Componentwise localization of solutions to systems of operator inclusions via Harnack type inequalities

Abstract


We establish compression-expansion type fixed point theorems for systems of operator inclusions with decomposable multivalued maps. The approach is vectorial allowing to localize individually the components of solutions and to obtain multiple solutions with multiplicity not necessarily concerned with all components of the solution. A general scheme of applicability of the theory is elaborated based on Harnack type inequalities and illustrated on systems of differential inclusions with one-dimentional ϕ-Laplacian.

Authors

Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania

Jorge Rodríguez-López
Departamento de Estatística, Análise Matemática e Optimización, CITMAga, Universidade de Santiago de Compostela, 15782, Facultade de Matemáticas, Campus Vida, Santiago, Spain.

Keywords

Compression-expansion fixed point theorem; Harnack type inequality; positive solution; operator inclusion; nonlinear system; ϕ-Laplacian

Paper coordinates

R. Precup, J. Rodríguez-López, Componentwise localization of solutions to systems of operator inclusions via Harnack type inequalities, Quaestiones Mathematicae, 46 (2023) no. 7, pp. 1481-1496, https://doi.org/10.2989/16073606.2022.2107959

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Componentwise localization of solutions to systems of operator inclusions via Harnack type inequalities

Componentwise localization of solutions to systems of operator inclusions via Harnack type inequalities

Radu Precup Department of Mathematics, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania, E-mail: r.precup@math.ubbcluj.ro  and  Jorge Rodríguez-López Departamento de Estatística, Análise Matemática e Optimización, Instituto de Matemáticas, Universidade de Santiago de Compostela, 15782, Facultade de Matemáticas, Campus Vida, Santiago, Spain, E-mail: jorgerodriguez.lopez@usc.es
Abstract.

We establish compression-expansion type fixed point theorems for systems of operator inclusions with decomposable multivalued maps. The approach is vectorial allowing to localize individually the components of solutions and to obtain multiple solutions with multiplicity not necessarily concerned with all components of the solution. A general scheme of applicability of the theory is elaborated based on Harnack type inequalities and illustrated on systems of differential inclusions with one-dimentional ϕ-Laplacian.

Mathematics Subject Classification: 47H10; 34A60; 34B18

Keywords and phrases: Compression–expansion fixed point theorem, Harnack type inequality, positive solution, operator inclusion, nonlinear system, ϕ-Laplacian.

1. Introduction

The main goal of this paper is to discuss the existence, componentwise localization and multiplicity of positive solutions for systems of differential inclusions with one-dimensional ϕ-Laplacian, namely

(1.1) {(ϕ1(u1))G1(t,u1,u2)(ϕ2(u2))G2(t,u1,u2),

for t(0,1), subject to the boundary conditions

u1(0)=u2(0)=u1(1)=u2(1)=0,

where for each i{1,2}, ϕi:(ai,ai)(bi,bi) (0<ai,bi+) is an increasing homeomorphism with ϕi(0)=0 and Gi is a multivalued map. In particular, the homeomorphisms ϕ1,ϕ2 can be one of the following

ϕ:,ϕ(x)=|x|p2x;
ϕ:(1,1),ϕ(x)=x1+x2;
ϕ:(1,1),ϕ(x)=x1x2,

corresponding to the one-dimensional p-Laplace operator, the mean curvature operator in the Euclidian and Minkowski space, respectively. Such type of equations involving the ϕ-Laplacian has been investigated in a large number of papers using fixed point methods, degree theory, upper and lower solution techniques and variational methods. We refer the reader to the papers [1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 17, 18, 24], to the survey work [16], and the bibliographies therein.

It is worth noting that in our work on system (1.1), the two homeomorphisms ϕ1,ϕ2 can be of different types. Also, for a solution u=(u1,u2), different localizations of the components u1 and u2 are possible, allowing to obtain multiple solution results where the multiplicity concerns either only one of the components or both of them.

More general, we shall discuss systems of operator inclusions in Banach spaces, of the form

(1.2) {L1(u1)F1(u1,u2)L2(u2)F2(u1,u2),

where, in this abstract setting, the ‘boundary conditions’ are simulated by the belonging of ui to the domain D(Li) of the not necessarily linear operator Li (i=1,2), while the ‘positivity’ of solutions means their belonging to a given cone.

