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saqarT velos mecn ierebaTa erovnuli ak ad em iis moam be, t. 5, #3, 20 11 BULL ET IN OF THE GEORGIAN NATIONAL ACADEM Y OF SCIENCE S, vol.

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3, 2011 Mathematics Positive Solutions of Two-Point Boundary Value Problems for Higher Order Nonlinear Singular Differential Equations Ivan Kiguradze Academy Member, A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University ABSTRACT.
For higher order nonlinear differential equations with singularities with respect to the time and phase variables, the sufficient conditions for the existence of positive solutions of the Dirichlet and Focal boundary value problems are established.

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Key words: nonlinear differential equation, strong singularity with respect to the time variable, strong singularity with respect to the phase variable, Dirichlet problem, focal boundary value problem, positive solution.
In an open interval  ,a b , we consider the nonlinear differential equation with the boundary conditions of one of the following two types:  mu 2   f  t u ,   i u  1   a    0,  i u  1   b    0  i   1, , m  and  i u  1   a    0,  u m i   1   b    0  i    .

, m 1, (1) (21) (22) (21) is called the conjugate boundary conditions, or the Dirichlet conditions, and (22) is called the focal boundary conditions, or the Nicoletti conditions.
The problems of the type (1), (21) and (1), (22) take central place in the theory of boundary value problems for ordinary differential equations and are the subject of numerous investigations (see [1-11] and the references therein).

In the present paper we give new sufficient conditions for the existence of positive solutions of the above-mentioned problems.

Throughout the paper, it is assumed that m is an arbitrary natural number, and f :   a b ,   0, ous function.
  is a continu- R  A function  tiable and at every point of the interval   0, u   : ,a b satisfies this equation.

a b   is said to be a positive solution of equation (1), if it is 2m-times continuously differen- , A positive solution of equation (1) is said to be a positive solution of problem (1), (21) (of problem (1), (22)), if it satisfies the boundary conditions (21) (the boundary conditions (22)), where by    ju  a  and    ju b  there are  © 2011 Bull.

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6 Ivan Kiguradze understood, respectively, the right and the left limits of the function  ju at the points a and b.

We are especially interested in the case where equation (1) has singularities with respect to the time and phase variables.
Following [1], [5] we will say that equation (1) with respect to the time variable has strong singularities (strong singularity) at the points a and b (at the point a), if 2 m  1   t a  f   t x dt ,  t 0  a 2 m  1   t a  f   t x dt , b  t 0 If, however, there is a set I  a b ,  0 such that    t 0  a     for a 2 m  1  t a  f   t x dx ,        t 0  b , x  0. k x  f  t x ,     for arbitrary t  I and k  , 0 0 lim sup  0 x then we will say that equation (1) has strong singularity with respect to the phase variable.
Unlike the previous well-known results (see [2, 3, 8, 10, 11]), the theorems below on the existence and uniqueness of a positive solution of problem (1), (21) (of problem (1), (22)) cover the case where equation (1), along with strong singularities with respect to the time variable at the points a and b (at the point a), has strong singularity with respect to the phase variable as well.
By m mC 2 ,   a b we denote the space of 2m-times continuously differentiable functions, satisfying the condition ,   2  m    t u dt   .

b  a First we consider problem (1), (21).

The following theorem is valid.

Theorem 1 (A principle of a priori boundedness).

Let in the domain   a b  ,  0,  the inequality  m    1 f  t x ,   h t x   (3) be fulfilled, where   0,1 is a constant, and     h a b   is a continuous function, not identically equal to 0, : , zero.
Let, moreover, there exist a positive number r such that for any continuous function  :  a b ,     0,1 everyy positive solution of the differential equation   nu     t f  t u ,      1 m  1    t h t u     , belonging to the space m mC 2 ,   a b and satisfying the boundary conditions (21), admits the estimate ,   2  m    t u dt r .

b  a (4) (5) Then problem (1), (21) in the space m mC 2 ,   a b has at least one positive solution.
,   On the basis of this theorem, the following theorem on the solvability and unique solvability of problem (1), (21) is proved.

Theorem 2.

Let in the domain   a b  ,  0,  the inequality    h t x      1 m f  t x ,        1 2 m   t a   1 2 m   b t        x h t x 0   0   q t x ,  (6) Bull.

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3, 2011 Positive Solutions of Two-Point Boundary Value Problems ... 7 be satisfied, where   ,   0   0,1 and 0 are constants, h and h 0 :    a b   are continuous functions, , 0,     is a continuous and nonincreasing in the second argument function.

If, moreover,, 0,1  0,   and q :   a b  ,  0, and  m   4    2 m  2 ,   1 !

!
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   h t  0, b  a    t a b t    1       0    m  1 2      h t dt 0   ,    t a b t       m  1 2 q t  ,   t a m    b t m  x dt    for x  0 , b  a (7) (8) then problem (1), (21) in the space m mC 2 ,   Theorem 3.

