Multidimensional Impulse Inequalities and General Bihari-Type Inequalities for Discontinuous

Multidimensional Impulse Inequalities and General Bihari-Type Inequalities for Discontinuous Functions with Retardation

Sergiy Borysenko* Angela Gallo ** Anna Maria Piccirillo***

*Dept.Differential Equations, National Tech. Univ. of Ukraine,“KPI” 37-Peremogj Prosp.–03056 Kyiv-Ukraine

**Dept. of Mathematic and Appl. “R.Caccioppoli” Univ. of Naples ”Federico II”-Claudio str,21- 80125 Naples – Italy

*** Dept. of Civil Engineering, Second Univ. of Naples-Roma str.,21-81031 – Aversa (CE)-Italy

MSC: 34B15 ,26D15 ,26D20

Keywords and phrases: inequalities, discontinuous functions, retardation, impulse integral inequalities

Abstract

In this article we investigate some impulsive integro-functional inequalities for functions of n independent variables. The problem of reducing multidimensional integro-sum functional inequalities to one-dimensional inequalities is considered too (by using conditions of solvability Chapligin’ problem for impulsive integral inequalities). Some new analogies Wendroff-type inequalities for discontinuous functions with finite jumps are founded.

1. Introduction

The theory of ordinary impulsive differential systems last 20 years have a strong development which explaining the appearance the new problems in investigations real processes in physics, biology, chemistry, electronics and many other branches of our life and activity. The integral representations of the solutions of ordinary impulsive differential systems usually have as continuous also discrete part and may be written in such form

x(t) = (t) +  k(t,s,x(s)) ds +  i(t, i) Ii ( u(i -0) ) (A)

t0 t0  tk t where (t) characterise initial perturbations, k(t,s,x) describe continuous perturbations

(right- hand side of system for t  i ) and Ii characterise the value of impulse perturbations n of the system ( x R ) , in fixed moments of time  i  , i some functions.

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Corresp. authors: *Tel.+380442417645 ** Tel.+390817683548- fax +390817683643 *** Tel.+390815010237.

E-mail addresses:*[email protected] **angallo @ unina.it ***annamaria .piccirillo@ unina2.it The representation (A) is preserve for the case, when impulse perturbations of the system   0   1  ..., lim i x   take place not only in fixed moments of time  i : i , but also on some

 i x: 0 x   1 x  ..., lim i x   uniformly for x hypersurfaces i In this case I is a moment of time when solution x(t) of the impulsive system meet with i (x)( for some conditions this meeting take place only once. If we investigate some real processes, which possible describe by impulsive differential system, we must use information about values of initial, continuous and discrete perturbations

( for example estimates of ||||, ||k||, || Ii ||). Then for ||x|| we obtain inequality, which describe by

u(t )   * (t) +  k*(t,s,u(s)) ds +  i * (t, i) Ii* ( u(i -0) ), (B)

t0 t0  ti t where u, *, k*, i*, Ii* are nonnegative functions and comparison impulsive equation

v(t ) =  * (t) +  k*(t,s,v(s)) ds +  i * (t, i) Ii* ( v(i -0) ) (C)

t0 t0  ti t

Equation (C) and inequality (B) have strong relations. For some conditions see [36]

u(t)  v* (t), where v* (t) some solution of the equation (C). In continuous case (Ii  0) the theory of integral inequalities based on fundamental results of Bellman [6] , Bihari [7] ,and its numerous generalizations [1] – [5], [22],[23], [25], [26], [30] - [32], [35], [38] - [42] and many others. Investigations of discrete inequalities (k 0) and its applications see classical monograph [2] and its references. The generalization of the method of integral inequalities for discontinuous functions and its applications for qualitative analysis impulsive systems: existence, uniqueness, boundedness, comparison, stability, etc. we refer to the results [5], [8] – [21] , [24], [27], [33], [34], [36], [37]. The investigation of periodic boundary value problems, where use the representation (A) see in [28], [29] (also references). In paper [11] studied Wendroff type impulsive integral inequality (B) , when t = (t1 , t2 ), t0 = m 1 2 (t01,t02 ) , k = f(t1 , t2 )  u (t1 , t2 ) , m0, i = (i ,i ) ,later in monograph [36] were investigated different kinds of multidimensional impulsive inequalities (B), for function u(t) m 1 n of n-independent variables t = (t1 ,…., tn ), k = f(t1 ,…, tn ) u (t1 ,…, tn ) i = (i ,… ,i ) with assumptions that i Ii= i u( ti -0), i = const. 0, i= 1,2,…..