If the operators Li(i=1,2) are invertible, as is the case of the boundary value problem related to system (1.1), then (1.2) is equivalent to the fixed point problem

(1.3) {u1L11F1(u1,u2)u2L21F2(u1,u2),

with decomposable multivalued maps L11F1 and L21F2.

Thus, it is natural to develop a theory for abstract systems of inclusions, of the form

(1.4) {u1N1(u1,u2)u2N2(u1,u2),

in a Banach space X, where N=(N1,N2):AX22X2 is a multivalued operator such that each of its components is a decomposable map, that is, the composition of two multivalued maps

Ni(u)=Ψi(u,Φi(u)),u=(u1,u2)A(i=1,2).

Our theoretical strategy is to make the reverse path, from abstract to concrete. Thus, first, in Section 2, we develop a fixed point theory with localization on components for the general system (1.4); next, in Section 3, we apply this theory to systems of type (1.2); and, finally, in Section 4, we particularize even more to obtain results for the considered boundary value problem related to system (1.1). Of course, one may use the abstract theory to obtain similar results for system (1.1) subject to some other boundary conditions. Also the results can be immediately extended to systems of more than two inclusions. Our restriction to systems of two inclusions is only to simplify the presentation.

The common approach to systems is to look at them as generalizations of equations. On the contrary, in our vectorial approch, a system is seen as a particular equation that has the splitting property. Consequently, a larger diversity of results may be expected in the case of systems.

This paper extends for inclusions the theory established in [9], [20], [21] and [22] for equations, and in [13] for inclusions with convex-valued maps. The extension is not trivial since, as explain in [19], several difficulties arise when treating compositions of multivalued maps. One of them consists in guaranteeing continuity properties for the maps, another one concerns the geometric properties of their values. For example, even if one of the maps is single-valued (but nonlinear) and the values of the other one are convex, the values of the composed map can be non-convex, contrary to the hypotheses of basic fixed point theorems for multivalued maps.

2. Existence results for operator systems with decomposable multivalued maps

Let X be a Banach space and consider the fixed point problem

{u1N1(u1,u2)u2N2(u1,u2),

where N=(N1,N2):AX22A is a multivalued operator such that each of its components is a decomposable map, that is, the composition of two upper semicontinuous (usc, for short) multivalued maps:

Ni(u)=Ψi(u,Φi(u)),u=(u1,u2)A(i=1,2).

2.1. A Schauder type fixed point theorem

First, we present a generalization of Kakutani’s fixed point theorem for a system of two decomposable maps. For the case of a single equation, the reader may find a similar result in [19].

Theorem 2.1.

Let A be a nonempty, convex and compact subset of X2, Y1, Y2 Banach spaces and N=(N1,N2):AX22A, (N1,N2)(u)=(Ψ1(u,Φ1(u)),Ψ2(u,Φ2(u))), a multivalued operator such that for each i{1,2},

  1. (i)

    Φi:A2Yi is usc with compact convex values;

  2. (ii)

    Ψi:A×Yi2X is usc with closed convex values.

Then N has a fixed point in A.

Proof.

Consider the set B=A×co¯Φ1(A)×co¯Φ2(A). Since A is compact and Φ1 is usc with compact values, co¯Φ1(A) is compact (similarly, co¯Φ2(A) is also compact). Hence, B is the Cartesian product of compact convex sets, so it is a compact convex subset of X2×Y2.

Now, we associate to N the map Π:B2B given by

Π(u1,u2,v1,v2)=Ψ1(u1,u2,v1)×Ψ2(u1,u2,v2)×Φ1(u1,u2)×Φ2(u1,u2).

Note that Π is usc with closed convex values and thus the Bohnenblust–Karlin fixed point theorem ensures that Π has a fixed point (u1,u2,v1,v2)B, that is,

u1Ψ1(u1,u2,v1),u2Ψ2(u1,u2,v2),v1Φ1(u1,u2),v2Φ2(u1,u2).

Therefore, the pair (u1,u2)A is a fixed point of N. ∎

Corollary 2.2.

Let C be a nonempty, convex and closed subset of X2, Y1, Y2 Banach spaces and N=(N1,N2):CX22C, (N1,N2)(u)=(Ψ1(u,Φ1(u)),Ψ2(u,Φ2(u))), a multivalued operator such that N(C) is relatively compact and for each i{1,2},

  1. (i)

    Φi:C2Yi is usc with compact convex values;

  2. (ii)

    Ψi:C×Yi2X is usc with closed convex values.

Then N has a fixed point in C.

