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volume 3, issue 1, article 14, 2002.

Received 17 May, 2001;

accepted 29 October, 2001.

Communicated by:B.G. Pachpatte

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Journal of Inequalities in Pure and Applied Mathematics

LOWER AND UPPER SOLUTIONS METHOD FOR FIRST ORDER DIFFERENTIAL INCLUSIONS WITH NONLINEAR

BOUNDARY CONDITIONS

M. BENCHOHRA AND S.K. NTOUYAS

Département de Mathématiques, Université de Sidi Bel Abbès BP 89, 22000 Sidi Bel Abbès, Algérie.

EMail:[email protected] Department of Mathematics, University of Ioannina 451 10 Ioannina, Greece.

EMail:[email protected]

URL:http://www.uoi.gr/schools/scmath/math/staff/snt/snt.htm

c

2000School of Communications and Informatics,Victoria University of Technology ISSN (electronic): 1443-5756

074-01

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The Lower and Upper Solutions Method for First Order Differential Inclusions with Nonlinear Boundary Conditions

M. BenchohraandS.K. Ntouyas

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J. Ineq. Pure and Appl. Math. 3(1) Art. 14, 2002

http://jipam.vu.edu.au

Abstract

In this paper a fixed point theorem for condensing maps combined with upper and lower solutions are used to investigate the existence of solutions for first order differential inclusions with general nonlinear boundary conditions.

2000 Mathematics Subject Classification:34A60

Key words: Initial value problem, Convex multivalued map, Differential inclusions, Nonlinear boundary conditions, Condensing map, Fixed point, Trunca- tion map, Upper and lower solutions.

1. Introduction

This paper is concerned with the existence of solutions for the boundary multivalued problem:

(1.1) y⁰(t)∈F(t, y(t)), for a.e. t∈J = [0, T]

(1.2) L(y(0), y(T)) = 0

where F : J ×R→2^R is a compact and convex valued multivalued map and L:R²→Ris a continuous single-valued map.

The method of upper and lower solutions has been successfully applied to study the existence of multiple solutions for initial and boundary value problems of first and second order.

This method has been used only in the context of single-valued differential equations. We refer to the books of Bernfeld-Lakshmikantham [4], Heikkila- Lakshmikantham [13], Ladde- Lakshmikantham-Vatsala [16], to the thesis of De Coster [7], to the papers of Carl-Heikkila- Kumpulainen [6], Cabada [5], Frigon [9], Frigon-O’Regan [10], Heikkila-Cabada [12], Lakshmikantham - Leela [17], Nkashama [20] and the references therein.

Using this method the authors obtained in [2] and [3] existence results for differential inclusions with periodic boundary conditions, for first and second order respectively.

In this paper we establish an existence result for the problem (1.1) – (1.2).

Our approach is based on the existence of upper and lower solutions and on a fixed point theorem for condensing maps due to Martelli [19].

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2. Preliminaries

We will briefly recall some basic definitions and facts from multivalued analysis that we will use in the sequel.

AC(J,R)is the space of all absolutely continuous functionsy:J→R. Condition

y≤y¯ if and only if y(t)≤y(t)¯ for all t∈J

defines a partial ordering in AC(J,R). If α, β ∈ AC(J,R) and α ≤ β, we denote

[α, β] ={y∈AC(J,R) :α≤y≤β}.

W^1,1(J,R) denotes the Banach space of functions y : J→R which are absolutely continuous and whose derivativey⁰ (which exists almost everywhere) is an element ofL¹(J,R)with the norm

kyk_W^1,1 =kyk_L¹ +ky⁰k_L¹ for ally∈W^1,1(J,R).

Let (X,|·|) be a normed space. A multivalued map G : X→2^X is convex (closed) valued if G(x) is convex (closed) for all x ∈ X. G is bounded on bounded sets if G(B) = ∪x∈BG(x) is bounded in X for all bounded subsets B ofX (i.e. sup_x∈B{sup{|y| : y ∈ G(x)}} < ∞). Gis called upper semicontinuous (u.s.c.) onXif for eachx0 ∈Xthe setG(x0)is a nonempty, closed subset ofX, and if for each open setV ofX containingG(x₀), there exists an open neighbourhoodU ofx₀ such thatG(U)⊆V.

Gis said to be completely continuous ifG(B)is relatively compact for every bounded subsetB ⊂X.

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If the multivalued mapGis completely continuous with nonempty compact values, thenGis u.s.c. if and only ifGhas a closed graph (i.e. xn→x∗, yn→y∗, y_n∈G(x_n)implyy∗ ∈G(x∗)).

Ghas a fixed point if there isx∈Xsuch thatx∈G(x).

In the followingCC(X)denotes the set of all nonempty compact and convex subsets ofX.