The results of [17], [18] ,[20], [27] generalize the investigations from [36] on the case when m m i Ii= i u ( ti -0), m0, and kernel k = f  u Without consideration remain the cases , when function u(t) of two (many) independent m variables satisfies inequality (B) with k(t,s,u) = f(t) W [u] , Ii= i u ( ti -0), where W some functions, which include also retardation for t in u = u( (t)) , ((t)  t). Our article devoted to investigations impulsive integral inequalities for discontinuous functions of n-independent variables (n 2). In Section 2 we consider impulsive integro-functional inequality for discontinuous functions of n-independent variables. By using the results [11] , we reduce , under some assumptions, multidimensional inequality to one-dimensional. In Section 3 for n=2 we obtain new analogies of Bihari result for Wendroff type inequality with different kinds of W and discontinuities of function u(t,x) in fixed points (ti , xi ) non- Lipschitz type. Our results are based on the investigations [1 – 42].

2. Multidimensional Inequalities. The problem of reducing

Following the investigation of papers [3] ,[11] ,[42], let us consider n-dimensional

n Euclidean space R with points t  t1,..., tn  , t0  t10 ,..., tn0  and natural ordering

t0  t  ti0  ti for i  1,..., n .

By using term integro-sum inequality [11] , [36] (or impulse integral inequality [5],[24]) for one-dimensional inequality such type

ut  t Kt, s,usds  t, ti i uti  0 t  t0  0, ti1  ti i  1,2,..., lim ti   , (1)   i t t t t0 0 i

where ut is a piecewise continuous function with 1-st kind discontinuity points ti ; we consider n-dimensional impulse integral inequalities the most general view with delay (integro-functional inequality) such type :

t1 tn ut ,..., t   t ,..., t  .... Kt ,..., t , s ,..., s , u s ,..., s ds ...ds  1 n 1 n   1 n 1 n 1 n 1 n c1 cn (2) 1 n 1 n 1 n   t1 ,..., tn , i ,..., i i u i  0,..., i  0 , c1   i  t1,..., cn   i  t n t0 ti t

Let us suppose that following assumptions are fulfilled:

(H 1 ) ut1,..., tn  is real valued nonnegative discontinuous function with finite type of

discontinuities in the points

1 n 1 n 1 n 1 n  1 ,..., 1 ,  2 ,..., 2 ,....:u j  0,..., j  0 u j  0,..., j  0, j  1,2,... (3) Remark 2.1 If ut1,..., tn  satisfies(3), we say that function ut1,...,tn  have finite jump in

1 n the points  j   j ,..., j  j  1,2,.. .

(H 2 ) In finite bounded domain c,T   c1,T1 ,...,cn ,Tn the quantity of  j  is bounded and

for infinite domain c,  the set of points  j is N, the set of natural numbers.

(H 3 ) The functions t  0 , i t,   0 ,i u  0 and also f , i are continuous

functions,i are continuous and nondecreasing at u . y (H 4 ) The kernel Kt, s, y  0 and nondecreasing at with fixed t , s .

c   i   i1 , i  0,1,2,... lim i   (H 5 ) i . n n (H 6 )  s , s belong to the space T of continuous functions F : R  R which satisfy such conditions [3] : n i) Fx  F1 x,...., Fn x , Fi : R  R , i  1,..., n

lim Fi x  i  1,..., n ii) x  iii) Fx  x . In the following, analogously in [11] (see also [36] , [42] we denote by

1 n t  t1,...,tn , t*  t2 ,..., tn , t  t1, t *, u  max0, ut, ut  sup u s, , si ci , ti , i   i ,..., i ,

* 2 n 1 *  i   i ,..., i , i   i , i  .

