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Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions

  • Xinan Hao EMAIL logo and Huaqing Wang
Published/Copyright: June 7, 2018
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Open Mathematics
From the journal Open Mathematics Volume 16 Issue 1

Abstract

In this paper, the existence of positive solutions for systems of semipositone singular fractional differential equations with a parameter and integral boundary conditions is investigated. By using fixed point theorem in cone, sufficient conditions which guarantee the existence of positive solutions are obtained. An example is given to illustrate the results.

Keywords: Positive solutions; Semipositone; Singular fractional differential systems; Integral boundary conditions
MSC 2010: 26A33; 34A08; 34B18

1 Introduction

The subject of fractional calculus has gained considerable popularity and importance during the past decades, mainly due to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. In recent years, fractional differential equations have been widely used in optics and thermal systems, electromagnetics, control engineering and robotic, and many other fields, see [1, 2, 3, 4, 5, 6] and the references therein. The research on fractional differential equations is very important in both theory and applications. By using nonlinear analysis tools, some scholars established the existence, uniqueness, multiplicity and qualitative properties of solutions, we refer the readers to [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and the references therein for fractional differential equations, and [21, 22, 33] for fractional differential systems.

Boundary value problems (BVPs for short) with integral boundary conditions for ordinary differential equations represent a very interesting and important class of problems, and arise in the study of various biological, physical and chemical processes [34, 35, 36, 37], such as heat conduction, thermo-elasticity, chemical engineering, underground water flow, and plasma physics. The existence of solutions or positive solutions for such class of problems has attracted much attention (see [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52] and the references therein).

In this paper, we study the systems of semipositone singualr fractional differential equations with a parameter and integral boundary conditions

D0+α−β(D0+βu(t))+λf1(t,u(t),D0+βu(t),v(t))=0,0<t<1,D0+γ−δ(D0+δv(t))+λf2(t,u(t),v(t),D0+δv(t))=0,0<t<1,D0+βu(0)=D0+β+1u(0)=0,D0+β+1u(1)=∫01D0+βu(s)dA(s),D0+δv(0)=D0+δ+1v(0)=0,D0+δ+1v(1)=∫01D0+δv(s)dB(s),(1)

where D0+α−β,D0+β,D0+γ−δ and D0+δ are the standard Riemann-Liouville fractional derivatives, λ > 0 is a parameter, 2 < α, γ ≤ 3, 0 < β, δ < 1, α − β > 2, γ − δ > 2. f1, f2 : (0,1) × [0,+∞)3 → (−∞,+∞) are continuous and may be singular at t = 0,1. A and B are nondecreasing functions of bunded variations, ∫01D0+βu(s)dA(s) and ∫01D0+δv(s)dB(s) are Riemann-Stieltjes integrals.

The study of nonlinear fractional differential systems is important as this kind of systems occur in various problems of applied mathematics. Recently, Wu et al. [26] considered the fractional differential systems involving nonlocal boundary conditions

D0+αu(t)+f(t,u(t),v(t))=0,0<t<1,1<α≤2,D0+βv(t)+g(t,u(t),v(t))=0,0<t<1,1<β≤2,u(0)=0,u(1)=∫01u(s)dA(s),v(0)=0,v(1)=∫01v(s)dB(s),(2)

where D0+α and D0+β are the standard Riemann-Liouville fractional derivatives, A, B are nondecreasing functions of bunded variations, ∫01u(s)dA(s) and ∫01v(s)dB(s) are Riemann-Stieltjes integrals, f(t, x, y), g(t, x, y) : (0, 1) × (0, ∞)2 → [0, ∞) are two continuous functions and may be singular at t = 0, 1 and x = y = 0. The existence of positive solutions is established by the upper and lower solutions technique and Schauder fixed point theorem. For the special boundary conditions u(1) = ∫01ϕ(s)u(s)ds, v(1) = ∫01φ(s)v(s)ds, where ϕ, φ ∈ L(0, 1) are nonnegative, Liu et al. [27] investigated the existence of a pair of positive solutions for nonlocal fractional differential systems (2) by constructing two cones and computing the fixed point index in product cone. For the case f = a(t)f͠(t, u(t)), g = b(t)g͠(t, v(t)), u(1) = ∫01ϕ(s)u(s)ds, v(1) = ∫01φ(s)v(s)ds, Yang [28] established sufficient conditions for the existence and nonexistence of positive solutions to fractional differential systems (2) by the Banach fixed point theorem, nonlinear differentiation of Leray-Schauder type and the fixed point theorems of cone expansion and compression of norm type.

In [29], Henderson, Luca and Tudorache discussed the systems of nonlinear fractional differential equations with integral boundary conditions

D0+αu(t)+λf(t,u(t),v(t))=0,0<t<1,n−1<α≤n,D0+βv(t)+μg(t,u(t),v(t))=0,0<t<1,m−1<β≤m,u(0)=u′(0)=⋯=u(n−2)(0)=0,u(1)=∫01u(s)dA(s),v(0)=v′(0)=⋯=v(m−2)(0)=0,v(1)=∫01v(s)dB(s)(3)

where D0+α and D0+β are the standard Riemann-Liouville fractional derivatives, f, g : [0, 1]×[0, ∞)2 → [0, ∞) are continuous. Under different combinations of superlinearity and sublinearity of the functions f and g, various existence and nonexistence results for positive solutions are derived in terms of different value of λ and μ via the Guo-Krasnosel’skii fixed point theorem. For the multi-point boundary conditions u(1) = ∑i=1paiu(ξi),v(1)=∑i=1qbiv(ηi), Henderson and Luca [30] proved the existence theorems for the positive solutions with respect to a cone for nonlinear fractional differential systems (3) by the Guo-Krasnosel’skii fixed point theorem. Under sufficient conditions on functions f and g, the authors [31] studied the existence and multiplicity of positive solutions of nonlinear fractional differential systems (3) when λ = μ = 1 and f = f͠(t, v), g = g͠(t, u) by using the Guo-Krasnosel’skii fixed point theorem and some theorems from the fixed point index theory.

