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Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence

  • Yuhua Long EMAIL logo and Baoling Zeng
Published/Copyright: December 29, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

In this paper, we study second-order nonlinear discrete Robin boundary value problem with parameter dependence. Applying invariant sets of descending flow and variational methods, we establish some new sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions of the system when the parameter belongs to appropriate intervals. In addition, an example is given to illustrate our results.

Keywords: Sign-changing solution; Difference equation; Robin boundary value problem; Invariant sets of descending flow
MSC 2010: 39A11

1 Introduction

Throughout this paper, let N, Z and R denote the sets of all nature numbers, integers and real numbers, respectively. We consider the following second-order nonlinear difference equation with Robin boundary value problem (BVP for short)

−Δ2x(k−1)=αf(k,x(k)),k∈[1,T],x(0)=Δx(T)=0, (1)

where T ≥ 2 is a given integer, [1,T] = :{1, 2, ⋯, T}, parameter α > 0, f:[1,T]× R → R is continuous in the second variable, Δ denotes the forward difference operator defined by Δ x(k) = x(k+1)−x(k), Δ2 x(k) = Δ(Δ x(k)).

Discrete nonlinear equations with parameter dependence play an important role in describing many physical problems, such as nonlinear elasticity theory or mechanics and engineering topics [1, 2]. In recent years, some authors also contributed to the study of (1) and obtained some interesting results. For example, when α = 1, Jiang and Zhou [3] employed strongly monotone operator and critical point theory to establish the existence of nontrivial positive solutions. By virtue of variational methods and critical point theory, Guo and Song [4] investigated the existence of positive solutions. Zhang and Xu [5] obtained the existence and uniqueness theorems of nontrivial solutions. For results on nonlinear difference equations with other boundary value problems, we can see [6, 7] and references therein. With reference to the sign-changing solution, many scholars studied it for differential equations by a variety of methods and techniques, such as Leray-Schauder degree theory [8], fixed point index theory [9], topological degree theory [10], critical point theory and invariant sets of descending flow [11, 12, 13] etc. However, to the author’s knowledge, there were few papers [14] that considered the sign-changing solutions for discrete boundary problems. One of the reasons is the fact that the traditional pasting techniques of differential equations are not applicable in the world of difference equations. The corresponding anchoring techniques are rather complicated, see the detailed discussion in [15].

View from above reasons and motivated by [14], the purpose of this paper is to apply invariant sets of descending flow and variational techniques to get some sufficient conditions for the existence of sign-changing solutions, negative solutions and positive solutions to (1).

In the following, we first consider the linear eigenvalue problem corresponding to (1)

−Δ2x(k−1)=λx(k),k∈[1,T],x(0)=Δx(T)=0, (2)

Let λk be eigenvalues of (2) and {zk}k=1T be the corresponding eigenvectors of {λk}k=1T, then

λk=4sin2⁡(2k−12T+1⋅π2),zk(j)=sin⁡(2k−1)πj2T+1,k,j=1,2,…,T.

Obviously, λ1=4sin2⁡π2(2T+1)>0 and z1(j)=sin⁡jπ2T+1>0 for j ∈ [1,T].

In this paper, we focus on the following assumptions:

  1. f0=maxk∈[1,T]lim supu→0|f(k,u)u|<λ1.

  2. lim|u|→∞f(k,u)u=r for k ∈ [1,T], where r ∈ (0,+ ∞) is a constant, or r = + ∞, ν > 2 and C > 0 satisfy |f(k,u)| ≤ C(1+|u|ν−1),k ∈ [1,T], u ∈ R.

  3. Either (i) lim|u|→∞[uf(k,u)−2F(k,u)]=−∞, uniformly for k ∈ [1,T],

    or (ii) lim|u|→∞[uf(k,u)−2F(k,u)]=+∞, uniformly for k ∈ [1,T],

where F(k,u)=∫0uf(k,s)ds.

Our results read as follows:

Theorem 1.1

Suppose (J1), (J2) hold and r > λ2α . If α∈12,+∞ and r is not an eigenvalue of (2), then (1) has at least three nontrivial solutions, one sign-changing, one positive and one negative.

Theorem 1.2

Suppose (J1)–(J3) hold and r > λ2α . If r is an eigenvalue of (2), then (1) has at least three nontrivial solutions, one sign-changing, one positive and one negative.