An upper semi-continuous mapG :X→2^X is said to be condensing [19] if for any bounded subset B ⊆ X, with µ(B) 6= 0,we haveµ(G(B)) < µ(B), whereµdenotes the Kuratowski measure of noncompacteness [1]. We remark that a compact map is the easiest example of a condensing map. For more details on multivalued maps see the books of Deimling [8] and Hu and Papageorgiou [15].

The multivalued mapF : J→CC(R)is said to be measurable, if for every y ∈R, the functiont7−→d(y, F(t)) = inf{|y−z|:z ∈F(t)}is measurable.

Definition 2.1. A multivalued mapF :J×R→2^Ris said to be anL¹-Carathéodory if

(i) t7−→F(t, y)is measurable for eachy∈R;

(ii) y7−→F(t, y)is upper semicontinuous for almost allt∈J; (iii) For eachk > 0, there existsh_k ∈L¹(J,R+)such that

kF(t, y)k= sup{|v|:v ∈F(t, y)} ≤h_k(t) for all |y| ≤k

and for almost all t∈J.

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So let us start by defining what we mean by a solution of problem (1.1) – (1.2).

Definition 2.2. A functiony∈AC(J,R)is said to be a solution of (1.1) – (1.2) if there exists a function v ∈ L¹(J,R) such that v(t) ∈ F(t, y(t)) a.e. onJ, y⁰(t) =v(t)a.e. onJ andL(y(0), y(T)) = 0.

The following concept of lower and upper solutions for (1.1) – (1.2) has been introduced by Halidias and Papageorgiou in [14] for second order multivalued boundary value problems. It will be the basic tools in the approach that follows.

Definition 2.3. A functionα ∈AC(J,R)is said to be a lower solution of (1.1) – (1.2) if there exists v1 ∈ L¹(J,R) such that v1(t) ∈ F(t, α(t)) a.e. on J, α⁰(t)≤v₁(t)a.e. onJ andL(α(0), α(T))≤0.

Similarly, a function β ∈ AC(J,R) is said to be an upper solution of (1.1) – (1.2) if there exists v2 ∈ L¹(J,R) such that v2(t) ∈ F(t, β(t)) a.e. on J, β⁰(t)≥v₂(t)a.e. onJ andL(β(0), β(T))≥0.

For the multivalued mapF and for eachy ∈C(J,R)we defineS_F,y¹ by S_F,y¹ ={v ∈L¹(J,R) :v(t)∈F(t, y(t))for a.e.t∈J}.

Our main result is based on the following:

Lemma 2.1. [18]. LetI be a compact real interval andX be a Banach space.

LetF :I×X→CC(X); (t, y)→F(t, y)measurable with respect totfor any y ∈ X and u.s.c. with respect to y for almost each t ∈ I and S_F,y¹ 6= ∅ for any y ∈ C(I, X)and let Γ be a linear continuous mapping fromL¹(I, X)to C(I, X), then the operator

Γ◦S_F¹ :C(I, X)→CC(C(I, X)), y 7−→(Γ◦S_F¹)(y) := Γ(S_F,y¹ )

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is a closed graph operator inC(I, X)×C(I, X).

Lemma 2.2. [19]. LetG:X→CC(X)be an u.s.c. condensing map. If the set M :={v ∈X :λv∈G(v)for someλ >1}

is bounded, thenGhas a fixed point.

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3. Main Result

We are now in a position to state and prove our existence result for the problem (1.1) – (1.2).

Theorem 3.1. SupposeF :J×R→CC(R)is anL¹-Carathéodory multivalued map. In addition assume the following conditions

(H1) there existαandβ inW^1,1(J,R) lower and upper solutions respectively for the problem (1.1) – (1.2) such that α≤β,

(H2) Lis a continuous single-valued map in(x, y)∈[α(0), β(0)]×[α(T), β(T)]

and nonincreasing iny∈[α(T), β(T)],

are satisfied. Then the problem (1.1) – (1.2) has at least one solution y ∈ W^1,1(J,R)such that

α(t)≤y(t)≤β(t) for all t ∈J.

Proof. Transform the problem into a fixed point problem. Consider the follow- ing modified problem (see [5])

(3.1) y⁰(t) +y(t)∈F1(t, y(t)), a.e. t∈J,

(3.2) y(0) =τ(0, y(0)−L(y(0), y(T)))

whereF₁(t, y) = F(t, τ(t, y)) +τ(t, y), τ(t, y) = max{α(t),min{y, β(t)}}

andy(t) = τ(t, y(t)).

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Remark 3.1. (i) Notice that F₁ is anL¹-Carathéodory multivalued map with compact convex values and there existsφ∈L¹(J,R⁺)such that

kF1(t, y(t))k ≤φ(t) + max(sup

t∈J

|α(t)|,sup

t∈J

|β(t)|)

for a.e. t∈J and all y∈C(J,R).