In [32], Wang et al. investigated the fractional differential systems involving integral boundary conditions arising from the study of HIV infection models

D0+αu(t)+λf(t,u(t),D0+βu(t),v(t))=0,0<t<1,2<α≤3,D0+γv(t)+λg(t,u(t))=0,0<t<1,2<γ≤3,D0+βu(0)=D0+β+1u(0)=0,D0+βu(1)=∫01D0+βu(s)dA(s),v(0)=v′(0)=0,v(1)=∫01v(s)dB(s),(4)

where λ > 0 is a parameter, 0 < β < 1, α-β > 2, D0+α,D0+β and D0+γ are the standard Riemann-Liouville fractional derivatives, A, B are nondecreasing functions of bunded variations, ∫01D0+βu(s)dA(s) and ∫01v(s)dB(s) are Riemann-Stieltjes integrals, f : (0, 1) × [0, +∞)3 → (−∞, +∞) and g : (0, 1) × [0, +∞) → (-∞, +∞) are two continuous functions and may be singular at t = 0, 1. By using the fixed point theorem in cone, existence results of positive solutions for systems (4) are established.

In [33], Jiang, Liu and Wu considered the following semipositone singular fractional differential systems:

D0+αu(t)+p(t)f(t,u(t),v(t))−q1(t)=0,0<t<1,2<α≤3,D0+βv(t)+q(t)g(t,u(t),v(t))−q2(t)=0,0<t<1,2<β≤3,u(0)=u′(0)=0,u′(1)=∫01u(s)dA(s),v(0)=v′(0)=0,v′(1)=∫01v(s)dB(s),(5)

where D0+α and D0+β are the standard Riemann-Liouville fractional derivatives, f, g : [0, 1]×[0, ∞)2 → [0, ∞) are continuous, q1, q2 : (0, 1)→ [0, +∞) are Lebesgue integrable, A, B are suitable functions of bounded variation, ∫01u(t)dA(t) and ∫01v(t)dB(t) involving Stieltjes integrals with signed measures. The existence and multiplicity of positive solutions to systems (5) are obtained by using a well known fixed point theorem.

It should be noted that the nonlinearity in most of the previous works needs to be nonnegative to get the positive solutions [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. When the nonlinearity is allowed to take on both positive and negative values, such problems are called semipositone problems in the literature. Motivated by the works mentioned above, we consider the semipositone singular fractional differential systems (1). The existence of positive solutions is established by applying the fixed point theorem in cone. In comparison with previous works, this paper has several new features. Firstly, nonlinearities are allowed to change sign and tend to negative infinity. Secondly, systems (1) involves a parameter and f1, f2 involve fractional derivatives of unknown functions. Finally, the nonlocal conditions are given by Riemann-Stieltjes integrals, which include two-point, three-point, multi-point and some nonlocal conditions as special cases.

2 Preliminaries and lemmas

For convenience of the reader, we present here some necessary definitions and properties about fractional calculus theory.

Definition 2.1

The Riemann-Liouville fractional integral of order α of a function u : (0, +∞) → (−∞, +∞) is given by

I0+αu(t)=1Γ(α)∫0t(t−s)α−1u(s)ds,

provided the right-hand side is pointwise defined on (0, +∞).

Definition 2.2

([1, 2]). The Riemann-Liouville fractional derivative of order α > 0 of a continuous function u : (0, +∞) → (−∞, +∞) is given by

D0+αu(t)=1Γ(n−α)ddtn∫0t(t−s)n−α−1u(s)ds,

where n = [α]+1, [α] denotes the integer part of the number α, provided the right-hand side is pointwise defined on (0, +∞).

Lemma 2.3

([1, 2]). If u ∈ L(0, 1), ρ > σ > 0 and n is a natural number, then

I0+ρI0+σu(t)=I0+ρ+σu(t),D0+σI0+ρu(t)=I0+ρ−σu(t),D0+σI0+σu(t)=u(t),ddtn(D0+σu(t))=D0+n+σu(t).

Lemma 2.4

([1, 2]). Assume that u ∈ C(0, 1) ∩ L(0, 1) with a fractional derivative of order α > 0 that belongs to C(0, 1) ∩ L(0, 1). Then

I0+αD0+αu(t)=u(t)+c1tα−1+c2tα−2+⋯+cNtα−N,

for some c1, c2, ⋯, cN ∈ (−∞, +∞), where N is the smallest integer greater than or equal to α.

Lemma 2.5

([33]). Given h ∈ C(0, 1) ∩ L(0, 1), then the BVP

D0+α−βx(t)+h(t)=0,0<t<1,2<α−β≤3,x(0)=x′(0)=0,x′(1)=0,

has a unique solution

x(t)=∫01G1∗(t,s)h(s)ds,

where

G1∗(t,s)=1Γ(α−β)tα−β−1(1−s)α−β−2,0≤t≤s≤1,tα−β−1(1−s)α−β−2−(t−s)α−β−1,0≤s≤t≤1,

and the Green functionG1∗ (t, s) has the following properties:

  1. G1∗(t, s) > 0, t, s ∈(0, 1).

  2. k1(t) G1∗(1, s) ≤ G1∗(t, s) ≤ G1∗(1, s), t, s ∈ [0, 1].

  3. G1∗(t,s)≤k1(t)Γ(α−β),t,s∈[0,1],where k1(t) = tα−β−1.

By Lemma 2.4, the unique solution of the BVP

D0+α−βx(t)=0,0<t<1,x(0)=x′(0)=0,x′(1)=1,

is γ1(t)=tα−β−1α−β−1. By [33], the Green function for the BVP

D0+α−βx(t)+h(t)=0,0<t<1,x(0)=x′(0)=0,x′(1)=∫01x(s)dA(s),

is given by

G1(t,s)=G1∗(t,s)+γ1(t)1−ȷ1ℓ1(s),(6)

where

ȷ1=∫01γ1(t)dA(t)≠1,ℓ1(s)=∫01G1∗(t,s)dA(t),s∈[0,1].

Similarly, the Green function for the BVP

D0+γ−δy(t)+h(t)=0,0<t<1,y(0)=y′(0)=0,y′(1)=∫01y(s)dB(s),

is given by

G2(t,s)=G2∗(t,s)+γ2(t)1−ȷ2ℓ2(s),(7)

where

G2∗(t,s)=1Γ(γ−δ)tγ−δ−1(1−s)γ−δ−2,0≤t≤s≤1,tγ−δ−1(1−s)γ−δ−2−(t−s)γ−δ−1,0≤s≤t≤1,ȷ2=∫01γ2(t)dB(t)≠1,ℓ2(s)=∫01G2∗(t,s)dB(t),γ2(t)=tγ−δ−1γ−δ−1.

