## Existence result of second-order differential equations with integral boundary conditions at resonance.(English)Zbl 1180.34016

The paper concerns the boundary value problem
$x''(t)=f(t,x(t),x'(t)), \quad 0<t<1,$
$x'(0)=\int^1_0h(t)x'(t)\,dt, \qquad x'(1)=\int^1_0g(t)x'(t)\,dt.$
It is assumed that the functions $$h$$ and $$g$$ are continuous and nonnegative and such that the conditions $$\int^1_0h(t)\,dt=1$$ and $$\int^1_0g(t)\,dt=1$$ hold. The linear operator $$L,$$ $$Lx:=x''$$, defined on the subspace of functions that belong to the Sobolev space $$W^{2,1}(0,1)$$ and satisfy the boundary conditions is a Fredholm operator with index zero when the functions $$h$$ and $$g$$ satisfy a condition formulated in the preliminary part of the paper. Moreover, its kernel is two-dimensional.
Next, a fixed point theorem due to J. Mawhin [Topological degree methods in nonlinear boundary value problems. Regional Conference Series in Mathematics. No. 40. R.I.: The American Mathematical Society (1979; Zbl 0414.34025)] is recalled. The existence of at least one solution of the boundary value problem is proved when certain conditions are satisfied. They are formulated in the main result which proof is based on Mawhin’s fixed point theorem. An example is presented.

### MSC:

 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations

Zbl 0414.34025
Full Text:

