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An Extreme Problem for a Volterra Type Integral Inclusion

DOI: 10.4236/oalib.1105605    91 Downloads   223 Views  
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ABSTRACT

In the work, we have studied the dependencies of the solutions to integral in-clusions from perturbation and investigated an extremal problem for integral inclusions. We obtained necessary and sufficient minimum conditions for ex-tremal problems of Volterra type convex inclusions. We also studied a non-convex extremal problem for the Volterra type inclusion. We obtained a high order necessary condition in the extremal problem for the Volterra type inclu-sion.

1. Dependence of the Solution to the Integral Inclusion from Perturbation

Let R n be the n-dimensional Euclidean space. The set of all nonempty compact (convex compact) subsets in R n we will designate as c o m p R n ( c o n v R n ) ; k : [ t 0 , T ] 2 M n is the continuous matrix function, wherewith M n being the set of all square n × n matrices of real elements ( b i j ); z : [ t 0 , T ] R n the continuous function; F : [ t 0 , T ] × R n c o m p R n the setvalued mapping.

Assume that if a vector is multiplied by a matrix, then the vector is a row vector, if a matrix is multiplied by a vector, then the vector is a column vector.

Let us consider a problem for inclusion

u ( t ) F ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) ) (1)

The function u ( ) L 1 n [ t 0 , T ] satisfying (1) we will call the solution to problem (1) (see [1] ).

Let a = max t , s [ t 0 , T ] k ( t , s ) = max t , s [ t 0 , T ] i = 1 n j = 1 n | k i , j ( t , s ) | , if k : [ t 0 , T ] 2 M n is the continuous matrix function.

Theorem 1. Let k : [ t 0 , T ] 2 M n be the continuous matrix function, z : [ t 0 , T ] R n the continuous function, F : [ t 0 , T ] × R n c o m p R n the multivalued mapping, t F ( t , x ) is measurable on t, and there exists a summable function M ( t ) > 0 such that ρ x ( F ( t , x ) , F ( t , x 1 ) ) M ( t ) | x x 1 | for

x , x 1 R n . Moreover, let ρ ( ) L 1 [ t 0 , T ] and u ¯ ( ) L 1 n [ t 0 , T ] be such that

d ( u ¯ ( t ) , F ( t , t 0 t k ( t , s ) u ¯ ( s ) d s + z ( t ) ) ) ρ ( t ) for t [ t 0 , T ] . Then there exists such a solution u ( ) L 1 n [ t 0 , T ] to problem (1) that

| t 0 t k ( t , s ) u ( s ) d s t 0 t k ( t , s ) u ¯ ( s ) d s | a t 0 t e m ( t ) m ( s ) ρ ( s ) d s ,

| u ( t ) u ¯ ( t ) | ρ ( t ) + a M ( t ) t 0 t e m ( t ) m ( s ) ρ ( s ) d s

for t [ t 0 , T ] , where m ( t ) = a t 0 t M ( s ) d s .

2. On Subdifferential of the Integral Functional

Let f : [ t 0 , T ] × R n ( , + ] is the normal convex integrant (see [2] ).

Let consider a subdifferential of the integral functional

J ( u ( ) ) = t 0 T f ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) ) d t

in L 1 n [ t 0 , T ] .

Theorem 2. If k : [ t 0 , T ] 2 M n be the continuous matrix function, z : [ t 0 , T ] R n the continuous function, f : [ t 0 , T ] × R n ( , + ] is the

normal convex integrant and function f ( t , t 0 t k ( t , s ) u ¯ ( s ) d s + z ( t ) + x ) is

summable for x R n , | x | δ , where u ¯ ( ) L 1 n [ t 0 , T ] , then J ( u ¯ ( ) ) is nonempty and υ * L n [ t 0 , T ] belongs to J ( u ¯ ( ) ) if and only if, there exist

u * ( ) L 1 n [ t 0 , T ] , u * ( t ) f ( t , t 0 t k ( t , s ) u ¯ ( s ) d s + z ( t ) ) , such, that υ * ( s ) = s T k ( t , s ) t u * ( t ) d t , where k ( τ , t ) t is the transpose of the matrix k ( τ , t ) .

