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 .

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 .

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 ) ) } .

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.

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] ).

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 ] .

The author declares no conflicts of interest regarding the publication of this paper.

Sadygov, M.A. (2019) An Extreme Problem for a Volterra Type Integral Inclusion. Open Access Library Journal, 6: e5605. https://doi.org/10.4236/oalib.1105605