### References:

 [1] Gallardo, J. M., Second order differential operators with integral boundary conditions and generation of semigroups, Rocky Mountain J. Math., 30, 1265-1292 (2000) · Zbl 0984.34014 [2] Karakostas, G. L.; Tsamatos, P. Ch., Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems, Electron. J. Differential Equations, 30, 1-17 (2002) · Zbl 0998.45004 [3] Lomtatidze, A.; Malaguti, L., On a nonlocal boundary-value problems for second-order nonlinear singular differential equations, Georgian Math. J., 7, 133-154 (2000) · Zbl 0967.34011 [4] Ewing, R. E.; Lin, T., A class of parameter estimation techniques for fluid flow in porous media, Adv. Water Resour., 14, 89-97 (1991) [5] Formaggia, L.; Nobile, F.; Quarteroni, A.; Veneziani, A., Multiscale modelling of the circulatory system: A preliminary analysis, Comput. Vis. Sci., 2, 75-83 (1999) · Zbl 1067.76624 [6] Shi, P., Weak solution to evolution problem with a nonlocal constraint, SIAM J. Anal., 24, 46-58 (1993) · Zbl 0810.35033 [7] Formaggia, L.; Nobile, F.; Quarteroni, A.; Veneziani, A., Multiscale modelling of the circulatory system: A preliminary analysis, Comput. Vis. Sci., 2, 75-83 (1999) · Zbl 1067.76624 [8] Taylor, C.; Hughes, T.; Zarins, C., Finite element modeling of blood flow in arteries, Comput. Methods Appl. Mech. Engrg., 158, 155-196 (1998) · Zbl 0953.76058 [9] Womersley, J. R., Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, J. Physiol., 127, 553-563 (1955) [10] Nicoud, F.; Schfonfeld, T., Integral boundary conditions for unsteady biomedical CFD applications, Internat. J. Numer. Methods Fluids, 40, 457-465 (2002) · Zbl 1061.76517 [11] Batten, G. W., Second-order correct boundary conditions for the numerical solution of the mixed boundary problem for parabolic equations, Math. Comp., 17, 405-413 (1963) · Zbl 0133.38601 [12] Bouziani, A.; Benouar, N. E., Mixed problem with integral conditions for a third order parabolic equation, Kobe J. Math., 15, 47-58 (1998) · Zbl 0921.35068 [13] Feng, M.; Ji, D.; Ge, W., Positive solutions for a class of boundary value problem with integral boundary conditions in Banach spaces, J. Comput. Appl. Math., 222, 351-363 (2008) · Zbl 1158.34336 [14] Zhang, X.; Feng, M.; Ge, W., Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces, Nonlinear Anal., 69, 3310-3321 (2008) · Zbl 1159.34020 [15] Ahmad, B.; Alsaedi, A.; Alghamdi, B. S., Analytic approximation of solutions of the forced duffing equation with integral boundary conditions, Nonlinear Anal. Real World Appl., 9, 1727-1740 (2008) · Zbl 1154.34311 [16] Zhang, X.; Liu, L., A necessary and sufficient condition of positive solutions for fourth order multi-point boundary value problem with $$p$$-Laplacian, Nonlinear Anal., 68, 3127-3137 (2008) · Zbl 1143.34016 [17] Infante, G.; Webb, J. R.L., Nonlinear non-local boundary-value problems and perturbed hammerstein integral equations, Proc. Edinb. Math. Soc., 49, 637-656 (2006) · Zbl 1115.34026 [18] Ma, R.; Castaneda, N., Existence of solutions of nonlinear $$m$$-point boundary value problems, J. Math. Anal. Appl., 256, 556-567 (2001) · Zbl 0988.34009 [19] Webb, J. R.L.; Infante, G., Positive solutions of nonlocal boundary value problems involving integral conditions, Nonlinear Differential Equations Appl., 15, 45-67 (2008) · Zbl 1148.34021 [20] Yang, Z., Positive solutions to a system of second-order nonlocal boundary value problems, Nonlinear Anal., 62, 1251-1265 (2005) · Zbl 1089.34022 [21] Gupta, C. P., A second-order $$m$$-point boundary value problems at resonance, Nonlinear Anal., 24, 1483-1489 (1995) · Zbl 0824.34023 [22] Feng, W.; Webb, J. R.L., Solvability of three-point boundary value problems at resonance, Nonlinear Anal., 30, 3227-3238 (1997) · Zbl 0891.34019 [23] Liu, B.; Yu, J., Solvability of multi-point boundary value problem at resonance (III), Appl. Math. Comput., 129, 119-143 (2002) · Zbl 1054.34033 [24] Gupta, C. P., Solvability of multi-point boundary value problem at resonance, Results Math., 28, 270-276 (1995) · Zbl 0843.34023 [25] Gupta, C. P., Existence theorems for a second-order $$m$$-point boundary value problem at resonance, Int. J. Math. Sci., 18, 705-710 (1995) · Zbl 0839.34027 [26] Feng, W.; Webb, J. R.L., Solvability of $$m$$-point boundary value problems with nonlinear growth, J. Math. Anal. Appl., 212, 467-480 (1997) · Zbl 0883.34020 [27] Liu, B., Solvability of multi-point boundary value problem at resonance (II), Appl. Math. Comput., 136, 353-377 (2003) · Zbl 1053.34016 [28] Liu, B.; Yu, J., Solvability of multi-point boundary value problem at resonance (I), Indian J. Pure Appl. Math., 33, 475-494 (2002) · Zbl 1021.34013 [29] Su, J.; Zhao, L., Multiple periodic solutions of ordinary differential equations with double resonance, Nonlinear Anal. (2008) [30] Rachankove, I.; Staněk, S., Topological degree method in functional boundary value problems at resonance, Nonlinear Anal., 27, 271-285 (1996) · Zbl 0853.34062 [31] Mawhin, J., Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS-NSF Regional Conf. Ser. in Math. (1979), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0414.34025 [32] Ma, R. Y., Multiplicity results for a third order boundary value problem at resonance, Nonlinear Anal., 32, 493-499 (1998) · Zbl 0932.34014 [33] Nagle, R. K.; Pothoven, K. L., On a third-order nonlinear boundary value problems at resonance, J. Math. Anal. Appl., 195, 148-159 (1995) · Zbl 0847.34026 [34] Gupta, C. P., On a third-order boundary value problem at resonance, Differential Integral Equations, 2, 1-12 (1989) · Zbl 0722.34014 [35] Du, Z.; Lin, X.; Ge, W., On a third order multi-point boundary value problem at resonance, J. Math. Anal. Appl., 302, 217-229 (2005) · Zbl 1072.34012 [36] Du, Z.; Lin, X.; Ge, W., Some higher order multi-point boundary value problem at resonance, J. Comput. Appl. Math., 177, 55-65 (2005) · Zbl 1059.34010 [37] Kuo, C., Solvability of a nonlinear two-point boundary value problems at resonance, J. Differential Equations, 140, 1-9 (1997) · Zbl 0887.34016 [38] Kuo, C., Solvability of a nonlinear two-point boundary value problem at resonance II, Nonlinear Anal., 54, 565-573 (2003) · Zbl 1054.34031 [39] Krasnosel’skii, A. M.; Mawhin, J., On some higher order boundary value problems at resonance, Nonlinear Anal., 24, 1141-1148 (1995) · Zbl 0829.34016
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