Proof.

It suffices to take the compact and convex set A=co¯N(C)C and to apply Theorem 2.1. ∎

2.2. A vector version of Krasnosel’skiĭ’s fixed point theorem in cones

Now we present the vector version of Krasnosel’skiĭ’s fixed point theorem in cones for the class of decomposable maps. Note that this type of componentwise localization of positive fixed points for systems was initiated in [20, 21] for single-valued compact maps and continued in [13] by a similar result for usc multivalued maps. Also, a Krasnosel’skiĭ type fixed point theorem in cones for a single decomposable map was proved in [5] (see also [23] for a fixed point index theory for decomposable maps).

Recall that a closed convex subset K of a Banach space X is a cone if it satisfies that K(K)={0} and λxK for every xK and for all λ0. Any cone K induces a partial order relation in X, denoted by , namely xy if yxK. One says that xy if yxK{0}.

In the sequel, let K1 and K2 be two cones of the Banach space X and so K:=K1×K2 is a cone of X2. For r,R+2, r=(r1,r2), R=(R1,R2), we denote

(Ki)ri,Ri :={uKi:riuRi}(i=1,2),
Kr,R :={u=(u1,u2)K:riuiRifor i=1,2}.

We look for fixed points of an operator N=(N1,N2):K2K, whose components N1 and N2 are of the form

Ni(u)=Ψi(u,Φi(u)),uK(i=1,2).

Here we assume that for a Banach space Y and two closed convex sets KY1,KY2Y, the following conditions on Ψi and Φi are satisfied for each i{1,2}:

  1. (HΦ)

    Φi:K2KYi is usc with compact convex values;

  2. (HΨ)

    Ψi:K×KYi2Ki is usc with closed convex values.

Now we state and prove the main result of this section.

Theorem 2.3.

Let αi,βi>0 with αiβi, ri=min{αi,βi} and Ri=max{αi,βi} for i=1,2. Assume that N:Kr,R2K,

N(u)=(Ψ1(u,Φ1(u)),Ψ2(u,Φ2(u))),

N(Kr,R) is relatively compact and for each i{1,2}, Φi and Ψi satisfy (HΦ) and (HΨ), respectively. In addition assume that the following conditions

ui Ni(u1,u2)+μhi for any (u1,u2)Kr,R with ui=αi and μ>0;
λui Ni(u1,u2) for any (u1,u2)Kr,R with ui=βi and λ>1

hold for some hiKi{0}(i=1,2). Then N has a fixed point u=(u1,u2)K with riuiRi for i=1,2.

Proof.

We consider four cases which cover all the possible combinations of compression–expansion conditions for N1 and N2.

Case 1: Assume first that αi<βi for both i=1,2 (compression for both N1 and N2). Then ri=αi and Ri=βi for i=1,2. Define the map N~=(N~1,N~2):K2K by

N~i(u1,u2)=μ(u1,u2)Ni(δ1(u1)u1u1,δ2(u2)u2u2)+(1μ(u1,u2))hi,

where μ(u1,u2)=min{u1/r1,u2/r2,1} and δi(ui)=max{min{ui,Ri},ri} for i=1,2. The map N~ is a decomposable map, namely, N~i(u1,u2)=Ψ~i(u1,u2,Φ~i(u1,u2)) (i=1,2), with

Φ~i(u1,u2) =Φi(δ1(u1)u1u1,δ2(u2)u2u2),
Ψ~i(u1,u2,v) =μ(u1,u2)Ψi(δ1(u1)u1u1,δ2(u2)u2u2,v)+(1μ(u1,u2))hi.

Observe that Φ~i and Ψ~i satisfy conditions (HΦ) and (HΨ), respectively. Moreover, note that N~(K) is relatively compact since its values belong to the compact set

C=co¯(N(Kr,R){h}),

where h=(h1,h2). Therefore, Corollary 2.2 applies and guarantees the existence of a fixed point of N~ in CK. It can be shown in a similar way to the proof of [13, Theorem 2] (or [20, 21]) that the fixed point u belongs to Kr,R. Hence, u is also a fixed point of the operator N which is located in Kr,R.

Case 2: Assume that α1<β1 (compression for N1) and β2<α2 (expansion for N2). Note that this case can be reduced to the previous one by considering the operator N=(N1,N2):K2K given by

N1(u1,u2) =N1(u1,(R2u2+r2u21)u2),
N2(u1,u2) =(R2u2+r2u21)1N2(u1,(R2u2+r2u21)u2).