Lemma 2.6

([33]). The Green functionG2∗(t, s) has the following properties:

  1. G2∗(t, s) > 0, t, s ∈(0, 1).

  2. k2(t) G2∗(1, s) ≤ G2∗(t, s) ≤ G2∗(1, s), t, s ∈ [0, 1].

  3. G2∗(t,s)≤k2(t)Γ(γ−δ),t,s∈[0,1],where k2(t) = tγ−δ−1.

Lemma 2.7

([33]). Letȷ1, ȷ2 ∈ [0, 1) andℓ1(s), ℓ2(s) ≥ 0 for s ∈ [0, 1], the functions G1(t, s) and G2(t, s) given by(6)and(7)satisfy:

  1. Gi(t, s) ≥ Gi∗(t, s) > 0, t, s ∈ (0, 1), i = 1, 2.

  2. ki(t) Gi∗(1, s) ≤ Gi(t, s) ≤ ρiGi∗(1, s), t, s ∈[0, 1], i = 1, 2.

  3. G1(t,s)≤ρ1Γ(α−β)k1(t),G2(t,s)≤ρ2Γ(γ−δ)k2(t),t,s∈[0,1],where

    ρ1=1+∫01dA(t)1−ȷ1,ρ2=1+∫01dB(t)1−ȷ2.

Now let us consider the following modified problem of systems (1)

D0+α−βω(t)+λf1(t,I0+βω(t),ω(t),I0+δz(t))=0,0<t<1,D0+γ−δz(t)+λf2(t,I0+βω(t),I0+δz(t),z(t))=0,0<t<1,ω(0)=ω′(0)=0,ω′(1)=∫01ω(s)dA(s),z(0)=z′(0)=0,z′(1)=∫01z(s)dB(s).(8)

Lemma 2.8

If (ω, z) ∈ C[0, 1] × C[0, 1] is a positive solution of systems(8), then(I0+βω,I0+δz)is a positive solution of systems(1).

Proof

Suppose (ω, z) ∈ C[0, 1] × C[0, 1] is a positive solution of systems (8), denote u(t)=I0+βω(t),v(t)=I0+δz(t), then

D0+α−β(D0+βu(t))=D0+α−β(D0+βI0+βω(t))=D0+α−βω(t)=−λf1(t,I0+βω(t),ω(t),I0+δz(t))=−λf1(t,u(t),D0+βu(t),v(t)),D0+γ−δ(D0+δv(t))=D0+γ−δ(D0+δI0+δz(t))=D0+γ−δz(t)=−λf2(t,I0+βω(t),I0+δz(t),z(t))=−λf2(t,u(t),v(t),D0+δv(t)),
D0+βu(0)=ω(0)=0,D0+β+1u(0)=ddt(D0+βu(t))|t=0=ω′(0)=0,D0+β+1u(1)=ddt(D0+βu(t))|t=1=ω′(1)=∫01ω(s)dA(s)=∫01D0+βu(s)dA(s),D0+δv(0)=z(0)=0,D0+δ+1v(0)=ddt(D0+δv(t))|t=0=z′(0)=0,D0+δ+1v(1)=ddt(D0+δv(t))|t=1=z′(1)=∫01z(s)dB(s)=∫01D0+δv(s)dB(s).

On the other hand, if ω(t) > 0, z(t) > 0, by Definition 2.1, we have u(t) > 0, v(t) > 0, t ∈(0, 1), then (u, v) = (I0+βω,I0+δz) is a positive solution of systems (1).

We impose the following assumptions:

  1. A, B are increasing functions of bounded variations such that ℓ1(s) ≥ 0, ℓ2(s) ≥ 0 for s ∈ [0, 1] and 0 ≤ ȷ1, ȷ2 < 1.

  2. f1, f2 : (0, 1) × [0, +∞)3 → (−∞, +∞) are continuous and satisfy

    −q1(t)≤f1(t,u1,u2,u3)≤p1(t)g1(t,u1,u2,u3),−q2(t)≤f2(t,v1,v2,v3)≤p2(t)g2(t,v1,v2,v3),

    where gi ∈ C([0, 1] × [0, +∞)3, [0, +∞)), qi, pi ∈ C((0, 1), [0, +∞)) and

    0<∫01pi(s)ds<+∞,0<∫01qi(s)ds<+∞,i=1,2.

  3. There exists a constant

    r>max2ρ12Γ(α−β)∫01q1(s)ds,2ρ22Γ(γ−δ)∫01q2(s)ds

    such that

    f1(t,u1,u2,u3)≥0,(t,u1,u2,u3)∈(0,1)×0,r2Γ(β+1)×0,r2×[0,+∞),f2(t,v1,v2,v3)≥0,(t,v1,v2,v3)∈(0,1)×[0,+∞)×0,r2Γ(δ+1)×0,r2.

    Define a modified function [z(t)]* for any z ∈ C[0, 1] by

    [z(t)]∗=z(t),z(t)≥0,0,z(t)<0.

    Next we consider the following systems:

    D0+α−βx(t)+λ[f1(t,I0+β[x(t)−a(t)]∗,[x(t)−a(t)]∗,I0+δ[y(t)−b(t)]∗)+q1(t)]=0,0<t<1,D0+γ−δy(t)+λ[f2(t,I0+β[x(t)−a(t)]∗,I0+δ[y(t)−b(t)]∗,[y(t)−b(t)]∗)+q2(t)]=0,0<t<1,x(0)=x′(0)=0,x′(1)=∫01x(s)dA(s),y(0)=y′(0)=0,y′(1)=∫01y(s)dB(s),(9)

    where a(t) = λ ∫01G1(t, s)q1(s)ds and b(t) = λ ∫01G2(t, s)q2(s)ds are the solutions of the following BVPs (10) and (11), respectively,

    D0+α−βa(t)+λq1(t)=0,0<t<1,a(0)=a′(0)=0,a′(1)=∫01a(s)dA(s),(10)
    D0+γ−δb(t)+λq2(t)=0,0<t<1,b(0)=b′(0)=0,b′(1)=∫01b(s)dB(s).(11)

Lemma 2.9

([33]). Assume that condition (H1) holds. Then the positive solutions a(t) and b(t) of BVPs(10)and(11)satisfy

a(t)≤λρ1Γ(α−β)k1(t)∫01q1(t)dt,b(t)≤λρ2Γ(γ−δ)k2(t)∫01q2(t)dt,t∈[0,1].

