Existence of solutions for functional boundary value problems of second-order nonlinear differential equations system at resonance

Abstract

In this paper, by using the coincidence degree theory due to Mawhin and constructing suitable operators, we study the solvability for functional boundary value problems of second-order nonlinear differential equations system at resonance with dim Ker L = 3 and 4, respectively.

Nonlinear differential equations involving mixed fractional derivatives with functional boundary data

“…At present, fractional differential equations involving different kinds of fractional derivatives (Caputo, Riemann-Liouville, Hadamard-type, and conformable-type to name a few) supplemented with a variety of boundary conditions have been investigated by many researchers, and one can find many interesting results on the topic in the related literature. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] However, among several types of fractional differential equations found in the literature, it is imperative to mention that the Caputo and Riemann-Liouville derivatives are studied separately in many cases. Moreover, the recent papers on the study of boundary value problems at resonance having mixed type fractional-order derivatives is not satisfactory, and the topic has not been extensively studied so far (see other studies [18][19][20][21][22][23][24][25][26][27][28][29][30][31] ).…”

“…At present, fractional differential equations involving different kinds of fractional derivatives (Caputo, Riemann–Liouville, Hadamard‐type, and conformable‐type to name a few) supplemented with a variety of boundary conditions have been investigated by many researchers, and one can find many interesting results on the topic in the related literature 1–17 . However, among several types of fractional differential equations found in the literature, it is imperative to mention that the Caputo and Riemann–Liouville derivatives are studied separately in many cases.…”

Existence of solutions for functional boundary value problems at resonance on the half-line

“…Very recently, the attention has shifted to problems on the infinite interval with linear functional conditions. There are a few papers to investigate the functional boundary value problems on the finite interval (see [7][8][9][10][11]). For example, in [8], Kosmatov and Jiang investigated second order functional problems with resonance of dimension one ⎧ ⎨ ⎩ x (t) = f (t, x(t), x (t)), 0 < t < 1, which improves the results of [6] and [7] in that respect as well and generalizes a number of recent works about two-point, three-point, multi-point, and integral boundary value problems.…”