# Nonzero Solutions for Nonlinear Systems of Fourth-Order Boundary Value Problems

Authors: Ravi P. Agarwal; Mohamed Jleli; Bessem Samet
Journal: Journal of Mathematics
 (2021)

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## Abstract

This study is devoted to the investigation of nonlinear systems of fourth-order boundary value problems. Namely, using some techniques from matrix analysis and ordinary differential equations, a Lyapunov-type inequality providing a necessary condition for the existence of nonzero solutions is obtained. Next, an estimate involving generalized eigenvalues is derived as an application of our main result.


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## Body

## 1. Introduction

In this study, we investigate the system of differential equations(1)y4t=ρtμt,y,z,0<t<1,z4t=σtξt,y,z,0<t<1,subjected to the boundary conditions(2)y0=y′0=y″1=y31=0,z0=z′0=z″1=z31=0,where ρ,σ:0,1⟶ℝ and μ,ξ:0,1×C0,1×C0,1⟶ℝ are the continuous functions with μ⋅,0,0=ξ⋅,0,0≡0. Clearly, y,z≡0,0 is a trivial solution to (1) and (2). The aim of this study is to obtain necessary conditions for the existence of nonzero solutions to the considered problem. Namely, we establish new Lyapunov-type inequalities [1] for (1) and (2) under reasonable conditions on the nonlinearities μt,y,z and ξt,y,z. Our approach is based essentially on matrix analysis and some arguments from ordinary differential equations.Fourth-order differential equations are useful in modeling many phenomena from physics ([2–7] and the references therein), which makes the study of such equations particularly interesting. In the literature, several contributions have been devoted to the investigation of sufficient conditions ensuring the existence of solutions to fourth-order boundary value problems ([2, 8–19] and the references therein). The study of necessary conditions for the existence of nontrivial solutions to fourth-order differential equations via Lyapunov-type inequalities has been investigated by some authors [20–22]. For instance, in [20], among other results, it was shown that, if y is a nontrivial solution to(3)y4t+λtyt=0,a<t<b,ya=y′a=y″a=y″b=0,where λ:a,b⟶ℝ is continuous, then(4)∫abλ+sds>5129b−a3,where λ+t=maxλt,0.For recent contributions related to Lyapunov-type inequalities, see e.g., [23–29] and the references therein.This study is organized as follows. The next section is devoted to some preliminaries. In Section3, the obtained results as well as their proofs are presented. Finally, some applications to generalized eigenvalue problems are given in Section 4.