Then from the inequality

t ut  t Kt, s, usds  t,  u  0   i i i t  t t0 0 i

we have

t ut  t Kt, s, usds  t,  uu  0    i i i c ci t t * 1 *   t Kt, s, us1, s ds  i t, i i u   0,  0     i i  c ci t t ~ * ~ 1 *   t Kt, s, u s1, s ds  i t, i i u   0,  0 , .    i i  c ci t

~ ~  t  sup v , Kt, s,   sup Kv, s, p , i q, i  sup i w, i  . where vi ti vi ti wq So, denoted by p t2 tn ~ K t, s , u s , s*  .... Kt, s, u s , s* ds ...ds , 1 1   1 2 n c2 cn  t, 1 , *  u   1  0, *  0  ~ t, 1 , *  u   1  0, *  0 , i  i i  i    i i   i  i i  i    i i  2 c2i t2 ...... n cn i tn

* * for every fixed t and  i the function ut1, t *satisfies one-dimensional impulse integral inequality

t1 * 1 * 1 * ut1,t *   t K t, s1,u s1 , t ds1  i t, , i i u  0,  0   i  i i  . (4) c 1 1 c1i t1

Let t*  t1* ,....,tn*  be an arbitrary point so that ti*  ci i  1,2,..., n. Thanking to the assumptions (H 3 ) ,

(H 4 ), for every ti ci ,ti*  we obtain such integro-sum inequality

t1 * 1 * 1 * ut1,t *   t*  K t* , s1, us1, s ds1  i t* , , i i u   0,  0   i  i i  . (5) c  1 t c1 1 i 1

Impulse integral inequality (5) is a one-dimensional inequality similar to (1). By using the result from [36] (Theorem 3.1.1, p.174) about solvability of Chaplygin problem for inequality (5) we obtain such estimate:

u t  v t    t*    , (6)

v t where  t*    is an arbitrary solution of impulse integral comparison majorant equation:

t1 vt   t  K t , s , v ds   t ,  v  0 *  * 1 1  i * i i i (7) c  1 t c1 1 i 1

1 1 continuous in each interval  i , i1 , i  N . Finally, since the point t* is arbitrary, we obtain

ut  ut  v t  t t  c . (8)

Remark 2.2 If in (2)  t  t , t  t the results of above investigations coincide with the results [11], (§3, p.1640-1641).

3. General Bihari theorems for Wendroff type impulse inequalities with delay 2 Now we consider the space R . Next statement is valid .

2 Theorem 3.1 Let us suppose that a nonnegative function ut, xdefined in the domain D  R :

    D   Dk j , Dk j  t, x:t tk1 ,tk , x x j1 , x j  , k  1,2,..., j  1,2,.. , continuous in D with exception   k, j1  of ti , xi points of finite jump : uti  0, xi  0  uti  0, xi  0 i  1,2,... and satisfies the impulse integro-functional inequality

t x ut, x  t, x qt, x f ,W u  , d d   u m t  0, x  0  1 2  i i i , (9) t ,x t ,x t,x t0 x0 0 0 i i where qt, x  1 , t, x  0 t, x D is nondecreasing with respect to t, x : p1  p2 , q1  q2  p1, q1   p2 , q2  at p1 , q1 , p2 , q2  D ; m  const.  0 ,  i  const.  0 i  N ; function f is nonnegative and satisfies such condition

f t, x  0 , t, x Di j i  j (i  1,2,... j  1,2,...) ; function W u belongs to the class K of functions such that

i1 ) W    W  W   ;

i2 ) W : 0,   0,  , W 0  0 ;

i3 ) W is nondecreasing i  s  i  1,2 ( i : R  R ,  i s  s , lim  i s  ) 4 ) i   T, s  ;

t , x  t , x if t  t , x  x i  1,2,... lim ti   , lim xi   here  i i   i1 i1  i i1 i i1 and i i . Then for every t, x D it results in :

 t x  1  f ,  ut, x  t, x qt, x  i  W  1  , 2 , q 1  , 2 d d (10)  ,   ti xi 

with ti  t  ti1 , xi  x  xi1 , t x f , 1 W  1  , 2 , q 1  , 2 d d  Dom i  (11)  , ti xi

v v d d  0 v  ,  i v  , i  1,2,..  W    W   1 ci

A  ti xi    m1 m 1 f , ci  1  i  ti , xi q ti , xi  i1 W  1  , 2 , q 1  , 2 d d     ,   ti1 xi1  (12) 0  m  1 and A  1 for  i  1,2,.. . m  1 and A  m

Proof. We prove the statement by using inductive method for the each domain Di i of continuous function ut, xand knowing the estimate for previously domain Di1i1 . It is obvious that

t x ut, x f , u m t  0, x  0 1 qt, x W u  , d d   i i  t, x  , 1 2  i t, x t ,x t ,x t,x t0 x0 0 0 i i (13)  t x m  f ,  ut  0, x  0    m1  i i    qt, x 1 W u 1  , 2 d d   i ti , xi    .  ,  ti , xi   t ,x t ,x t,x     t0 x0 0 0 i i 

Let us introduce

t x m f ,  ut  0, x  0 * m1  i i  u t, x  1 W u 1  , 2 d d   i ti , xi   , (14)  ,  ti , xi  t ,x t ,x t,x   t0 x0 0 0 i i

* u t0 , x0   1 .