Each component of N can be written as the composition Ni(u)=Ψi(u,Φi(u)), where

Φi(u1,u2) =Φi(u1,(R2u2+r2u21)u2)(i=1,2),
Ψ1(u1,u2,v) =Ψ1(u1,(R2u2+r2u21)u2,v),
Ψ2(u1,u2,v) =(R2u2+r2u21)1Ψ2(u1,(R2u2+r2u21)u2,v).

Notice that the map N is in case 1 and thus it has a fixed point vKr,R. Then the point u=(u1,u2) defined as u1=v1 and u2=(R2u2+r2u21)v2 is a fixed point of the operator N.

Case 3: Assume that β1<α1 (expansion for N1) and α2<β2 (compression for N2). This case is completely analogous to the previous one.

Case 4: Assume that βi<αi for i=1,2 (expansion for both N1 and N2). This situation reduces to case 1 by taking the decomposable map N=(N1,N2) defined as

N1(u1,u2) =(R1u1+r1u11)1N1((R1u1+r1u11)u1,u2),
N2(u1,u2) =(R2u2+r2u21)1N2(u1,(R2u2+r2u21)u2).

Thus, the proof is finished. ∎

In particular, if Ψ(u,v)=Ψ(v), then from Theorem 2.3 we obtain the following compression–expansion fixed point theorem for inclusion systems involving compositions of the form ΨΦ.

Theorem 2.4.

Let αi,βi>0 with αiβi, ri=min{αi,βi} and Ri=max{αi,βi} for i=1,2. Assume that N:Kr,R2K,

N=(Ψ1Φ1,Ψ2Φ2),

N(Kr,R) is relatively compact and for each i{1,2}, Φi and Ψi satisfy (HΦ) and (HΨ), respectively. In addition assume that the following conditions

ui Ni(u1,u2)+μhi for any (u1,u2)Kr,R with ui=αi and μ>0;
λui Ni(u1,u2) for any (u1,u2)Kr,R with ui=βi and λ>1

hold for some hiKi{0}(i=1,2). Then N has a fixed point u=(u1,u2)K with riuiRi for i=1,2.

As a straightforward consequence we obtain the following result.

Corollary 2.5.

Let αi,βi>0 with αiβi, ri=min{αi,βi} and Ri=max{αi,βi} for i=1,2. Assume that N:Kr,R2K,

N=(Ψ1Φ1,Ψ2Φ2),

N(Kr,R) is relatively compact and for each i{1,2}, Φi and Ψi satisfy (HΦ) and (HΨ), respectively. In addition assume that the following conditions hold:

y ui for all yNi(u1,u2),(u1,u2)Kr,R with ui=αi;
y ui for all yNi(u1,u2),(u1,u2)Kr,R with ui=βi.

Then N has a fixed point u=(u1,u2)K with riuiRi for i=1,2.

3. Positive solutions for systems of operator inclusions

In this section, we apply the general fixed point theorems established from above to the following system of operator inclusions

(3.1) {Li(ui)Fi(u1,u2)uiD(Li)(i=1,2),

where for each i, Li:D(Li)XY is a not necessarily linear map which is invertible, Fi:X×X2Y is a set-valued map, and X,Y are Banach spaces with continuous embedding XY.

Equivalently, we deal with the following inclusion system of type (1.4),

(3.2) uiLi1Fi(u1,u2),uiX(i=1,2).

We look for positive solutions for (3.1), that is, solutions u=(u1,u2) with uiK0iX, where K0iY is a cone for every i=1,2. We use the same symbol to denote the ordering in Y induced either by K01 or K02. Moreover, we assume that Li1(K0i)K0i for every i{1,2}.

Let P be a cone in X and for each i{1,2}, consider the following basic assumptions:

  1. (H1)

    Li1:K0iD(Li) can be written as the composition Li1=TiSi, where

    1. (a)

      Si:K0iP is a continuous linear operator which maps bounded sets into relatively compact sets and SiFi has closed and convex values;

    2. (b)

      Ti:PD(Li) is a continuous map.

  2. (H2)

    Si and Ti are positive and increasing, i.e.,

    0uvimplies0Si(u)Si(v)and0Ti(u)Ti(v).

    In addition, 0D(Li) and Li(0)=0.

