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An Existence Result on Noncompact Intervals for Second Order Functional Differential Inclusions
Journal of Mathematical Analysis and Applications 248, 520᎐531 Ž2000. doi:10.1006rjmaa.2000.6926, available online at http:rrwww.idealibrary.com on
An Existence Result on Noncompact Intervals for Second Order Functional Differential Inclusions M. Benchohra Department of Mathematics, Uni¨ esity of Sidi Bel Abbes, BP 89, 22000 Sidi Bel Abbes, Algeria E-mail: [email protected]
and S. K. Ntouyas Department of Mathematics, Uni¨ ersity of Ioannina, 451 10 Ioannina, Greece E-mail: [email protected] Submitted by J. Henderson Received January 5, 2000
In this paper we investigate the existence of mild solutions on an infinite interval for second order functional differential inclusions in Banach spaces. We shall make use of a theorem of Ma, which is an extension to multivalued maps on locally convex topological spaces of Schaefer’s theorem. 䊚 2000 Academic Press Key Words: initial value problems; convex multivalued map; mild solution; functional differential inclusion; existence; fixed point; abstract space.
1. INTRODUCTION Existence of solutions on compact intervals for functional differential equations has received much attention in recent years. We refer for instance to the books of Erbe et al. w6x, Henderson w11x, the survey paper of Ntouyas w20x, the papers of Hristova and Bainov w12x, Nieto et al. w19x, and Liz and Nieto w17x, and the references cited therein. Methods used are usually the topological transversality theory of Dugundji and Granas w5x and the monotone iterative method combined with upper and lower solutions w15x. 520 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
FUNCTIONAL DIFFERENTIAL INCLUSIONS
521
For other results, see for instance the papers of Avgerinos and Papageorgiou w1x, Papageorgiou w21, 22x, for differential inclusions on compact intervals, and Benchohra w2x for results for first order differential inclusions on unbounded intervals. Additional results for differential equations in abstract spaces, with properties of semigroup theory, can be found in the book of Goldstein w9x. In this paper we shall prove a theorem which assures the existence of mild solutions defined on an unbounded real interval J for the initial value problem ŽIVP for short. of the second order functional differential inclusion t g J s w 0, ⬁ .
yY y Ay g F Ž t , yt . , y0 s ,
X
y Ž 0. s ,
Ž 1. Ž 2.
where F : J = C Ž J 0 , E . ª 2 E Žhere J 0 s wyr, 0x. is a bounded, closed, convex multivalued map, g C Ž J 0 , E ., g E, and A is the infinitesimal generator of a strongly continuous cosine family C Ž t . : t g R4 in a real Banach space E with the norm < ⭈ <. For any continuous function y defined on the interval wyr, ⬁. and any t g J, we denote by yt the element of C Ž J 0 , E . defined by yt Ž . s y Ž t q . ,
g J0 .
Here yt Ž⭈. represents the history of the state from time t y r, up to the present time t. The method we are going to use is to reduce the existence of solutions to problem Ž1. ᎐ Ž2. to the search for fixed points of a suitable multivalued map on the Frechet space C Žwyr, ⬁., E .. In order to prove the existence of ´ fixed points, we shall rely on a theorem due to Ma w18x, which is an extension to multivalued maps between locally convex topological spaces of Schaefer’s theorem w23x. In w3x we studied existence results on noncompact intervals for first order differential inclusions. Here we extend the results of w3x to second order functional differential inclusions on unbounded real intervals.
2. PRELIMINARIES In this section, we introduce notations, definitions, and preliminary facts from multivalued analysis which are used throughout this paper. Jm is the compact real interval w0, m x Ž m g N ..
522
BENCHOHRA AND NTOUYAS
C Ž J, E . is the linear metric Frechet space of continuous functions from ´ J into E with the metric Žsee Dugundji and Granas w5x. d Ž y, z . s
⬁
2ym 5 y y z 5 m
Ý
1 q 5 y y z5m
ms0
for each y, z g C Ž J , E . ,
where 5 y 5 m [ sup y Ž t . : t g Jm 4 . B Ž E . denotes the Banach space of bounded linear operators from E into E. A measurable function y : J ª E is Bochner integrable if and only if < y < is Lebesgue integrable. For properties of Bochner integral we refer to Yosida w26x. L1 Ž J, E . denotes the Banach space of continuous functions y : J ª E which are Bochner integrable normed by 5 y 5 L1 s
⬁
H0
y Ž t . dt
for all y g L1 Ž J , E . .
Up denotes the neighbourhood of 0 in C Ž J, E . defined by Up [ y g C Ž J , E . : 5 y 5 m F p for each m g N 4 . The convergence in C Ž J, E . is the uniform convergence on compact intervals, i.e., y j ª y in C Ž J, E . if and only if for each m g N, 5 y j y y 5 m ª 0 in C Ž Jm , E . as j ª ⬁. M : C Ž J, E . is a bounded set if and only if there exists a positive function g C Ž J, R . such that yŽ t. F Ž t.
for all t g J and all y g M.
The Ascoli᎐Arzela theorem says that a set M : C Ž J, E . is compact if and only if for each m g N, M is a compact set in the Banach space Ž C Ž Jm , E ., 5 ⭈ 5 m .. We say that a family C Ž t . : t g R4 of operators in B Ž E . is a strongly continuous cosine family if Ži. C Ž0. s I Ž I is the identity operator in E ., Žii. C Ž t q s . q C Ž t y s . s 2C Ž t .C Ž s . for all s, t g R, Žiii. the map t ¬ C Ž t . y is strongly continuous for each y g E; The strongly continuous sine family SŽ t . : t g R4 , associated to the given strongly continuous cosine family C Ž t . : t g R4 , is defined by SŽ t . y s
t
H0 C Ž s . y ds,
y g E, t g R.