If a b has at least one positive solution.


,   m    1 f    t x ,   f  t y ,          1 2 m   t a   1 2 m   b t       x  y  for a   t b x ,   y 0 , where  is a nonnegative constant, satisfying inequality (7), then problem (1), (22) in the space most one positive solution.

m mC 2 ,    a b has at ,  Let us now consider problem (1), (22).

Assume   max t x ; 0  t y ,   x f f ,   : x 0   y  x for a   t b , 0  x 0  .

x The following theorems are valid.


Theorem 4.

(A principle of a priori boundedness).

Let b  t 0  f  t x ; 0 ,  x dt   for any t 0    a b , , x 0  0 and x  x 0 ,   and let in the domain  h a b   is a continuous function, not identically equal to zero.
Let, moreover, there exist a positive number r such that every  inequality (3) be fulfilled, where is a constant and  a b  0,1  0, 0,      : , , positive solution of problem (4), (22), belonging to the space m mC 2 ,   a b , admits estimate (5).

Then problem (1), (22) ,   in the space m mC 2 ,  Theorem 5.

Let in the domain     a b has at least one positive solution.
,  a b  ,  0,  the inequality    h t x      1 m f  t x ,    2 m   t a    x h t x 0   0   q t x ,  ,   0   0,1 , 0 are constants, h and h 0 :    a b   are continuous functions, and , 0,  be fulfilled, where    q a b  , 0,  :    0,1     is a continuous and nonincreasing in the second argument function.

If, moreover,, 0,   h t  0, b  a   t a  1   0    m  1 2      h t dt 0   , Bull.

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3, 2011 m    t a  1 2 q t  ,   t a m   x dt b  a   for x  0 , (9) 8 Ivan Kiguradze and inequality (7) is fulfilled, then problem (1), (22) in the space m mC 2 ,   Theorem 6.

If a b has at least one positive solution.

,   m    1 f    t x ,   f  t y ,      2 m   t a   x  y  for a   t b x ,   y 0, where  is a nonnnegative constant, satisfying inequality (7), then problem (1), (22) in the space most one positive solution.
m mC 2 ,    a b has at ,  As examples, let us consider the differential equations  mu 2  m     1     p t u    p t u 1     p t u 2    ,  mu 2  m     1      p t u q t u  ,   ,  (10) (11) where   and 0,1   0  are constants,    q a b  , :  0,    is the continuous and nonincreasing in the second argument function.

0, p p p , 1 2 , :     a b   are continuous functions, and 0, , From Theorems 2 and 3 follow the following corollaries.

Corollary 1.
Corollary 1.

Let   p t       1 2 m   t a   1 2 m   b t      for a   t b , (12) where  is the nonnegative constant satisfying inequality (7).

If, moreover,,   p t 1  0, b  a    t a b t     1         m  1 2      p t dt 1   ,    t a b t      1      m  1 2   p t dt 2   , b  a then the problem (10), (21) in the space Corollary 2.

Let in the domain   a b  , m mC 2 ,   a b has at least one positive solution.
,    0,  the inequality  q t x ( , )    q t 0 (13) be fulfilled, where q 0 :     a b   is the continuous function, not identically equal to zero.

If, moreover, the , 0, conditions (7), (8) and (12) are fulfilled, then the problem (11), (21) in the space solution.

m mC 2 ,   From Theorems 5 and 6 follow the following corollaries.

Corollary 3.

Let a b has a unique positive ,     p t     t a 2m for a   t b , (14) where  is a nonnnegative constant satisfying inequality (7).
If, moreover,, Bull.

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3, 2011 Positive Solutions of Two-Point Boundary Value Problems ... 9   p t 1  0, b  a   t a    1    m  1 2      p t dt 1   ,   t a  m   1   1 2   p t dt 2   , b  a then the problem (10), (22) in the space Corollary 4.

Let in the domain  , m mC 2 ,   a b has at least one positive solution.
,    a b   0,  inequality (13) be fulfilled, where  q 0 :   a b   is a continu- , 0,   ous function, not identically equal to zero.

If, moreover, the conditions (7), (9) and (14) are fulfilled, then the problem (11), (22) in the space Remark.

Let m mC 2 ,   a b has a unique positive solution.
,     p t       1 2 m   t a   1 2 m   b t  ,      q t x ,   exp       t a   b t m  m  x            p t     t a  , 2 m  q t x ,   exp   t a m  x              where  is a positive number satisfying inequality (7).

Then all conditions of Corollary 2 (of Corollary 4) are fulfilled.
On the other hand, in the case under consideration, equation (11) along with strong singularities with respect to the time variable at the points a and b (at the point a) has strong singularity with respect to the phase variable as well.

Acknowledgement.

This work is supported by the Shota Rustaveli National Science Foundation (Project No.

GNSF/ ST09_175_3-101).


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Received July, 2011 Bull.

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