  3. (H3)

    There exists ψiK0i{0} such that for every uK0iX, one has

    uuψi.
  4. (H4)

    There exists φiK0i{0} such that for every uK0iX with LiuK0i, one has

    uφiu(abstract Harnack inequality).
  5. (H5)

    Fi:X22Y is usc, maps bounded sets into bounded sets and

    Fi((K01×K02)X2)K0i.
  6. (H6)

    For each vector λ=(λ1,λ2)(0,+)2, there exist elements Fi¯(λ),Fi¯(λ)K0i such that

    Fi¯(λ)yFi¯(λ) for every yFi(u1,u2)

    and every (u1,u2) such that ujK0jX and λjφjujλjψj, j=1,2.

For each i{1,2}, we consider the cone in X

Ki={uK0iX:uφiu}

and the product cone in X2, K=K1×K2.

Let Ni:X22X be the operators

(3.3) Ni(u1,u2)=Li1Fi(u1,u2),i=1,2.

Note that we can write N=(N1,N2)=(Ψ1Φ1,Ψ2Φ2), where

(3.4) Ψi=Ti,Φi=SiFifor i=1,2.

For r,R(0,+)2, r=(r1,r2),R=(R1,R2), the notation Kr,R stands for

Kr,R={uK:riuiRi for i=1,2},

and from (H3) and (H4), for every (u1,u2)Kr,R one has

rjφjujφjujujψjRjψj,j=1,2.

Observe that clearly for i=1,2, the map Ψi is usc with convex and compact values since it is a single-valued continuous map. Moreover, Φi is usc (as the composition of a continuous single-valued map and an usc multivalued map) and Φi(Kr,R) is relatively compact since Fi(Kr,R) is bounded and Si maps bounded sets into relatively compact sets.

Lemma 3.1.

Assume that conditions (H1)-(H5) hold. Then for N=(N1,N2), one has

N(K)K.
Proof.

Let us see that Ni(K)Ki for i=1,2. Indeed, if uK and vNi(u)=Li1Fi(u), then u1K01X, u2K02X and thus, by (H5), Fi(u)K0i. Next, by (H2), we obtain that vLi1Fi(u)K0iD(Li)K0iX. Finally, LivFi(u) and Fi(u)K0i imply that vφiv, as a consequence of the abstract Harnack inequality in (H4). Therefore, vKi, as wished. ∎

Now we present the following general existence and localization result.

Theorem 3.2.

Assume that for each i{1,2}, conditions (H1)-(H6) hold and there exist αi,βi>0 with αiβi such that

(3.5) Li1Fi¯(u1,u2) uifor all uKr,R with ui=αi,
(3.6) ui Li1Fi¯(u1,u2)for all uKr,R with ui=βi,

where ri=min{αi,βi} and Ri=max{αi,βi}.

Then problem (3.1) has at least one positive solution uK with

riuiRi(i=1,2).
Proof.

Let us apply Corollary 2.5 to the operator N defined in (3.3) and the cone K considered above. Note that N is a decomposable map with Φi and Ψi as in (3.4).

First, we prove that

(3.7) Li1yuifor all yFi(u) and all uKr,R with ui=αi.

By condition (H6), Fi¯(u1,u2)y for all yFi(u1,u2) and all uKr,R. Hence, by (H2), Li1Fi¯(u1,u2)Li1y for all yFi(u1,u2) and all uKr,R. Now condition (3.5) implies (3.7).

Secondly, we show that

(3.8) uiLi1yfor all yFi(u) and all uKr,R with ui=βi.

By conditions (H2) and (H6), Li1yLi1Fi¯(u1,u2) for all yFi(u1,u2), uKr,R. Then condition (3.6) implies (3.8).

Therefore, Corollary 2.5 ensures the existence of at least one fixed point for N located in Kr,R, which is a positive solution of the system (3.1). ∎

It is said that the norm of the Banach space X is monotone with respect to the ordering given by the cones K0i (i=1,2) if 0xy implies xy. In that case, conditions (3.5) and (3.6) hold provided that (3.9) and (3.10) below are satisfied.

Corollary 3.3.

Assume that, for each i{1,2}, conditions (H1)-(H6) hold, the norm is monotone with respect to the ordering and there exist αi,βi>0 with αiβi such that

(3.9) Li1Fi¯(u1,u2) >αifor all uKr,R with ui=αi,
(3.10) Li1Fi¯(u1,u2) <βifor all uKr,R with ui=βi,

where ri=min{αi,βi} and Ri=max{αi,βi}.