FUNCTIONAL DIFFERENTIAL INCLUSIONS
523
The infinitesimal generator A : E ª E of a cosine family C Ž t . : t g R4 is defined by Ay s
d2 dt 2
C Ž 0 . y.
For more details on strongly continuous cosine and sine families, we refer the reader to the book of Goldstein w9x and to the papers of Fattorini w7, 8x and Travis and Webb w24, 25x. Let Ž X, 5 ⭈ 5. be a Banach space. A multivalued map G : X ª X is convex Žclosed. valued if GŽ x . is convex Žclosed. for all x g X. G is bounded on bounded sets if GŽ B . s Dx g B GŽ x . is bounded in X for any bounded set B of X Ži.e., sup x g B sup5 y 5 : y g GŽ x .44 - ⬁.. G is called upper semicontinuous Žu.s.c.. on X if for each x# g X the set GŽ x#. is a nonempty, closed subset of X, and if for each open set B of X containing GŽ x#., there exists an open neighbourhood V of x# such that GŽ V . : B. G is said to be completely continuous if GŽ B . is relatively compact for every bounded subset B : X. If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph Ži.e., x n ª x#, yn ª y#, yn g Gx n and imply y# g Gx#.. G has a fixed point if there is x g X such that x g Gx. In the following BCC Ž X . denotes the set of all nonempty bounded, closed, and convex subsets of X. A multivalued map G : J ª BCC Ž E . is said to be measurable if for each x g E the function Y : J ª R defined by Y Ž t . s d Ž x, G Ž t . . s inf < x y z < : z g G Ž t . 4 belongs to L1 Ž J, R .. For more details on multivalued maps see the books of Deimling w4x and Hu and Papageorgiou w13x. Let us list the following hypotheses ŽH1. A is an infinitesimal generator of a given strongly continuous and bounded cosine family C Ž t . : t g J 4 ; ŽH2. F : J = C Ž J 0 , E . ª BCC Ž E .; Ž t, u. ¬ F Ž t, u. is measurable with respect to t for each u g C Ž J 0 , E ., u.s.c. with respect to u for each t g J, and for each fixed u g C Ž J 0 , E . the set SF , u s g g L1 Ž J , E . : g Ž t . g F Ž t , y . for a.e. t g J 4 is nonempty;
524
BENCHOHRA AND NTOUYAS
ŽH3. 5 F Ž t, u.5 [ sup< ¨ < g F Ž t, u.4 F pŽ t . Ž5 u 5. for almost all t g J and all u g C Ž J 0 , E ., where p g L1 Ž J, Rq . and : Rqª Ž0, ⬁. is continuous and increasing with m
Mm
H0
p Ž s . ds -
d
⬁
Hc
for each m g N ;
Ž .
where c s M 5 5 q Mm < < and M s sup5 C Ž t .5; t g J 4 ; ŽH4. for each neighbourhood Up of 0, u g Up and t g J the set
½
C Ž t . Ž 0. q S Ž t . q
t
H0 S Ž t y s . g Ž s . ds : g g S
F, u
5
is relatively compact. Remark 2.1. If dim E - ⬁ and J is a compact real interval, then for each u g C Ž J, E . SF, u / ⭋ Žsee Lasota and Opial w16x.. DEFINITION 2.1. A continuous solution y Ž t . of the integral inclusion y Ž t . g C Ž t . Ž 0. q S Ž t . q
t
H0 S Ž t y s . F Ž s, y . ds s
Ž 3.
with y 0 s , yX Ž0. s is called a mild solution on wyr, ⬁. of Ž1. ᎐ Ž2.. The following lemmas are crucial in the proof of our main theorem. LEMMA 2.1 w16x. Let I be a compact real space. Let F be a multi¨ alued map satisfying continuous mapping from L1 Ž I, X . to C Ž I, X .. ⌫ ( SF : C Ž I, X . ª BCC Ž C Ž I, X . . ,
inter¨ al and X be a Banach Ž H 2. and let ⌫ be a linear Then the operator
y ¬ Ž ⌫ ( SF . Ž y . [ ⌫ Ž SF , y .
is a closed graph operator in C Ž I, X . = C Ž I, X .. LEMMA 2.2 w18x. Let X be a locally con¨ ex space and N : X ª 2 X be a compact con¨ ex ¨ alued, u.s.c. multi¨ alued map such that for each closed neighbourhood Up of 0 N ŽUp . is a relati¨ ely compact set for each p g N. If the set ⍀ [ y g X : y g Ny for some ) 1 4 is bounded, then N has a fixed point.
525
FUNCTIONAL DIFFERENTIAL INCLUSIONS
3. MAIN RESULT Now, we are able to state and prove our main theorem. THEOREM 3.1. Assume that F satisfies ŽH1. ᎐ ŽH4.. Then the IVP Ž1. ᎐ Ž2. has at least one mild solution on wyr, ⬁.. Proof. Let C Žwyr, ⬁., E . be the Frechet space endowed with the semi´ norms 5 y 5 m [ sup y Ž t . : t g w yt , m x 4 ,
for y g C Ž w yr , ⬁ . , E . , m g N.
Transform the problem into a fixed point problem. Consider the multivalued map, N : C Žwyr, ⬁., E . ª 2 C Žwyr, ⬁., E . defined by
¡
~
Ny [ h g C Ž w yr , ⬁ . , E . :
¢
¡ Ž t . ,
hŽ t . s
~
¦
if t g J 0
C Ž t . Ž 0. q S Ž t .
