Further results on the investigation of solutions of integral boundary value problems

Abstract

ABSTRACT. We give a new approach for the investigation of existence and construction of an approximate solutions of nonlinear non-autonomous systems of ordinary differential equations under nonlinear integral boundary conditions depending on the derivative. The constructivity of a suggested technique is shown on the example of non-linear integral boundary value problem with two solutions.

Periodic boundary value problems for higher‐order fractional differential systems

“…[1][2][3][4] The most reasonable alternative for FDS is to study periodic boundary value problem (PBVP) on finite intervals. In this paper, we continue our extension 5,6 of a numerical-analytic technique for ordinary differential equations suggested in previous works [7][8][9][10][11] to the investigation of existence and approximate construction of solutions for FDS of the Caputo type of higher orders subjected to periodic boundary conditions. Then scalar FDE with asymptotically constant nonlinearities are investigated leading to Landesman-Lazer-type conditions.…”

Successive approximations and interval halving for integral boundary value problems

“…To replace the boundary conditions (1.5) by certain linear two-point linear separated ones, similarly to [6], [10], [11], [9] we apply a certain "freezing" technique. Namely, we introduce the vectors of parameterś D col.´1;´2; :::;´n/; D col. 1 ; 2 ; :::; n /; Á D col.Á 1 ; Á 2 ; :::; Á n / (3.6)…”

“…The validity of assertion 1 is verified by direct computation. To obtain the other required properties, similarly to [6], [9] we will prove that under the condition assumed for fixed´2 D a ; 2 D aCb 2 and t 2 h a; According to the recurrence relation (3.10) of Lemma 2, using the Lipschitz condition f 2 Lip.K x ; D x / and estimation (3.11), for m D 1 it follows from (5.12) and (5.10) that…”

“…The aim of this paper is to show how a natural interval halving and parametrization technique can help to essentially improve the sufficient convergence conditions mentioned in papers [6], [9].…”

“…Note that in [6], [9], [5], [10] where the numerical-analytic approach was applied, instead of the previous inequality the convergence condition 2r.Q/ < 1: has appeared. Thus, the convergence condition is weakened by its half.…”