Initial value problem for first-order integro-differential equation of Volterra type on time scales

Existence of positive periodic solutions for the impulsive Lotka–Volterra cooperative population model with time-delay and harvesting control on time scales

“…It follows from Lemma 2.4 in [36] that (¯ ) is relatively compact in X. The proof of Lemma 2.9 is complete.…”

Some new results for BVPs of first-order nonlinear integro-differential equations of volterra type

“…Taking into account that the family of BVP (1.2) is equivalent to the family of problem x = Tx, our problem is reduced to show that T has a least one fixed point. For this purpose, we apply Schaefer's Theorem by showing that all potential solutions of 14) are bounded a priori, with the bound being independent of l. With this in mind, let x be a solution of (3.14). Note that x is also a solution of (3.6).…”

“…For the recent developments involving existence of solutions to BVPs for integro-differential equations and impulsive integro-differential equations we can refer to [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. So far the main method appeared in the references to guarantee the existence of solutions is the method of upper and low solutions.…”

Extremal Solutions of Periodic Boundary Value Problems for First-Order Impulsive Integrodifferential Equations of Mixed-Type on Time Scales

“…Also, similarly to the above we can show that {v Δ n (t)} is uniformly bounded. Using Lemma 2.4 [12], we know that there exist u * ,v * such that lim n→∞ u n (t) = u * (t),lim n→∞ v n (t) = v * (t) uniformly on J T . Taking limits as n → ∞, by (3.11), we have that …”