upon substituting z = 1 in (18) above. Finally, the lemma holds if (20) is rearranged and simplified. □
Corollary 6 From the numerical results (Tables 1-6 in the appendix), it is clear that
Lemma 7 (First Criterion) A maximizer of the group is the solution for the policy map-constraint-occupation rate problem such that
Proof. We seek a solution for such that
where is a real valued semi-linear continuous function with respect to all its arguments. (22) is equivalent to
By Lemma 1, T is necessarily compact. Let T be a fixed point of G(.). Then (23) reduces to
Consider a differentiable arc in the plane such that points on .
By multivariate chain rule on the left hand side of (24), we have
Assuming that solves (21) and combining (24) and (25) and rearranging, we have
So5 that the couple differential equations
constitute in general a solution for for when T is a fixed point of . □
Lemma 8 (Second Criterion) Any continuously differentiable solution for satisfying the first criterion above must coincide with the original solution along the base curve .
Proof. Since (27) and (28) are coupled systems, only in rare cases analytic solution exists in closed form. However, if we specify an initial value for , and , then the existence of a unique solution pair and is guaranteed. A solution that did not pass through the origin leading to cannot be a solution for since it is nowhere differentiable around . □
Lemma 9 (Optimality Criterion) Suppose so that the constraint dependent occupation rate . A solution that passes through the origin for is optimal a.s.
Proof. Given that , we have . By the numerical approximation (Tables 1-6 in the appendix) . Given that is dependent, it is then trivial. □
3. Scope for Future Work
There is a scope in extending our results to some special cases of the problem solved in this work. For instance, when the function is independent of N or when and are linear or even non-linear combination of and T and N. The author are grateful to all literature sources used.
There is no competing interest of any kind within the authorship of this work.
SS: Drafted the entire manuscript, provided the introductory chapter (Section 1) and proved Lemmas 1, 2, 3, 5, 7, 8 and 9 together with Corollary 6.
HM: Participated in the sequence alignment of the manuscript and provided the numerical simulations.
MLM: Participated in the design of the manuscript and proved lemma 4 under my supervision.
All authors read and approved the final manuscript.
The authors are grateful to Dr. Babangida A. Albaba; the current Rector of the Katsina State Institute of Technology and Management (KSITM) for spearing time to go through the entire thesis leading to this manuscript and for making valuable suggestions.
For a numerical approximation, we study the model in (11) under various sizes of constraint numbers c and varying occupation rate for . The following numerical results are obtained.
Table 1. E[N] when c = 0.
Table 2. E[N] when c = 10.
Table 3. E[N] when c = 23.
Table 4. E[N] when r = 0.5.
Table 5. E[N] when r = 0.75.
Table 6. E[N] when r = 0.9.
Conflicts of Interest
The authors declare no conflicts of interest.
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