Theorem 3. If k : [ t 0 , T ] 2 M n be measurable bounded matrix function, z : [ t 0 , T ] R n be measurable bounded function, f : [ t 0 , T ] × R n ( , + ] is

the normal convex integrant and function f ( t , t 0 t k ( t , s ) u ¯ ( s ) d s + z ( t ) + x ) is

summable for x R n , | x | δ , where u ¯ ( ) L 1 n [ t 0 , T ] , then J ( u ¯ ( ) ) is nonempty and functional υ * L n [ t 0 , T ] belongs to J ( u ¯ ( ) ) if and only if,

there exist u * ( ) L 1 n [ t 0 , T ] , u * ( t ) f ( t , t 0 t k ( t , s ) u ¯ ( s ) d s + z ( t ) ) , such, that υ * ( s ) = s T k ( t , s ) t u * ( t ) d t .

3. On Subdifferential of the Terminal Functional

Let k : [ t 0 , T ] 2 M n continuous matrix function, z : [ t 0 , T ] R n continuous function, φ : R n ( , + ] proper convex function in R n . Consider a subdifferential of the terminal functional F ( u ( ) ) = φ ( t 0 T k ( T , s ) u ( s ) d s + z ( T ) ) in L 1 n [ t 0 , T ] , where z ( ) C n [ t 0 , T ] .

Theorem 4. If φ -proper convex function in R n and continuous in the

point t 0 T k ( T , s ) u ¯ ( s ) d s + z ( T ) , then

F ( u ¯ ( ) ) = { b k ( T , s ) : b φ ( t 0 T k ( T , s ) u ¯ ( s ) d s + z ( T ) ) } .

4. Convex Extremal Problem for Integral Inclusions

Let k : [ t 0 , T ] 2 M n be the continuous matrix function, z : [ t 0 , T ] R n the continuous function. Hereafter we will assume that f : [ t 0 , T ] × R n ( , + ] is the normal convex integrant, φ : R n ( , + ] the convex function. Let t 0 < T , F : [ t 0 , T ] × R n c o m p R n { } is the multivalued mapping.

The problem of minimization of the functional

J ( u ) = φ ( t 0 T k ( T , s ) u ( s ) d s + z ( T ) ) + t 0 T f ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) ) d t (2)

is considered under the following constraints

u ( t ) F ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) ) , (3)

where t [ t 0 , T ] , u ( ) L 1 n [ t 0 , T ] .

Introducing the notation ω ( t , x , z ) = { 0 , z F ( t , x ) + , z F ( t , x ) we have that problem (2) and (3) is equivalent to the minimization of the functional

J 1 ( u ) = φ ( t 0 T k ( T , s ) u ( s ) d s + z ( T ) ) + t 0 T f ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) ) d t + t 0 T ω ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) , u ( t ) ) d t

among all functions u ( ) L 1 n [ t 0 , T ] .

Let the mapping t g r F t = { ( x , y ) : y F ( t , x ) } be measurable on [ t 0 , T ] , the set g r F t be closed and convex for almost all t [ t 0 , T ] and F ( t , x ) be compact for all ( t , x ) . From here it follows that ω ( t , x , z ) is a convex normal integrant on [ t 0 , T ] × ( R n × R n ) .

Let us consider the following functional

S ( u , υ ) = φ ( t 0 T k ( T , s ) u ( s ) d s + z ( T ) ) + t 0 T f ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) ) d t + t 0 T ω ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) , u ( t ) + υ ( t ) ) d t ,

where υ ( ) L 1 n [ t 0 , T ] . Let h ( υ ) = inf u L 1 n [ t 0 , T ] S ( u , υ ) . The problem (2) and (3) is called stable, if h ( 0 ) is finite and function h is subdifferentiable at zero (see [3] ).

Lemma 1. Let F : [ t 0 , T ] × R n c o m p R n { } ; the mapping t F ( t , x ) be measurable on [ t 0 , T ] ; the mapping x F ( t , x ) be closed and convex for almost all t [ t 0 , T ] , i.e. g r F t be closed and convex for almost all t [ t 0 , T ] ; there exist such a summable function λ ( t ) that F ( t , x ) λ ( t ) ( 1 + | x | ) for x R n ; there exist a solution u 0 ( t ) to the problem

u 0 ( t ) F ( t , t 0 t k ( t , s ) u 0 ( s ) d s + z ( t ) ) such that x 0 ( t ) = t 0 t k ( t , s ) u 0 ( s ) d s + z ( t ) belongs to d o m F t = { x : F ( t , x ) } coupled with some ε tube, i.e. { x : | x 0 ( t ) x | ε } d o m F t ; f : [ t 0 , T ] × R n ( , + ] the normal convex integrant; φ : R n ( , + ] the convex function and inf u L 1 n [ t 0 , T ] J 1 ( u ) is finite; the function f ( t , t 0 t k ( t , s ) u 0 ( s ) d s + z ( t ) + y ) be summarized for y R n , | y | < r , where r > 0 , and function φ ( ) be continuous at the point t 0 T k ( T , s ) u 0 ( s ) d s + z ( T ) . Then the function h is subdifferentiable at zero, i.e. problem (2) and (3) is stable.