The remainder of this paper is organized as follows. After introducing some notations and preliminary results in Section 2, we complete the proof of Theorem 1.1 and give an example to illustrate our result in Section 3.

2 Variational structure and preliminary results

Given m ≥ 0, let G = {x:[0,T+1]→ R|x(0) = Δ x(T) = 0} be a T-dimensional Hilbert space which is equipped with the inner product

x,ym=∑k=1T[Δx(k−1)Δy(k−1)+mx(k)y(k)],forallx,y∈G,

then the induced norm ∥⋅∥m is

∥x∥m=(∑k=1T[|Δx(k−1)|2+m|x(k)|2])12,forallx∈G.

Let H be the T-dimensional Hilbert space equipped with the usual inner product (⋅,⋅) and norm ∥⋅∥. It is easy to see that G is isomorphic to H, ∥⋅∥m and ∥⋅∥ are equivalent. Denote x+ = max{x,0},x− = min{x,0}. Then for any x ∈ H, 〈 ⋅,⋅〉 m ≥ 0.

Define functional I:H→ ℝ as

I(x)=12∑k=1T|Δx(k−1)|2−α∑k=1TF(k,x(k)). (3)

For any x = (x(1), x(2), ⋯, x(T))⊤ ∈ H, I(x) can be rewritten as

I(x)=12(Ax,x)−α∑k=1TF(k,x(k)), (4)

here ατ is the transpose of the vector α on H, A is T× T matrix

2−10⋅⋅⋅00−12−1⋅⋅⋅000−12⋅⋅⋅00⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅000⋅⋅⋅2−1000⋅⋅⋅−11.

Remark 2.1

In fact, many existing results are applicable. Namely, one can apply numerous results for the variational formulation (4), see [16, 17, 18].

For m ≥ 0, consider BVP

−Δ2x(k−1)+mx(k)=h(k),k∈[1,T],x(0)=Δx(T)=0, (5)

where h:[1,T]→R. It is not hard to see that (5) and the system of linear algebra equations (A+mI)x = h are equivalent, then the unique solution of (5) can be expressed by

x=(A+mI)−1h. (6)

On the other side, we have

Lemma 2.2

The unique solution of (5) is

x(k)=∑s=1TGm(k,s)h(s),k∈[0,T+1],

here Gm(k,s) can be written as

Gm(k,s)=(pT−s−ps−T−pT+1−s+ps−T−1)(pk−p−k)W,0≤k≤s≤T+1,(pT−k−pk−T−pT+1−k+pk−T−1)(ps−p−s)W,0≤s≤k≤T+1,

W = (pT+1−p−T−1−pT+p−T)(p−1−p), p=2+m+4m+m22.

Proof

First study the homogeneous equation of (5)

−Δ2x(k−1)+mx(k)=0,k∈[1,T],x(0)=Δx(T)=0, (7)

then the corresponding characteristic equation of (7) is p2−(2+m)p+1 = 0. Consider m > 0 (m = 0 is trivial), then (2+m)2−4 > 0, which means we have

p1=2+m+4m+m22,p2=2+m−4m+m22.

Then two independent solutions of (7) can be expressed by x1(k) = p1k and x2(k) = p2k . Therefore, the general solution of (5) is x(k) = a1(k) p1k +a2(k) p2k .

The next step is to determine coefficients a1(k) and a2(k). Using the method of variation of constant, we get the general solution of (5) as

x(k)=a1(0)+∑s=1kp2sh(s)p2−p1p1k+a2(0)−∑s=1kp1sh(s)p2−p1p2k.

From initial conditions, we find a1(0) = −a2(0) and

a1(0)=∑s=1T(p1Tp2s−p1sp2T−p2sp1T+1+p1sp2T+1)(p1T+1−p2T+1−p1T+p2T)(p2−p1)h(s).

Write W=(p1T+1−p2T+1−p1T+p2T)(p2−p1),p1=p2−1=p, then

x(k)=1W[∑s=1T(pT−s−ps−T−pT+1−s+ps−T−1)(pk−p−k)+∑s=1k(Pk−s−Ps−k)(pT+1−p−T−1−pT+p−T)]h(s)

which means the proof of Lemma 2.2 is completed. □

Remark 2.3

From Lemma 2.2, for k,s ∈ [1,T], there holds Gm(k,s) = Gm(s,k) > 0. For any x ∈ H, k ∈ [1,T], define Km, fm, Am:H → H as follows

(Kmx)(k)=∑s=1TGm(k,s)x(s),(fmx)(k)=f(k,x(k))+mx(k),Am=Kmfm

where Am:H → H is a completely continuous operator. Combining (6) with Lemma 2.2, we get Km = (A+mI)−1.