¢
q
t
H0 S Ž t y s . g Ž s . ds,
if t g J
¥,
§
where g g SF , y s g g L1 Ž J , E . : g Ž t . g F Ž t , yt . for a.e. t g J 4 . Remark 3.1. It is clear that the fixed points of N are mild solutions to Ž1. ᎐ Ž2.. We shall show that N ŽUq . is relatively compact for each neighbourhood Uq of 0 g C Ž J, E . with q g N and the multivalued map N has bounded, closed, and convex values and it is u.s.c. The proof will be given in several steps. Step 1.
Ny is con¨ ex for each y g C Ž J, E ..
Indeed, if h1 , h 2 belong to Ny, then there exist g 1 , g 2 g SF, y such that for each t g J we have h1 Ž t . s C Ž t . Ž 0 . q S Ž t . q
t
H0 S Ž t y s . g Ž s . ds 1
526
BENCHOHRA AND NTOUYAS
and h 2 Ž t . s C Ž t . Ž 0. q S Ž t . q
t
H0 S Ž t y s . g Ž s . ds. 2
Let 0 F ␣ F 1. Then for each t g J we have
Ž ␣ h1 q Ž 1 y ␣ . h 2 . Ž t . s C Ž t . Ž 0 . q S Ž t . q
t
H0 S Ž t y s .
␣ g 1 Ž s . q Ž 1 y ␣ . g 2 Ž s . ds.
Since SF, y is convex Žbecause F has convex values. then
␣ h1 q Ž 1 y ␣ . h 2 g Ny. Step 2.
N ŽUq . is bounded in C Ž J, E . for each q g N.
Indeed, it is enough to show that there exists a positive constant l such that for each h g Ny, y g Uq one has 5 h 5 ⬁ F l . If h g Ny, then there exists g g SF, y such that for each t g J we have h Ž t . s C Ž t . Ž 0. q S Ž t . q
t
H0 S Ž t y s . g Ž s . ds.
By ŽH3. we have for each t g Jm that hŽ t . F C Ž t .
Ž 0 . q S Ž t . . < < q
t
S Ž t y s . g Ž s . ds
H0
F M 5 5 q Mm < < q mM sup Ž y . yg w0, q x
t
žH
/
p Ž s . ds .
0
Then for each h g N ŽUq . we have 5 h 5 ⬁ F M 5 5 q Mm < < q mM sup
tg w0, m x
Step 3.
t
žH
/
p Ž s . ds max
0
sup Ž y . [ l .
ygUq yg w0, q x
For each q g N, N ŽUq . is equicontinuous for Uq g C Ž J, E ..
Let t 1 , t 2 g Jm , 0 - t 1 - t 2 , and Uq be a neighbourhood of 0 in C Ž J, E . for q g N. For each y g Uq and h g Ny, there exists g g SF, y such that h Ž t . s C Ž t . Ž 0. q S Ž t . q
t
H0 S Ž t y s . g Ž s . ds.
527
FUNCTIONAL DIFFERENTIAL INCLUSIONS
Thus h Ž t 2 . y h Ž t1 . F C Ž t 2 . Ž 0. y C Ž t1 . Ž 0. q S Ž t 2 . y S Ž t1 . t2
q
H0
q
Ht
t2
S Ž t 2 y s . y S Ž t 1 y s . g Ž s . ds S Ž t 1 y s . g Ž s . ds
1
F C Ž t 2 . Ž 0. y C Ž t1 . Ž 0. q S Ž t 2 . y S Ž t1 . q M Ž t 2 y t1 .
m
H0
g Ž s . ds q Mm
t2
Ht
g Ž s . ds.
1
As t 2 ª t 1 the right-hand side of the above inequality tends to zero. The equicontinuity for the cases t 1 - t 2 F 0 and t 1 F 0 F t 2 follows from the uniform continuity of on the interval J 0 and from the relation h Ž t 2 . y h Ž t1 . s h Ž t 2 . y Ž t1 . F h Ž t 2 . y h Ž 0. q Ž 0. y Ž t1 . , respectively. As a consequence of Step 2, Step 3, and ŽH4. together with the Ascoli᎐Arzela theorem we can conclude that N ŽUq . is relatively compact in C Ž J, E .. Step 4.
N has a closed graph.
Let yn ª y#, h n g Nyn , and h n ª h#. We shall prove that h# g Ny#. h n g Nyn means that there exists g n g SF, y n, such that h n Ž t . s C Ž t . Ž 0. q S Ž t . q
t
H0 S Ž t y s . g Ž s . ds. n
We must prove that there exists g# g SF, y# such that h# Ž t . s C Ž t . Ž 0 . q S Ž t . q
t
H0 S Ž t y s . g# Ž s . ds.
Ž 4.
The idea is then to use the facts that Ži. Žii.
h n ª h#; h n C Ž t . Ž0. y SŽ t . g ⌫ Ž SF, y n . where
⌫ : L1 Ž J , E . ª C Ž J , E .
defined by Ž ⌫g . Ž t . [
t
H0 S Ž t y s . g Ž s . ds.
528
BENCHOHRA AND NTOUYAS
If ⌫ ( SF is a closed graph operator, we would be done. But we don’t know whether ⌫ ( S F is a closed graph operator. So, we cut the functions yn , h n y C Ž t . Ž0. y sŽ t . , g n and we consider them defined on the interval w k, k q 1x for any k g N j 04 . Then, using Lemma 2.1, in this case we are able to affirm that Ž4. is true on the compact interval w k, k q 1x, i.e., h0 Ž t .
w k , kq1 x
s C Ž t . Ž 0. q S Ž t . q
t
H0 S Ž t y s . g#Ž s . ds k
k for a suitable L1-selection g# of F Ž t, y# t . on the interval w k, k q 1x. k At this point we can paste the functions g# obtaining the selection g# defined by
for t g w k, k q 1 . .
k g# Ž t . s g# Ž t.
We obtain then that g# is a L1-selection and Ž4. will be satisfied. We give now the details. Clearly we have that
Ž h n y C Ž t . Ž 0. y S Ž t . . y Ž h# y C Ž t . Ž 0. y S Ž t . .