## 2. Some Preliminaries

First, we fix some notations. We denote by≤ℝ2 the partial order in the Euclidean space ℝ2 defined as(5)u⟶≤ℝ2v⟶⇔ui≤vi,i=1,2,for every u⟶=u1u2,v⟶=v1v2∈ℝ2.We denote byM2+ the set of square matrices having nonnegative coefficients, i.e.,(6)M2+=mij1≤i,j≤2:mij≥0,1≤i,j≤2.ForC∈M2+, the trace of C is denoted by TraceC, the determinant of C is denoted by detC, and the spectral radius of C is denoted by ρC, i.e.,(7)ρC=maxλiC:i=1,2,where λiC are the complex eigenvalues of C.We equipM2+ with the norm ⋅ defined as(8)C=supu⟶∈ℝ2,u⟶2≠0Cu⟶2u⟶2,C∈M2+,where ⋅2 is the Euclidean norm in ℝ2.The following lemmas will be useful later.Lemma 1.
Letu⟶,v⟶∈ℝ2 with(9)0⟶≤ℝ2u⟶≤ℝ2v⟶,where 0⟶ is the zero vector in ℝ2. Then, u⟶2≤v⟶2.Proof.
The result follows from the fact that(10)P=u⟶∈ℝ2:u⟶≥ℝ20⟶is a normal cone in ℝ2 with normal constant equal to 1 (e.g., [30]).Lemma 2 (See e.g., [31]).
LetC∈M2+. If ρC<1, then(11)limn⟶∞Cn=0.Lemma 3 (See [25]).
LetC∈M2+. Then,(12)ρC=TraceC+TraceC2−4detC2.Lemma 4 (See [12]).
Letx∈C40,1∩C10,1∩C30,1 be a solution to(13)x4t=ft,0<t<1,x0=x′0=x″1=x31=0,where f∈C0,1. Then,(14)xt=∫01Ht,τfτdτ,0≤t≤1,where(15)0≤Ht,τ=3t−ττ26,if0≤τ≤t≤1,3τ−tt26,if0≤t≤τ≤1.Lemma 5.
For all0<τ<1,(16)max0≤t≤1Ht,τ=H1,τ=3−ττ26.Proof.
Fix0<τ<1. Since H⋅,τ is nondecreasing in τ,1, we deduce that(17)Ht,τ≤H1,τ,for allt∈τ,1.
Similarly, it can be easily shown thatH⋅,τ is nondecreasing in 0,τ, which yields(18)Ht,τ≤Hτ,τ=τ33,for allt∈0,τ.
Combining (17) with (18), for all 0≤t≤1, we obtain(19)Ht,τ≤max3−ττ26,τ33=3−ττ26.
Hence, (16) is proved.Throughout this study, we denote by⋅∞ the norm in C0,1 defined as(20)ϑ∞=max0≤t≤1ϑt,ϑ∈C0,1.


## 3. Results and Proofs

We investigate (1) and (2) under the following assumptions:(A1)ρ,σ:0,1⟶ℝ are continuous(A2)μ,ξ:0,1×C0,1×C0,1⟶ℝ are continuous(A3)μ⋅,0,0=ξ⋅,0,0≡0 (where 0 is the zero function)(A4) For allt,y,z∈0,1×C0,1×C0,1,(21)μt,y,z≤μ11ty∞+μ12tz∞,ξt,y,z≤ξ21ty∞+ξ22tz∞,where μ11,μ12,ξ21,ξ22:0,1⟶0,∞ are the continuous functions.By solution to (1) and (2), we mean a pair of functions y,z, y,z∈C40,1∩C10,1∩C30,1, satisfying (1) and the initial conditions (2). A solution y,z to (1) and (2) is said to be nontrivial, if y,z≡0,0.Forδ,η∈C0,1, let(22)Jδη=∫013−ττ2δτητdτ.Theorem 1.
If (1) and (2) have a nontrivial solution, then(23)Jρμ11+Jσξ22+Jρμ11−Jσξ222+4Jσξ21Jρμ12≥12.Proof.
Lety,z be a nontrivial solution to (1) and (2), and suppose that(24)Jρμ11+Jσξ22+Jρμ11−Jσξ222+4Jσξ21Jρμ12<12.
By Lemma4, y,z∈C0,1×C0,1 is a nontrivial solution to the system of integral equations:(25)yt=∫01Ht,τρτμτ,y,zdτzt=∫01Ht,τστξτ,y,zdτ,0≤t≤1.
Using (A4) and (16), for all 0≤t≤1, we obtain(26)yt≤∫01Ht,τρτμτ,y,zdτ≤∫013−ττ26ρτμ11τy∞+μ12τz∞dτ=∫013−ττ26ρτμ11τdτy∞+∫013−ττ26ρτμ12τdτz∞,which leads to(27)y∞≤∫013−ττ26ρτμ11τdτy∞+∫013−ττ26ρτμ12τdτz∞.
Similarly, by (A4) and (16), we obtain(28)z∞≤∫013−ττ26στξ21τdτy∞+∫013−ττ26στξ22τdτz∞.
Combining (27) with (28), we deduce that(29)0⟶≤ℝ2ϕy,z⟶≤ℝ2Cϕy,z⟶,where ϕy,z⟶=y∞z∞ and(30)C=16Jρμ11Jρμ12Jσξ21Jσξ22.
Next, using Lemma3 and (24), we deduce that(31)ρC<1.
On the other hand, using Lemma1 and (29), we obtain(32)ϕy,z⟶2≤Cϕy,z⟶2≤Cϕy,z⟶2.
Sincey,z is nontrivial, then ϕy,z⟶≠0⟶, and the above inequality leads to(33)C≥1.
But by Lemma2 and (31), we know that(34)limn⟶∞Cn=0,which contradicts (33). This proves (23).Next, we discuss some particular cases of Theorem1.


