solutions to the initial value problems (IVP) of first-order differential equations with respect to non-additive measure (more precisely, distorted Lebesgue measure). It particularly occurs in the mathematical modeling of biology. We begin by expressing the differential equation in terms of ordinary derivative and the derivative with respect to the distorted Lebesgue measure. Then, by using the fixed point theorem on cones, we construct an operator and prove the existence of positive increasing solutions on cones in semi-order Banach spaces. In addition, we also use Picard’s-Lindel\”of theorem to prove the existence and uniqueness of the solution of the equation. Second, we investigate the existence of a solution to the boundary value problem (BVP) on cones with integral boundary conditions of a mix-order differential equation with respect to non-additive measures. Moreover, the Krasnoselskii fixed point theorem is also applied to both BVP and IVP and obtains at least one positive increasing solution. Examples with graphs are provided to validate the results.
03 May 2022Submitted to Mathematical Methods in the Applied Sciences
04 May 2022Submission Checks Completed
04 May 2022Assigned to Editor
10 May 2022Reviewer(s) Assigned
08 Jun 2022Editorial Decision: Revise Minor
08 Jun 20221st Revision Received
09 Jun 2022Submission Checks Completed
09 Jun 2022Assigned to Editor
10 Jun 2022Reviewer(s) Assigned
12 Jul 2022Review(s) Completed, Editorial Evaluation Pending
14 Jul 2022Editorial Decision: Revise Major
17 Jul 20222nd Revision Received
18 Jul 2022Submission Checks Completed
18 Jul 2022Assigned to Editor
18 Jul 2022Reviewer(s) Assigned
29 Jul 2022Review(s) Completed, Editorial Evaluation Pending
30 Jul 2022Editorial Decision: Accept