Remark 2.4

According to Lemma 2.2, it is not difficult to see that {x(k)}k=0T+1 is a solution of (1) if and only if x={x(k)}k=1T∈H is a fixed point of Am.

Lemma 2.5

The functional I defined by (3) is Fréchet differentiable on H and I′(x) has the expression I′(x) = x−Km fmx for x ∈ H.

Proof

For any x,y ∈ H, using the mean value theorem, it follows I(x+y)−I(x)=12∑k=1T|Δy(k−1)|2+∑k=1T[Δx(k−1)Δy(k−1)−αf(k,x(k)+θ(k)y(k))y(k)],

here θ(k) ∈ (0,1), k ∈ [1,T]. As f is continuous in x, we find

I(x+y)−I(x)−x,ym+∑k=1T(αf(k,x(k))+mx(k))y(k)=∥y∥mo(1)

which leads to

lim∥y∥m→01∥y∥m(α∑k=1T[f(k,y(k)−f(k,x(k)+θ(k)y(k)))y(k)]+12∥y∥m2−12m∥y∥2)=0,

thus I is Fréchet differentiable on H and

I′(x),ym=x,ym−∑k=1T(αf(k,x(k))+mx(k))y(k). (8)

On the other side, for all x = {x(k)} ∈ H, z = {z(k)} ∈ H, there holds

∑k=1TΔ2x(k−1)y(k)=∑k=1T[Δx(k)y(k)−Δx(k−1)y(k)]=−∑k=1TΔx(k−1)Δy(k−1).

Making use of the definition of inner product and Lemma 2.2, we get

x−Kmfmx,ym=x,ym−∑k=1T(αf(k,x(k))+mx(k))y(k)

then 〈I′(x),y〉m = 〈x−Kmfmx,y〉m for all x,y ∈ H, i.e., IȲ(x) = x−Km fmx. This completes the proof of Lemma 2.5. □

Remark 2.6

According to Lemma 2.5 and Remark 2.4, we find out that critical points of I defined on H are precisely solutions of (1).

Definition 2.7

([19]). Let I be a C1 functional defined on E. I is said to satisfy Palais-Smale condition ((PS)condition for short) if any sequence {un} ⊂ E for which I(un) is bounded and I′(un) → 0(n → ∞) possesses a convergent subsequence in E.

Definition 2.8

([20]). Assume I be a C1 functional defined on E. If any sequence {un} such that I(un) is bounded and (1+∥un∥m)∥I′(un)∥m → 0 as n → ∞ has a convergent subsequence in E, then we say that I satisfies the Cerami condition ((C) condition for short).

Lemma 2.9

([21]). Let H be a Hilbert space, there are two open convex subsets B1 and B2 on H with Am(∂ B1) ⊂ B1, Am(∂ B2) ⊂ B2 and B1 ⋂ B2 ≠ ∅. If I ∈ C1(H,R) satisfies the (PS) condition and I′(x) = x−Amx for all x ∈ H. Assume there is a path g:[0,1] → H such that

g(0)∈B1∖B2,g(1)∈B2∖B1,

and

infu∈B1¯⋂B2¯I(x)>supτ∈[0,1]I(g(τ)),

then I has at least four critical points, one in H ∖(B1 ⋃ B2), one in B1 ∖ B2, one in B1 ⋂ B2, and one in B2 ∖ B1.

Remark 2.10

By Theorem 5.1 [20], we can replace (PS) condition by weaker (C) condition in Lemma 2.9.

In this paper, we will analyse the properties of the flow, pay close attention to the direction and the destination to which the flow goes, and seek the limit along the flow. We are interested in those points in H across which the flow does not go to infinity and work for seeking such points in H. If we have such a point, then the flow curve crossing it goes ultimately to a critical point. It seems that one would obtain many critical points if he or she is given many such points. However, even if there may be many such points, we cannot get more than one critical point in general since the different flow curves may ultimately go to the same critical point. In order to get more critical points, we will define the concept of invariant set of descending flow and then we will divide the whole space H into several invariant subsets of descending flow. In this way, we can get more than one critical point.