### 3.1. Nonlinearities Involving Trigonometric Functions

Consider the system of differential equations(35)y4t=ρtsinyt+zt,0<t<1,z4t=σtarctanyt+zt,0<t<1,under the boundary conditions (2), where ρ,σ∈C0,1. Observe that (35) is a particular case of (1) with(36)μt,y,z=sinyt+zt,t,y,z∈0,1×C0,1×C0,1,ξt,y,z=arctanyt+zt,t,y,z∈0,1×C0,1×C0,1.Obviously,μ and ξ satisfy (A3). Moreover, for all t,y,z∈0,1×C0,1×C0,1,(37)μt,y,z=sinyt+zt≤yt+zt≤y∞+z∞,ξt,y,z=arctanyt+zt≤yt+zt≤y∞+z∞.Then, (A4) is satisfied with(38)μ11=μ12=ξ21=ξ22=1.Hence, by Theorem1, we deduce the following.Corollary 1.
If (35) and (2) have a nontrivial solution, then(39)Jρ1+Jσ1+Jρ1−Jσ12+4Jσ1Jρ1≥12.


### 3.2. Nonlocal Source Terms

Consider the system of differential equations(40)y4t=ρt∫0tt−sα−1ys+zsds,0<t<1,z4t=σt∫0tt−sβ−1ys+zsds,0<t<1,under the boundary condition (16), where ρ,σ∈C0,1 and α,β>0. Problem (40) is a particular case of (1) with(41)μt,y,z=∫0tt−sα−1ys+zsds,if0<t≤1,0,ift=0,ξt,y,z=∫0tt−sβ−1ys+zsdsif0<t≤1,0,ift=0,for all y,z∈C0,1. Obviously, μ and ξ satisfy (A3). Moreover, for all t,y,z∈0,1×C0,1×C0,1,(42)μt,y,z≤∫0tt−sα−1dsy∞+z∞=tααy∞+z∞.and similarly(43)ξt,y,z≤tββy∞+z∞.Then, (A4) is satisfied with(44)μ11t=μ12t=tαα,0≤t≤1,ξ21t=ξ22t=tββ,0≤t≤1.Then, by Theorem 1, we deduce the following.Corollary 2.
If (39) and (2) have a nontrivial solution, then(45)Jρtαα+Jσtββ+Jρtαα−Jσtββ2+4JσtββJρtαα≥12.Example 1.
Consider the system of differential equation (35) with ρ≡1 and σt=9/43−t for all t∈0,1. Elementary calculations show that(46)Jρ1=Jσ1=34,Jρ1+Jσ1+Jρ1−Jσ12+4Jσ1Jρ1=3<12.
Hence, by Corollary1, we deduce that (35) and (2) have no nontrivial solution.


## 3.1. Nonlinearities Involving Trigonometric Functions

Consider the system of differential equations(35)y4t=ρtsinyt+zt,0<t<1,z4t=σtarctanyt+zt,0<t<1,under the boundary conditions (2), where ρ,σ∈C0,1. Observe that (35) is a particular case of (1) with(36)μt,y,z=sinyt+zt,t,y,z∈0,1×C0,1×C0,1,ξt,y,z=arctanyt+zt,t,y,z∈0,1×C0,1×C0,1.Obviously,μ and ξ satisfy (A3). Moreover, for all t,y,z∈0,1×C0,1×C0,1,(37)μt,y,z=sinyt+zt≤yt+zt≤y∞+z∞,ξt,y,z=arctanyt+zt≤yt+zt≤y∞+z∞.Then, (A4) is satisfied with(38)μ11=μ12=ξ21=ξ22=1.Hence, by Theorem1, we deduce the following.Corollary 1.
If (35) and (2) have a nontrivial solution, then(39)Jρ1+Jσ1+Jρ1−Jσ12+4Jσ1Jρ1≥12.