Let υ R n . Assume

ω 0 ( t , x , υ ) = inf z R n { ( z | υ ) + ω ( t , x , z ) } = inf { ( z | υ ) : z F ( t , x ) } ,

where inf = + .

Theorem 5. Let F : [ t 0 , T ] × R n c o m p R n { } ; the mapping t F ( t , x ) be measurable on [ t 0 , T ] ; the mapping x F ( t , x ) be closed and convex for almost all t [ t 0 , T ] ; f be the normal convex integrant on [ t 0 , T ] × R n ; φ the convex function on R n ; k : [ t 0 , T ] 2 M n the continuous matrix function; z : [ t 0 , T ] R n the continuous function. For the function u ¯ ( ) L 1 n [ t 0 , T ] to minimize the functional (2) among all the solutions to the problem (3), it is sufficient that there exist u 1 * ( ) , u 2 * ( ) L 1 n [ t 0 , T ] and b R n such that

1) u 1 * ( t ) f ( t , t 0 t k ( t , s ) u ¯ ( s ) d s + z ( t ) ) ,

2) b φ ( z ( T ) + t 0 T k ( T , s ) u ¯ ( s ) d s ) ,

3) u 2 * ( t ) ω 0 ( t , t 0 t k ( t , s ) u ¯ ( s ) d s + z ( t ) , t T k ( τ , t ) t ( u 1 * ( τ ) + u 2 * ( τ ) ) d τ + K ( T , t ) t b ) ,

4) ω 0 ( t , z ( t ) + t 0 t k ( t , s ) u ¯ ( s ) d s , t T k ( τ , t ) t ( u 1 * ( τ ) + u 2 * ( τ ) ) d τ + K ( T , t ) t b ) = ( u ¯ ( t ) | t T k ( τ , t ) t ( u 1 * ( τ ) + u 2 * ( τ ) ) d τ + K ( T , t ) t b ) + ω ( t , z ( t ) + t 0 t k ( t , s ) u ¯ ( s ) d s , u ¯ ( t ) ) ,

and if for u 0 ( t ) = u ¯ ( t ) the condition of lemma 1 is satisfied, then conditions 1) - 4) become necessary.

5. Nonconvex Extremal Problem for Integral Inclusions

Let k : [ t 0 , T ] 2 M n be the continuous matrix function; z : [ t 0 , T ] R n the continuous function, i.e. z ( ) C n [ t 0 , T ] . Hereafter we will assume that f : [ t 0 , T ] × R n × R n ( , + ] is the normal integrant and φ : R n ( , + ] is the function. Let t 0 < T , F : [ t 0 , T ] × R n c o m p R n be the multivalued mapping.

We consider the following problem of minimization of the functional

J ( u ) = φ ( t 0 T k ( T , s ) u ( s ) d s + z ( T ) ) + t 0 T f ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) , u ( t ) ) d t , (4)

under the following constraints

u ( t ) F ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) ) , (5)

where t [ t 0 , T ] , u ( ) L 1 n [ t 0 , T ] .

Let ψ ( s , x , y ) = inf { | z y | : z F ( s , x ) } and consider the minimization of the functional

J r ( u ) = φ ( t 0 T k ( T , s ) u ( s ) d s + z ( T ) ) + t 0 T f ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) , u ( t ) ) d t + r t 0 T ψ ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) , u ( t ) ) d t

among all the functions u ( ) L 1 n [ t 0 , T ] .

Theorem 6. If u ¯ ( ) L 1 n [ t 0 , T ] is the solution to the problem (4) and (5), F : [ t 0 , T ] × R n c o m p R n { } and t F ( t , x ) are measurable on t, x ¯ ( t ) = t 0 t k ( t , s ) u ¯ ( s ) d s + z ( t ) , there exist k ( ) L 1 [ t 0 , T ] , M ( ) L 1 [ t 0 , T ] ,

k 1 > 0 , k 2 > 0 and α > 0 such that B ( x ¯ ( t ) , α ) d o m F t = { x R n : F ( t , x ) } at t [ t 0 , T ] and

| φ ( z ) φ ( u ) | k 2 | z u | ,

| f ( t , x 1 , y 1 ) f ( t , x 2 , y 2 ) | k ( t ) | x 1 x 2 | + k 1 | y 1 y 2 | ,