(ii) By the definition ofτ it is clear thatα(0) ≤y(0)≤β(0).

Clearly a solution to (3.1) – (3.2) is a fixed point of the operatorN :C(J,R)

→2^C(J,R)defined by

N(y) :=

h∈C(J,R) :h(t) = y(0) + Z t

0

[v(s) +y(s)−y(s)]ds, v ∈S˜_F,y¹

where

S˜_F,y¹ ={v ∈S_F,y¹ :v(t)≥v₁(t)a.e. on A₁ and v(t)≤v₂(t)a.e. on A₂}, S_F,y¹ ={v ∈L¹(J,R) :v(t)∈F(t, y(t))for a.e.t∈J},

A₁ ={t∈J :y(t)< α(t)≤β(t)}, A₂ ={t∈J :α(t)≤β(t)< y(t)}.

Remark 3.2. (i) For each y ∈ C(J,R) the setS_F,y¹ is nonempty (see Lasota and Opial [18]).

(ii) For eachy∈ C(J,R)the setS˜_F,y¹ is nonempty. Indeed, by (i) there exists v ∈S_F,y¹ . Set

w=v₁χ_A₁ +v₂χ_A₂ +vχ_A₃,

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where

A3 ={t ∈J :α(t)≤y(t)≤β(t)}.

Then by decomposabilityw∈S˜_F,y¹ .

We shall show that N is a completely continuous multivalued map, u.s.c.

with convex closed values. The proof will be given in several steps.

Step 1: N(y)is convex for eachy∈C(J,R).

Indeed, ifh, hbelong toN(y), then there existv ∈S˜_F,y¹ andv ∈S˜_F,y¹ such that

h(t) =y(0) + Z t

0

[v(s) +y(s)−y(s)]ds, t∈J and

h(t) = y(0) + Z t

0

[v(s) +y(s)−y(s)]ds, t∈J.

Let0≤k ≤1.Then for eacht ∈Jwe have [kh+ (1−k)h](t) =y(0) +

Z t

0

[kv(s) + (1−k)v(s) +y(s)−y(s)]ds.

SinceS˜_F,y¹ is convex (becauseF has convex values) then kh+ (1−k)h∈G(y).

Step 2: N sends bounded sets into bounded sets inC(J,R).

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Let B_r := {y ∈ C(J,R) : kyk∞ ≤ r}, (kyk∞ := sup{|y(t)| : t ∈ J}) be a bounded set inC(J,R)and y ∈ Br, then for each h ∈ N(y)there exists v ∈S˜_F,y¹ such that

h(t) = y(0) + Z t

0

Thus for eacht ∈J we get

|h(t)| ≤ |y(0)|+ Z t

0

[|v(s)|+|y(s)|+|y(s)|]ds

≤ max(α(0), β(0)) +kφrk_L¹ +Tmax(r,sup

t∈J

|α(t)|,sup

t∈J

|β(t)|) +T r.

Step 3: N sends bounded sets inC(J,R)into equicontinuous sets.

Letu₁, u₂ ∈J, u₁ < u₂,B_r:={y∈C(J,R) :kyk∞≤r}be a bounded set in C(J,R)andy∈B_r. For eachh∈N(y)there existsv ∈S˜_F,y¹ such that

h(t) = y(0) + Z t

0

We then have

|h(u₂)−h(u₁)|

≤ Z u2

u1

[|v(s) +y(s)|+|y(s)|]ds

≤ Z u2

u1

|φ_r(s)|ds+ (u₂−u₁) max(r,sup

t∈J

|α(t)|,sup

t∈J

|β(t)|) +r(u₂−u₁).

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As a consequence of Step 2, Step 3 together with the Ascoli-Arzela theorem we can conclude that N :C(J,R)→2^C(J,^R⁾is a compact multivalued map, and therefore, a condensing map.

Step 4: N has a closed graph.

Letyn→y0, hn∈N(yn)andhn→h0. We shall prove thath0 ∈N(y0).

h_n∈N(y_n)means that there existsv_n∈S˜_F,y_n such that

h_n(t) =y(0) + Z t

0

[v_n(s) +y_n(s)−y_n(s)]ds, t∈J.

We must prove that there existsv0 ∈S˜_F,y¹

0 such that h₀(t) =y(0) +

Z t

0

[v₀(s) +y₀(s)−y₀(s)]ds, t∈J.

Consider the linear continuous operatorΓ :L¹(J,R)→C(J,R)defined by (Γv)(t) =

Z t

0

v(s)ds.

We have

h_n−y(0)− Z t

0

[y_n(s)−y_n(s)]ds

−

h₀−y(0) + Z t

0

[y₀(s)−y₀(s)]ds

_∞

→0.