Then problem (3.1) has at least one positive solution uK with

riuiRi(i=1,2).

A number of particular examples of differential problems which illustrate the applicability of the previous theory can be seen in [9] (see also [22]). Roughly speaking, the abstract theory works provided that a Harnack type inequality can be obtained for the problem.

4. Application to ϕ-Laplacian systems of inclusions

The aim of this section is to derive sufficient conditions for the existence and localization of positive solutions for systems of the form

(4.1) {(ϕ1(u1))G1(t,u1,u2) in (0,1)(ϕ2(u2))G2(t,u1,u2) in (0,1)u1(0)=u1(1)=0=u2(0)=u2(1),

where for each i{1,2}, ϕi:(ai,ai)(bi,bi) (0<ai,bi+) is an increasing homeomorphism such that ϕi(0)=0 and Gi:(0,1)×+22+ is an usc multivalued map with closed convex values, which maps bounded sets into bounded sets.

Problem (4.1) can be studied by means of the abstract scheme described in Section 3. Here X=C[0,1], Y=L(0,1), = the sup-norm in X, P is the positive cone of C[0,1], K0i is the positive cone of L(0,1), ψi1 and

Li(w)(t)=(ϕi(w(t))),Fi(u1,u2)=Gi(,u1(),u2())(i=1,2).

Note that, for each wL(0,1) and each i{1,2}, one has

(4.2) Li1(w)(t)=t1ϕi1(0sw(τ)𝑑τ)𝑑s

and Li1=TiSi, with

Ti(w)(t)=t1ϕi1(w(s))𝑑s,Si(w)(t)=0tw(τ)𝑑τ.

In the sequel, we assume that Gi is such that SiFi is an usc multivalued map with closed and convex values. Clearly, conditions (H1), (H2), (H3) and (H5) hold.

On the other hand, condition (H4) holds here thanks to the following Harnack type inequality proved in [10] for the differential operator Lu:=(ϕ(u)) (where ϕ:(a,a)(b,b) is an increasing homeomorphism with ϕ(0)=0) and the boundary conditions u(0)=u(1)=0.

Proposition 4.1.

For each c(0,1) and any uC1[0,1] with u(0)=u(1)=0, u(t)(a,a) for every t[0,1], ϕuW1,1(0,1) and (ϕ(u))0 on [0,1], the following inequality holds:

u(t)(1c)ufor all t[0,c].

Hence, condition (H4) holds by taking the function φi given by

φi(t)={1ci, for t[0,ci]0, for t(ci,1],

where 0<ci<1 is fixed.

Assume that for each i{1,2}, the multivalued map Gi satisfies the following condition:

  1. (C)

    There exist continuous functions fi,hi:[0,1]×+2+ such that

    1. (i)

      fi and hi are nondecreasing in the second and third variables on +;

    2. (ii)

      for every (t,x,y)[0,1]×+2,

      fi(t,x,y)zhi(t,x,y)<bifor all zGi(t,x,y).

Finally, hypothesis (H6) is satisfied for

Fi¯(λ1,λ2)(t) =fi(t,φ1(t)λ1,φ2(t)λ2),
Fi¯(λ1,λ2)(t) =hi(t,λ1,λ2).

Therefore, as a consequence of Corollary 3.3, we obtain the following result concerning the existence of positive solutions for (4.1).

Theorem 4.2.

Assume that, for each i{1,2}, condition (C) holds and there exist αi,βi>0 with αiβi such that

(4.3) c11ϕ11(0c1f1(τ,(1c1)α1,φ2(τ)r2)𝑑τ)𝑑s>α1,
(4.4) c21ϕ21(0c2f2(τ,φ1(τ)r1,(1c2)α2)𝑑τ)𝑑s>α2,
(4.5) 01ϕ11(0sh1(τ,β1,R2)𝑑τ)𝑑s<β1,
(4.6) 01ϕ21(0sh2(τ,R1,β2)𝑑τ)𝑑s<β2,

where ri=min{αi,βi} and Ri=max{αi,βi}.

Then problem (4.1) has at least one positive solution u=(u1,u2)K with

riuiRi(i=1,2).
Proof.

Let us apply Corollary 3.3 with the operators Li1 as defined in (4.2).