⬁
ª 0,
as n ª ⬁. Now, we consider for all k g N j 04 , the mapping SFk : C Ž w k, k q 1 x , E . ª L1 Ž w k , k q 1 x , E . k 1 Žw u ¬ SF, k, k q 1x, E . : f Ž t . g F Ž t, uŽ t .. for a.e. t g w k, k q u[ fgL 1x4 . Also, we consider the linear continuous operators
⌫k : L1 Ž w k , k q 1 x , E . ª C Ž w k, k q 1 x , E . g ¬ ⌫k Ž g . Ž t . s
t
H0 S Ž t y s . g Ž s . ds.
From Lemma 2.1, it follows that ⌫k ( SFk is a closed graph operator for all k g N j 04 . Moreover, we have that
Ž h n Ž t . y C Ž t . Ž 0. y S Ž t . . w k , kq1x g ⌫k Ž SFk , y
n
..
Since yn ¬ y#, it follows from Lemma 2.1 that
Ž h# Ž t . y C Ž t . Ž 0 . y S Ž t . .
w k , kq1 x
s
t
H0 S Ž t y s . g#Ž s . ds
k k for some g# g SF, y# . So the function g# defined on J by k g# Ž t . s g# Ž t.
for t g w k , k q 1 .
is in SF, y# since g#Ž t . g F Ž t, y#. for a.e. t g J.
k
529
FUNCTIONAL DIFFERENTIAL INCLUSIONS
Step 5. The set ⍀ [ y g C Ž J , E . : y g Ny, for some ) 1 4 is bounded. Let y g ⍀. Then y g Ny for some ) 1. Thus there exists g g SF, y such that y Ž t . s y1 C Ž t . Ž 0 . q y1 S Ž t . q y1
t
H0 S Ž t y s . g Ž s . ds,
t g J.
This implies by ŽH3. that for each t g Jm we have t
y Ž t . F M 5 5 q Mm < < q Mm
H0 p Ž s . Ž 5 y 5. ds. s
We consider the function defined by
Ž t . s sup y Ž s . : yr F s F t 4 ,
0 F t F m.
Let tU g wyr, t x be such that Ž t . s < y Ž tU .<. If tU g w0, m x, by the previous inequality we have for t g w0, m x t
Ž t . F M 5 5 q MM 5 < q Mm
H0 p Ž s . Ž 5 y 5. ds
F M 5 5 q Mm < < q Mm
s
t
H0 p Ž s . Ž Ž s . . ds.
If tU g J 0 then Ž t . s 5 5 and the previous inequality holds since M G 1. Let us take the right-hand side of the above inequality as ¨ Ž t .. Then we have ¨ Ž 0 . s M 5 5 q Mm < < s c
and
Ž t . F ¨ Ž t . , t g Jm .
Using the increasing character of we get X
¨ Ž t . F Mmp Ž t . Ž Ž t . . ,
t g Jm .
This implies for each t g Jm that ¨Ž .
H Ž0.t ¨
du
Ž u.
F Mm
m
H0
p Ž s . ds -
⬁
du
H Ž0. Ž u . . ¨
530
BENCHOHRA AND NTOUYAS
This inequality implies that there exists a constant b such that ¨ Ž t . F b, t g Jm , and hence Ž t . F b, t g Jm . Since for every t g w0, m x, 5 yt 5 F Ž t ., we have 5 y 5 ⬁ [ sup y Ž t . : yr F t F m4 F b, where b depends only m and on the functions p and . This shows that ⍀ is bounded. Set X [ C Žwyr, ⬁., E .. As a consequence of Lemma 2.2 we deduce that N has a fixed point which is a mild solution of Ž1. ᎐ Ž2. on wyr, ⬁..
REFERENCES 1. E. P. Avgerinos and N. S. Papageorgiou, On quasilinear evolution inclusions, Glas. Mat. Ser. III 28 Ž48., No. 1 Ž1993., 35᎐52. 2. M. Benchohra, Existence of mild solutions on infinite intervals to first order initial value problems for a class of differential inclusions in Banach spaces, Discuss. Math. Differential Incl. 19 Ž1999., 85᎐95. 3. M. Benchohra and S. K. Ntouyas, Existence results on noncompact intervals for functional differential and integrodifferential inclusions in Banach spaces, submitted for publication. 4. K. Deimling, ‘‘Multivalued Differential Equations,’’ de Gruyter, BerlinrNew York, 1992. 5. J. Dugundji and A. Granas, ‘‘Fixed Point Theory,’’ Monografie Mat., PWN, Warsaw, 1982. 6. L. H. Erbe, Q. Kong, and B. G. Zhang, ‘‘Oscillation Theory for Functional Differential Equations,’’ Pure and Applied Mathematics, Academic Press, San Diego, 1994. 7. H. O. Fattorini, Ordinary differential equations in linear topological spaces, I, J. Differential Equations 5 Ž1968., 72᎐105. 8. H. O. Fattorini, Ordinary differential equations in linear topological spaces, II, J. Differential Equations 6 Ž1969., 50᎐70. 9. J. A. Goldstein, ‘‘Semigroups of Linear Operators and Applications,’’ Oxford Univ. Press, New York, 1985. 10. S. Heikkila and V. Lakshmikantham, ‘‘Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations,’’ Dekker, New York, 1994. 11. J. Henderson, ‘‘Boundary Value Problems for Functional Differential Equations,’’ World Scientific, Singapore, 1995. 12. S. G. Hristova and D. D. Bainov, Application of the monotone iterative techniques of Lakshmikantham to the solution of the initial value problem for functional differential equations, J. Math. Phys. Sci. 24 Ž1990., 405᎐413. 13. Sh. Hu and N. Papageorgiou, ‘‘Handbook of Multivalued Analysis,’’ Kluwer Academic, DordrechtrBostonrLondon, 1997. 14. T. Kusano and S. Oharu, Semilinear evolution equations with singularities in ordered Banach spaces, Differential Integral Equations 5, No. 6 Ž1992., 1383᎐1405. 15. G. S. Ladde, V. Laksmikantham and A. S. Vatsala, ‘‘Monotone Iterative Techniques for Nonlinear Differential Equations,’’ Pitman, Boston, 1985. 16. A. Lasota and Z. Opial, An application of the Kakutani-Ky-Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 Ž1965., 781᎐786.