ρ X ( F ( t , x 1 ) , F ( t , x 2 ) ) M ( t ) | x 1 x 2 |

for z , u B ( x ¯ ( T ) , α ) , x 1 , x 2 B ( x ¯ ( t ) , α ) , y 1 , y 2 R n . Then there exist a number r 0 > 0 such that u ¯ ( t ) minimizes the functional J r ( u ) in D for

r r 0 , where D = { u ( ) L 1 n [ t 0 , T ] : u ( ) u ¯ ( ) L 1 n [ t 0 , T ] α β } , β > ( 1 + a ( e m ( T ) + a e m ( T ) t 0 T M ( t ) d t ) ) ( a t 0 T M ( t ) d t + 1 ) + a , m ( t ) = a t 0 t M ( s ) d s .

Theorem 7. Let the condition of the theorem 6 be satisfied and the function u ¯ ( t ) among all solutions to the problem (5) minimizes the functional (4). Then there exists u * ( ) L 1 n [ t 0 , T ] and b R n such that

1) ( u * ( t ) , t T k ( τ , t ) t u * ( τ ) d τ K ( T , t ) t b ) C ( f ( t , t 0 t k ( t , s ) u ¯ ( s ) d s + z ( t ) , u ¯ ( t ) ) + r ψ ( t , t 0 t k ( t , s ) u ¯ ( s ) d s + z ( t ) , u ¯ ( t ) ) ) ,

2) b C φ ( z ( T ) + t 0 T k ( T , s ) u ¯ ( s ) d s ) ,

where C g ( x ¯ ) is Clarke subdifferential of the function g at the point x ¯ (see [4] ).

6. A Higher Order Necessary Condition in the Extremal Problem for the Volterra Type Inclusion

Consider the problem (4) and (5), where f ( t , x , y ) = f ( t , x ) . Assume

ψ ( s , x , y ) = inf { | z y | : z F ( s , x ) } .

We consider the following problem of minimization of the function

J r ( u ) = φ ( t 0 T k ( T , s ) u ( s ) d s + z ( T ) ) + t 0 T f ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) ) d t + r ( ( t 0 T ψ ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) , u ( t ) ) d t ) β + u ¯ ( ) u ( ) L 1 n β ν ( t 0 T ψ ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) , u ( t ) ) d t ) ν )

among all functions u ( ) L 1 n [ t 0 , T ] .

Let u ¯ ( ) L 1 n [ t 0 , T ] be the solution to the problem (4) and (5). Let x ¯ ( t ) = t 0 t k ( t , s ) u ¯ ( s ) d s + z ( t ) .

Theorem 8. Let F : [ t 0 , T ] × R n c o m p R n be the multivalued mapping, the mapping t F ( t , x ) be measurable on [ t 0 , T ] ; f be the normal integrant on [ t 0 , T ] × R n ; φ the function in R n ; k : [ t 0 , T ] 2 M n the continuous matrix function; z : [ t 0 , T ] R n the continuous function, and there exists a summable function M ( t ) > 0 such that ρ x ( F ( t , x ) , F ( t , x 1 ) ) M ( t ) | x x 1 | for x , x 1 R n ; there exist k 1 ( ) L 1 [ t 0 , T ] , k 1 ( t ) > 0 and number k 2 > 0 such that

| f ( t , x 1 ) f ( t , x 2 ) | k 1 ( t ) | x 1 x 2 | ν ( | x 2 x ¯ ( t ) | β ν + | x 1 x 2 | β ν )

for x 1 , x 2 R n ,

| φ ( x ) φ ( y ) | k 2 | x y | ν ( | y x ¯ ( T ) | β ν + | x y | β ν )

for x , y R n (see [5] ). If the function u ¯ ( t ) among all solutions of the problem (5) minimizes the functional (4), then there exists a number r 0 > 0 such that u ¯ ( t ) minimizes the functional J r ( u ) in L 1 n [ t 0 , T ] for r r 0 .

Let g : [ t 0 , T ] × R n ( , + ] be the normal integrant on [ t 0 , T ] × R n ; e : R n ( , + ] the function.

Let assume

S ( u ) = e ( t 0 T k ( T , s ) u ( s ) d s + z ( T ) ) + t 0 T g ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) ) d t ,

H r ( u ) = J ( u ) S ( u ) + r ( ( t 0 T ψ ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) , u ( t ) ) d t ) β + u ¯ ( ) u ( ) L 1 n β ν ( t 0 T ψ ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) , u ( t ) ) d t ) ν )

= φ ( t 0 T k ( T , s ) u ( s ) d s + z ( T ) ) e ( t 0 T k ( T , s ) u ( s ) d s + z ( T ) ) + t 0 T f ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) ) d t t 0 T g ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) ) d t + r ( ( t 0 T ψ ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) , u ( t ) ) d t ) β + u ¯ ( ) u ( ) L 1 n β ν ( t 0 T ψ ( t , t 0 t k ( t , s ) u ( s ) d s + z ( t ) , u ( t ) ) d t ) ν ) .