Lemma 2.10

If (x, y) ∈ C[0, 1] × C[0, 1] with x(t) > a(t), y(t) > b(t) for any t ∈(0, 1) is a positive solution of systems(9), then (ω(t) = x(t) − a(t), z(t) = y(t) − b(t)) is a positive solution of systems(8), and(u(t)=I0+βω(t),v(t)=I0+δz(t))is a positive solution of systems(1).

Proof

In fact, if (x, y) ∈ C[0, 1] × C[0, 1] is a solution of systems (9) with x(t) > a(t), y(t) > b(t), then from systems (9) and the definition of []*, we get

D0+α−βω(t)=D0+α−β(x(t)−a(t))=D0+α−βx(t)−D0+α−βa(t)=−λ[f1(t,I0+β[x(t)−a(t)]∗,[x(t)−a(t)]∗,I0+δ[y(t)−b(t)]∗)+q1(t)]−[−λq1(t)]=−λf1(t,I0+β[x(t)−a(t)]∗,[x(t)−a(t)]∗,I0+δ[y(t)−b(t)]∗)=−λf1(t,I0+βω(t),ω(t),I0+δz(t)),
ω(0)=x(0)−a(0)=0,ω′(0)=x′(0)−a′(0)=0,ω′(1)=x′(1)−a′(1)=∫01x(s)dA(s)−∫01a(s)dA(s)=∫01ω(s)dA(s),

and

D0+γ−δz(t)=D0+γ−δ(y(t)−b(t))=D0+γ−δy(t)−D0+γ−δb(t)=−λ[f2(t,I0+β[x(t)−a(t)]∗,I0+δ[y(t)−b(t)]∗,[y(t)−b(t)]∗)+q2(t)]−[−λq2(t)]=−λf2(t,I0+β[x(t)−a(t)]∗,I0+δ[y(t)−b(t)]∗,[y(t)−b(t)]∗)=−λf2(t,I0+βω(t),I0+δz(t),z(t)),
z(0)=y(0)−b(0)=0,z′(0)=y′(0)−b′(0)=0,z′(1)=y′(1)−b′(1)=∫01y(s)dB(s)−∫01b(s)dB(s)=∫01z(s)dB(s).

So (ω, z) is a positive solution of systems (8). It follows from Lemma 2.8 that (u(t)=I0+βω(t),v(t)=I0+βz(t)) is a positive solution of systems (1).

Let X = C[0, 1] × C[0, 1], then X is a Banach space with the norm

∥(u,v)∥1=∥u∥+∥v∥,∥u∥=max0≤t≤1|u(t)|,∥v∥=max0≤t≤1|v(t)|,(u,v)∈X.

Let

P=(u,v)∈X:u(t)≥ρ1−1k1(t)∥u∥,v(t)≥ρ2−1k2(t)∥v∥,t∈[0,1],

then P is a cone of X. Define an operator A : P → X by

A(x,y)=(A1(x,y),A2(x,y)),

where A1, A2 : P → C[0, 1] are defined by

A1(x,y)(t)=λ∫01G1(t,s)(f1(s,I0+β[x(s)−a(s)]∗,[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗)+q1(s))ds,A2(x,y)(t)=λ∫01G2(t,s)(f2(s,I0+β[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗,[y(s)−b(s)]∗)+q2(s))ds.

Clearly, if (x, y) ∈ P is a fixed point of A, then (x, y) is a solution of systems (9).

Lemma 2.11

Assume that conditions (H1) – (H3) hold, then A : P → P is a completely continuous operator.

Proof

For any (x, y) ∈ P, there exists a constant L > 0 such that ∥(x, y)∥1 ≤ L, then

[x(s)−a(s)]∗≤x(s)≤∥x∥≤∥(x,y)∥1≤L,s∈[0,1],[y(s)−b(s)]∗≤y(s)≤∥y∥≤∥(x,y)∥1≤L,s∈[0,1],
I0+β[x(s)−a(s)]∗=∫0s(s−t)β−1[x(t)−a(t)]∗Γ(β)dt≤LΓ(β+1),I0+δ[y(s)−b(s)]∗=∫0s(s−t)δ−1[y(t)−b(t)]∗Γ(δ)dt≤LΓ(δ+1).

It follows from Lemma 2.7 that

A1(x,y)(t)≤λ∫01ρ1G1∗(1,s)(p1(s)g1(s,I0+β[x(s)−a(s)]∗,[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗)+q1(s))ds≤λρ1(M+1)∫01G1∗(1,s)(p1(s)+q1(s))ds<+∞,A2(x,y)(t)≤λ∫01ρ2G2∗(1,s)(p2(s)g2(s,I0+β[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗,[y(s)−b(s)]∗)+q2(s))ds≤λρ2(M+1)∫01G2∗(1,s)(p2(s)+q2(s))ds<+∞,

where

M=maxmaxt∈[0,1],0≤u1≤LΓ(β+1),0≤u2≤L,0≤u3≤LΓ(δ+1)g1(t,u1,u2,u3),maxt∈[0,1],0≤v1≤LΓ(β+1),0≤v2≤LΓ(δ+1),0≤v3≤Lg2(t,v1,v2,v3).

Thus, A : P → X is well defined.

Next, we prove A(P) ⊂ P. Denote

F1(s)=f1(s,I0+β[x(s)−a(s)]∗,[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗)+q1(s),s∈[0,1],F2(s)=f2(s,I0+β[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗,[y(s)−b(s)]∗)+q2(s),s∈[0,1].

For any (x, y) ∈ P, we have

∥A1(x,y)∥=max0≤t≤1|A1(x,y)(t)|≤λρ1∫01G1∗(1,s)F1(s)ds.

So

A1(x,y)(t)≥λ∫01k1(t)G1∗(1,s)F1(s)ds≥ρ1−1k1(t)∥A1(x,y)∥,t∈[0,1].

Similarly, A2(x,y)(t)≥ρ2−1k2(t)∥A2(x,y)∥,t∈[0,1]. Thus, A(P) ⊂ P.

According to the Ascoli-Arzela theorem and the Lebesgue dominated convergence theorem, we can easily get that A : P → P is completely continuous. □

Lemma 2.12

([53]). Let P be a cone in Banach space E, Ω1and Ω2are bounded open sets in E, θ ∈ Ω1, Ω1 ⊂ Ω2, A : P ∩ (Ω2∖Ω1) → P is a completely continuous operator. If the following conditions are satisfied:

∥Ax∥≤∥x∥,∀x∈P∩∂Ω1,∥Ax∥≥∥x∥,∀x∈P∩∂Ω2,or∥Ax∥≥∥x∥,∀x∈P∩∂Ω1,∥Ax∥≤∥x∥,∀x∈P∩∂Ω2,

then A has at least one fixed point in P ∩(Ω2∖Ω1).