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17. E. Liz and J. J. Nieto, Periodic BVPs for a class of functional differential equations, J. Math. Anal. Appl. 200 Ž1996., 680᎐686. 18. T. W. Ma, Topological degrees for set-valued compact vector fields in locally convex spaces, Dissertationes Math. 92 Ž1972., 1᎐43. 19. J. J. Nieto, Y. Jiang, and Y. Jurang, Monotone iterative method for functional differential equations, Nonlinear Anal. 32 Ž1998., 741᎐749. 20. S. K. Ntouyas, Initial and boundary value problems for functional differential equations via the topological transversality method: A survey, Bull. Greek Math. Soc. 40 Ž1998., 3᎐41. 21. N. S. Papageorgiou, Mild solutions of semilinear evolution inclusions, Indian J. Pure Appl. Math. 26, No. 3 Ž1995., 189᎐216. 22. N. S. Papageorgiou, Boundary value problems for evolution inclusions, Comment. Math. Uni¨ . Carolin. 29 Ž1988., 355᎐363. ¨ 23. H. Schaefer, Uber die methode der a priori schranken, Math. Ann. 129 Ž1955., 415᎐416. 24. C. C. Travis and G. F. Webb, Second order differential equations in Banach spaces, in ‘‘Proc. Int. Symp. on Nonlinear Equations in Abstract Spaces,’’ pp. 331᎐361, Academic Press, New York, 1978. 25. C. C. Travis and G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Hungar. 32 Ž1978., 75᎐96. 26. K. Yosida, ‘‘Functional Analysis,’’ 6th ed., Springer-Verlag, Berlin, 1980.
An Existence Result on Noncompact Intervals for Second Order Functional Differential Inclusions M. Benchohra Department of Mathematics, Uni¨ esity of Sidi Bel Abbes, BP 89, 22000 Sidi Bel Abbes, Algeria E-mail: [email protected]
and S. K. Ntouyas Department of Mathematics, Uni¨ ersity of Ioannina, 451 10 Ioannina, Greece E-mail: [email protected] Submitted by J. Henderson Received January 5, 2000
In this paper we investigate the existence of mild solutions on an infinite interval for second order functional differential inclusions in Banach spaces. We shall make use of a theorem of Ma, which is an extension to multivalued maps on locally convex topological spaces of Schaefer’s theorem. 䊚 2000 Academic Press Key Words: initial value problems; convex multivalued map; mild solution; functional differential inclusion; existence; fixed point; abstract space.
1. INTRODUCTION Existence of solutions on compact intervals for functional differential equations has received much attention in recent years. We refer for instance to the books of Erbe et al. w6x, Henderson w11x, the survey paper of Ntouyas w20x, the papers of Hristova and Bainov w12x, Nieto et al. w19x, and Liz and Nieto w17x, and the references cited therein. Methods used are usually the topological transversality theory of Dugundji and Granas w5x and the monotone iterative method combined with upper and lower solutions w15x. 520 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
FUNCTIONAL DIFFERENTIAL INCLUSIONS
521
For other results, see for instance the papers of Avgerinos and Papageorgiou w1x, Papageorgiou w21, 22x, for differential inclusions on compact intervals, and Benchohra w2x for results for first order differential inclusions on unbounded intervals. Additional results for differential equations in abstract spaces, with properties of semigroup theory, can be found in the book of Goldstein w9x. In this paper we shall prove a theorem which assures the existence of mild solutions defined on an unbounded real interval J for the initial value problem ŽIVP for short. of the second order functional differential inclusion t g J s w 0, ⬁ .
yY y Ay g F Ž t , yt . , y0 s ,
X
y Ž 0. s ,
Ž 1. Ž 2.
where F : J = C Ž J 0 , E . ª 2 E Žhere J 0 s wyr, 0x. is a bounded, closed, convex multivalued map, g C Ž J 0 , E ., g E, and A is the infinitesimal generator of a strongly continuous cosine family C Ž t . : t g R4 in a real Banach space E with the norm < ⭈ <. For any continuous function y defined on the interval wyr, ⬁. and any t g J, we denote by yt the element of C Ž J 0 , E . defined by yt Ž . s y Ž t q . ,
g J0 .
Here yt Ž⭈. represents the history of the state from time t y r, up to the present time t. The method we are going to use is to reduce the existence of solutions to problem Ž1. ᎐ Ž2. to the search for fixed points of a suitable multivalued map on the Frechet space C Žwyr, ⬁., E .. In order to prove the existence of ´ fixed points, we shall rely on a theorem due to Ma w18x, which is an extension to multivalued maps between locally convex topological spaces of Schaefer’s theorem w23x. In w3x we studied existence results on noncompact intervals for first order differential inclusions. Here we extend the results of w3x to second order functional differential inclusions on unbounded real intervals.
2. PRELIMINARIES In this section, we introduce notations, definitions, and preliminary facts from multivalued analysis which are used throughout this paper. Jm is the compact real interval w0, m x Ž m g N ..