Theorem 9. Let F : [ t 0 , T ] × R n c o m p R n the multivalued mapping, the mapping t F ( t , x ) be measurable on [ t 0 , T ] ; f be the normal integrant on [ t 0 , T ] × R n ; φ the function in R n ; k : [ t 0 , T ] 2 M n the continuous matrix function; z : [ t 0 , T ] R n the continuous function and there exist a summable function M ( t ) > 0 such that ρ x ( F ( t , x ) , F ( t , x 1 ) ) M ( t ) | x x 1 | for x , x 1 R n ; there exist the normal integrant g : [ t 0 , T ] × R n ( , + ] , the functions e : R n ( , + ] , k 1 ( ) L 1 [ t 0 , T ] , k 1 ( t ) > 0 and number k 2 > 0 such that

| f ( t , x 1 ) g ( t , x 1 ) f ( t , x 2 ) + g ( t , x 2 ) | k 1 ( t ) | x 1 x 2 | ν ( | x 2 x ¯ ( t ) | β ν + | x 1 x 2 | β ν )

for x 1 , x 2 R n , where x ¯ ( t ) = t 0 t k ( t , s ) u ˜ ( s ) d s + z ( t ) ,

| φ ( x ) e ( x ) φ ( y ) + e ( y ) | k 2 | x y | ν ( | y x ¯ ( T ) | β ν + | x y | β ν )

for x , y R n and let u ˜ ( ) L 1 n [ t 0 , T ] solutions of the problem (4)-(5). Then there exist a number r 0 > 0 such that u ˜ ( t ) minimizes the functional H r ( u ) in u { υ L 1 n [ t 0 , T ] : S ( w υ ) S ( u ˜ ) } for r r 0 , where w υ solutions to the problem (5), which satisfy the main theorem 1 for u ¯ ( ) = υ ( t ) .

It’s possible to get the local variant of theorems 8 and 9 analogical to theorem 6.

Let assume

J r { β } + ( u ¯ ; u ) = lim λ 0 ¯ 1 λ β ( J r ( u ¯ + λ u ) J r ( u ¯ ) ) ,

J r { β } ( u ¯ ; u ) = lim _ λ 0 1 λ β ( J r ( u ¯ + λ u ) J r ( u ¯ ) )

for u ( ) L 1 n [ t 0 , T ] .

Corollary 1. If the condition of theorem 8 is satisfied, then there exist a number r 0 > 0 such that J r { β } + ( u ¯ ; u ) J r { β } ( u ¯ ; u ) 0 for r r 0 and u ( ) L 1 n [ t 0 , T ] .

Let assume

E r ( u ) = φ { β } ( t 0 T k ( T , s ) u ¯ ( s ) d s + z ( T ) ; t 0 T k ( T , s ) u ( s ) d s ) + t 0 T f { β } + ( t , t 0 t k ( t , s ) u ¯ ( s ) d s + z ( t ) ; t 0 t k ( t , s ) u ( s ) d s ) d t + r ( ( t 0 T ψ { 1 } + ( t , t 0 t k ( t , s ) u ¯ ( s ) d s + z ( t ) , u ¯ ( t ) ; t 0 t k ( t , s ) u ( s ) d s , u ( t ) ) d t ) β + u ( ) L 1 n β ν ( t 0 T ψ { 1 } + ( t , t 0 t k ( t , s ) u ¯ ( s ) d s + z ( t ) , u ¯ ( t ) ; t 0 t k ( t , s ) u ( s ) d s , u ( t ) ) d t ) ν ) .

Theorem 10. If the condition of theorem 8 is satisfied, then there exist a number r 0 > 0 such that E r ( u ) 0 for r r 0 and u ( ) L 1 n [ t 0 , T ] .

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Sadygov, M. (2019) An Extreme Problem for a Volterra Type Integral Inclusion. Open Access Library Journal, 6, 1-8. doi: 10.4236/oalib.1105605.

References

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[3] Ekeland, I. and Temam, R. (1979) Convex Analysis and Variational Problems. Mir, Moscow, 309 p.
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[5] Sadygov, M.A. (2014) Subdifferential of High Orders and Optimization. LAP Lambert Academic Publishing, Saarbrucken, 359 p.

  
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