3 Main results

Theorem 3.1

Assume that conditions (H1) – (H3) are satisfied. Further assume that the following condition holds:

(H4) There exists [a, b] ⊂(0, 1) such that

limu2→+∞mint∈[a,b]u1,u3≥0f1(t,u1,u2,u3)u2=+∞,limv3→+∞mint∈[a,b]v1,v2≥0f2(t,v1,v2,v3)v3=+∞.

Then there existsλ > 0 such that for any 0 < λ < λ, systems(1)have at least one positive solution (u, v).

Proof

Let

Ω1={(x,y)∈X:∥(x,y)∥1<r},λ¯=min1,r2ρ1(g1∗+1)∫01G1∗(1,s)[p1(s)+q1(s)]ds,r2ρ2(g2∗+1)∫01G2∗(1,s)[p2(s)+q2(s)]ds,

where

g1∗=maxg1(s,u1,u2,u3):0≤s≤1,0≤u1≤rΓ(β+1),0≤u2≤r,0≤u3≤rΓ(δ+1),g2∗=maxg2(s,v1,v2,v3):0≤s≤1,0≤v1≤rΓ(β+1),0≤v2≤rΓ(δ+1),0≤v3≤r.

Suppose 0 < λ < λ, then for any (x, y) ∈ P ∩ ∂Ω1, s ∈[0, 1], we have

0≤[x(s)−a(s)]∗≤x(s)≤∥x∥≤r,0≤[y(s)−b(s)]∗≤y(s)≤∥y∥≤r,I0+β[x(s)−a(s)]∗=∫0s(s−τ)β−1[x(τ)−a(τ)]∗Γ(β)dτ≤rΓ(β+1),I0+δ[y(s)−b(s)]∗=∫0s(s−τ)δ−1[y(τ)−b(τ)]∗Γ(δ)dτ≤rΓ(δ+1).

Thus,

∥A1(x,y)∥≤λ∫01ρ1G1∗(1,s)[f1(s,I0+β[x(s)−a(s)]∗,[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗)+q1(s)]ds≤λρ1∫01G1∗(1,s)[p1(s)g1(s,I0+β[x(s)−a(s)]∗,[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗)+q1(s)]ds<λ¯ρ1(g1∗+1)∫01G1∗(1,s)[p1(s)+q1(s)]ds≤r2,

and

∥A2(x,y)∥≤λ∫01ρ2G2∗(1,s)[f2(s,I0+β[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗,[y(s)−b(s)]∗)+q1(s)]ds≤λρ2∫01G2∗(1,s)[p2(s)g2(s,I0+β[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗,[y(s)−b(s)]∗)+q2(s)]ds<λ¯ρ2(g2∗+1)∫01G2∗(1,s)[p2(s)+q2(s)]ds≤r2.

So

∥A(x,y)∥1=∥A1(x,y)∥+∥A2(x,y)∥≤r=∥(x,y)∥1,∀(x,y)∈P∩∂Ω1.(12)

On the other hand, let ki~=mint∈[a,b]ki(t) and L~>max4ρ1λk1~2∫abG1∗(1,s)ds,4ρ2λk2~2∫abG2∗(1,s)ds. By (H4), there exists N > 0 such that

f1(t,u1,u2,u3)≥L~u2,t∈[a,b],u2≥N,u1,u3≥0,f2(t,v1,v2,v3)≥L~v3,t∈[a,b],v3≥N,v1,v2≥0.

Let

R>max2r,4ρ1Nk1~,4ρ2Nk2~,Ω2={(x,y)∈P:∥(x,y)∥1<R}.

For any (x, y) ∈ P ∩ ∂Ω2, ∥(x, y)∥1 = R, we have ∥x∥≥R2 or ∥y∥≥R2. If ∥x∥≥R2, we deduce

x(t)−a(t)≥ρ1−1k1(t)∥x∥−ρ1Γ(α−β)k1(t)∫01q1(s)ds=k1(t)∥x∥ρ1−ρ1Γ(α−β)∫01q1(s)ds≥k1~R2ρ1−R4ρ1=k1~4ρ1R≥N,t∈[a,b],

then

f1(s,I0+β[x(s)−a(s)]∗,[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗)=f1(s,I0+β(x(s)−a(s)),(x(s)−a(s)),I0+δ[y(s)−b(s)]∗)≥L~[x(s)−a(s)]≥k1~L~R4ρ1.

Thus

∥A1(x,y)∥=maxt∈[0,1]λ∫01G1(t,s)[f1(s,I0+β[x(s)−a(s)]∗,[x(s)−a(s)]∗,I0+δ[y(s)−b(s)]∗)+q1(s)]ds≥maxt∈[0,1]λ∫abk1(t)G1∗(1,s)L~(x(s)−a(s))ds≥λk1~L~∫abG1∗(1,s)k1~4ρ1Rds=λk1~2L~4ρ1R∫abG1∗(1,s)ds≥R=∥(x,y)∥1.

If ∥y∥≥R2, in a similar manner, we have y(t)−b(t)≥k2~4ρ2R≥N,t∈[a,b], and

∥A2(x,y)∥≥λk2~2L~4ρ2R∫abG2∗(1,s)ds≥R=∥(x,y)∥1.

Thus

∥A(x,y)∥1=∥A1(x,y)∥+∥A2(x,y)∥≥∥(x,y)∥1,∀(x,y)∈P∩∂Ω2.(13)

By (12), (13) and Lemma 2.12, A has a fixed point (x͠, y͠) with r ≤ ∥(x͠, y͠)∥1 ≤ R. Now, we will prove x͠(t) > a(t), y͠(t) ≥ b(t)(or x͠(t) ≥ a(t), y͠(t) > b(t)), t ∈ (0, 1). We shall divide the proof into three cases: (i) ∥x~∥≥r2,∥y~∥≥r2;(ii)∥x~∥>r2,∥y~∥<r2;(iii)∥x~∥<r2,∥y~∥>r2.

  1. If ∥x͠∥ ≥ r2, from (H3) and Lemma 2.9, we get

    x~(t)≥ρ1−1k1(t)∥x~∥≥ρ1−1k1(t)r2>ρ1−1k1(t)ρ12Γ(α−β)∫01q1(s)ds=ρ1Γ(α−β)k1(t)∫01q1(s)ds≥a(t),t∈(0,1).