522
BENCHOHRA AND NTOUYAS
C Ž J, E . is the linear metric Frechet space of continuous functions from ´ J into E with the metric Žsee Dugundji and Granas w5x. d Ž y, z . s
⬁
2ym 5 y y z 5 m
Ý
1 q 5 y y z5m
ms0
for each y, z g C Ž J , E . ,
where 5 y 5 m [ sup y Ž t . : t g Jm 4 . B Ž E . denotes the Banach space of bounded linear operators from E into E. A measurable function y : J ª E is Bochner integrable if and only if < y < is Lebesgue integrable. For properties of Bochner integral we refer to Yosida w26x. L1 Ž J, E . denotes the Banach space of continuous functions y : J ª E which are Bochner integrable normed by 5 y 5 L1 s
⬁
H0
y Ž t . dt
for all y g L1 Ž J , E . .
Up denotes the neighbourhood of 0 in C Ž J, E . defined by Up [ y g C Ž J , E . : 5 y 5 m F p for each m g N 4 . The convergence in C Ž J, E . is the uniform convergence on compact intervals, i.e., y j ª y in C Ž J, E . if and only if for each m g N, 5 y j y y 5 m ª 0 in C Ž Jm , E . as j ª ⬁. M : C Ž J, E . is a bounded set if and only if there exists a positive function g C Ž J, R . such that yŽ t. F Ž t.
for all t g J and all y g M.
The Ascoli᎐Arzela theorem says that a set M : C Ž J, E . is compact if and only if for each m g N, M is a compact set in the Banach space Ž C Ž Jm , E ., 5 ⭈ 5 m .. We say that a family C Ž t . : t g R4 of operators in B Ž E . is a strongly continuous cosine family if Ži. C Ž0. s I Ž I is the identity operator in E ., Žii. C Ž t q s . q C Ž t y s . s 2C Ž t .C Ž s . for all s, t g R, Žiii. the map t ¬ C Ž t . y is strongly continuous for each y g E; The strongly continuous sine family SŽ t . : t g R4 , associated to the given strongly continuous cosine family C Ž t . : t g R4 , is defined by SŽ t . y s
t
H0 C Ž s . y ds,
y g E, t g R.
FUNCTIONAL DIFFERENTIAL INCLUSIONS
523
The infinitesimal generator A : E ª E of a cosine family C Ž t . : t g R4 is defined by Ay s
d2 dt 2
C Ž 0 . y.
For more details on strongly continuous cosine and sine families, we refer the reader to the book of Goldstein w9x and to the papers of Fattorini w7, 8x and Travis and Webb w24, 25x. Let Ž X, 5 ⭈ 5. be a Banach space. A multivalued map G : X ª X is convex Žclosed. valued if GŽ x . is convex Žclosed. for all x g X. G is bounded on bounded sets if GŽ B . s Dx g B GŽ x . is bounded in X for any bounded set B of X Ži.e., sup x g B sup5 y 5 : y g GŽ x .44 - ⬁.. G is called upper semicontinuous Žu.s.c.. on X if for each x# g X the set GŽ x#. is a nonempty, closed subset of X, and if for each open set B of X containing GŽ x#., there exists an open neighbourhood V of x# such that GŽ V . : B. G is said to be completely continuous if GŽ B . is relatively compact for every bounded subset B : X. If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph Ži.e., x n ª x#, yn ª y#, yn g Gx n and imply y# g Gx#.. G has a fixed point if there is x g X such that x g Gx. In the following BCC Ž X . denotes the set of all nonempty bounded, closed, and convex subsets of X. A multivalued map G : J ª BCC Ž E . is said to be measurable if for each x g E the function Y : J ª R defined by Y Ž t . s d Ž x, G Ž t . . s inf < x y z < : z g G Ž t . 4 belongs to L1 Ž J, R .. For more details on multivalued maps see the books of Deimling w4x and Hu and Papageorgiou w13x. Let us list the following hypotheses ŽH1. A is an infinitesimal generator of a given strongly continuous and bounded cosine family C Ž t . : t g J 4 ; ŽH2. F : J = C Ž J 0 , E . ª BCC Ž E .; Ž t, u. ¬ F Ž t, u. is measurable with respect to t for each u g C Ž J 0 , E ., u.s.c. with respect to u for each t g J, and for each fixed u g C Ž J 0 , E . the set SF , u s g g L1 Ž J , E . : g Ž t . g F Ž t , y . for a.e. t g J 4 is nonempty;
524
BENCHOHRA AND NTOUYAS
ŽH3. 5 F Ž t, u.5 [ sup< ¨ < g F Ž t, u.4 F pŽ t . Ž5 u 5. for almost all t g J and all u g C Ž J 0 , E ., where p g L1 Ž J, Rq . and : Rqª Ž0, ⬁. is continuous and increasing with m
Mm
H0
p Ž s . ds -
d
⬁
Hc
for each m g N ;
Ž .
where c s M 5 5 q Mm < < and M s sup5 C Ž t .5; t g J 4 ; ŽH4. for each neighbourhood Up of 0, u g Up and t g J the set
½
C Ž t . Ž 0. q S Ž t . q
t
H0 S Ž t y s . g Ž s . ds : g g S
F, u
5
is relatively compact. Remark 2.1. If dim E - ⬁ and J is a compact real interval, then for each u g C Ž J, E . SF, u / ⭋ Žsee Lasota and Opial w16x.. DEFINITION 2.1. A continuous solution y Ž t . of the integral inclusion y Ž t . g C Ž t . Ž 0. q S Ž t . q
t
H0 S Ž t y s . F Ž s, y . ds s
Ž 3.
with y 0 s , yX Ž0. s is called a mild solution on wyr, ⬁. of Ž1. ᎐ Ž2.. The following lemmas are crucial in the proof of our main theorem. LEMMA 2.1 w16x. Let I be a compact real space. Let F be a multi¨ alued map satisfying continuous mapping from L1 Ž I, X . to C Ž I, X .. ⌫ ( SF : C Ž I, X . ª BCC Ž C Ž I, X . . ,
inter¨ al and X be a Banach Ž H 2. and let ⌫ be a linear Then the operator
y ¬ Ž ⌫ ( SF . Ž y . [ ⌫ Ž SF , y .
is a closed graph operator in C Ž I, X . = C Ž I, X .. LEMMA 2.2 w18x. Let X be a locally con¨ ex space and N : X ª 2 X be a compact con¨ ex ¨ alued, u.s.c. multi¨ alued map such that for each closed neighbourhood Up of 0 N ŽUp . is a relati¨ ely compact set for each p g N. If the set ⍀ [ y g X : y g Ny for some ) 1 4 is bounded, then N has a fixed point.