3 Proof of main result

Let convex cones Λ = {x ∈ H:x ≥ 0} and −Λ = {x ∈ H:x ≤ 0}. The distance respecting to ∥⋅∥m in H is written by distm. For arbitrary ε > 0, we denote

Bε+={x∈H:distm(x,Λ)<ε},Bε−={x∈H:distm(x,−Λ)<ε}.

Notice that Bε+ and Bε− are open convex subsets on H with Bε+⋂Bε−≠∅ and H∖(Bε+⋃Bε−) contains only sign-changing functions.

Lemma 3.1

Suppose one of the following condition holds.

  1. r = + ∞ or

  2. r < + ∞ is not an eigenvalue of (2), here r is defined by (J2).

Then the functional I defined by (3) satisfies (PS) condition for all α∈12,+∞.

Proof

  1. Assume r = + ∞. Let {xn} ⊂ H be a (PS) sequence. Since H is a finite dimensional space, we only need to show {xn} is bounded. If r = + ∞, choosing a constant y > 0, for all (k,u) ∈ [1,T]× R, we have F(k,u) ≥ λT u2 −y. Then

    I(xn)=12(Axn,xn)−α∑k=1TF(k,xn(k))≤12−αλT∥xn∥2+Tαy, (9)

    thus ∥xn∥2≤2I(xn)−2Tαy(1−2α)λT is bounded for α>12.

  2. suppose r < + ∞ is not an eigenvalue of (2). We are now ready to prove that {xn} is bounded. Arguing by contradiction, we suppose there is a subsequence of {xn} with ρn = ∥xn∥ → + ∞ as n → ∞ and for each k ∈ [1,T], either {xn(k)} is bounded or xn(k) → + ∞. Put yn=xnρn. Clearly, ∥yn∥ = 1. Then there have a subsequence of {yn} and y ∈ H satisfying that yn → y as n → ∞. Write dn = f(1,xn(1))xn(1)yn(1),...,f(T,xn(T))xn(T)yn(T)⊤.

    Since lim|u|→+∞f(k,u)u=r for all k ∈ [1,T] and I′(xn) = xn−Km fmxn, we get

    I′(xn)ρn=yn−1ρnK0f0xn=yn−K0dn→y−K0ry.

    For I′(xn)ρn→0 as n → ∞, we have y−K0ry → 0. In view of Lemma 2.5, we find that r is an eigenvalue of matrix A, which contradicts the assumption. So {xn} is bounded and the proof is finished. □

Lemma 3.2

I satisfies (C) condition under (J3).

Proof

First assume (J3)(i) is satisfied. There exists a constant M1 > 0 such that {xn} ⊂ H with I(xn) ≤ M1 and (1+∥xn∥m)∥I′(xn)∥m ≤ M1, then

−3M1≤α∑k=1T[xn(k)f(k,xn(k))−2F(k,xn(k))]. (10)

We claim {xn} is bounded. Actually, if {xn} is unbounded, it possesses a subsequence of {xn} and some k0 ∈ [1,T] satisfying |xn(k0)| → + ∞ as n → ∞. According to (J3)(i), we get

xn(k0)f(k0,xn(k0))−2F(k0,xn(k0))→−∞asn→∞,

and there is a constant M2 > 0 such that uf(k,u)−2F(k,u) ≤ M2 for any k ∈ [1,T] and u ∈ R. Therefore,

∑k=1T[xn(k)f(k,xn(k))−2F(k,xn(k))]→−∞

which contradicts (10). So our claim is proved and I satisfies the (C) condition.

Finally, assume (J3)(ii) hold. In a similar way as above, we find that I satisfies (C) condition. Then Lemma 3.2 is verified. □

Lemma 3.3

If (J1) and (J2) hold, there exist m ≥ 0 and ε0 > 0 such that for 0 < ε < ε0, we have:

  1. if x∈Bε− is a nontrivial critical point of I and Am(∂Bε−)⊂Bε−, then x is a negative solution of (1);

  2. if x∈Bε+ is a nontrivial critical point of I and Am(∂Bε+)⊂Bε+, then x is a positive solution of (1).