Then we obtain:

ut, x  t, xqt, xu * t, x, * * u 1 t, 2 x   1 t, 2 xq 1 t, 2 xu  1 t, 2 x   1 t, 2 xq 1 t, 2 xu t, x, * uti  0, xi  0  ti , xi qti , xi u ti  0, xi  0, (15) * W u  1 t, 2 x  W  1 t, 2 xq 1 t, 2 xu  1 t, 2 x * *  W  1 t, 2 xq 1 t, 2 xu t, x W 1 t, 2 xq 1 t, 2 xW u t, x.

From (14), (15) it follows t x f , u * t, x  1 W   , q  ,  W u* ,d d   , 1 2 1 2 t0 x0 (16) m    m1 t , x q m t , x u * t  0, x  0.  i i i i i i i t0 ,x0 ti ,xi t,x

Let us consider the domain D11 ; so inequality (16) reduces itself in the next one:

t x * f , * u t, x  1 W  1  , 2 q 1  , 2  W u ,d d . (17)  , t0 x0

By using results [ 3 ], [ 34 ] we obtain such estimate:

 t x  * 1  f ,  u t, x  0  W 1  , 2 q 1  , 2  d d, (18)   ,   t0 x0 

v d 1 where  0 v  ,  0 is the inverse of  0 and t  t0 , x  x0 such that  W   1

t x f , 1 W 1  , 2 q 1  , 2  d d  Dom 0  . (19)  , t0 x0

Thus from (15) , (17), (18), (19) it follows

 t x  1  f ,  ut, x  t, xqt, x 0  W 1  , 2 q 1  , 2  d d   ,  (20)  t0 x0 

t, x D11 and satisfying (19) .

By following the scheme from [11] ( see also [15], [34],[36] ) based on inductive method, we consider domain D2 2 . Then t1 x1 f , u * t, x  1 W   , q  ,  W u* ,d d   , 1 2 1 2 t0 x0 t x f ,  W  , q  ,  W u * ,d d   , 1 2 1 2 t1 x1 m1 m *m  1 t1 , x1 q t1 , x1 u t1  0, x1  0 

 t1 x1  1  f ,   0  W  1  , 2 q 1  , 2  d d   ,  t0 x0  (21) t x f , *  W 1  , 2 q 1  , 2 W u ,d d   , t1 x1 m   t1 x1  m1 m  1  f ,   1 t1 , x1 q t1 , x1  0  W 1  , 2 q 1  , 2  d d    ,   t0 x0  t x f ,  c  W   , q  , W u * ,d d , 1  , 1 2 1 2 t1 x1 where

 t1 x1  m1 m 1  f ,  c1  1 1 t1 , x1 q t1 , x1   0  W  1  , 2 q 1  , 2  d d,  ,  (22) t0 x0  if 0  m  1 ;

m   t1 x1  m1 m  1  f ,  c1  1 1 t1 , x1 q t1 , x1   0  W  1  , 2 q 1  , 2  d d   ,  (23)  t0 x0  if m  1 .

' 1 Let us point out that the inequality  0 v  0 guarantees  0 exists and is increasing function. By using (11) with i  1 from (21) we obtain:

 t x  * 1  f ,  u t, x  1  W 1  , 2 q 1  , 2  d d with t1  t  t 2 , x1  x  x2 ;   ,   t1 x1 

so, ut, x satisfies (10) for t, x D2 2 .

Now let us suppose that (10) holds for t, x Dk k ; then for t, x Dk 1k1 we have t x * f , * u t, x ck  W1 , 2 q1 , 2 W u ,d d , tk  t  tk1 , xk  x  xk1 (24)  , tk xk

where ck is defined from (12). On this step we complete the proof.