525
FUNCTIONAL DIFFERENTIAL INCLUSIONS
3. MAIN RESULT Now, we are able to state and prove our main theorem. THEOREM 3.1. Assume that F satisfies ŽH1. ᎐ ŽH4.. Then the IVP Ž1. ᎐ Ž2. has at least one mild solution on wyr, ⬁.. Proof. Let C Žwyr, ⬁., E . be the Frechet space endowed with the semi´ norms 5 y 5 m [ sup y Ž t . : t g w yt , m x 4 ,
for y g C Ž w yr , ⬁ . , E . , m g N.
Transform the problem into a fixed point problem. Consider the multivalued map, N : C Žwyr, ⬁., E . ª 2 C Žwyr, ⬁., E . defined by
¡
~
Ny [ h g C Ž w yr , ⬁ . , E . :
¢
¡ Ž t . ,
hŽ t . s
~
¦
if t g J 0
C Ž t . Ž 0. q S Ž t .
¢
q
t
H0 S Ž t y s . g Ž s . ds,
if t g J
¥,
§
where g g SF , y s g g L1 Ž J , E . : g Ž t . g F Ž t , yt . for a.e. t g J 4 . Remark 3.1. It is clear that the fixed points of N are mild solutions to Ž1. ᎐ Ž2.. We shall show that N ŽUq . is relatively compact for each neighbourhood Uq of 0 g C Ž J, E . with q g N and the multivalued map N has bounded, closed, and convex values and it is u.s.c. The proof will be given in several steps. Step 1.
Ny is con¨ ex for each y g C Ž J, E ..
Indeed, if h1 , h 2 belong to Ny, then there exist g 1 , g 2 g SF, y such that for each t g J we have h1 Ž t . s C Ž t . Ž 0 . q S Ž t . q
t
H0 S Ž t y s . g Ž s . ds 1
526
BENCHOHRA AND NTOUYAS
and h 2 Ž t . s C Ž t . Ž 0. q S Ž t . q
t
H0 S Ž t y s . g Ž s . ds. 2
Let 0 F ␣ F 1. Then for each t g J we have
Ž ␣ h1 q Ž 1 y ␣ . h 2 . Ž t . s C Ž t . Ž 0 . q S Ž t . q
t
H0 S Ž t y s .
␣ g 1 Ž s . q Ž 1 y ␣ . g 2 Ž s . ds.
Since SF, y is convex Žbecause F has convex values. then
␣ h1 q Ž 1 y ␣ . h 2 g Ny. Step 2.
N ŽUq . is bounded in C Ž J, E . for each q g N.
Indeed, it is enough to show that there exists a positive constant l such that for each h g Ny, y g Uq one has 5 h 5 ⬁ F l . If h g Ny, then there exists g g SF, y such that for each t g J we have h Ž t . s C Ž t . Ž 0. q S Ž t . q
t
H0 S Ž t y s . g Ž s . ds.
By ŽH3. we have for each t g Jm that hŽ t . F C Ž t .
Ž 0 . q S Ž t . . < < q
t
S Ž t y s . g Ž s . ds
H0
F M 5 5 q Mm < < q mM sup Ž y . yg w0, q x
t
žH
/
p Ž s . ds .
0
Then for each h g N ŽUq . we have 5 h 5 ⬁ F M 5 5 q Mm < < q mM sup
tg w0, m x
Step 3.
t
žH
/
p Ž s . ds max
0
sup Ž y . [ l .
ygUq yg w0, q x
For each q g N, N ŽUq . is equicontinuous for Uq g C Ž J, E ..
Let t 1 , t 2 g Jm , 0 - t 1 - t 2 , and Uq be a neighbourhood of 0 in C Ž J, E . for q g N. For each y g Uq and h g Ny, there exists g g SF, y such that h Ž t . s C Ž t . Ž 0. q S Ž t . q
t
H0 S Ž t y s . g Ž s . ds.
527
FUNCTIONAL DIFFERENTIAL INCLUSIONS
Thus h Ž t 2 . y h Ž t1 . F C Ž t 2 . Ž 0. y C Ž t1 . Ž 0. q S Ž t 2 . y S Ž t1 . t2
q
H0
q
Ht
t2
S Ž t 2 y s . y S Ž t 1 y s . g Ž s . ds S Ž t 1 y s . g Ž s . ds
1
F C Ž t 2 . Ž 0. y C Ž t1 . Ž 0. q S Ž t 2 . y S Ž t1 . q M Ž t 2 y t1 .
m
H0
g Ž s . ds q Mm
t2
Ht
g Ž s . ds.
1
As t 2 ª t 1 the right-hand side of the above inequality tends to zero. The equicontinuity for the cases t 1 - t 2 F 0 and t 1 F 0 F t 2 follows from the uniform continuity of on the interval J 0 and from the relation h Ž t 2 . y h Ž t1 . s h Ž t 2 . y Ž t1 . F h Ž t 2 . y h Ž 0. q Ž 0. y Ž t1 . , respectively. As a consequence of Step 2, Step 3, and ŽH4. together with the Ascoli᎐Arzela theorem we can conclude that N ŽUq . is relatively compact in C Ž J, E .. Step 4.