Proof

  1. According to (J1) and (J2), for all u ≠ 0 and k ∈ [1,T], there exists m ≥ 0 such that

    u(αf(k,u)+mu)>0. (11)

    Let y = Am(x) and x+ = max{x,0}, x− = min{x,0} for x ∈ H. Since

    ∥x∥m2=∑k=1T[|Δx(k−1)|2+m|x(k)|2]=xτAx+m∥x∥2=λ∥x∥2+m∥x∥2=(λ+m)∥x∥2,

    it follows λ1+m∥x∥≤∥x∥m≤λT+m∥x∥ and

    ∥x+∥=infz∈−Λ∥x−z∥≤1m+λ1infz∈−Λ∥x−z∥m=1m+λ1distm(x,−Λ). (12)

    By (J1) and (J2), there exist constants τ > 0, C > 0 and ν > 2 such that

    |αf(k,u)+mu|≤(m+λ1−τ)|u|+αC|u|ν−1,∀(k,u)∈[1,T]×R. (13)

    Choosing a constant D > 0, since x ∈ H, we have

    |x|ν:=(∑k=1T|x(k)|ν)1ν≤Dmin{∥x∥,∥x∥m},∀x∈H. (14)

    It is obvious that |x|2 = ∥x∥. Moreover, y+ = y−y− and y− ∈ −Λ imply distm(y,−Λ) ≤ ∥y−y−∥m = ∥y+∥m. Making use of (12), (15) and (14), we get

    distm(y,−Λ)∥y+∥m≤(m+λ1−τm+λ1distm(x,−Λ)+C1(distm(x,−Λ))ν−1)∥y+∥m,

    here C1=αCDν(m+λ1)ν−1. Hence

    distm(y,−Λ)≤m+λ1−τm+λ1distm(x,−Λ)+C1(distm(x,−Λ))ν−1.

    Let C1(distm(x,−Λ))ν−2=τ2(m+λ1), there holds

    distm(Am(x),−Λ)≤2(m+λ1)−τ2(m+λ1)distm(x,−Λ). (15)

    Since 2(m+λ1)−τ2(m+λ1)<1, we obtain

    Am(x)∈Bε−,∀u∈Bε−.

    If x∈Bε− is a nontrivial critical point of I, it is clear that I′(x) = x−Amx = 0. It follows from (15) that x ∈ −Λ ∖{0}. Combining (11) and remark 2.3, we have x(k) < 0. Consequently, x is a negative solution of (1).

  2. can be discussed similarly, we only need to change y+ to y− to prove (ii). For simplicity, we omit its proof. □

Lemma 3.4

Suppose z1,z2 be eigenvectors corresponding to eigenvalues λ1,λ2 of (2) and u ∈ H2 = span{z1,z2}. If r>λ2α, then I(x) → − ∞ as ∥x∥m → + ∞.

Proof

  1. If r = + ∞. From (9), for any x ∈ H, we have I(x) → −∞ as ∥x∥m → + ∞.

  2. Assume r∈λ2α,+∞. For u ∈ H2, x = ε1 z1+ε2 z2. In general, we can suppose (z1,z2) = 0. Thus ∥x∥2=x,x=ε1z1+ε2z2,ε1z1+ε2z2=ε12∥z1∥2+ε22∥z2∥2 and there exists ε satisfying 0 < ε < minr−λ1α,r−λ2α. From lim|x|→+∞f(k,x)x=r, for any k ∈ [1,T] and x ∈ R, there exists a constant ζ > 0 such that F(k,x)≥r−ε2x2−ζ.

    Then for x ∈ H2, it follows

    I(x)≤12(λ1−αr+αε)ε12∥z1∥2+12(λ2−αr+αε)ε22∥z2∥2+Tζα.

    That is, I(x) → −∞ as ∥x∥m → + ∞ for λ1−α r+α ε < 0 and λ2−α r+α ε < 0. □

Now we are in the position to prove Theorem 1.1 by using Lemma 2.9.

Proof of Theorem 1.1

From (15), we get αF(k,x)+m2|x|2≤(m+λ1−τ)12|x|2+αCν|x|ν, which together with (14) gives that

I(x)≥τ2(m+λ1)∥x∥m2−αCDνν∥x∥mν.