Remark 2.3 The result of theorem 2.1 is a new analogy Bihari statement for discontinuous functions of two independent variables with nonLipschitz type discontinuities and retardation.

Remark 2.4

1. If  i  0, qt, x  1, W u  u , ,   1  , 2   , ,we obtain classical result by Wendroff ( see [22] );

2. If  i  0 from Theorem 2.1 Akinyele result ( [3], th.4 as n  2 ) follows ;

3. If qt, x  1, W u  u , ,  , , m  1 we obtain Borysenko result ( [11], theorem p. 1638);

4. If qt, x  1, W u  u , ,  , we have : a)result Borysenko-Iovane ( [17], th.2.1 ), b)if

qt, x  1, W u  u m m  1, m  0 Theorem 2.2 in [17], c) if W u  u m Theorem 2.3 in [17], d)

if W u  u Theorem 3.1 in [17], e) if W u  u m Theorem 3.2 in [17] ;

B For one-dimensional case (n=1) from Theorem 2.1 it follows :

1. Agarwal results ([2], Corollary 4.12 for m=1, similar th. 4.21 for m  1 ) ;

2. classical Bellman results [5, p58], Bihari [7];

3. Borysenko, Gallo, Toscano [9,10 lemma 1, lemma 2 ] if in Theorem 2.1 we assume

q  1, W u  u m , q  1, t  t ;

5. Gallo, Piccirillo [18, th. 2.1] , if W u  u ; [18, th. 2.2], if W u  u m ; Borysenko [36, th.3.71,p.232], if m=1; Borysenko, Samoilenko [34, th.3.7.6 (g=0)], if m=1; Iovane [20,

th.2.1], if q  1, W u  u m ; [20, th.2.2], if W u  u m . By using the results of investigations by Yeh, Shih [40] and Akinyle [3], we consider the class F’ of functions f such that a) f x is nonnegative, continuous and non decreasing as x  0 ; b) f : 0, 0,  , f x  0 , if x  0 , f 0  0 ; c) x  1 , y  0 x 1 f y  f x 1 y and obtain the following statement

Theorem 3.2 Let us suppose that function ut, x in the domain D satisfies inequality (9) and conditions of Theorem 3.1 ( with exception of conditions about function W u . Then if W  F’, we

* * obtain for arbitrary t0  t  t , x0  x  x following estimate:

 t x  1   1  , 2   ut, x  t, x qt, x  i  f , q 1  , 2  d d , (24)  ,   ti xi 

v v d d where ti  t  ti1 , xi  x  xi1 ,  0 v  ,  i v  , i  1,2,.. ,  W    W   1 ci

A   ti xi    m1 m  1  1  , 2   ci  1  i  ti , xi q ti , xi  i1 f , q 1  , 2  d d      ,    ti1 xi1  (25)  A  1 if 0  m  1 with  i  1,2,..  A  m if m  1 and t * , x* are chosen such that

 t x  * *   1  , 2  1  t , x  supt, x: f , q 1  , 2  d d  Dom j , j  0,1,2,.. .   ,   t j x j 

Proof. It is obvious that

* W u  1 t, 2 x  W[ 1 t, 2 xq 1 t, 2 x u t, x] 

 1 t, 2 xq 1 t, 2 x *  W  1 t, 2 xq 1 t, 2 x u t, x (26)  1 t, 2 xq 1 t, 2 x *   1 t, 2 xq 1 t, 2 x.W u t, x where u * t, x is defined by (14). Then

t x   ,  u * t, x  1 f , q  ,  1 2 W u * ,d d   1 2 , t0 x0 (27) m    m1 t , x q m t , x u * t  0, x  0.  i i i i i i i t0 ,x0 ti ,xi t,x

Later for inequality (27) we use procedure of inductive method similar as for inequality (16) in order to conclude the proof of Theorem 2.2.

Remark 3.3 Theorem 3.2 (similar as Theorem 3.1) is a new analogy Bihari result [7] for discontinuous functions.

Remark 3.4 By using the results of Section 2 we can prolong the results of Th. 3.1, 3.2 on n- dimensional case of impulsive integral inequalities.

Remark 3.5 If  i  0 Theorem 3.2 is similar to: a) result by Akinyle [3, Th.5, if n  2 ] , b)result by Yeh, Shih for n  2 [40, Th.3]; for m  1 the estimate (24) is more exact than (7.10) one in [36, Th. 3.7.2] .

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