    Similarly, if ∥y͠∥ ≥ r2 we obtain y͠(t) > b(t), t ∈ (0, 1).

  2. For ∥x͠∥ > r2, similar to (i), we have x͠(t) > a(t), t ∈ (0, 1). For ∥y͠∥ < r2, we have

    0≤[y~(s)−b(s)]∗≤y~(s)≤∥y~∥≤r2,0≤I0+β[x~(s)−a(s)]∗=I0+β(x~(s)−a(s))≤RΓ(β+1),0≤I0+δ[y~(s)−b(s)]∗≤r2Γ(δ+1).

    It follows from (H2) that

    f2(t,v1,v2,v3)≥0,(t,v1,v2,v3)∈(0,1)×0,RΓ(β+1)×0,r2Γ(δ+1)×0,r2,

    then

    y~(t)=λ∫01G2(t,s)[f2(s,I0+β[x~(s)−a(s)]∗,I0+δ[y~(s)−b(s)]∗,[y~(s)−b(s)]∗)+q2(s)]ds≥λ∫01G2(t,s)q2(s)ds=b(t),t∈(0,1).

  3. If ∥x͠∥ < r2 and ∥y͠∥ > r2, similar to (ii), we have x͠(t) ≥ a(t), y͠(t) > b(t), t ∈ (0, 1).

So by Lemma 2.10 we know that (u¯(t),v¯(t))=(I0+β(x~(t)−a(t)),I0+δ(y~(t)−b(t))) is a positive solution of systems (1). □

4 An Example

Example 4.1

Consider the following problem:

D0+52(D0+14u(t))+λπt(1−t)(u(t)−90)2+(D0+14u(t)−3)4+v2(t)685t4−148=0,t∈(0,1),D0+94(D0+18v(t))+λΓ(94)1−t4u2(t)+(v(t)−30)2+(D0+18v(t)−3)230t−112=0,t∈(0,1),D0+14u(0)=D0+54u(0)=0,D0+54u(1)=9697D0+14u116,D0+18v(0)=D0+98v(0)=0,D0+98v(1)=4041D0+18v116.(14)

Problem (14) can be regarded as a problem of the form (1) with α=114,β=14,γ=198,δ=18,

A(t)=0,t∈0,116,9697,t∈116,1,B(t)=0,t∈0,116,4041,t∈116,1,
f1(t,u1,u2,u3)=πt341−t(u1−90)2+(u2−3)4+u32685−π48t(1−t),f2(t,v1,v2,v3)=Γ(94)t1−t4v12+(v2−30)2+(v3−3)230−Γ(94)121−t4.

Evidently, f1, f2 : (0, 1) × [0, +∞)3 → (−∞, +∞) are continuous and singular at t = 0, 1,

By direct calculations, we get

k1(t)=t32,k2(t)=t54,γ1(t)=23t32,γ2(t)=45t54,ρ1=2,ρ2=2,ȷ1=∫01γ1(t)dA(t)=9697×13×11632=197,ℓ1(s)=9697G1∗116,s≥0,ȷ2=∫01γ2(t)dB(t)=4041×45×11654=141,ℓ2(s)=4041G2∗116,s≥0,

so condition (H1) is satisfied. Let

q1(t)=π48t(1−t),p1(t)=πt341−t,g1(t,u2,u2,u3)=(u1−90)2+(u2−3)4+u32685,q2(t)=Γ(94)121−t4,p2(t)=Γ(94)t1−t4,g2(t,v2,v2,v3)=v12+(v2−30)2+(v3−3)230,

then

∫01q1(t)dt=π48B12,12=ππ48,∫01q2(t)dt=Γ(94)9,∫01p1(t)dt=πB14,12,∫01p2(t)dt=Γ94B34,12,

so condition (H2) is satisfied. On the other hand,

max2ρ12Γ(α−β)∫01q1(s)ds,2ρ22Γ(γ−δ)∫01q2(s)ds=89.

Select r = 2, then

f1(t,u1,u2,u3)≥πt(1−t)(u1−90)2+(u2−3)4+u32685−148>0,(t,u1,u2,u3)∈(0,1)×0,1Γ(54)×[0,1]×[0,+∞),
f2(t,v1,v2,v3)≥Γ(94)1−t4v12+(v2−30)2+(v3−3)230−112>0,(t,v1,v2,v3)∈(0,1)×[0,+∞)×0,1Γ(98)×[0,1].

Hence, condition (H3) is satisfied. In addition, for any [a, b] ⊂ (0, 1),

limu2→+∞mint∈[a,b]u1,u3≥0⁡f1(t,u1,u2,u3)u2=+∞,limv3→+∞mint∈[a,b]v1,v2≥0⁡f2(t,v1,v2,v3)v3=+∞,

condition (H4) is satisfied. Hence, problem (14) has at least one positive solution by Theorem 3.1.

Acknowledgement

The authors would like to thank the referees for their pertinent comments and valuable suggestions. This work is supported financially by the National Natural Science Foundation of China (11501318) and the China Postdoctoral Science Foundation (2017M612230).

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Received: 2017-11-19
Accepted: 2018-04-19
Published Online: 2018-06-07