N has a closed graph.
Let yn ª y#, h n g Nyn , and h n ª h#. We shall prove that h# g Ny#. h n g Nyn means that there exists g n g SF, y n, such that h n Ž t . s C Ž t . Ž 0. q S Ž t . q
t
H0 S Ž t y s . g Ž s . ds. n
We must prove that there exists g# g SF, y# such that h# Ž t . s C Ž t . Ž 0 . q S Ž t . q
t
H0 S Ž t y s . g# Ž s . ds.
Ž 4.
The idea is then to use the facts that Ži. Žii.
h n ª h#; h n C Ž t . Ž0. y SŽ t . g ⌫ Ž SF, y n . where
⌫ : L1 Ž J , E . ª C Ž J , E .
defined by Ž ⌫g . Ž t . [
t
H0 S Ž t y s . g Ž s . ds.
528
BENCHOHRA AND NTOUYAS
If ⌫ ( SF is a closed graph operator, we would be done. But we don’t know whether ⌫ ( S F is a closed graph operator. So, we cut the functions yn , h n y C Ž t . Ž0. y sŽ t . , g n and we consider them defined on the interval w k, k q 1x for any k g N j 04 . Then, using Lemma 2.1, in this case we are able to affirm that Ž4. is true on the compact interval w k, k q 1x, i.e., h0 Ž t .
w k , kq1 x
s C Ž t . Ž 0. q S Ž t . q
t
H0 S Ž t y s . g#Ž s . ds k
k for a suitable L1-selection g# of F Ž t, y# t . on the interval w k, k q 1x. k At this point we can paste the functions g# obtaining the selection g# defined by
for t g w k, k q 1 . .
k g# Ž t . s g# Ž t.
We obtain then that g# is a L1-selection and Ž4. will be satisfied. We give now the details. Clearly we have that
Ž h n y C Ž t . Ž 0. y S Ž t . . y Ž h# y C Ž t . Ž 0. y S Ž t . .
⬁
ª 0,
as n ª ⬁. Now, we consider for all k g N j 04 , the mapping SFk : C Ž w k, k q 1 x , E . ª L1 Ž w k , k q 1 x , E . k 1 Žw u ¬ SF, k, k q 1x, E . : f Ž t . g F Ž t, uŽ t .. for a.e. t g w k, k q u[ fgL 1x4 . Also, we consider the linear continuous operators
⌫k : L1 Ž w k , k q 1 x , E . ª C Ž w k, k q 1 x , E . g ¬ ⌫k Ž g . Ž t . s
t
H0 S Ž t y s . g Ž s . ds.
From Lemma 2.1, it follows that ⌫k ( SFk is a closed graph operator for all k g N j 04 . Moreover, we have that
Ž h n Ž t . y C Ž t . Ž 0. y S Ž t . . w k , kq1x g ⌫k Ž SFk , y
n
..
Since yn ¬ y#, it follows from Lemma 2.1 that
Ž h# Ž t . y C Ž t . Ž 0 . y S Ž t . .
w k , kq1 x
s
t
H0 S Ž t y s . g#Ž s . ds
k k for some g# g SF, y# . So the function g# defined on J by k g# Ž t . s g# Ž t.
for t g w k , k q 1 .
is in SF, y# since g#Ž t . g F Ž t, y#. for a.e. t g J.
k
529
FUNCTIONAL DIFFERENTIAL INCLUSIONS
Step 5. The set ⍀ [ y g C Ž J , E . : y g Ny, for some ) 1 4 is bounded. Let y g ⍀. Then y g Ny for some ) 1. Thus there exists g g SF, y such that y Ž t . s y1 C Ž t . Ž 0 . q y1 S Ž t . q y1
t
H0 S Ž t y s . g Ž s . ds,
t g J.
This implies by ŽH3. that for each t g Jm we have t
y Ž t . F M 5 5 q Mm < < q Mm
H0 p Ž s . Ž 5 y 5. ds. s
We consider the function defined by
Ž t . s sup y Ž s . : yr F s F t 4 ,
0 F t F m.
Let tU g wyr, t x be such that Ž t . s < y Ž tU .<. If tU g w0, m x, by the previous inequality we have for t g w0, m x t
Ž t . F M 5 5 q MM 5 < q Mm
H0 p Ž s . Ž 5 y 5. ds
F M 5 5 q Mm < < q Mm
s
t
H0 p Ž s . Ž Ž s . . ds.
If tU g J 0 then Ž t . s 5 5 and the previous inequality holds since M G 1. Let us take the right-hand side of the above inequality as ¨ Ž t .. Then we have ¨ Ž 0 . s M 5 5 q Mm < < s c
and
Ž t . F ¨ Ž t . , t g Jm .
Using the increasing character of we get X
¨ Ž t . F Mmp Ž t . Ž Ž t . . ,
t g Jm .
This implies for each t g Jm that ¨Ž .
H Ž0.t ¨
du
Ž u.
F Mm
m
H0
p Ž s . ds -
⬁
du
H Ž0. Ž u . . ¨
530
BENCHOHRA AND NTOUYAS
This inequality implies that there exists a constant b such that ¨ Ž t . F b, t g Jm , and hence Ž t . F b, t g Jm . Since for every t g w0, m x, 5 yt 5 F Ž t ., we have 5 y 5 ⬁ [ sup y Ž t . : yr F t F m4 F b, where b depends only m and on the functions p and . This shows that ⍀ is bounded. Set X [ C Žwyr, ⬁., E .. As a consequence of Lemma 2.2 we deduce that N has a fixed point which is a mild solution of Ž1. ᎐ Ž2. on wyr, ⬁..
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