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From Lemma2.1, it follows thatΓ◦S˜_F¹ is a closed graph operator.

Also from the definition ofΓwe have that

hn(t)−y(0)− Z t

0

[y_n(s)−y0(s)]ds

∈Γ S˜_F,y¹

n

. Sincey_n→y₀ it follows from Lemma2.1that

h₀(t) = y(0) + Z t

0

[v₀(s) +y₀(s)−y₀(s)]ds, t∈J for somev₀ ∈S˜_F,y¹

0.

Next we shall show thatN has a fixed point, by proving that Step 5: The set

M :={v ∈C(J,R) :λv ∈N(v)for someλ >1}

is bounded.

Lety ∈ M thenλy ∈ N(y)for someλ > 1. Thus there existsv ∈ S˜_F,y¹ such that

y(t) = λ⁻¹y(0) +λ⁻¹ Z t

0

Thus

|y(t)| ≤ |y(0)|+ Z t

0

|v(s) +y(s)−y(s)|ds, t∈J.

From the definition ofτ there existsφ ∈L¹(J,R⁺)such that

kF(t, y(t))k= sup{|v|:v ∈F(t, y(t))} ≤φ(t)for eachy∈C(J,R),

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|y(t)| ≤max(α(0), β(0))+kφk_L¹+T max(sup

t∈J

|α(t)|,sup

t∈J

|β(t)|)+

Z t

0

|y(s)|ds.

Set

z₀ = max(α(0), β(0)) +kφk_L¹ +Tmax(sup

t∈J

|α(t)|,sup

t∈J

|β(t)|).

Using the Gronwall’s Lemma ([11, p. 36]) we get for eacht∈J

|y(t)| ≤ z₀+z₀ Z t

0

e^t−sds

≤ z₀+z₀(e^t−1).

Thus

kyk∞ ≤z0+z0(e^T −1).

This shows thatM is bounded.

Hence, Lemma 2.2 applies andN has a fixed point which is a solution to problem (3.1) – (3.2).

Step 6: We shall show that the solutionyof (3.1)-(3.2) satisfies α(t)≤y(t)≤β(t) for all t ∈J.

Letybe a solution to (3.1) – (3.2). We prove that α(t)≤y(t) for all t ∈J.

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Suppose not. Then there existt₁, t₂ ∈J, t₁ < t₂ such thatα(t₁) = y(t₁)and α(t)> y(t) for all t∈(t₁, t₂).

In view of the definition ofτ one has

y⁰(t) +y(t)∈F(t, α(t)) +α(t) a.e. on (t₁, t₂).

Thus there exists v(t) ∈ F(t, α(t))a.e. onJ withv(t) ≥ v₁(t)a.e. on(t₁, t₂) such that

y⁰(t) +y(t) =v(t) +α(t) a.e. on (t1, t2).

An integration on(t₁, t], witht∈(t₁, t₂)yields

y(t)−y(t₁) = Z t

t1

[v(s) + (α−y)(s)]ds

>

Z t

t1

v(s)ds.

Sinceαis a lower solution to (1.1) – (1.2), then α(t)−α(t₁)≤

Z t

t1

v₁(s)ds, t∈(t₁, t₂).

It follows from the factsy(t1) =α(t1), v(t)≥v1(t)that α(t)< y(t) for all t ∈(t₁, t₂)

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which is a contradiction, sincey(t)< α(t) for all t∈(t₁, t₂). Consequently α(t)≤y(t) for all t ∈J.

Analogously, we can prove that

y(t)≤β(t) for all t ∈J.

This shows that the problem (3.1) – (3.2) has a solution in the interval[α, β].

Finally, we prove that every solution of (3.1) – (3.2) is also a solution to (1.1) – (1.2). We only need to show that

α(0) ≤y(0)−L(y(0), y(T))≤β(0).

Notice first that we can prove that

α(T)≤y(T)≤β(T).

Suppose now thaty(0)−L(y(0), y(T))< α(0). Theny(0) =α(0)and y(0)−L(α(0), y(T))< α(0).

SinceLis nonincreasing iny, we have

α(0) ≤α(0)−L(α(0), α(T))≤α(0)−L(α(0), y(T))< α(0) which is a contradiction.

Analogously we can prove that

y(0)−L(τ(0), τ(T))≤β(0).

Thenyis a solution to (1.1) – (1.2).

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Remark 3.3. Observe that ifL(x, y) =ax−by−c,we obtain that Theorem3.1 gives an existence result for the problem

y⁰(t)∈F(t, y(t)), for a.e. t∈J = [0, T] ay(0)−by(T) = c

witha, b ≥0, a+b > 0which includes the periodic case (a =b = 1, c = 0) and the initial and the terminal problem.

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