First, let us show that condition (3.9) is satisfied for i=1 (the case i=2 is analogous). Since for each nonnegative function wL(0,1),

L11(w)=01ϕ11(0sw(τ)𝑑τ)𝑑s,

by the monotonicity assumptions on f1, we have that for each uKr,R with u1=α1,

L11F1¯(u1,u2) 01ϕ11(0sf1(τ,φ1(τ)α1,φ2(τ)r2)𝑑τ)𝑑s
c11ϕ11(0sf1(τ,φ1(τ)α1,φ2(τ)r2)𝑑τ)𝑑s
c11ϕ11(0c1f1(τ,(1c1)α1,φ2(τ)r2)𝑑τ)𝑑s.

Hence, applying (4.3), we deduce that condition (3.9) in Corollary 3.3 holds.

Finally, we check that condition (3.10) is also satisfied for i=2 (the case i=1 is similar). We have that for each uKr,R with u2=β2,

L21F2¯(u1,u2)01ϕ21(0sh2(τ,R1,β2)𝑑τ)𝑑s,

and so the conclusion follows from (4.6). ∎

Note that, in particular, if c1=c2=:c and ϕ1 and ϕ2 are odd homeomorphisms, then conditions (4.3)-(4.6) hold if the following inequalities are satisfied:

(1c)ϕ11(cminτ[0,c]f1(τ,(1c)α1,(1c)r2)) >α1,
(1c)ϕ21(cminτ[0,c]f2(τ,(1c)r1,(1c)α2)) >α2,
ϕ11(maxτ[0,1]h1(τ,β1,R2)) <β1,
ϕ21(maxτ[0,1]h2(τ,R1,β2)) <β2.

Next we give an example of application of Theorem 4.2, where the operators associated to the two equations of the system have different behaviors: compression for one of them and expansion for the other one.

Example 4.3.

Consider the system

(4.7) {(u1+u2)G1(u,v) in (0,1)v′′G2(u,v) in (0,1)u(0)=u(1)=0=v(0)=v(1),

where the usc multivalued maps G1 and G2 are given by

(4.8) G1(u,v)={12([u3,u4]eu+cos2v)if u[0,1]12(e1+cos2v)if u>1

and

G2(u,v)=[1+sin2u,2]v2.

One may easily verify that condition (C) holds for the functions fi and hi (i=1,2) defined as

f1(u)={min{12u3eu,12e}if u[0,1],12eif u>1,
h1(u)=4/5,f2(v)=v2,h2(v)=2v2.

Moreover, choosing c=1/2, straightforward computations show that we can take α1=1/50, β1=2, α2=18 and β2=1/3. Therefore, according to Theorem 4.2, problem (4.7) has at least one positive solution (u,v) such that

150u2and13v18.
Remark 4.1 (Asymptotic conditions).

As shown in Example 4.3, it is meaningful the simple case where condition (C) is given by functions of the form fi(t,u1,u2)=fi(ui) and hi(t,u1,u2)=hi(ui) for i=1,2.

In this case, the existence of the numbers αi is guaranteed by the following asymptotic behavior at zero or infinity:

lim supλ0+(1c)ϕi1(cfi((1c)λ))λ>1orlim supλ+(1c)ϕi1(cfi((1c)λ))λ>1.

Similarly, the existence of the numbers βi can be obtained from the following asymptotic behavior at zero or infinity:

lim infλ0+ϕi1(hi(λ))λ<1orlim infλ+ϕi1(hi(λ))λ<1.
Remark 4.2 (Multiple solutions).

Multiplicity results can be immediately established if several pairs of numbers (α1,β1) or (α2,β2) as in (4.3)-(4.6) exist. Note that we may obtain multiple solutions with multiplicity not necessarily concerned with all components of the solution, as shown in the following example.

Example 4.4.

Consider the system (4.7) with G1 as defined in (4.8) and

G2(u,v)=[1+sin2u,2]v2+19v3.

To check condition (C), take f1 and h1 as in Example 4.3,

f2(v)=v2+19v3andh2(v)=2v2+19v3.

Again, with c=1/2, one may easily verify that conditions (4.3)-(4.6) hold by taking α1=1/50, β1=2 and as pair (α2,β2), any one of the following two pairs (1/500,1/4), (20,1/3).

Thus Theorem 4.2 applied twice ensures the existence of at least two positive solutions (u1,v1) and (u2,v2) such that

150u1,u22,1500v114and13v220.

Acknowledgements

Jorge Rodríguez-López was partially supported by Xunta de Galicia ED431C 2019/02.

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