## Existence and multiplicity of positive solutions to nonlinear first-order PBVPs on time scales.(English)Zbl 1134.34016

Summary: We consider the following nonlinear first-order periodic boundary value problems on time scales
$\begin{cases} x^\Delta(t)+ p(t)x(\sigma(t))=f(x(t)), &t\in[0,T]_{\mathbb T},\\ x(0)=x(\sigma(T)). \end{cases}$
Some new existence and multiplicity criteria of positive solutions are established by using several well-known fixed point theorems.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 39A10 Additive difference equations
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### References:

 [1] Cabada, A., The method of lower and upper solutions for nth-order periodic boundary value problems, J. appl. math. stoch. anal., 7, 33-47, (1994) · Zbl 0801.34026 [2] Cabada, A.; Lois, S., Maximum principles for fourth and sixth order periodic boundary value problems, Nonlinear anal., 29, 1161-1171, (1997) · Zbl 0886.34018 [3] Cabada, A.; Nieto, J.J., Extremal solutions of second order nonlinear periodic boundary value problems, Appl. math. comput., 40, 135-145, (1990) · Zbl 0723.65056 [4] Lakshmikantham, V., Periodic boundary value problems of first and second order differential equations, J. appl. math. simul., 2, 131-138, (1989) · Zbl 0712.34058 [5] Lakshmikantham, V.; Leela, S., Remarks on first and second order periodic boundary value problems, Nonlinear anal., 8, 281-287, (1984) · Zbl 0532.34029 [6] Leela, S.; Oguztoreli, M.N., Periodic boundary value problem for differential equations with delay and monotone iterative method, J. math. anal. appl., 122, 301-307, (1987) · Zbl 0616.34062 [7] Li, Y., Positive solutions of fourth-order periodic boundary value problems, Nonlinear anal., 54, 1069-1078, (2003) · Zbl 1030.34025 [8] Li, Y., Positive solutions of higher-order periodic boundary value problems, Comput. math. appl., 48, 153-161, (2004) · Zbl 1062.34021 [9] Peng, S., Positive solutions for first order periodic boundary value problem, Appl. math. comput., 158, 345-351, (2004) · Zbl 1082.34510 [10] Rachunkova, I.; Tvrdy, M.; Vrkoc, I., Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. differential equations, 176, 445-469, (2001) · Zbl 1004.34008 [11] Tisdell, C.C., Existence of solutions to first-order periodic boundary value problems, J. math. anal. appl., 323, 2, 1325-1332, (2006) · Zbl 1109.34016 [12] Wan, Z.; Chen, Y., Remarks on the periodic boundary value problems for first-order differential equations, Comput. math. appl., 37, 49-55, (1999) · Zbl 0936.34013 [13] Atici, F.M.; Cabada, A., Existence and uniqueness results for discrete second-order periodic boundary value problems, Comput. math. appl., 45, 1417-1427, (2003) · Zbl 1057.39008 [14] Cabada, A.; Otero-Espinar, V., Optimal existence results for n-th order periodic boundary value difference problems, J. math. anal. appl., 247, 67-86, (2000) · Zbl 0962.39006 [15] Cabada, A.; Otero-Espinar, V., Comparison results for n-th order periodic difference equations, Nonlinear anal., 47, 2395-2406, (2001) · Zbl 1042.39505 [16] Sun, J.P., Positive solution for first-order discrete periodic boundary value problem, Appl. math. lett., 19, 1244-1248, (2006) · Zbl 1180.39023 [17] Agarwal, R.P.; Bohner, M., Basic calculus on time scales and some of its applications, Results math., 35, 3-22, (1999) · Zbl 0927.39003 [18] Bohner, M.; Peterson, A., Dynamic equations on time scales: an introduction with applications, (2001), Birkhäuser Boston · Zbl 0978.39001 [19] Hilger, S., Analysis on measure chains-A unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001 [20] Kaymakcalan, B.; Lakshmikantham, V.; Sivasundaram, S., Dynamic systems on measure chains, (1996), Kluwer Academic Publishers Boston · Zbl 0869.34039 [21] Cabada, A., Extremal solutions and green’s functions of higher order periodic boundary value problems in time scales, J. math. anal. appl., 290, 35-54, (2004) · Zbl 1056.39018 [22] Dai, Q.; Tisdell, C.C., Existence of solutions to first-order dynamic boundary value problems, Int. J. differ. equ., 1, 1-17, (2006) · Zbl 1116.39009 [23] Gulsan Topal, S., Second-order periodic boundary value problems on time scales, Comput. math. appl., 48, 637-648, (2004) · Zbl 1068.34016 [24] Sun, J.P.; Li, W.T., Positive solution for system of nonlinear first-order PBVPs on time scales, Nonlinear anal., 62, 131-139, (2005) · Zbl 1071.34017 [25] Sun, J.P.; Li, W.T., Existence of solutions to nonlinear first-order PBVPs on time scales, Nonlinear anal., 67, 883-888, (2007) · Zbl 1120.34314 [26] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press NewYork · Zbl 0661.47045 [27] Leggett, R.W.; Williams, L.R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana univ. math. J., 28, 673-688, (1979) · Zbl 0421.47033 [28] Tisdell, C.C.; Drabek, P.; Henderson, J., Multiple solutions to dynamic equations on time scales, Comm. appl. nonlinear anal., 11, 4, 25-42, (2004) · Zbl 1082.34055
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