## 3.2. Nonlocal Source Terms

Consider the system of differential equations(40)y4t=ρt∫0tt−sα−1ys+zsds,0<t<1,z4t=σt∫0tt−sβ−1ys+zsds,0<t<1,under the boundary condition (16), where ρ,σ∈C0,1 and α,β>0. Problem (40) is a particular case of (1) with(41)μt,y,z=∫0tt−sα−1ys+zsds,if0<t≤1,0,ift=0,ξt,y,z=∫0tt−sβ−1ys+zsdsif0<t≤1,0,ift=0,for all y,z∈C0,1. Obviously, μ and ξ satisfy (A3). Moreover, for all t,y,z∈0,1×C0,1×C0,1,(42)μt,y,z≤∫0tt−sα−1dsy∞+z∞=tααy∞+z∞.and similarly(43)ξt,y,z≤tββy∞+z∞.Then, (A4) is satisfied with(44)μ11t=μ12t=tαα,0≤t≤1,ξ21t=ξ22t=tββ,0≤t≤1.Then, by Theorem 1, we deduce the following.Corollary 2.
If (39) and (2) have a nontrivial solution, then(45)Jρtαα+Jσtββ+Jρtαα−Jσtββ2+4JσtββJρtαα≥12.Example 1.
Consider the system of differential equation (35) with ρ≡1 and σt=9/43−t for all t∈0,1. Elementary calculations show that(46)Jρ1=Jσ1=34,Jρ1+Jσ1+Jρ1−Jσ12+4Jσ1Jρ1=3<12.
Hence, by Corollary1, we deduce that (35) and (2) have no nontrivial solution.


## 4. Generalized Eigenvalues Problems

We say thate=eij1≤i,j≤2 is a generalized eigenvalue of the system of differential equations(47)y4t=e11yt+e12zt,0<t<1,z4t=e21yt+e22zt,0<t<1,subjected to the boundary condition (2), if (2) and (44) admit a nonzero solution ye,ze. Notice that (44) is a particular case of (1) with(48)ρt=1,μt,y,z=e11yt+e12zt,t,y,z∈0,1×C0,1×C0,1,σt=1,ξt,y,z=e21yt+e22zt,t,y,z∈0,1×C0,1×C0,1.Moreover, (A2)–(A4) are satisfied with(49)μ11=e11,μ12=e12,ξ21=e21,ξ22=e22.Hence, by Theorem1, if e is a generalized eigenvalue of (2) and (44), then(50)J1e11+J1e22+J1e11−J1e222+4J1e21J1e12≥12.On the other hand, forλ∈ℝ, we have(51)J1λ=λ∫013−ττ2dτ=3λ4.Hence, (50) reduces to(52)e11+e22+e11−e222+4e21e12≥16.Therefore, the following result follows.Corollary 3.
Ife=eij1≤i,j≤2 is a generalized eigenvalue of (43) and (2), then(53)e11+e22+e11−e222+4e21e12≥16.


## 5. Conclusion

Using some techniques from matrix analysis and ordinary differential equations, a necessary condition for the existence of nonzero solutions to (1) and (2) is obtained (Theorem 1). As particular cases of (1), we discussed nonlinearities involving trigonometric functions (Corollary 1) and nonlocal source terms (Corollary 2). Finally, we applied our main result to obtain an estimate involving generalized eigenvalues (Corollary 3).



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