© 2018 Hao and Wang, published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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  17. A further study on ordered regular equivalence relations in ordered semihypergroups
  18. On the structure vector field of a real hypersurface in complex quadric
  19. Rank relations between a {0, 1}-matrix and its complement
  20. Lie n superderivations and generalized Lie n superderivations of superalgebras
  21. Time parallelization scheme with an adaptive time step size for solving stiff initial value problems
  22. Stability problems and numerical integration on the Lie group SO(3) × R3 × R3
  23. On some fixed point results for (s, p, α)-contractive mappings in b-metric-like spaces and applications to integral equations
  24. On algebraic characterization of SSC of the Jahangir’s graph 𝓙n,m
  25. A greedy algorithm for interval greedoids
  26. On nonlinear evolution equation of second order in Banach spaces
  27. A primal-dual approach of weak vector equilibrium problems
  28. On new strong versions of Browder type theorems
  29. A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices
  30. Restriction conditions on PL(7, 2) codes (3 ≤ |𝓖i| ≤ 7)
  31. Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents
  32. Introduction to disoriented knot theory
  33. Restricted triangulation on circulant graphs
  34. Boundedness control sets for linear systems on Lie groups
  35. Chen’s inequalities for submanifolds in (κ, μ)-contact space form with a semi-symmetric metric connection
  36. Disjointed sum of products by a novel technique of orthogonalizing ORing
  37. A parametric linearizing approach for quadratically inequality constrained quadratic programs
  38. Generalizations of Steffensen’s inequality via the extension of Montgomery identity
  39. Vector fields satisfying the barycenter property
  40. On the freeness of hypersurface arrangements consisting of hyperplanes and spheres
  41. Biderivations of the higher rank Witt algebra without anti-symmetric condition
  42. Some remarks on spectra of nuclear operators
  43. Recursive interpolating sequences
  44. Involutory biquandles and singular knots and links
  45. Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉
  46. Topological entropy for positively weak measure expansive shadowable maps
  47. Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values
  48. On 𝓠-regular semigroups
  49. One kind power mean of the hybrid Gauss sums
  50. A reduced space branch and bound algorithm for a class of sum of ratios problems
  51. Some recurrence formulas for the Hermite polynomials and their squares
  52. A relaxed block splitting preconditioner for complex symmetric indefinite linear systems
  53. On f - prime radical in ordered semigroups
  54. Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions
  55. Disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators
  56. A stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment
  57. Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod-Haldane functional response
  58. The points and diameters of quantales
  59. Directed colimits of some flatness properties and purity of epimorphisms in S-posets
  60. Super (a, d)-H-antimagic labeling of subdivided graphs
  61. On the power sum problem of Lucas polynomials and its divisible property
  62. Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential
  63. On generalized P-reducible Finsler manifolds
  64. On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences
  65. On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque Lp,α spaces
  66. On the different kinds of separability of the space of Borel functions
  67. Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers
  68. Functional analysis method for the M/G/1 queueing model with single working vacation
  69. Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions
  70. The existence of solutions to certain type of nonlinear difference-differential equations
  71. Domination in 4-regular Knödel graphs
  72. Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay
  73. Algebras of right ample semigroups
  74. Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains
  75. Nontrivial periodic solutions to delay difference equations via Morse theory
  76. A note on the three-way generalization of the Jordan canonical form
  77. On some varieties of ai-semirings satisfying xp+1 ≈ x
  78. Abstract-valued Orlicz spaces of range-varying type
  79. On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums
  80. Arithmetic of generalized Dedekind sums and their modularity
  81. Multipreconditioned GMRES for simulating stochastic automata networks
  82. Regularization and error estimates for an inverse heat problem under the conformable derivative
  83. Transitivity of the εm-relation on (m-idempotent) hyperrings
  84. Learning Bayesian networks based on bi-velocity discrete particle swarm optimization with mutation operator
  85. Simultaneous prediction in the generalized linear model
  86. Two asymptotic expansions for gamma function developed by Windschitl’s formula
  87. State maps on semihoops
  88. 𝓜𝓝-convergence and lim-inf𝓜-convergence in partially ordered sets
  89. Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation
  90. New topology in residuated lattices
  91. Optimality and duality in set-valued optimization utilizing limit sets
  92. An improved Schwarz Lemma at the boundary
  93. Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit
  94. Toeplitz matrices whose elements are coefficients of Bazilevič functions
  95. Epi-mild normality
  96. Nonlinear elastic beam problems with the parameter near resonance
  97. Orlicz difference bodies
  98. The Picard group of Brauer-Severi varieties
  99. Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
  100. Weak group inverse
  101. Infinite growth of solutions of second order complex differential equation
  102. Semi-Hurewicz-Type properties in ditopological texture spaces
  103. Chaos and bifurcation in the controlled chaotic system
  104. Translatability and translatable semigroups
  105. Sharp bounds for partition dimension of generalized Möbius ladders
  106. Uniqueness theorems for L-functions in the extended Selberg class
  107. An effective algorithm for globally solving quadratic programs using parametric linearization technique
  108. Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
  109. On categorical aspects of S -quantales
  110. On the algebraicity of coefficients of half-integral weight mock modular forms
  111. Dunkl analogue of Szász-mirakjan operators of blending type
  112. Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
  113. Global stability of a distributed delayed viral model with general incidence rate
  114. Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
  115. Boundary value problems of a discrete generalized beam equation via variational methods
  116. Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
  117. Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
  118. Spectrum of free-form Sudoku graphs
  119. Regularity of fuzzy convergence spaces
  120. The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
  121. On further refinements for Young inequalities
  122. Pretty good state transfer on 1-sum of star graphs
  123. On a conjecture about generalized Q-recurrence
  124. Univariate approximating schemes and their non-tensor product generalization
  125. Multi-term fractional differential equations with nonlocal boundary conditions
  126. Homoclinic and heteroclinic solutions to a hepatitis C evolution model
  127. Regularity of one-sided multilinear fractional maximal functions
  128. Galois connections between sets of paths and closure operators in simple graphs
  129. KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
  130. θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
  131. An integral that counts the zeros of a function
  132. On rough sets induced by fuzzy relations approach in semigroups
  133. Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
  134. The fourth order strongly noncanonical operators
  135. Topical Issue on Cyber-security Mathematics
  136. Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
  137. Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
  138. On dynamic network security: A random decentering algorithm on graphs
https://doi.org/10.1515/math-2018-0055
Keywords for this article
Positive solutions; Semipositone; Singular fractional differential systems; Integral boundary conditions
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Articles in the same Issue