It follows from (12) that ∥x±∥≤1m+λ1distm(x,∓Λ)≤1m+λ1ε0 for any x∈Bε+¯⋂Bε−¯. Then there is c0 > −∞ such that infx∈Bε+¯⋂Bε−¯I(x)=c0. Moreover, in view of Lemma 3.4, we can choose R > 2ε0 such that I(x) < c0−1 for all x ∈ H2 and ∥x∥m = R. To apply Lemma 2.9, we define a path g:[0,1] → H2 as

g(s)=Rz1cos⁡(πs)+z2sin⁡(πs)∥z1cos⁡(πs)+z2sin⁡(πs)∥m.

By direct computation, we get

g(0)=Rz1∥z1∥m∈Bε+∖Bε−,g(1)=−Rz1∥z1∥m∈Bε−∖Bε+,infx∈Bε+¯⋂Bε−¯I(x)>supτ∈[0,1]I(g(τ)).

Combining Lemmas 3.1, 3.3 and 2.9, we find there is a critical point in H∖(Bε+¯⋃Bε−¯) corresponding to a sign-changing solution of (1). Moreover, we also have a critical point in Bε+∖Bε−¯(Bε−∖Bε+¯) corresponding to a positive solution(a negative solution) of (1). The proof is completed. □

Remark 3.5

By Lemma 3.2 and Remark 2.10, we find the proof of Theorem 1.2 is analogous to the proof of Theorem 1.1 and we therefore omit it.

Finally, we exhibit an example to illustrate Theorem 1.1.

Example 3.6

Consider (1) with α = 2 and f(k,x)=|x|−n|x|+1mx. Here m>2sin2⁡3π2(2T+1), 0 < n < 4sin2⁡π2(2T+1)m. Then

F(k,u)=mx22−m(n+1)(x−ln(1+x)),x≥0,mx22+m(n+1)(x+ln(1−x)),x<0.

By direct computation, we have lim|x|→∞[2F(k,x)−xf(k,x)]=−∞ uniformly for k ∈ [1,T] and λ1 = 4sin2⁡π2(2T+1),λ2=4sin2⁡3π2(2T+1). In addition, f0=maxk∈[1,T]lim supx→0|f(k,x)x|=mn<λ1 and lim|x|→∞f(k,x)x=r=m>λ22. Then (1) satisfies conditions of Theorem 1.1, thus it has at least a positive solution, a negative solution and a sign-changing solution.

For the case T = 2, here m>2sin2⁡3π10≈1.28,0<n<4sin2⁡π10m≈0.36m, thus we can choose m = 1.5 and n = 0.1. After not very complicated calculation, we find (0, 0.161, 0.298,0.298), (0,8.73,−5.77,−5.77), (0, −0.1616, −0.2975,−0.2975) and (0,−8.73,5.77,5.77) are nontrivial approximate solutions of (1).

Acknowledgement

This work is supported by the Program for Changjiang Scholars and Innovative Research Team in University (IRT16R16).

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Received: 2017-8-2
Accepted: 2017-10-24
Published Online: 2017-12-29

© 2017 Long and Zeng

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

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https://doi.org/10.1515/math-2017-0129
Keywords for this article
Sign-changing solution; Difference equation; Robin boundary value problem; Invariant sets of descending flow
Creative Commons
BY-NC-ND 4.0

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  253. Special Issue on Recent Developments in Differential Equations
  254. Inverse problem for a physiologically structured population model with variable-effort harvesting
  255. Special Issue on Recent Developments in Differential Equations
  256. Existence of solutions for delay evolution equations with nonlocal conditions
  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
  260. Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application
  261. Special Issue on Recent Developments in Differential Equations
  262. Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
  263. Special Issue on Recent Developments in Differential Equations
  264. Integro-differential systems with variable exponents of nonlinearity
  265. Special Issue on Recent Developments in Differential Equations
  266. Elliptic operators on refined Sobolev scales on vector bundles
  267. Special Issue on Recent Developments in Differential Equations
  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
  272. Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain
  273. Topical Issue on Metaheuristics - Methods and Applications
  274. Learnheuristics: hybridizing metaheuristics with machine learning for optimization with dynamic inputs
  275. Topical Issue on Metaheuristics - Methods and Applications
  276. Nature–inspired metaheuristic algorithms to find near–OGR sequences for WDM channel allocation and their performance comparison
  277. Topical Issue on Cyber-security Mathematics
  278. Monomial codes seen as invariant subspaces
  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
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