  1. Regular Articles
  2. Algebraic proofs for shallow water bi–Hamiltonian systems for three cocycle of the semi-direct product of Kac–Moody and Virasoro Lie algebras
  3. On a viscous two-fluid channel flow including evaporation
  4. Generation of pseudo-random numbers with the use of inverse chaotic transformation
  5. Singular Cauchy problem for the general Euler-Poisson-Darboux equation
  6. Ternary and n-ary f-distributive structures
  7. On the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics
  8. Evaluation of integrals with hypergeometric and logarithmic functions
  9. Bounded solutions of self-adjoint second order linear difference equations with periodic coeffients
  10. Oscillation of first order linear differential equations with several non-monotone delays
  11. Existence and regularity of mild solutions in some interpolation spaces for functional partial differential equations with nonlocal initial conditions
  12. The log-concavity of the q-derangement numbers of type B
  13. Generalized state maps and states on pseudo equality algebras
  14. Monotone subsequence via ultrapower
  15. Note on group irregularity strength of disconnected graphs
  16. On the security of the Courtois-Finiasz-Sendrier signature
  17. A further study on ordered regular equivalence relations in ordered semihypergroups
  18. On the structure vector field of a real hypersurface in complex quadric
  19. Rank relations between a {0, 1}-matrix and its complement
  20. Lie n superderivations and generalized Lie n superderivations of superalgebras
  21. Time parallelization scheme with an adaptive time step size for solving stiff initial value problems
  22. Stability problems and numerical integration on the Lie group SO(3) × R3 × R3
  23. On some fixed point results for (s, p, α)-contractive mappings in b-metric-like spaces and applications to integral equations
  24. On algebraic characterization of SSC of the Jahangir’s graph 𝓙n,m
  25. A greedy algorithm for interval greedoids
  26. On nonlinear evolution equation of second order in Banach spaces
  27. A primal-dual approach of weak vector equilibrium problems
  28. On new strong versions of Browder type theorems
  29. A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices
  30. Restriction conditions on PL(7, 2) codes (3 ≤ |𝓖i| ≤ 7)
  31. Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents
  32. Introduction to disoriented knot theory
  33. Restricted triangulation on circulant graphs
  34. Boundedness control sets for linear systems on Lie groups
  35. Chen’s inequalities for submanifolds in (κ, μ)-contact space form with a semi-symmetric metric connection
  36. Disjointed sum of products by a novel technique of orthogonalizing ORing
  37. A parametric linearizing approach for quadratically inequality constrained quadratic programs
  38. Generalizations of Steffensen’s inequality via the extension of Montgomery identity
  39. Vector fields satisfying the barycenter property
  40. On the freeness of hypersurface arrangements consisting of hyperplanes and spheres
  41. Biderivations of the higher rank Witt algebra without anti-symmetric condition
  42. Some remarks on spectra of nuclear operators
  43. Recursive interpolating sequences
  44. Involutory biquandles and singular knots and links
  45. Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉
  46. Topological entropy for positively weak measure expansive shadowable maps
  47. Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values
  48. On 𝓠-regular semigroups
  49. One kind power mean of the hybrid Gauss sums
  50. A reduced space branch and bound algorithm for a class of sum of ratios problems
  51. Some recurrence formulas for the Hermite polynomials and their squares
  52. A relaxed block splitting preconditioner for complex symmetric indefinite linear systems
  53. On f - prime radical in ordered semigroups
  54. Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions
  55. Disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators
  56. A stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment
  57. Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod-Haldane functional response
  58. The points and diameters of quantales
  59. Directed colimits of some flatness properties and purity of epimorphisms in S-posets
  60. Super (a, d)-H-antimagic labeling of subdivided graphs
  61. On the power sum problem of Lucas polynomials and its divisible property
  62. Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential
  63. On generalized P-reducible Finsler manifolds
  64. On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences
  65. On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque Lp,α spaces
  66. On the different kinds of separability of the space of Borel functions
  67. Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers
  68. Functional analysis method for the M/G/1 queueing model with single working vacation
  69. Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions
  70. The existence of solutions to certain type of nonlinear difference-differential equations
  71. Domination in 4-regular Knödel graphs
  72. Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay
  73. Algebras of right ample semigroups
  74. Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains
  75. Nontrivial periodic solutions to delay difference equations via Morse theory
  76. A note on the three-way generalization of the Jordan canonical form
  77. On some varieties of ai-semirings satisfying xp+1 ≈ x
  78. Abstract-valued Orlicz spaces of range-varying type
  79. On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums
  80. Arithmetic of generalized Dedekind sums and their modularity
  81. Multipreconditioned GMRES for simulating stochastic automata networks
  82. Regularization and error estimates for an inverse heat problem under the conformable derivative
  83. Transitivity of the εm-relation on (m-idempotent) hyperrings
  84. Learning Bayesian networks based on bi-velocity discrete particle swarm optimization with mutation operator
  85. Simultaneous prediction in the generalized linear model
  86. Two asymptotic expansions for gamma function developed by Windschitl’s formula
  87. State maps on semihoops
  88. 𝓜𝓝-convergence and lim-inf𝓜-convergence in partially ordered sets
  89. Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation
  90. New topology in residuated lattices
  91. Optimality and duality in set-valued optimization utilizing limit sets
  92. An improved Schwarz Lemma at the boundary
  93. Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit
  94. Toeplitz matrices whose elements are coefficients of Bazilevič functions
  95. Epi-mild normality
  96. Nonlinear elastic beam problems with the parameter near resonance
  97. Orlicz difference bodies
  98. The Picard group of Brauer-Severi varieties
  99. Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
  100. Weak group inverse
  101. Infinite growth of solutions of second order complex differential equation
  102. Semi-Hurewicz-Type properties in ditopological texture spaces
  103. Chaos and bifurcation in the controlled chaotic system
  104. Translatability and translatable semigroups
  105. Sharp bounds for partition dimension of generalized Möbius ladders
  106. Uniqueness theorems for L-functions in the extended Selberg class
  107. An effective algorithm for globally solving quadratic programs using parametric linearization technique
  108. Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
  109. On categorical aspects of S -quantales
  110. On the algebraicity of coefficients of half-integral weight mock modular forms
  111. Dunkl analogue of Szász-mirakjan operators of blending type
  112. Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
  113. Global stability of a distributed delayed viral model with general incidence rate
  114. Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
  115. Boundary value problems of a discrete generalized beam equation via variational methods
  116. Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
  117. Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
  118. Spectrum of free-form Sudoku graphs
  119. Regularity of fuzzy convergence spaces
  120. The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
  121. On further refinements for Young inequalities
  122. Pretty good state transfer on 1-sum of star graphs
  123. On a conjecture about generalized Q-recurrence
  124. Univariate approximating schemes and their non-tensor product generalization
  125. Multi-term fractional differential equations with nonlocal boundary conditions
  126. Homoclinic and heteroclinic solutions to a hepatitis C evolution model
  127. Regularity of one-sided multilinear fractional maximal functions
  128. Galois connections between sets of paths and closure operators in simple graphs
  129. KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
  130. θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
  131. An integral that counts the zeros of a function
  132. On rough sets induced by fuzzy relations approach in semigroups
  133. Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
  134. The fourth order strongly noncanonical operators
  135. Topical Issue on Cyber-security Mathematics
  136. Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
  137. Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
  138. On dynamic network security: A random decentering algorithm on graphs
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